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\input texinfo    @c -*-texinfo-*-
@setfilename gmp.info
@include version.texi
@settitle GNU MP @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
@end iftex

@copying
This manual describes how to install and use the GNU multiple precision
arithmetic library, version @value{VERSION}.

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002
Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.1 or any later
with the Front-Cover Texts being A GNU Manual'', and with the Back-Cover
Texts being You have freedom to copy and modify this GNU Manual, like GNU
software''.  A copy of the license is included in @ref{GNU Free Documentation
@end copying

@c  Texinfo version 4.2 or up will be needed to process this into .info files.
@c
@c  The supplied texinfo.tex (or newer) should be used when processing into
@c  .dvi etc.
@c
@c  The version number and edition number are taken from version.texi provided
@c  by automake (note it's regenerated only if you configure with
@c  --enable-maintainer-mode).
@c
@c  in the "Compatibility" section) obviously must be looked at manually
@c  though.
@c
@c  "cindex" entries have been made for function categories and programming
@c  topics.  Minutiae like particular systems and processors mentioned in
@c  various places have been left out so as not to bury important topics under
@c  a lot of junk.  "mpn" functions aren't in the concept index because a
@c  beginner looking for "GCD" or something is only going to be confused by
@c  pointers to low level routines.

@dircategory GNU libraries
@direntry
* gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
@end direntry

@c  html <meta name=description content="...">
@documentdescription
How to install and use the GNU multiple precision arithmetic library, version @value{VERSION}.
@end documentdescription

@c smallbook
@finalout
@setchapternewpage on

@ifnottex
@node Top, Copying, (dir), (dir)
@top GNU MP
@end ifnottex

@iftex
@titlepage
@title GNU MP
@subtitle The GNU Multiple Precision Arithmetic Library
@subtitle Edition @value{EDITION}
@subtitle @value{UPDATED}

@author by Torbj@"orn Granlund, Swox AB
@email{tege@@swox.com}

@c Include the Distribution inside the titlepage so
@c that headings are turned off.

@tex
\global\parindent=0pt
\global\parskip=8pt
\global\baselineskip=13pt
@end tex

@page
@vskip 0pt plus 1filll
@end iftex

@insertcopying
@ifnottex
@sp 1
@end ifnottex

@iftex
@end titlepage
@end iftex

@ifnothtml
@contents
@end ifnothtml

* Copying::                    GMP Copying Conditions (LGPL).
* Introduction to GMP::        Brief introduction to GNU MP.
* Installing GMP::             How to configure and compile the GMP library.
* GMP Basics::                 What every GMP user should know.
* Reporting Bugs::             How to usefully report bugs.
* Integer Functions::          Functions for arithmetic on signed integers.
* Rational Number Functions::  Functions for arithmetic on rational numbers.
* Floating-point Functions::   Functions for arithmetic on floats.
* Low-level Functions::        Fast functions for natural numbers.
* Random Number Functions::    Functions for generating random numbers.
* Formatted Output::           @code{printf} style output.
* Formatted Input::            @code{scanf} style input.
* C++ Class Interface::        Class wrappers around GMP types.
* BSD Compatible Functions::   All functions found in BSD MP.
* Custom Allocation::          How to customize the internal allocation.
* Language Bindings::          Using GMP from other languages.
* Algorithms::                 What happens behind the scenes.
* Internals::                  How values are represented behind the scenes.

* Contributors::	       Who brings your this library?
* References::                 Some useful papers and books to read.
* Concept Index::
* Function Index::

@c  @m{T,N} is $T$ in tex or @math{N} otherwise.  This is an easy way to give
@c  different forms for math in tex and info.  Commas in N or T don't work,
@c  but @C{} can be used instead.  \, works in info but not in tex.
@iftex
@macro m {T,N}
@tex$\T\$@end tex
@end macro
@end iftex
@ifnottex
@macro m {T,N}
@math{\N\}
@end macro
@end ifnottex

@macro C {}
,
@end macro

@c  @ms{V,N} is $V_N$ in tex or just vn otherwise.  This suits simple
@c  subscripts like @ms{x,0}.
@iftex
@macro ms {V,N}
@tex$\V\_{\N\}$@end tex
@end macro
@end iftex
@ifnottex
@macro ms {V,N}
\V\\N\
@end macro
@end ifnottex

@c  @nicode{S} is plain S in info, or @code{S} elsewhere.  This can be used
@c  when the quotes that @code{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.  Doesn't work as @nicode{'\\0'}
@c  though (gives two backslashes in tex).
@ifinfo
@macro nicode {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nicode {S}
@code{\S\}
@end macro
@end ifnotinfo

@c  @nisamp{S} is plain S in info, or @samp{S} elsewhere.  This can be used
@c  when the quotes that @samp{} gives in info aren't wanted, but the
@c  fontification in tex or html is wanted.
@ifinfo
@macro nisamp {S}
\S\
@end macro
@end ifinfo
@ifnotinfo
@macro nisamp {S}
@samp{\S\}
@end macro
@end ifnotinfo

@c  Usage: @GMPtimes{}
@c  Give either \times or the word "times".
@tex
\gdef\GMPtimes{\times}
@end tex
@ifnottex
@macro GMPtimes
times
@end macro
@end ifnottex

@c  Usage: @GMPmultiply{}
@c  Give * in info, or nothing in tex.
@tex
\gdef\GMPmultiply{}
@end tex
@ifnottex
@macro GMPmultiply
*
@end macro
@end ifnottex

@c  Usage: @GMPabs{x}
@c  Give either |x| in tex, or abs(x) in info or html.
@tex
\gdef\GMPabs#1{|#1|}
@end tex
@ifnottex
@macro GMPabs {X}
@abs{}(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPfloor{x}
@c  Give either \lfloor x\rfloor in tex, or floor(x) in info or html.
@tex
\gdef\GMPfloor#1{\lfloor #1\rfloor}
@end tex
@ifnottex
@macro GMPfloor {X}
floor(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPceil{x}
@c  Give either \lceil x\rceil in tex, or ceil(x) in info or html.
@tex
\gdef\GMPceil#1{\lceil #1 \rceil}
@end tex
@ifnottex
@macro GMPceil {X}
ceil(\X\)
@end macro
@end ifnottex

@c  Math operators already available in tex, made available in info too.
@c  For example @bmod{} can be used in both tex and info.
@ifnottex
@macro bmod
mod
@end macro
@macro gcd
gcd
@end macro
@macro ge
>=
@end macro
@macro le
<=
@end macro
@macro log
log
@end macro
@macro min
min
@end macro
@macro rightarrow
->
@end macro
@end ifnottex

@c  New math operators.
@c  @abs{} can be used in both tex and info, or just \abs in tex.
@tex
\gdef\abs{\mathop{\rm abs}}
@end tex
@ifnottex
@macro abs
abs
@end macro
@end ifnottex

@c  @cross{} is a \times symbol in tex, or an "x" in info.  In tex it works
@c  inside or outside .
@tex
\gdef\cross{\ifmmode\times\else$\times$\fi}
@end tex
@ifnottex
@macro cross
x
@end macro
@end ifnottex

@c  @times{} made available as a "*" in info and html (already works in tex).
@ifnottex
@macro times
*
@end macro
@end ifnottex

@c  Usage: @W{text}
@c  Like @w{} but working in math mode too.
@tex
\gdef\W#1{\ifmmode{#1}\else\w{#1}\fi}
@end tex
@ifnottex
@macro W {S}
@w{\S\}
@end macro
@end ifnottex

@c  Usage: \GMPdisplay{text}
@c  Put the given text in an @display style indent, but without turning off
@c  paragraph reflow etc.
@tex
\gdef\GMPdisplay#1{%
\noindent
#1\par}
@end tex

@c  Usage: \GMPhat
@c  A new \hat that will work in math mode, unlike the texinfo redefined
@c  version.
@tex
\gdef\GMPhat{\mathaccent"705E}
@end tex

@c  Usage: \GMPraise{text}
@c  For use in a  math expression as an alternative to "^".  This is good
@c  for @code{} in an exponent, since there seems to be no superscript font
@c  for that.
@tex
\gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}}
@end tex

@c  Usage: @texlinebreak{}
@c  A line break as per @*, but only in tex.
@iftex
@macro texlinebreak
@*
@end macro
@end iftex
@ifnottex
@macro texlinebreak
@end macro
@end ifnottex

@c  Usage: @maybepagebreak
@c  Allow tex to insert a page break, if it feels the urge.
@c  Normally blocks of @deftypefun/funx are kept together, which can lead to
@c  some poor page break positioning if it's a big block, like the sets of
@c  division functions etc.
@tex
\gdef\maybepagebreak{\penalty0}
@end tex
@ifnottex
@macro maybepagebreak
@end macro
@end ifnottex

@node Copying, Introduction to GMP, Top, Top
@comment  node-name, next, previous,  up
@unnumbered GNU MP Copying Conditions
@cindex Copying conditions
@cindex Conditions for copying GNU MP

This library is @dfn{free}; this means that everyone is free to use it and
free to redistribute it on a free basis.  The library is not in the public
domain; it is copyrighted and there are restrictions on its distribution, but
these restrictions are designed to permit everything that a good cooperating
citizen would want to do.  What is not allowed is to try to prevent others
from further sharing any version of this library that they might get from
you.@refill

Specifically, we want to make sure that you have the right to give away copies
of the library, that you receive source code or else can get it if you want
it, that you can change this library or use pieces of it in new free programs,
and that you know you can do these things.@refill

To make sure that everyone has such rights, we have to forbid you to deprive
anyone else of these rights.  For example, if you distribute copies of the GNU
MP library, you must give the recipients all the rights that you have.  You
must make sure that they, too, receive or can get the source code.  And you
must tell them their rights.@refill

Also, for our own protection, we must make certain that everyone finds out
that there is no warranty for the GNU MP library.  If it is modified by
someone else and passed on, we want their recipients to know that what they
have is not what we distributed, so that any problems introduced by others
will not reflect on our reputation.@refill

The precise conditions of the license for the GNU MP library are found in the
Lesser General Public License version 2.1 that accompanies the source code,
see @file{COPYING.LIB}.  Certain demonstration programs are provided under the
terms of the plain General Public License version 2, see @file{COPYING}.

@node Introduction to GMP, Installing GMP, Copying, Top
@comment  node-name,  next,  previous,  up
@chapter Introduction to GNU MP
@cindex Introduction

GNU MP is a portable library written in C for arbitrary precision arithmetic
on integers, rational numbers, and floating-point numbers.  It aims to provide
the fastest possible arithmetic for all applications that need higher
precision than is directly supported by the basic C types.

Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits.  GMP is designed to
give good performance for both, by choosing algorithms based on the sizes of
the operands, and by carefully keeping the overhead at a minimum.

The speed of GMP is achieved by using fullwords as the basic arithmetic type,
by using sophisticated algorithms, by including carefully optimized assembly
code for the most common inner loops for many different CPUs, and by a general
emphasis on speed (as opposed to simplicity or elegance).

There is carefully optimized assembly code for these CPUs:
@cindex CPUs supported
ARM,
DEC Alpha 21064, 21164, and 21264,
AMD 29000,
AMD K6, K6-2 and Athlon,
Hitachi SuperH and SH-2,
HPPA 1.0, 1.1 and 2.0,
Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86,
Intel IA-64, i960,
Motorola MC68000, MC68020, MC88100, and MC88110,
Motorola/IBM PowerPC 32 and 64,
National NS32000,
IBM POWER,
MIPS R3000, R4000,
SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC,
DEC VAX,
and
Zilog Z8000.
Some optimizations also for
Cray vector systems,
Clipper,
IBM ROMP (RT),
and
Pyramid AP/XP.

@cindex Mailing lists
There are two public mailing lists of interest.  One for general questions and
discussions about usage of the GMP library and one for discussions about
@uref{http://swox.com/mailman/listinfo/}.  These lists are @strong{not} for
bug reports.

The proper place for bug reports is @email{bug-gmp@@gnu.org}.  See
@ref{Reporting Bugs} for info about reporting bugs.

@cindex Web page
For up-to-date information on GMP, please see the GMP web pages at

@display
@uref{http://swox.com/gmp/}
@end display

@display
@uref{ftp://ftp.gnu.org/gnu/gmp}
@end display

Many sites around the world mirror @samp{ftp.gnu.org}, please use a mirror
near you, see @uref{http://www.gnu.org/order/ftp.html} for a full list.

@section How to use this Manual

Everyone should read @ref{GMP Basics}.  If you need to install the library
yourself, then read @ref{Installing GMP}.  If you have a system with multiple
ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used
on applications.

The rest of the manual can be used for later reference, although it is
probably a good idea to glance through it.

@node Installing GMP, GMP Basics, Introduction to GMP, Top
@comment  node-name,  next,  previous,  up
@chapter Installing GMP
@cindex Installing GMP
@cindex Configuring GMP
@cindex Building GMP

GMP has an autoconf/automake/libtool based configuration system.  On a
Unix-like system a basic build can be done with

@example
./configure
make
@end example

@noindent
Some self-tests can be run with

@example
make check
@end example

@noindent
And you can install (under @file{/usr/local} by default) with

@example
make install
@end example

If you experience problems, please report them to @email{bug-gmp@@gnu.org}.
See @ref{Reporting Bugs}, for information on what to include in useful bug
reports.

* Build Options::
* ABI and ISA::
* Notes for Package Builds::
* Notes for Particular Systems::
* Known Build Problems::

@node Build Options, ABI and ISA, Installing GMP, Installing GMP
@section Build Options
@cindex Build options

All the usual autoconf configure options are available, run @samp{./configure
--help} for a summary.  The file @file{INSTALL.autoconf} has some generic
installation information too.

@table @asis
@item Non-Unix Systems

@samp{configure} requires various Unix-like tools.  On an MS-DOS system DJGPP
can be used, and on MS Windows Cygwin or MINGW can be used,

@display
@uref{http://www.cygnus.com/cygwin}
@uref{http://www.delorie.com/djgpp}
@uref{http://www.mingw.org}
@end display

Microsoft also publishes an Interix Services for Unix'' which can be used to
build GMP on Windows (with a normal @samp{./configure}), but it's not free
software.

The @file{macos} directory contains an unsupported port to MacOS 9 on Power
Macintosh, see @file{macos/README}.  Note that MacOS X Darwin'' should use
the normal Unix-style @samp{./configure}.

It might be possible to build without the help of @samp{configure}, certainly
all the code is there, but unfortunately you'll be on your own.

@item Build Directory

To compile in a separate build directory, @command{cd} to that directory, and
prefix the configure command with the path to the GMP source directory.  For
example

@example
cd /my/build/dir
/my/sources/gmp-@value{VERSION}/configure
@end example

Not all @samp{make} programs have the necessary features (@code{VPATH}) to
support this.  In particular, SunOS and Slowaris @command{make} have bugs that
make them unable to build in a separate directory.  Use GNU @command{make}

@item @option{--disable-shared}, @option{--disable-static}

By default both shared and static libraries are built (where possible), but
one or other can be disabled.  Shared libraries result in smaller executables
and permit code sharing between separate running processes, but on some CPUs
are slightly slower, having a small cost on each function call.

@item Native Compilation, @option{--build=CPU-VENDOR-OS}

For normal native compilation, the system can be specified with
@samp{--build}.  By default @samp{./configure} uses the output from running
@samp{./config.guess}.  On some systems @samp{./config.guess} can determine
the exact CPU type, on others it will be necessary to give it explicitly.  For
example,

@example
./configure --build=ultrasparc-sun-solaris2.7
@end example

In all cases the @samp{OS} part is important, since it controls how libtool
generates shared libraries.  Running @samp{./config.guess} is the simplest way
to see what it should be, if you don't know already.

@item Cross Compilation, @option{--host=CPU-VENDOR-OS}

When cross-compiling, the system used for compiling is given by @samp{--build}
and the system where the library will run is given by @samp{--host}.  For
example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,

@example
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
@end example

Compiler tools are sought first with the host system type as a prefix.  For
example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain
@command{ranlib}.  This makes it possible for a set of cross-compiling tools
to co-exist with native tools.  The prefix is the argument to @samp{--host},
and this can be an alias, such as @samp{m68k-linux}.  But note that tools
don't have to be setup this way, it's enough to just have a @env{PATH} with a
suitable cross-compiling @command{cc} etc.

Compiling for a different CPU in the same family as the build system is a form
of cross-compilation, though very possibly this would merely be special
options on a native compiler.  In any case @samp{./configure} avoids depending
on being able to run code on the build system, which is important when
creating binaries for a newer CPU since they very possibly won't run on the
build system.

In all cases the compiler must be able to produce an executable (of whatever
format) from a standard C @code{main}.  Although only object files will go to
make up @file{libgmp}, @samp{./configure} uses linking tests for various
purposes, such as determining what functions are available on the host system.

Currently a warning is given unless an explicit @samp{--build} is used when
cross-compiling, because it may not be possible to correctly guess the build
system type if the @env{PATH} has only a cross-compiling @command{cc}.

Note that the @samp{--target} option is not appropriate for GMP.  It's for use
when building compiler tools, with @samp{--host} being where they will run,
and @samp{--target} what they'll produce code for.  Ordinary programs or
libraries like GMP are only interested in the @samp{--host} part, being where
they'll run.  (Some past versions of GMP used @samp{--target} incorrectly.)

@item CPU types

In general, if you want a library that runs as fast as possible, you should
configure GMP for the exact CPU type your system uses.  However, this may mean
the binaries won't run on older members of the family, and might run slower on
other members, older or newer.  The best idea is always to build GMP for the
exact machine type you intend to run it on.

The following CPUs have specific support.  See @file{configure.in} for details
of what code and compiler options they select.

@itemize @bullet

@c Keep this formatting, it's easy to read and it can be grepped to
@c automatically test that CPUs listed get through ./config.sub

@item
Alpha:
@nisamp{alpha},
@nisamp{alphaev5},
@nisamp{alphaev56},
@nisamp{alphapca56},
@nisamp{alphapca57},
@nisamp{alphaev6},
@nisamp{alphaev67},
@nisamp{alphaev68}

@item
Cray:
@nisamp{c90},
@nisamp{j90},
@nisamp{t90},
@nisamp{sv1}

@item
HPPA:
@nisamp{hppa1.0},
@nisamp{hppa1.1},
@nisamp{hppa2.0},
@nisamp{hppa2.0n},
@nisamp{hppa2.0w}

@item
MIPS:
@nisamp{mips},
@nisamp{mips3},
@nisamp{mips64}

@item
Motorola:
@nisamp{m68k},
@nisamp{m68000},
@nisamp{m68010},
@nisamp{m68020},
@nisamp{m68030},
@nisamp{m68040},
@nisamp{m68060},
@nisamp{m68302},
@nisamp{m68360},
@nisamp{m88k},
@nisamp{m88110}

@item
POWER:
@nisamp{power},
@nisamp{power1},
@nisamp{power2},
@nisamp{power2sc}

@item
PowerPC:
@nisamp{powerpc},
@nisamp{powerpc64},
@nisamp{powerpc401},
@nisamp{powerpc403},
@nisamp{powerpc405},
@nisamp{powerpc505},
@nisamp{powerpc601},
@nisamp{powerpc602},
@nisamp{powerpc603},
@nisamp{powerpc603e},
@nisamp{powerpc604},
@nisamp{powerpc604e},
@nisamp{powerpc620},
@nisamp{powerpc630},
@nisamp{powerpc740},
@nisamp{powerpc7400},
@nisamp{powerpc7450},
@nisamp{powerpc750},
@nisamp{powerpc801},
@nisamp{powerpc821},
@nisamp{powerpc823},
@nisamp{powerpc860},

@item
SPARC:
@nisamp{sparc},
@nisamp{sparcv8},
@nisamp{microsparc},
@nisamp{supersparc},
@nisamp{sparcv9},
@nisamp{ultrasparc},
@nisamp{ultrasparc2},
@nisamp{ultrasparc2i},
@nisamp{ultrasparc3},
@nisamp{sparc64}

@item
80x86 family:
@nisamp{i386},
@nisamp{i486},
@nisamp{i586},
@nisamp{pentium},
@nisamp{pentiummmx},
@nisamp{pentiumpro},
@nisamp{pentium2},
@nisamp{pentium3},
@nisamp{pentium4},
@nisamp{k6},
@nisamp{k62},
@nisamp{k63},
@nisamp{athlon}

@item
Other:
@nisamp{a29k},
@nisamp{arm},
@nisamp{clipper},
@nisamp{i960},
@nisamp{ns32k},
@nisamp{pyramid},
@nisamp{sh},
@nisamp{sh2},
@nisamp{vax},
@nisamp{z8k}
@end itemize

CPUs not listed will use generic C code.

@item Generic C Build

If some of the assembly code causes problems, or if otherwise desired, the
generic C code can be selected with CPU @samp{none}.  For example,

@example
./configure --host=none-unknown-freebsd3.5
@end example

Note that this will run quite slowly, but it should be portable and should at
least make it possible to get something running if all else fails.

@item @option{ABI}

On some systems GMP supports multiple ABIs (application binary interfaces),
meaning data type sizes and calling conventions.  By default GMP chooses the
best ABI available, but a particular ABI can be selected.  For example

@example
./configure --host=mips64-sgi-irix6 ABI=n32
@end example

See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what
applications need to do.

@item @option{CC}, @option{CFLAGS}

By default the C compiler used is chosen from among some likely candidates,
with @command{gcc} normally preferred if it's present.  The usual
@samp{CC=whatever} can be passed to @samp{./configure} to choose something
different.

For some systems, default compiler flags are set based on the CPU and
compiler.  The usual @samp{CFLAGS="-whatever"} can be passed to
@samp{./configure} to use something different or to set good flags for systems
GMP doesn't otherwise know.

The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure},
and can be found in each generated @file{Makefile}.  This is the easiest way
to check the defaults when considering changing or adding something.

Note that when @samp{CC} and @samp{CFLAGS} are specified on a system
supporting multiple ABIs it's important to give an explicit
@samp{ABI=whatever}, since GMP can't determine the ABI just from the flags and
won't be able to select the correct assembler code.

If just @samp{CC} is selected then normal default @samp{CFLAGS} for that
compiler will be used (if GMP recognises it).  For example @samp{CC=gcc} can
be used to force the use of GCC, with default flags (and default ABI).

@item @option{CPPFLAGS}

Any flags like @samp{-D} defines or @samp{-I} includes required by the
preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}.
Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but
preprocessing uses just @samp{CPPFLAGS}.  This distinction is because most
preprocessors won't accept all the flags the compiler does.  Preprocessing is
done separately in some configure tests, and in the @samp{ansi2knr} support
for K&R compilers.

@item C++ Support, @option{--enable-cxx}
C++ support in GMP can be enabled with @samp{--enable-cxx}, in which case a
C++ compiler will be required.  As a convenience @samp{--enable-cxx=detect}
can be used to enable C++ support only if a compiler can be found.  The C++
support consists of a library @file{libgmpxx.la} and header file
@file{gmpxx.h}.

A separate @file{libgmpxx.la} has been adopted rather than having C++ objects
within @file{libgmp.la} in order to ensure dynamic linked C programs aren't
bloated by a dependency on the C++ standard library, and to avoid any chance
that the C++ compiler could be required when linking plain C programs.

@file{libgmpxx.la} will use certain internals from @file{libgmp.la} and can
only be expected to work with @file{libgmp.la} from the same GMP version.
Future changes to the relevant internals will be accompanied by renaming, so a
mismatch will cause unresolved symbols rather than perhaps mysterious
misbehaviour.

In general @file{libgmpxx.la} will be usable only with the C++ compiler that
built it, since name mangling and runtime support are usually incompatible
between different compilers.

@item @option{CXX}, @option{CXXFLAGS}
When C++ support is enabled, the C++ compiler and its flags can be set with
variables @samp{CXX} and @samp{CXXFLAGS} in the usual way.  The default for
@samp{CXX} is the first compiler that works from a list of likely candidates,
with @command{g++} normally preferred when available.  The default for
@samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then
for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers
@samp{-g} or nothing.  Trying @samp{CFLAGS} this way is convenient when using
@samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will
usually suit @samp{g++}.

It's important that the C and C++ compilers match, meaning their startup and
runtime support routines are compatible and that they generate code in the
same ABI (if there's a choice of ABIs on the system).  @samp{./configure}
isn't currently able to check these things very well itself, so for that
reason @samp{--disable-cxx} is the default, to avoid a build failure due to a
compiler mismatch.  Perhaps this will change in the future.

Incidentally, it's normally not good enough to set @samp{CXX} to the same as
@samp{CC}.  Although @command{gcc} for instance recognises @file{foo.cc} as
C++ code, only @command{g++} will invoke the linker the right way when
building an executable or shared library from object files.

@item Temporary Memory, @option{--enable-alloca=<choice>}
@cindex Stack overflow segfaults
@cindex @code{alloca}

GMP allocates temporary workspace using one of the following three methods,
which can be selected with for instance
@samp{--enable-alloca=malloc-reentrant}.

@itemize @bullet
@item
@samp{alloca} - C library or compiler builtin.
@item
@samp{malloc-reentrant} - the heap, in a re-entrant fashion.
@item
@samp{malloc-notreentrant} - the heap, with global variables.
@end itemize

For convenience, the following choices are also available.
@samp{--disable-alloca} is the same as @samp{--enable-alloca=no}.

@itemize @bullet
@item
@samp{yes} - a synonym for @samp{alloca}.
@item
@samp{no} - a synonym for @samp{malloc-reentrant}.
@item
@samp{reentrant} - @code{alloca} if available, otherwise
@samp{malloc-reentrant}.  This is the default.
@item
@samp{notreentrant} - @code{alloca} if available, otherwise
@samp{malloc-notreentrant}.
@end itemize

@code{alloca} is reentrant and fast, and is recommended, but when working with
large numbers it can overflow the available stack space, in which case one of
the two malloc methods will need to be used.  Alternately it might be possible
to increase available stack with @command{limit}, @command{ulimit} or
@code{setrlimit}, or under DJGPP with @command{stubedit} or
@code{@w{_stklen}}.  Note that depending on the system the only indication of
stack overflow might be a segmentation violation.

@samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe,
but @samp{malloc-notreentrant} is faster and should be used if reentrancy is
not required.

The two malloc methods in fact use the memory allocation functions selected by
@code{mp_set_memory_functions}, these being @code{malloc} and friends by
default.  @xref{Custom Allocation}.

An additional choice @samp{--enable-alloca=debug} is available, to help when
debugging memory related problems (@pxref{Debugging}).

@item FFT Multiplication, @option{--disable-fft}

By default multiplications are done using Karatsuba, 3-way Toom-Cook, and
Fermat FFT.  The FFT is only used on large to very large operands and can be
disabled to save code size if desired.

@item Berkeley MP, @option{--enable-mpbsd}

The Berkeley MP compatibility library (@file{libmp}) and header file
(@file{mp.h}) are built and installed only if @option{--enable-mpbsd} is used.
@xref{BSD Compatible Functions}.

@item MPFR, @option{--enable-mpfr}
@cindex MPFR

The optional MPFR functions are built and installed only if
@option{--enable-mpfr} is used.  These are in a separate library
@file{libmpfr.a} and are documented separately too (@pxref{Introduction to
MPFR,, Introduction to MPFR, mpfr, MPFR}).

@item Assertion Checking, @option{--enable-assert}

This option enables some consistency checking within the library.  This can be
of use while debugging, @pxref{Debugging}.

@item Execution Profiling, @option{--enable-profiling=prof/gprof}

Profiling support can be enabled either for @command{prof} or @command{gprof}.
This adds @samp{-p} or @samp{-pg} respectively to @samp{CFLAGS}, and for some
systems adds corresponding @code{mcount} calls to the assembler code.
@xref{Profiling}.

@item @option{MPN_PATH}

Various assembler versions of each mpn subroutines are provided.  For a given
CPU, a search is made though a path to choose a version of each.  For example
@samp{sparcv8} has

@example
MPN_PATH="sparc32/v8 sparc32 generic"
@end example

which means look first for v8 code, then plain sparc32 (which is v7), and
finally fall back on generic C.  Knowledgeable users with special requirements
can specify a different path.  Normally this is completely unnecessary.

@item Documentation

The document you're now reading is @file{gmp.texi}.  The usual automake
targets are available to make PostScript @file{gmp.ps} and/or DVI
@file{gmp.dvi}.

HTML can be produced with @samp{makeinfo --html}, see @ref{makeinfo
html,Generating HTML,Generating HTML,texinfo,Texinfo}.  Or alternately
@samp{texi2html}, see @ref{Top,Texinfo to HTML,About,texi2html,Texinfo To
HTML}.

PDF can be produced with @samp{texi2dvi --pdf} (@pxref{PDF
Output,PDF,,texinfo,Texinfo}) or with @samp{pdftex}.

Some supplementary notes can be found in the @file{doc} subdirectory.

@end table

@need 2000
@node ABI and ISA, Notes for Package Builds, Build Options, Installing GMP
@section ABI and ISA
@cindex ABI
@cindex Application Binary Interface
@cindex ISA
@cindex Instruction Set Architecture

ABI (Application Binary Interface) refers to the calling conventions between
functions, meaning what registers are used and what sizes the various C data
types are.  ISA (Instruction Set Architecture) refers to the instructions and
registers a CPU has available.

Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
latter for compatibility with older CPUs in the family.  GMP supports some
CPUs like this in both ABIs.  In fact within GMP @samp{ABI} means a
combination of chip ABI, plus how GMP chooses to use it.  For example in some
32-bit ABIs, GMP may support a limb as either a 32-bit @code{long} or a 64-bit
@code{long long}.

By default GMP chooses the best ABI available for a given system, and this
generally gives significantly greater speed.  But an ABI can be chosen
explicitly to make GMP compatible with other libraries, or particular
application requirements.  For example,

@example
./configure ABI=32
@end example

In all cases it's vital that all object code used in a given program is
compiled for the same ABI.

Usually a limb is implemented as a @code{long}.  When a @code{long long} limb
is used this is encoded in the generated @file{gmp.h}.  This is convenient for
applications, but it does mean that @file{gmp.h} will vary, and can't be just
copied around.  @file{gmp.h} remains compiler independent though, since all
compilers for a particular ABI will be expected to use the same limb type.

Currently no attempt is made to follow whatever conventions a system has for
installing library or header files built for a particular ABI.  This will
probably only matter when installing multiple builds of GMP, and it might be
as simple as configuring with a special @samp{libdir}, or it might require
more than that.  Note that builds for different ABIs need to done separately,
with a fresh @command{./configure} and @command{make} each.

@table @asis
@sp 1
@need 1000
@item HPPA 2.0 (@samp{hppa2.0*})

@table @asis
@item @samp{ABI=2.0w}

The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up
when using @command{cc}.  @command{gcc} support for this is in progress.
Applications must be compiled with

@example
cc  +DD64
@end example

@item @samp{ABI=2.0n}

The 2.0n ABI means the 32-bit HPPA 1.0 ABI but with a 64-bit limb using
@code{long long}.  This is available on HP-UX 10 or up when using
@command{cc}.  No @command{gcc} support is planned for this.  Applications
must be compiled with

@example
cc  +DA2.0 +e
@end example

@item @samp{ABI=1.0}

HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI.
No special compiler options are needed for applications.
@end table

All three ABIs are available for CPUs @samp{hppa2.0w} and @samp{hppa2.0}, but
for CPU @samp{hppa2.0n} only 2.0n or 1.0 are allowed.

@sp 1
@need 1000
@item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]})

IRIX 6 supports the n32 and 64 ABIs and always has a 64-bit MIPS 3 or better
CPU.  In both these ABIs GMP uses a 64-bit limb.  A new enough @command{gcc}
is required (2.95 for instance).

@table @asis
@item @samp{ABI=n32}

The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a
@code{long long}.  Applications must be compiled with

@example
gcc  -mabi=n32
cc   -n32
@end example

@item @samp{ABI=64}

The 64-bit ABI is 64-bit pointers and integers.  Applications must be compiled
with

@example
gcc  -mabi=64
cc   -64
@end example
@end table

Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary
support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.

@sp 1
@need 1000
@item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630})

@table @asis
@item @samp{ABI=aix64}

The AIX 64 ABI uses 64-bit limbs and pointers and is available on systems
@samp{*-*-aix*}.  Applications must be compiled (and linked) with

@example
gcc  -maix64
xlc  -q64
@end example

@item @samp{ABI=32}

This is the basic 32-bit PowerPC ABI.  No special compiler options are needed
for applications.
@end table

@sp 1
@need 1000
@item Sparc V9 (@samp{sparcv9} and @samp{ultrasparc*})

@table @asis
@item @samp{ABI=64}

The 64-bit V9 ABI is available on Solaris 2.7 and up and GNU/Linux.  GCC 2.95
or up, or Sun @command{cc} is required.  Applications must be compiled with

@example
gcc  -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc   -xarch=v9
@end example

@item @samp{ABI=32}

On Solaris 2.6 and earlier, and on Solaris 2.7 with the kernel in 32-bit mode,
only the plain V8 32-bit ABI can be used, since the kernel doesn't save all
registers.  GMP still uses as much of the V9 ISA as it can in these
circumstances.  No special compiler options are required for applications,
though using something like the following requesting V9 code within the V8 ABI
is recommended.

@example
gcc  -mv8plus
cc   -xarch=v8plus
@end example

@command{gcc} 2.8 and earlier only supports @samp{-mv8} though.
@end table

Don't be confused by the names of these sparc @samp{-m} and @samp{-x} options,
they're called @samp{arch} but they effectively control the ABI.

On Solaris 2.7 with the kernel in 32-bit-mode, a normal native build will
reject @samp{ABI=64} because the resulting executables won't run.
@samp{ABI=64} can still be built if desired by making it look like a
cross-compile, for example

@example
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
@end example
@end table

@need 2000
@node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing GMP
@section Notes for Package Builds
@cindex Build notes for binary packaging
@cindex Packaged builds

GMP should present no great difficulties for packaging in a binary
distribution.

@cindex Libtool versioning
@cindex Shared library versioning
Libtool is used to build the library and @samp{-version-info} is set
appropriately, having started from @samp{3:0:0} in GMP 3.0.  The GMP 4 series
will be upwardly binary compatible in each release and will be upwardly binary
compatible with all of the GMP 3 series.  Additional function interfaces may
be added in each release, so on systems where libtool versioning is not fully
checked by the loader an auxiliary mechanism may be needed to express that a
dynamic linked application depends on a new enough GMP.

An auxiliary mechanism may also be needed to express that @file{libgmpxx.la}
(from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libgmp.la}
from the same GMP version, since this is not done by the libtool versioning,
nor otherwise.  A mismatch will result in unresolved symbols from the linker,

Using @samp{DESTDIR} or a @samp{prefix} override with @samp{make install} and
a shared @file{libgmpxx} may run into a libtool relinking problem, see
@ref{Known Build Problems}.

When building a package for a CPU family, care should be taken to use
@samp{--host} (or @samp{--build}) to choose the least common denominator among
the CPUs which might use the package.  For example this might necessitate
@samp{i386} for x86s, or plain @samp{sparc} (meaning V7) for SPARCs.

Users who care about speed will want GMP built for their exact CPU type, to
make use of the available optimizations.  Providing a way to suitably rebuild
a package may be useful.  This could be as simple as making it possible for a
user to omit @samp{--build} (and @samp{--host}) so @samp{./config.guess} will
detect the CPU.  But a way to manually specify a @samp{--build} will be wanted
for systems where @samp{./config.guess} is inexact.

Note that @file{gmp.h} is a generated file, and will be architecture and ABI
dependent.

@need 2000
@node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing GMP
@section Notes for Particular Systems
@cindex Build notes for particular systems
@cindex Particular systems
@cindex Systems
@table @asis

@c This section is more or less meant for notes about performance or about
@c build problems that have been worked around but might leave a user
@c scratching their head.  Fun with different ABIs on a system belongs in the
@c above section.

@item AIX 3 and 4

On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since
some versions of the native @command{ar} fail on the convenience libraries
used.  A shared build can be attempted with

@example
./configure --enable-shared --disable-static
@end example

Note that the @samp{--disable-static} is necessary because in a shared build
libtool makes @file{libgmp.a} a symlink to @file{libgmp.so}, apparently for
the benefit of old versions of @command{ld} which only recognise @file{.a},
but unfortunately this is done even if a fully functional @command{ld} is
available.

@item ARM

On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a
bug in unsigned division, giving wrong results for some operands.  GMP
@samp{./configure} will demand GCC 2.95.4 or later.

@item Compaq C++
Compaq C++ on OSF 5.1 has two flavours of @code{iostream}, a standard one and
an old pre-standard one (see @samp{man iostream_intro}).  GMP can only use the
standard one, which unfortunately is not the default but must be selected by
defining @code{__USE_STD_IOSTREAM}.  Configure with for instance

@example
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
@end example

@item Microsoft Windows
On systems @samp{*-*-cygwin*}, @samp{*-*-mingw*} and @samp{*-*-pw32*} by
default GMP builds only a static library, but a DLL can be built instead using

@example
./configure --disable-static --enable-shared
@end example

Static and DLL libraries can't both be built, since certain export directives
in @file{gmp.h} must be different.  @samp{--enable-cxx} cannot be used when
building a DLL, since libtool doesn't currently support C++ DLLs.  This might
change in the future.

@item Microsoft C
A MINGW DLL build of GMP can be used with Microsoft C.  Libtool doesn't
install @file{.lib} and @file{.exp} files, but they can be created with the
following commands, where @file{/my/inst/dir} is the install directory (with a
@file{lib} subdirectory).

@example
lib /machine:IX86 /def:_libs/libgmp-3.dll-def
cp libgmp-3.lib /my/inst/dir/lib
cp _libs/libgmp-3.dll-exp /my/inst/dir/lib/libgmp-3.exp
@end example

MINGW uses @samp{msvcrt.dll} for I/O, so applications wanting to use the GMP
I/O routines must be compiled with @samp{cl /MD} to do the same.  If one of
the other I/O choices provided by MS C is desired then the suggestion is to
use the GMP string functions and confine I/O to the application.

@item Motorola 68k CPU Types

@samp{m68k} is taken to mean 68000.  @samp{m68020} or higher will give a
performance boost on applicable CPUs.  @samp{m68360} can be used for CPU32
series chips.  @samp{m68302} can be used for Dragonball'' series chips,
though this is merely a synonym for @samp{m68000}.

@item OpenBSD 2.6

@command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it
unsuitable for @file{.asm} file processing.  @samp{./configure} will detect
the problem and either abort or choose another m4 in the @env{PATH}.  The bug
is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.

@item Power CPU Types

In GMP, CPU types @samp{power*} and @samp{powerpc*} will each use instructions
not available on the other, so it's important to choose the right one for the
CPU that will be used.  Currently GMP has no assembler code support for using
just the common instruction subset.  To get executables that run on both, the
current suggestion is to use the generic C code (CPU @samp{none}), possibly
with appropriate compiler options (like @samp{-mcpu=common} for
@command{gcc}).  CPU @samp{rs6000} (which is not a CPU but a family of
workstations) is accepted by @file{config.sub}, but is currently equivalent to
@samp{none}.

@item Sparc CPU Types

@samp{sparcv8} or @samp{supersparc} on relevant systems will give a
significant performance increase over the V7 code.

@item Sparc App Regs
@cindex Sparc
The GMP assembler code for both 32-bit and 64-bit Sparc clobbers the
application registers'' @code{g2}, @code{g3} and @code{g4}, the same way
that the GCC default @samp{-mapp-regs} does (@pxref{SPARC Options,,, gcc,
Using the GNU Compiler Collection (GCC)}).

This makes that code unsuitable for use with the special V9
@samp{-mcmodel=embmedany} (which uses @code{g4} as a data segment pointer),
and for applications wanting to use those registers for special purposes.  In
these cases the only suggestion currently is to build GMP with CPU @samp{none}
to avoid the assembler code.

@item SunOS 4

@command{/usr/bin/m4} lacks various features needed to process @file{.asm}
files, and instead @samp{./configure} will automatically use
@command{/usr/5bin/m4}, which we believe is always available (if not then use
GNU m4).

@item x86 CPU Types

@samp{i386} selects generic code which will run reasonably well on all x86
chips.

@samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for the intended
P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
P-III)@.  @samp{i386} is a better choice when making binaries that must run on
both.

@samp{pentium4} and an SSE2 capable assembler are important for best results
on Pentium 4.  The specific code is for instance roughly a 2@cross{} to
3@cross{} speedup over the generic @samp{i386} code.

@item x86 MMX and SSE2 Code

If the CPU selected has MMX code but the assembler doesn't support it, a
warning is given and non-MMX code is used instead.  This will be an inferior
build, since the MMX code that's present is there because it's faster than the
corresponding plain integer code.  The same applies to SSE2.

Old versions of @samp{gas} don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 doesn't (and unfortunately
there's no newer assembler for that system).

Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register
to register @code{movq} instructions, and so can't be used for MMX code.
Install a recent @command{gas} if MMX code is wanted on these systems.
@end table

@need 2000
@node Known Build Problems,  , Notes for Particular Systems, Installing GMP
@section Known Build Problems
@cindex Build problems known

@c This section is more or less meant for known build problems that are not
@c otherwise worked around and require some sort of manual intervention.

You might find more up-to-date information at @uref{http://swox.com/gmp/}.

@table @asis
The version of libtool currently in use rather aggressively strips compiler
options when linking a shared library.  This will hopefully be relaxed in the
future, but for now if this is a problem the suggestion is to create a little
script to hide them, and for instance configure with

@example
./configure CC=gcc-with-my-options
@end example

@item DJGPP
The DJGPP port of @command{bash} 2.03 is unable to run the @samp{configure}
script, it exits silently, having died writing a preamble to
@file{config.log}.  Use @command{bash} 2.04 or higher.

@samp{make all} was found to run out of memory during the final
@file{libgmp.la} link on one system tested, despite having 64Mb available.  A
separate @samp{make libgmp.la} helped, perhaps recursing into the various
subdirectories uses up memory.

@item @samp{DESTDIR} and shared @file{libgmpxx}
@cindex @samp{DESTDIR}
@samp{make install DESTDIR=/my/staging/area}, or the same with a @samp{prefix}
override, to install to a temporary directory is not fully supported by
current versions of libtool when building a shared version of a library which
depends on another being built at the same time, like @file{libgmpxx} and
@file{libgmp}.

The problem is that @file{libgmpxx} is relinked at the install stage to ensure
that if the system puts a hard-coded path to @file{libgmp} within
@file{libgmpxx} then that path will be correct.  Naturally the linker is
directed to look only at the final location, not the staging area, so if
@file{libgmp} is not already in that final location then the link will fail.

A workaround for this on SVR4 style systems, such as GNU/Linux, where paths
are not hard-coded, is to include the staging area in the linker's search
using @code{LD_LIBRARY_PATH}.  For example with @samp{--prefix=/usr} but
installing under @samp{/my/staging/area},

@example
LD_LIBRARY_PATH=/my/staging/area/usr/lib \
make install DESTDIR=/my/staging/area
@end example

@item GNU binutils @command{strip} prior to 2.12
@cindex Stripped libraries

@command{strip} from GNU binutils 2.11 and earlier should not be used on the
static libraries @file{libgmp.a} and @file{libmp.a} since it will discard all
but the last of multiple archive members with the same name, like the three
versions of @file{init.o} in @file{libgmp.a}.  Binutils 2.12 or higher can be
used successfully.

The shared libraries @file{libgmp.so} and @file{libmp.so} are not affected by
this and any version of @command{strip} can be used on them.

@item @command{make} syntax error

On certain versions of SCO OpenServer 5 and IRIX 6.5 the native @command{make}
is unable to handle the long dependencies list for @file{libgmp.la}.  The
symptom is a syntax error'' on the following line of the top-level
@file{Makefile}.

@example
libgmp.la: $(libgmp_la_OBJECTS)$(libgmp_la_DEPENDENCIES)
@end example

Either use GNU Make, or as a workaround remove
@code{$(libgmp_la_DEPENDENCIES)} from that line (which will make the initial build work, but if any recompiling is done @file{libgmp.la} might not be rebuilt). @item MacOS X and GCC Libtool currently only knows how to create shared libraries on MacOS X using the native @command{cc} (which is a modified GCC), not a plain GCC. A static-only build should work though (@samp{--disable-shared}). Also, libtool currently cannot build C++ shared libraries on MacOS X, so if @samp{--enable-cxx} is desired then @samp{--disable-shared} must be used. Hopefully this will be fixed in the future. @item NeXT prior to 3.3 The system compiler on old versions of NeXT was a massacred and old GCC, even if it called itself @file{cc}. This compiler cannot be used to build GMP, you need to get a real GCC, and install that. (NeXT may have fixed this in release 3.3 of their system.) @item POWER and PowerPC Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or PowerPC. If you want to use GCC for these machines, get GCC 2.7.2.1 (or later). @item Sequent Symmetry Use the GNU assembler instead of the system assembler, since the latter has serious bugs. @item Solaris 2.6 The system @command{sed} prints an error Output line too long'' when libtool builds @file{libgmp.la}. This doesn't seem to cause any obvious ill effects, but GNU @command{sed} is recommended, to avoid any doubt. @item Sparc Solaris 2.7 with gcc 2.95.2 in ABI=32 A shared library build of GMP seems to fail in this combination, it builds but then fails the tests, apparently due to some incorrect data relocations within @code{gmp_randinit_lc_2exp_size}. The exact cause is unknown, @samp{--disable-shared} is recommended. @item Windows DLL test programs When creating a DLL version of @file{libgmp}, libtool creates wrapper scripts like @file{t-mul} for programs that would normally be @file{t-mul.exe}, in order to setup the right library paths etc. This works fine, but the absence of @file{t-mul.exe} etc causes @command{make} to think they need recompiling every time, which is an annoyance when re-running a @samp{make check}. @end table @node GMP Basics, Reporting Bugs, Installing GMP, Top @comment node-name, next, previous, up @chapter GMP Basics @cindex Basics @strong{Using functions, macros, data types, etc.@: not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.} @menu * Headers and Libraries:: * Nomenclature and Types:: * Function Classes:: * Variable Conventions:: * Parameter Conventions:: * Memory Management:: * Reentrancy:: * Useful Macros and Constants:: * Compatibility with older versions:: * Demonstration Programs:: * Efficiency:: * Debugging:: * Profiling:: * Autoconf:: * Emacs:: @end menu @node Headers and Libraries, Nomenclature and Types, GMP Basics, GMP Basics @section Headers and Libraries @cindex Headers @cindex @file{gmp.h} All declarations needed to use GMP are collected in the include file @file{gmp.h}. It is designed to work with both C and C++ compilers. @example #include <gmp.h> @end example Note however that prototypes for GMP functions with @code{FILE *} parameters are only provided if @code{<stdio.h>} is included too. @example #include <stdio.h> #include <gmp.h> @end example Likewise @code{<stdarg.h>} (or @code{<varargs.h>}) is required for prototypes with @code{va_list} parameters, such as @code{gmp_vprintf}. And @code{<obstack.h>} for prototypes with @code{struct obstack} parameters, such as @code{gmp_obstack_printf}, when available. @cindex Libraries @cindex Linking All programs using GMP must link against the @file{libgmp} library. On a typical Unix-like system this can be done with @samp{-lgmp}, for example @example gcc myprogram.c -lgmp @end example GMP C++ functions are in a separate @file{libgmpxx} library. This is built and installed if C++ support has been enabled (@pxref{Build Options}). For example, @example g++ mycxxprog.cc -lgmpxx -lgmp @end example GMP is built using Libtool and an application can use that to link if desired, @pxref{Top,Shared library support for GNU,Introduction,libtool,GNU Libtool} If GMP has been installed to a non-standard location then it may be necessary to use @samp{-I} and @samp{-L} compiler options to point to the right directories, and some sort of run-time path for a shared library. Consult your compiler documentation, for instance @ref{Top,,Introduction,gcc,Using and Porting the GNU Compiler Collection}. @node Nomenclature and Types, Function Classes, Headers and Libraries, GMP Basics @section Nomenclature and Types @cindex Nomenclature @cindex Types @cindex Integer @tindex @code{mpz_t} @noindent In this manual, @dfn{integer} usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is @code{mpz_t}. Here are some examples of how to declare such integers: @example mpz_t sum; struct foo @{ mpz_t x, y; @}; mpz_t vec[20]; @end example @cindex Rational number @tindex @code{mpq_t} @noindent @dfn{Rational number} means a multiple precision fraction. The C data type for these fractions is @code{mpq_t}. For example: @example mpq_t quotient; @end example @cindex Floating-point number @tindex @code{mpf_t} @noindent @dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is @code{mpf_t}. @cindex Limb @tindex @code{mp_limb_t} @noindent A @dfn{limb} means the part of a multi-precision number that fits in a single machine word. (We chose this word because a limb of the human body is analogous to a digit, only larger, and containing several digits.) Normally a limb is 32 or 64 bits. The C data type for a limb is @code{mp_limb_t}. @node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics @section Function Classes @cindex Function classes There are six classes of functions in the GMP library: @enumerate @item Functions for signed integer arithmetic, with names beginning with @code{mpz_}. The associated type is @code{mpz_t}. There are about 150 functions in this class. @item Functions for rational number arithmetic, with names beginning with @code{mpq_}. The associated type is @code{mpq_t}. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately. @item Functions for floating-point arithmetic, with names beginning with @code{mpf_}. The associated type is @code{mpf_t}. There are about 60 functions is this class. @item Functions compatible with Berkeley MP, such as @code{itom}, @code{madd}, and @code{mult}. The associated type is @code{MINT}. @item Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions' names begin with @code{mpn_}. The associated type is array of @code{mp_limb_t}. There are about 30 (hard-to-use) functions in this class. @item Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers. @end enumerate @node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics @section Variable Conventions @cindex Variable conventions @cindex Conventions for variables GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last. GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, @code{mpz_mul}, can be used to square @code{x} and put the result back in @code{x} with @example mpz_mul (x, x, x); @end example Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details. A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example, @example void foo (void) @{ mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) @{ mpz_mul (n, @dots{}); mpz_fdiv_q (n, @dots{}); @dots{} @} mpz_clear (n); @} @end example @node Parameter Conventions, Memory Management, Variable Conventions, GMP Basics @section Parameter Conventions @cindex Parameter conventions @cindex Conventions for parameters When a GMP variable is used as a function parameter, it's effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated @code{const} to provoke a compiler error or warning on attempting to modify them. When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A @code{return} of an @code{mpz_t} etc doesn't return the object, only a pointer, and this is almost certainly not what's wanted. Here's an example accepting an @code{mpz_t} parameter, doing a calculation, and storing the result to the indicated parameter. @example void foo (mpz_t result, const mpz_t param, unsigned long n) @{ unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); @} int main (void) @{ mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; @} @end example @code{foo} works even if the mainline passes the same variable for @code{param} and @code{result}, just like the library functions. But sometimes it's tricky to make that work, and an application might not want to bother supporting that sort of thing. For interest, the GMP types @code{mpz_t} etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each @code{mpz_t} etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases. @need 1000 @node Memory Management, Reentrancy, Parameter Conventions, GMP Basics @section Memory Management @cindex Memory Management The GMP types like @code{mpz_t} are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn't have enough. @code{mpz_t} and @code{mpq_t} variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use @code{mpz_realloc2}, or clear variables no longer needed. @code{mpf_t} variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn't change. All memory is allocated using @code{malloc} and friends by default, but this can be changed, see @ref{Custom Allocation}. Temporary memory on the stack is also used (via @code{alloca}), but this can be changed at build-time if desired, see @ref{Build Options}. @node Reentrancy, Useful Macros and Constants, Memory Management, GMP Basics @section Reentrancy @cindex Reentrancy @cindex Thread safety @cindex Multi-threading GMP is reentrant and thread-safe, with some exceptions: @itemize @bullet @item If configured with @option{--enable-alloca=malloc-notreentrant} (or with @option{--enable-alloca=notreentrant} when @code{alloca} is not available), then naturally GMP is not reentrant. @item @code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the selected precision. @code{mpf_init2} can be used instead. @item @code{mpz_random} and the other old random number functions use a global random state and are hence not reentrant. The newer random number functions that accept a @code{gmp_randstate_t} parameter can be used instead. @item @code{mp_set_memory_functions} uses global variables to store the selected memory allocation functions. @item If the memory allocation functions set by a call to @code{mp_set_memory_functions} (or @code{malloc} and friends by default) are not reentrant, then GMP will not be reentrant either. @item If the standard I/O functions such as @code{fwrite} are not reentrant then the GMP I/O functions using them will not be reentrant either. @item It's safe for two threads to read from the same GMP variable simultaneously, but it's not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It's not safe for two threads to generate a random number from the same @code{gmp_randstate_t} simultaneously, since this involves an update of that variable. @item On SCO systems the default @code{<ctype.h>} macros use per-file static variables and may not be reentrant, depending whether the compiler optimizes away fetches from them. The GMP text-based input functions are affected. @end itemize @need 2000 @node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics @section Useful Macros and Constants @cindex Useful macros and constants @cindex Constants @deftypevr {Global Constant} {const int} mp_bits_per_limb @findex mp_bits_per_limb @cindex Bits per limb @cindex Limb size The number of bits per limb. @end deftypevr @defmac __GNU_MP_VERSION @defmacx __GNU_MP_VERSION_MINOR @defmacx __GNU_MP_VERSION_PATCHLEVEL @cindex Version number @cindex GMP version number The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. @end defmac @deftypevr {Global Constant} {const char * const} gmp_version @findex gmp_version The GMP version number, as a null-terminated string, in the form i.j'' or i.j.k''. This release is @nicode{"@value{VERSION}"}. @end deftypevr @node Compatibility with older versions, Demonstration Programs, Useful Macros and Constants, GMP Basics @section Compatibility with older versions @cindex Compatibility with older versions @cindex Upward compatibility This version of GMP is upwardly binary compatible with all 4.x and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions. @itemize @bullet @item @code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency with other @code{mpn} functions. @item @code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and 3.0.1, but in 3.1 reverted to the 2.x style. @end itemize There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4. Please see the GMP 2 manual for details. The Berkeley MP compatibility library (@pxref{BSD Compatible Functions}) is source and binary compatible with the standard @file{libmp}. @c @enumerate @c @item Integer division functions round the result differently. The obsolete @c functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv}, @c @code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the @c quotient towards @c @ifinfo @c @minus{}infinity). @c @end ifinfo @c @iftex @c @tex @c$-\infty$). @c @end tex @c @end iftex @c There are a lot of functions for integer division, giving the user better @c control over the rounding. @c @item The function @code{mpz_mod} now compute the true @strong{mod} function. @c @item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use @c @strong{mod} for reduction. @c @item The assignment functions for rational numbers do no longer canonicalize @c their results. In the case a non-canonical result could arise from an @c assignment, the user need to insert an explicit call to @c @code{mpq_canonicalize}. This change was made for efficiency. @c @item Output generated by @code{mpz_out_raw} in this release cannot be read @c by @code{mpz_inp_raw} in previous releases. This change was made for making @c the file format truly portable between machines with different word sizes. @c @item Several @code{mpn} functions have changed. But they were intentionally @c undocumented in previous releases. @c @item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui} @c are now implemented as macros, and thereby sometimes evaluate their @c arguments multiple times. @c @item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1 @c for 0^0. (In version 1, they yielded 0.) @c In version 1 of the library, @code{mpq_set_den} handled negative @c denominators by copying the sign to the numerator. That is no longer done. @c Pure assignment functions do not canonicalize the assigned variable. It is @c the responsibility of the user to canonicalize the assigned variable before @c any arithmetic operations are performed on that variable. @c Note that this is an incompatible change from version 1 of the library. @c @end enumerate @need 1000 @node Demonstration Programs, Efficiency, Compatibility with older versions, GMP Basics @section Demonstration programs @cindex Demonstration programs @cindex Example programs @cindex Sample programs The @file{demos} subdirectory has some sample programs using GMP. These aren't built or installed, but there's a @file{Makefile} with rules for them. For instance, @example make pexpr ./pexpr 68^975+10 @end example @noindent The following programs are provided @itemize @bullet @item @samp{pexpr} is an expression evaluator, the program used on the GMP web page. @item The @samp{calc} subdirectory has a similar but simpler evaluator using @command{lex} and @command{yacc}. @item The @samp{expr} subdirectory is yet another expression evaluator, a library designed for ease of use within a C program. See @file{demos/expr/README} for more information. @item @samp{factorize} is a Pollard-Rho factorization program. @item @samp{isprime} is a command-line interface to the @code{mpz_probab_prime_p} function. @item @samp{primes} counts or lists primes in an interval, using a sieve. @item @samp{qcn} is an example use of @code{mpz_kronecker_ui} to estimate quadratic class numbers. @item @cindex @code{perl} The @samp{perl} subdirectory is a comprehensive perl interface to GMP. See @file{demos/perl/INSTALL} for more information. Documentation is in POD format in @file{demos/perl/GMP.pm}. @end itemize @need 1000 @node Efficiency, Debugging, Demonstration Programs, GMP Basics @section Efficiency @cindex Efficiency @table @asis @item Small operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands. @item Static Linking On some CPUs, in particular the x86s, the static @file{libgmp.a} should be used for maximum speed, since the PIC code in the shared @file{libgmp.so} will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there's a gain to be had. @item Initializing and clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too. @item Reallocations An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing values will have its memory repeatedly @code{realloc}ed, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can be called to allocate the necessary space from the beginning (@pxref{Initializing Integers}). It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2} is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though. @item @code{2exp} functions It's up to an application to call functions like @code{mpz_mul_2exp} when appropriate. General purpose functions like @code{mpz_mul} make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. @item @code{ui} and @code{si} functions The @code{ui} functions and the small number of @code{si} functions exist for convenience and should be used where applicable. But if for example an @code{mpz_t} contains a value that fits in an @code{unsigned long} there's no need extract it and call a @code{ui} function, just use the regular @code{mpz} function. @item In-Place Operations @code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg} and @code{mpf_neg} are fast when used for in-place operations like @code{mpz_abs(x,x)}, since in the current implementation only a single field of @code{x} needs changing. On suitable compilers (GCC for instance) this is inlined too. @code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui} benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since usually only one or two limbs of @code{x} will need to be changed. The same applies to the full precision @code{mpz_add} etc if @code{y} is small. If @code{y} is big then cache locality may be helped, but that's all. @code{mpz_mul} is currently the opposite, a separate destination is slightly better. A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one limb, make a temporary copy of @code{x} before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. @code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make no attempt to recognise a copy of something to itself, so a call like @code{mpz_set(x,x)} will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable. @example if (x != y) mpz_set (x, y); @end example Note that it's never worth introducing extra @code{mpz_set} calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement. @item Divisibility Testing (Small Integers) @code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions for testing whether an @code{mpz_t} is divisible by an individual small integer. They use an algorithm which is faster than @code{mpz_tdiv_ui}, but which gives no useful information about the actual remainder, only whether it's zero (or a particular value). However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo @math{23@times{}29@times{}31 = 20677} and then test that. The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like @code{mpz_tdiv_ui}. If the quotient is only rarely wanted then it's probably best to just take a remainder and then go back and calculate the quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the remainder is zero). @item Rational Arithmetic The @code{mpq} functions operate on @code{mpq_t} values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end. The @code{mpq_numref} and @code{mpq_denref} macros give access to the numerator and denominator to do things outside the scope of the supplied @code{mpq} functions. @xref{Applying Integer Functions}. The canonical form for rationals allows mixed-type @code{mpq_t} and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than @code{mpq_add}. For example, @example /* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z); @end example @item Number Sequences Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui} are designed for calculating isolated values. If a range of values is wanted it's probably best to call to get a starting point and iterate from there. @item Text Input/Output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there's usually nothing of particular interest to be seen in the digits, so the base doesn't matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers. @c Reference: Knuth volume 2 section 4.1, page 184 of second edition. :-) @end table @node Debugging, Profiling, Efficiency, GMP Basics @section Debugging @cindex Debugging @table @asis @item Stack Overflow Depending on the system, a segmentation violation or bus error might be the only indication of stack overflow. See @samp{--enable-alloca} choices in @ref{Build Options}, for how to address this. In new enough versions of GCC, @samp{-fstack-check} may be able to ensure an overflow is recognised by the system before too much damage is done, or @samp{-fstack-limit-symbol} or @samp{-fstack-limit-register} may be able to add checking if the system itself doesn't do any (@pxref{Code Gen Options,, Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}). These options must be added to the @samp{CFLAGS} used in the GMP build (@pxref{Build Options}), adding them just to an application will have no effect. Note also they're a slowdown, adding overhead to each function call and each stack allocation. @item Heap Problems The most likely cause of application problems with GMP is heap corruption. Failing to @code{init} GMP variables will have unpredictable effects, and corruption arising elsewhere in a program may well affect GMP. Initializing GMP variables more than once or failing to clear them will cause memory leaks. In all such cases a malloc debugger is recommended. On a GNU or BSD system the standard C library @code{malloc} has some diagnostic facilities, see @ref{Allocation Debugging,,,libc,The GNU C Library Reference Manual}, or @samp{man 3 malloc}. Other possibilities, in no particular order, include @display @uref{http://www.inf.ethz.ch/personal/biere/projects/ccmalloc} @uref{http://quorum.tamu.edu/jon/gnu} @ (debauch) @uref{http://dmalloc.com} @uref{http://www.perens.com/FreeSoftware} @ (electric fence) @uref{http://packages.debian.org/fda} @uref{http://www.gnupdate.org/components/leakbug} @uref{http://people.redhat.com/~otaylor/memprof} @uref{http://www.cbmamiga.demon.co.uk/mpatrol} @end display The GMP default allocation routines in @file{memory.c} also have a simple sentinel scheme which can be enabled with @code{#define DEBUG} in that file. This is mainly designed for detecting buffer overruns during GMP development, but might find other uses. @item Stack Backtraces On some systems the compiler options GMP uses by default can interfere with debugging. In particular on x86 and 68k systems @samp{-fomit-frame-pointer} is used and this generally inhibits stack backtracing. Recompiling without such options may help while debugging, though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply. @item GNU Debugger A sample @file{.gdbinit} is included in the distribution, showing how to call some undocumented dump functions to print GMP variables from within GDB. Note that these functions shouldn't be used in final application code since they're undocumented and may be subject to incompatible changes in future versions of GMP. @item Source File Paths GMP has multiple source files with the same name, in different directories. For example @file{mpz}, @file{mpq}, @file{mpf} and @file{mpfr} each have an @file{init.c}. If the debugger can't already determine the right one it may help to build with absolute paths on each C file. One way to do that is to use a separate object directory with an absolute path to the source directory. @example cd /my/build/dir /my/source/dir/gmp-@value{VERSION}/configure @end example This works via @code{VPATH}, and might require GNU @command{make}. Alternately it might be possible to change the @code{.c.lo} rules appropriately. @item Assertion Checking The build option @option{--enable-assert} is available to add some consistency checks to the library (see @ref{Build Options}). These are likely to be of limited value to most applications. Assertion failures are just as likely to indicate memory corruption as a library or compiler bug. Applications using the low-level @code{mpn} functions, however, will benefit from @option{--enable-assert} since it adds checks on the parameters of most such functions, many of which have subtle restrictions on their usage. Note however that only the generic C code has checks, not the assembler code, so CPU @samp{none} should be used for maximum checking. @item Temporary Memory Checking The build option @option{--enable-alloca=debug} arranges that each block of temporary memory in GMP is allocated with a separate call to @code{malloc} (or the allocation function set with @code{mp_set_memory_functions}). This can help a malloc debugger detect accesses outside the intended bounds, or detect memory not released. In a normal build, on the other hand, temporary memory is allocated in blocks which GMP divides up for its own use, or may be allocated with a compiler builtin @code{alloca} which will go nowhere near any malloc debugger hooks. @item Maximum Debuggability To summarize the above, a GMP build for maximum debuggability would be @example ./configure --disable-shared --enable-assert \ --enable-alloca=debug --host=none CFLAGS=-g @end example For C++, add @samp{--enable-cxx CXXFLAGS=-g}. @item Checker The checker program (@uref{http://savannah.gnu.org/projects/checker}) can be used with GMP. It contains a stub library which means GMP applications compiled with checker can use a normal GMP build. A build of GMP with checking within GMP itself can be made. This will run very very slowly. Configure with @example ./configure --host=none-pc-linux-gnu CC=checkergcc @end example @samp{--host=none} must be used, since the GMP assembler code doesn't support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don't yet support the standard C++ library. @item Valgrind The valgrind program (@uref{http://devel-home.kde.org/~sewardj}) is a memory checker for x86s. It translates and emulates machine instructions to do strong checks for uninitialized data (at the level of individual bits), memory accesses through bad pointers, and memory leaks. Current versions (20020226 snapshot) don't support MMX or SSE, so GMP must be configured for an x86 without those (eg. plain @samp{i386}), or with a special @code{MPN_PATH} that excludes those subdirectories (@pxref{Build Options}). @item Other Problems Any suspected bug in GMP itself should be isolated to make sure it's not an application problem, see @ref{Reporting Bugs}. @end table @node Profiling, Autoconf, Debugging, GMP Basics @section Profiling @cindex Profiling Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought. Depending on the system, it may be possible to get a flat profile, meaning simple timer sampling of the program counter, with no special GMP build options, just a @samp{-p} when compiling the mainline. This is a good way to ensure minimum interference with normal operation. The necessary symbol type and size information exists in most of the GMP assembler code. The @samp{--enable-profiling} build option can be used to add suitable compiler flags, either for @command{prof} (@samp{-p}) or @command{gprof} (@samp{-pg}), see @ref{Build Options}. Which of the two is available and what they do will depend on the system, and possibly on support available in @file{libc}. For some systems appropriate corresponding @code{mcount} calls are added to the assembler code too. On x86 systems @command{prof} gives call counting, so that average time spent in a function can be determined. @command{gprof}, where supported, adds call graph construction, so for instance calls to @code{mpn_add_n} from @code{mpz_add} and from @code{mpz_mul} can be differentiated. On x86 and 68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are incompatible, so the latter is not used when @command{gprof} profiling is selected, which may result in poorer code generation. If @command{prof} profiling is selected instead it should still be possible to use @command{gprof}, but only the @samp{gprof -p} flat profile and call counts can be expected to be valid, not the @samp{gprof -q} call graph. @node Autoconf, Emacs, Profiling, GMP Basics @section Autoconf @cindex Autoconf detections Autoconf based applications can easily check whether GMP is installed. The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like @code{__gmpz}. The following therefore would be a simple test, @example AC_CHECK_LIB(gmp, __gmpz_init) @end example This just uses the default @code{AC_CHECK_LIB} actions for found or not found, but an application that must have GMP would want to generate an error if not found. For example, @example AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR( [GNU MP not found, see http://swox.com/gmp])]) @end example If functions added in some particular version of GMP are required, then one of those can be used when checking. For example @code{mpz_mul_si} was added in GMP 3.1, @example AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR( [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])]) @end example An alternative would be to test the version number in @file{gmp.h} using say @code{AC_EGREP_CPP}. That would make it possible to test the exact version, if some particular sub-minor release is known to be necessary. An application that can use either GMP 2 or 3 will need to test for @code{__gmpz_init} (GMP 3 and up) or @code{mpz_init} (GMP 2), and it's also worth checking for @file{libgmp2} since Debian GNU/Linux systems used that name in the past. For example, @example AC_CHECK_LIB(gmp, __gmpz_init, , [AC_CHECK_LIB(gmp, mpz_init, , [AC_CHECK_LIB(gmp2, mpz_init)])]) @end example In general it's suggested that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions. Occasionally an application will need or want to know the size of a type at configuration or preprocessing time, not just with @code{sizeof} in the code. This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or up is best for this, since prior versions needed certain @samp{-D} defines on systems using a @code{long long} limb. The following would suit Autoconf 2.50 or up, @example AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>]) @end example The optional @code{mpfr} functions are provided in a separate @file{libmpfr.a}, and this might be from GMP with @option{--enable-mpfr} or from MPFR installed separately. Either way @file{libmpfr} depends on @file{libgmp}, it doesn't stand alone. Currently only a static @file{libmpfr.a} will be available, not a shared library, since upward binary compatibility is not guaranteed. @example AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR( [Need MPFR either from GNU MP 4 or separate MPFR package. See http://www.mpfr.org or http://swox.com/gmp]) @end example @node Emacs, , Autoconf, GMP Basics @section Emacs @cindex Emacs @key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup, emacs, The Emacs Editor}). The GMP manual can be included in such lookups by putting the following in your @file{.emacs}, @c This isn't pretty, but there doesn't seem to be a better way (in emacs @c 21.2 at least). info-lookup->mode-value could be used for the "assoc"s, @c but that function isn't documented, whereas info-lookup-alist is. @c @example (eval-after-load "info-look" '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist)))) (setcar (nthcdr 3 mode-value) (cons '("(gmp)Function Index" nil "^ -.* " "\\>") (nth 3 mode-value))))) @end example The same can be done for MPFR, with @code{(mpfr)} in place of @code{(gmp)}. @node Reporting Bugs, Integer Functions, GMP Basics, Top @comment node-name, next, previous, up @chapter Reporting Bugs @cindex Reporting bugs @cindex Bug reporting If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find. Before you report a bug, check it's not already addressed in @ref{Known Build Problems}, or perhaps @ref{Notes for Particular Systems}. You may also want to check @uref{http://swox.com/gmp/} for patches for this release. Please include the following in any report, @itemize @bullet @item The GMP version number, and if pre-packaged or patched then say so. @item A test program that makes it possible for us to reproduce the bug. Include instructions on how to run the program. @item A description of what is wrong. If the results are incorrect, in what way. If you get a crash, say so. @item If you get a crash, include a stack backtrace from the debugger if it's informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}).

@item
Please do not send core dumps, executables or @command{strace}s.

@item
The configuration options you used when building GMP, if any.

@item
The name of the compiler and its version.  For @command{gcc}, get the version
with @samp{gcc -v}, otherwise perhaps @samp{what which cc}, or similar.

@item
The output from running @samp{uname -a}.

@item
The output from running @samp{./config.guess}, and from running
@samp{./configfsf.guess} (might be the same).

@item
If the bug is related to @samp{configure}, then the contents of
@file{config.log}.

@item
If the bug is related to an @file{asm} file not assembling, then the contents
of @file{config.m4} and the offending line or lines from the temporary
@file{mpn/tmp-<file>.s}.
@end itemize

Please make an effort to produce a self-contained report, with something
definite that can be tested or debugged.  Vague queries or piecemeal messages
are difficult to act on and don't help the development effort.

It is not uncommon that an observed problem is actually due to a bug in the
compiler; the GMP code tends to explore interesting corners in compilers.

If your bug report is good, we will do our best to help you get a corrected
version of the library; if the bug report is poor, we won't do anything about
it (except maybe ask you to send a better report).

If you think something in this manual is unclear, or downright incorrect, or if
the language needs to be improved, please send a note to the same address.

@node Integer Functions, Rational Number Functions, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter Integer Functions
@cindex Integer functions

This chapter describes the GMP functions for performing integer arithmetic.

GMP integers are stored in objects of type @code{mpz_t}.

* Initializing Integers::
* Assigning Integers::
* Simultaneous Integer Init & Assign::
* Converting Integers::
* Integer Arithmetic::
* Integer Division::
* Integer Exponentiation::
* Integer Roots::
* Number Theoretic Functions::
* Integer Comparisons::
* Integer Logic and Bit Fiddling::
* I/O of Integers::
* Integer Random Numbers::
* Integer Import and Export::
* Miscellaneous Integer Functions::

@node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Integer initialization functions
@cindex Initialization functions

The functions for integer arithmetic assume that all integer objects are
initialized.  You do that by calling the function @code{mpz_init}.  For
example,

@example
@{
mpz_t integ;
mpz_init (integ);
@dots{}
@dots{}
mpz_sub (integ, @dots{});

/* Unless the program is about to exit, do ... */
mpz_clear (integ);
@}
@end example

As you can see, you can store new values any number of times, once an
object is initialized.

@deftypefun void mpz_init (mpz_t @var{integer})
Initialize @var{integer}, and set its value to 0.
@end deftypefun

@deftypefun void mpz_init2 (mpz_t @var{integer}, unsigned long @var{n})
Initialize @var{integer}, with space for @var{n} bits, and set its value to 0.

@var{n} is only the initial space, @var{integer} will grow automatically in
the normal way, if necessary, for subsequent values stored.  @code{mpz_init2}
makes it possible to avoid such reallocations if a maximum size is known in
@end deftypefun

@deftypefun void mpz_clear (mpz_t @var{integer})
Free the space occupied by @var{integer}.  Call this function for all
@code{mpz_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpz_realloc2 (mpz_t @var{integer}, unsigned long @var{n})
Change the space allocated for @var{integer} to @var{n} bits.  The value in
@var{integer} is preserved if it fits, or is set to 0 if not.

This function can be used to increase the space for a variable in order to
avoid repeated automatic reallocations, or to decrease it to give memory back
to the heap.
@end deftypefun

@deftypefun void mpz_array_init (mpz_t @var{integer_array}[], size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}})
This is a special type of initialization.  @strong{Fixed} space of
@var{fixed_num_bits} bits is allocated to each of the @var{array_size}
integers in @var{integer_array}.

The space will not be automatically increased, unlike the normal
@code{mpz_init}, but instead an application must ensure it's sufficient for
any value stored.  The following space requirements apply to various
functions,

@itemize @bullet
@item
@code{mpz_abs}, @code{mpz_neg}, @code{mpz_set}, @code{mpz_set_si} and
@code{mpz_set_ui} need room for the value they store.

@item
room for the larger of the two operands, plus an extra
@code{mp_bits_per_limb}.

@item
@code{mpz_mul}, @code{mpz_mul_ui} and @code{mpz_mul_ui} need room for the sum
of the number of bits in their operands, but each rounded up to a multiple of
@code{mp_bits_per_limb}.

@item
@code{mpz_swap} can be used between two array variables, but not between an
array and a normal variable.
@end itemize

For other functions, or if in doubt, the suggestion is to calculate in a
regular @code{mpz_init} variable and copy the result to an array variable with
@code{mpz_set}.

@code{mpz_array_init} can reduce memory usage in algorithms that need large
arrays of integers, since it avoids allocating and reallocating lots of small
memory blocks.  There is no way to free the storage allocated by this
function.  Don't call @code{mpz_clear}!
@end deftypefun

@deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc})
Change the space for @var{integer} to @var{new_alloc} limbs.  The value in
@var{integer} is preserved if it fits, or is set to 0 if not.  The return
value is not useful to applications and should be ignored.

@code{mpz_realloc2} is the preferred way to accomplish allocation changes like
this.  @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that
@code{_mpz_realloc} takes the new size in limbs.
@end deftypefun

@node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Assignment Functions
@cindex Integer assignment functions
@cindex Assignment functions

These functions assign new values to already initialized integers
(@pxref{Initializing Integers}).

@deftypefun void mpz_set (mpz_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpz_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_set_d (mpz_t @var{rop}, double @var{op})
@deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op})
@deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op})
Set the value of @var{rop} from @var{op}.

@code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to
make it an integer.
@end deftypefun

@deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
Set the value of @var{rop} from @var{str}, a null-terminated C string in base
@var{base}.  White space is allowed in the string, and is simply ignored.  The
base may vary from 2 to 36.  If @var{base} is 0, the actual base is determined
from the leading characters: if the first two characters are 0x'' or 0X'',
hexadecimal is assumed, otherwise if the first character is 0'', octal is
assumed, otherwise decimal is assumed.

This function returns 0 if the entire string is a valid number in base
@var{base}.  Otherwise it returns @minus{}1.

[It turns out that it is not entirely true that this function ignores
white-space.  It does ignore it between digits, but not after a minus sign or
within or after 0x''.  We are considering changing the definition of this
function, making it fail when there is any white-space in the input, since
that makes a lot of sense.  Send your opinion of this change to
@email{bug-gmp@@gnu.org}.  Do you really want it to accept @nicode{"3 14"} as
meaning 314 as it does now?]
@end deftypefun

@deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2})
Swap the values @var{rop1} and @var{rop2} efficiently.
@end deftypefun

@node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Combined Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Integer init and assign

For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there.  These functions'
names have the form @code{mpz_init_set@dots{}}

Here is an example of using one:

@example
@{
mpz_t pie;
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
@dots{}
mpz_sub (pie, @dots{});
@dots{}
mpz_clear (pie);
@}
@end example

@noindent
Once the integer has been initialized by any of the @code{mpz_init_set@dots{}}
functions, it can be used as the source or destination operand for the ordinary
integer functions.  Don't use an initialize-and-set function on a variable

@deftypefun void mpz_init_set (mpz_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpz_init_set_ui (mpz_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpz_init_set_si (mpz_t @var{rop}, signed long int @var{op})
@deftypefunx void mpz_init_set_d (mpz_t @var{rop}, double @var{op})
Initialize @var{rop} with limb space and set the initial numeric value from
@var{op}.
@end deftypefun

@deftypefun int mpz_init_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base})
Initialize @var{rop} and set its value like @code{mpz_set_str} (see its
documentation above for details).

If the string is a correct base @var{base} number, the function returns 0;
if an error occurs it returns @minus{}1.  @var{rop} is initialized even if
an error occurs.  (I.e., you have to call @code{mpz_clear} for it.)
@end deftypefun

@node Converting Integers, Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Integer conversion functions
@cindex Conversion functions

This section describes functions for converting GMP integers to standard C
types.  Functions for converting @emph{to} GMP integers are described in
@ref{Assigning Integers} and @ref{I/O of Integers}.

@deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op})
Return the value of @var{op} as an @code{unsigned long}.

If @var{op} is too big to fit an @code{unsigned long} then just the least
significant bits that do fit are returned.  The sign of @var{op} is ignored,
only the absolute value is used.
@end deftypefun

@deftypefun {signed long int} mpz_get_si (mpz_t @var{op})
If @var{op} fits into a @code{signed long int} return the value of @var{op}.
Otherwise return the least significant part of @var{op}, with the same sign
as @var{op}.

If @var{op} is too big to fit in a @code{signed long int}, the returned
result is probably not very useful.  To find out if the value will fit, use
the function @code{mpz_fits_slong_p}.
@end deftypefun

@deftypefun double mpz_get_d (mpz_t @var{op})
Convert @var{op} to a @code{double}.
@end deftypefun

@deftypefun double mpz_get_d_2exp (signed long int *@var{exp}, mpz_t @var{op})
Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
raised to @var{exp}}, with @math{0.5@le{}@GMPabs{@var{d}}<1}, is a good
approximation to @var{op}.
@end deftypefun

@deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base may vary
from 2 to 36.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base})
+ 2}.  The two extra bytes are for a possible minus sign, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun

@deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n})
Return limb number @var{n} from @var{op}.  The sign of @var{op} is ignored,
just the absolute value is used.  The least significant limb is number 0.

@code{mpz_size} can be used to find how many limbs make up @var{op}.
@code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to
@code{mpz_size(@var{op})-1}.
@end deftypefun

@need 2000
@node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Integer arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun

@deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@deftypefunx void mpz_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_mul_si (mpz_t @var{rop}, mpz_t @var{op1}, long int @var{op2})
@deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_addmul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_submul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_submul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

@deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
@cindex Bit shift left
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.  This operation can also be defined as a left shift by @var{op2}
bits.
@end deftypefun

@deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpz_abs (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@need 2000
@node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions
@section Division Functions
@cindex Integer division functions
@cindex Division functions

Division is undefined if the divisor is zero.  Passing a zero divisor to the
division or modulo functions (including the modular powering functions
@code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by
zero.  This lets a program handle arithmetic exceptions in these functions the
same way as for normal C @code{int} arithmetic.

@c  Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line
@c  between each, and seem to let tex do a better job of page breaks than an
@c  @sp 1 in the middle of one big set.

@deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@end deftypefun

@deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@end deftypefun

@deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@maybepagebreak
@deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}})
@deftypefunx {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}})
@maybepagebreak
@deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{b}})
@cindex Bit shift right

@sp 1
Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder
@var{r}.  For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}.
The rounding is in three styles, each suiting different applications.

@itemize @bullet
@item
@code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will
have the opposite sign to @var{d}.  The @code{c} stands for ceil''.

@item
@code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and
@var{r} will have the same sign as @var{d}.  The @code{f} stands for
floor''.

@item
@code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign
as @var{n}.  The @code{t} stands for truncate''.
@end itemize

In all cases @var{q} and @var{r} will satisfy
@m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and
@var{r} will satisfy @math{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}.

The @code{q} functions calculate only the quotient, the @code{r} functions
only the remainder, and the @code{qr} functions calculate both.  Note that for
@code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or
results will be unpredictable.

For the @code{ui} variants the return value is the remainder, and in fact
returning the remainder is all the @code{div_ui} functions do.  For
@code{tdiv} and @code{cdiv} the remainder can be negative, so for those the
return value is the absolute value of the remainder.

The @code{2exp} functions are right shifts and bit masks, but of course
rounding the same as the other functions.  For positive @var{n} both
@code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp} are simple bitwise right
shifts.  For negative @var{n}, @code{mpz_fdiv_q_2exp} is effectively an
arithmetic right shift treating @var{n} as twos complement the same as the
bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp} effectively
treats @var{n} as sign and magnitude.
@end deftypefun

@deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}})
Set @var{r} to @var{n} @code{mod} @var{d}.  The sign of the divisor is
ignored; the result is always non-negative.

@code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the
remainder as well as setting @var{r}.  See @code{mpz_fdiv_ui} above if only
the return value is wanted.
@end deftypefun

@deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d})
@deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long @var{d})
@cindex Exact division functions
Set @var{q} to @var{n}/@var{d}.  These functions produce correct results only
when it is known in advance that @var{d} divides @var{n}.

These routines are much faster than the other division functions, and are the
best choice when exact division is known to occur, for example reducing a
rational to lowest terms.
@end deftypefun

@deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d})
@deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, unsigned long int @var{d})
@deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, unsigned long int @var{b})
Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of
@code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}.
@end deftypefun

@deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d})
@deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d})
@deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, unsigned long int @var{b})
Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the
case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}.
@end deftypefun

@need 2000
@node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions
@section Exponentiation Functions
@cindex Integer exponentiation functions
@cindex Exponentiation functions
@cindex Powering functions

@deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod})
@deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod})
Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp})
modulo @var{mod}}.

Negative @var{exp} is supported if an inverse @math{@var{base}^@W{-1} @bmod
@var{mod}} exists (see @code{mpz_invert} in @ref{Number Theoretic Functions}).
If an inverse doesn't exist then a divide by zero is raised.
@end deftypefun

@deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp})
@deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp})
Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}.  The case
@math{0^0} yields 1.
@end deftypefun

@need 2000
@node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions
@section Root Extraction Functions
@cindex Integer root functions
@cindex Root extraction functions

@deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n})
Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor@C{},} the truncated integer
part of the @var{n}th root of @var{op}.  Return non-zero if the computation
was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power.
@end deftypefun

@deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor@C{},} the truncated
integer part of the square root of @var{op}.
@end deftypefun

@deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op})
Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part
of the square root of @var{op}}, like @code{mpz_sqrt}.  Set @var{rop2} to the
remainder @m{(@var{op} - @var{rop1}^2),
@var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a
perfect square.

If @var{rop1} and @var{rop2} are the same variable, the results are
undefined.
@end deftypefun

@deftypefun int mpz_perfect_power_p (mpz_t @var{op})
Return non-zero if @var{op} is a perfect power, i.e., if there exist integers
@m{a,@var{a}} and @m{b,@var{b}}, with @m{b>1, @var{b}>1}, such that
@m{@var{op}=a^b, @var{op} equals @var{a} raised to the power @var{b}}.

Under this definition both 0 and 1 are considered to be perfect powers.
Negative values of @var{op} are accepted, but of course can only be odd
perfect powers.
@end deftypefun

@deftypefun int mpz_perfect_square_p (mpz_t @var{op})
Return non-zero if @var{op} is a perfect square, i.e., if the square root of
@var{op} is an integer.  Under this definition both 0 and 1 are considered to
be perfect squares.
@end deftypefun

@need 2000
@node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions
@section Number Theoretic Functions
@cindex Number theoretic functions

@deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps})
@cindex Prime testing functions
Determine whether @var{n} is prime.  Return 2 if @var{n} is definitely prime,
return 1 if @var{n} is probably prime (without being certain), or return 0 if
@var{n} is definitely composite.

This function does some trial divisions, then some Miller-Rabin probabilistic
primality tests.  @var{reps} controls how many such tests are done, 5 to 10 is
a reasonable number, more will reduce the chances of a composite being
returned as probably prime''.

Miller-Rabin and similar tests can be more properly called compositeness
tests.  Numbers which fail are known to be composite but those which pass
might be prime or might be composite.  Only a few composites pass, hence those
which pass are considered probably prime.
@end deftypefun

@deftypefun void mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the next prime greater than @var{op}.

This function uses a probabilistic algorithm to identify primes.  For
practical purposes it's adequate, the chance of a composite passing will be
extremely small.
@end deftypefun

@c mpz_prime_p not implemented as of gmp 3.0.

@c @deftypefun int mpz_prime_p (mpz_t @var{n})
@c Return non-zero if @var{n} is prime and zero if @var{n} is a non-prime.
@c This function is far slower than @code{mpz_probab_prime_p}, but then it
@c never returns non-zero for composite numbers.

@c (For practical purposes, using @code{mpz_probab_prime_p} is adequate.
@c The likelihood of a programming error or hardware malfunction is orders
@c of magnitudes greater than the likelihood for a composite to pass as a
@c prime, if the @var{reps} argument is in the suggested range.)
@c @end deftypefun

@deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@cindex Greatest common divisor functions
Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}.
The result is always positive even if one or both input operands
are negative.
@end deftypefun

@deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2})
Compute the greatest common divisor of @var{op1} and @var{op2}.  If
@var{rop} is not @code{NULL}, store the result there.

If the result is small enough to fit in an @code{unsigned long int}, it is
returned.  If the result does not fit, 0 is returned, and the result is equal
to the argument @var{op1}.  Note that the result will always fit if @var{op2}
is non-zero.
@end deftypefun

@deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b})
@cindex Extended GCD
Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in
addition set @var{s} and @var{t} to coefficients satisfying
@math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}.
@var{g} is always positive, even if one or both of @var{a} and @var{b} are
negative.

If @var{t} is @code{NULL} then that value is not computed.
@end deftypefun

@deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long @var{op2})
@cindex Least common multiple functions
Set @var{rop} to the least common multiple of @var{op1} and @var{op2}.
@var{rop} is always positive, irrespective of the signs of @var{op1} and
@var{op2}.  @var{rop} will be zero if either @var{op1} or @var{op2} is zero.
@end deftypefun

@deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
@cindex Modular inverse functions
Compute the inverse of @var{op1} modulo @var{op2} and put the result in
@var{rop}.  If the inverse exists, the return value is non-zero and @var{rop}
will satisfy @math{0 @le{} @var{rop} < @var{op2}}.  If an inverse doesn't exist
the return value is zero and @var{rop} is undefined.
@end deftypefun

@deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b})
@cindex Jacobi symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})}.  This is defined only for @var{b} odd.
@end deftypefun

@deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p})
Calculate the Legendre symbol @m{\left(a \over p\right),
(@var{a}/@var{p})}.  This is defined only for @var{p} an odd positive
prime, and for such @var{p} it's identical to the Jacobi symbol.
@end deftypefun

@deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b})
@deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b})
@deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b})
@deftypefunx int mpz_si_kronecker (long @var{a}, mpz_t @var{b})
@deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b})
@cindex Kronecker symbol functions
Calculate the Jacobi symbol @m{\left(a \over b\right),
(@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over
2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or
@m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even.

When @var{b} is odd the Jacobi symbol and Kronecker symbol are
identical, so @code{mpz_kronecker_ui} etc can be used for mixed
precision Jacobi symbols too.

@file{demos/qcn.c} which uses @code{mpz_kronecker_ui}.
@end deftypefun

@deftypefun {unsigned long int} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f})
Remove all occurrences of the factor @var{f} from @var{op} and store the
result in @var{rop}.  The return value is how many such occurrences were
removed.
@end deftypefun

@deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{op})
@cindex Factorial functions
Set @var{rop} to @var{op}!, the factorial of @var{op}.
@end deftypefun

@deftypefun void mpz_bin_ui (mpz_t @var{rop}, mpz_t @var{n}, unsigned long int @var{k})
@deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, unsigned long int @var{n}, @w{unsigned long int @var{k}})
@cindex Binomial coefficient functions
Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over
@var{k}} and store the result in @var{rop}.  Negative values of @var{n} are
supported by @code{mpz_bin_ui}, using the identity
@m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6
part G.
@end deftypefun

@deftypefun void mpz_fib_ui (mpz_t @var{fn}, unsigned long int @var{n})
@deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, unsigned long int @var{n})
@cindex Fibonacci sequence functions
@code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci
number.  @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to
@m{F_{n-1},F[n-1]}.

These functions are designed for calculating isolated Fibonacci numbers.  When
a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and
iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
similar.
@end deftypefun

@deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, unsigned long int @var{n})
@deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, unsigned long int @var{n})
@cindex Lucas number functions
@code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas
number.  @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1}
to @m{L_{n-1},L[n-1]}.

These functions are designed for calculating isolated Lucas numbers.  When a
sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and
iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or
similar.

The Fibonacci numbers and Lucas numbers are related sequences, so it's never
necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}.  The
formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers
Algorithm}, the reverse is straightforward too.
@end deftypefun

@node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Integer comparison functions
@cindex Comparison functions

@deftypefn Function int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefnx Function int mpz_cmp_d (mpz_t @var{op1}, double @var{op2})
@deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2})
@deftypefnx Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if
@math{@var{op1} < @var{op2}}.

Note that @code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate
their arguments more than once.
@end deftypefn

@deftypefn Function int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2})
@deftypefnx Function int mpz_cmpabs_d (mpz_t @var{op1}, double @var{op2})
@deftypefnx Function int mpz_cmpabs_ui (mpz_t @var{op1}, unsigned long int @var{op2})
Compare the absolute values of @var{op1} and @var{op2}.  Return a positive
value if @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if
@math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if
@math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}.

Note that @code{mpz_cmpabs_si} is a macro and will evaluate its arguments more
than once.
@end deftypefn

@deftypefn Macro int mpz_sgn (mpz_t @var{op})
@cindex Sign tests
@cindex Integer sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its argument
multiple times.
@end deftypefn

@node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions
@comment  node-name,  next,  previous,  up
@section Logical and Bit Manipulation Functions
@cindex Logical functions
@cindex Bit manipulation functions
@cindex Integer bit manipulation functions

These functions behave as if twos complement arithmetic were used (although
sign-magnitude is the actual implementation).  The least significant bit is
number 0.

@deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} logical-and @var{op2}.
@end deftypefun

@deftypefun void mpz_ior (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} inclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2})
Set @var{rop} to @var{op1} exclusive-or @var{op2}.
@end deftypefun

@deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op})
Set @var{rop} to the one's complement of @var{op}.
@end deftypefun

@deftypefun {unsigned long int} mpz_popcount (mpz_t @var{op})
If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is
the number of 1 bits in the binary representation.  If @math{@var{op}<0}, the
number of 1s is infinite, and the return value is @var{MAX_ULONG}, the largest
possible @code{unsigned long}.
@end deftypefun

@deftypefun {unsigned long int} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2})
If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return
the hamming distance between the two operands, which is the number of bit
positions where @var{op1} and @var{op2} have different bit values.  If one
operand is @math{@ge{}0} and the other @math{<0} then the number of bits
different is infinite, and the return value is @var{MAX_ULONG}, the largest
possible @code{unsigned long}.
@end deftypefun

@deftypefun {unsigned long int} mpz_scan0 (mpz_t @var{op}, unsigned long int @var{starting_bit})
@deftypefunx {unsigned long int} mpz_scan1 (mpz_t @var{op}, unsigned long int @var{starting_bit})
Scan @var{op}, starting from bit @var{starting_bit}, towards more significant
bits, until the first 0 or 1 bit (respectively) is found.  Return the index of
the found bit.

If the bit at @var{starting_bit} is already what's sought, then
@var{starting_bit} is returned.

If there's no bit found, then @var{MAX_ULONG} is returned.  This will happen
in @code{mpz_scan0} past the end of a positive number, or @code{mpz_scan1}
past the end of a negative.
@end deftypefun

@deftypefun void mpz_setbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
Set bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun void mpz_clrbit (mpz_t @var{rop}, unsigned long int @var{bit_index})
Clear bit @var{bit_index} in @var{rop}.
@end deftypefun

@deftypefun int mpz_tstbit (mpz_t @var{op}, unsigned long int @var{bit_index})
Test bit @var{bit_index} in @var{op} and return 0 or 1 accordingly.
@end deftypefun

@node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Integer input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream.  Passing a @code{NULL} pointer for a @var{stream} argument to any of
these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

@deftypefun size_t mpz_out_str (FILE *@var{stream}, int @var{base}, mpz_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base may vary from 2 to 36.

Return the number of bytes written, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpz_inp_str (mpz_t @var{rop}, FILE *@var{stream}, int @var{base})
Input a possibly white-space preceded string in base @var{base} from stdio
stream @var{stream}, and put the read integer in @var{rop}.  The base may vary
from 2 to 36.  If @var{base} is 0, the actual base is determined from the
leading characters: if the first two characters are 0x' or 0X', hexadecimal
is assumed, otherwise if the first character is 0', octal is assumed,
otherwise decimal is assumed.

Return the number of bytes read, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpz_out_raw (FILE *@var{stream}, mpz_t @var{op})
Output @var{op} on stdio stream @var{stream}, in raw binary format.  The
integer is written in a portable format, with 4 bytes of size information, and
that many bytes of limbs.  Both the size and the limbs are written in
decreasing significance order (i.e., in big-endian).

The output can be read with @code{mpz_inp_raw}.

Return the number of bytes written, or if an error occurred, return 0.

The output of this can not be read by @code{mpz_inp_raw} from GMP 1, because
of changes necessary for compatibility between 32-bit and 64-bit machines.
@end deftypefun

@deftypefun size_t mpz_inp_raw (mpz_t @var{rop}, FILE *@var{stream})
Input from stdio stream @var{stream} in the format written by
@code{mpz_out_raw}, and put the result in @var{rop}.  Return the number of
bytes read, or if an error occurred, return 0.

This routine can read the output from @code{mpz_out_raw} also from GMP 1, in
spite of changes necessary for compatibility between 32-bit and 64-bit
machines.
@end deftypefun

@need 2000
@node Integer Random Numbers, Integer Import and Export, I/O of Integers, Integer Functions
@comment  node-name,  next,  previous,  up
@section Random Number Functions
@cindex Integer random number functions
@cindex Random number functions

The random number functions of GMP come in two groups; older function
that rely on a global state, and newer functions that accept a state
Functions} for more information on how to use and not to use random
number functions.

@deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
Generate a uniformly distributed random integer in the range 0 to @m{2^n-1,
2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

@deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, mpz_t @var{n})
Generate a uniform random integer in the range 0 to @math{@var{n}-1},
inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{n})
Generate a random integer with long strings of zeros and ones in the
binary representation.  Useful for testing functions and algorithms,
since this kind of random numbers have proven to be more likely to
trigger corner-case bugs.  The random number will be in the range
0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization})
before invoking this function.
@end deftypefun

@deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{max_size} limbs.  The generated
random number doesn't satisfy any particular requirements of randomness.
Negative random numbers are generated when @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_urandomb} or
@end deftypefun

@deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size})
Generate a random integer of at most @var{max_size} limbs, with long strings
of zeros and ones in the binary representation.  Useful for testing functions
and algorithms, since this kind of random numbers have proven to be more
likely to trigger corner-case bugs.  Negative random numbers are generated
when @var{max_size} is negative.

This function is obsolete.  Use @code{mpz_rrandomb} instead.
@end deftypefun

@node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions
@section Integer Import and Export

@code{mpz_t} variables can be converted to and from arbitrary words of binary
data with the following functions.

@deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, int @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op})
@cindex Integer import
@cindex Import
Set @var{rop} from an array of word data at @var{op}.

The parameters specify the format of the data.  @var{count} many words are
read, each @var{size} bytes.  @var{order} can be 1 for most significant word
first or -1 for least significant first.  Within each word @var{endian} can be
1 for most significant byte first, -1 for least significant first, or 0 for
the native endianness of the host CPU.  The most significant @var{nails} bits
of each word are skipped, this can be 0 to use the full words.

There are no data alignment restrictions on @var{op}, any address is allowed.

Here's an example converting an array of @code{unsigned long} data, most
significant element first and host byte order within each value.

@example
unsigned long  a[20];
mpz_t          z;
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
@end example

This example assumes the full @code{sizeof} bytes are used for data in the
given type, which is usually true, and certainly true for @code{unsigned long}
everywhere we know of.  However on Cray vector systems it may be noted that
@code{short} and @code{int} are always stored in 8 bytes (and with
@code{sizeof} indicating that) but use only 32 or 46 bits.  The @var{nails}
feature can account for this, by passing for instance
@code{8*sizeof(int)-INT_BIT}.
@end deftypefun

@deftypefun void *mpz_export (void *@var{rop}, size_t *@var{count}, int @var{order}, int @var{size}, int @var{endian}, size_t @var{nails}, mpz_t @var{op})
@cindex Integer export
@cindex Export
Fill @var{rop} with word data from @var{op}.

The parameters specify the format of the data produced.  Each word will be
@var{size} bytes and @var{order} can be 1 for most significant word first or
-1 for least significant first.  Within each word @var{endian} can be 1 for
most significant byte first, -1 for least significant first, or 0 for the
native endianness of the host CPU.  The most significant @var{nails} bits of
each word are unused and set to zero, this can be 0 to produce full words.

The number of words produced is written to @code{*@var{count}}.  @var{rop}
must have enough space for the data, or if @var{rop} is @code{NULL} then a
result array of the necessary size is allocated using the current GMP
allocation function (@pxref{Custom Allocation}).  In either case the return
value is the destination used, @var{rop} or the allocated block.

If @var{op} is non-zero then the most significant word produced will be
non-zero.  If @var{op} is zero then the count returned will be zero and
nothing written to @var{rop}.  If @var{rop} is @code{NULL} in this case, no
block is allocated, just @code{NULL} is returned.

There are no data alignment restrictions on @var{rop}, any address is allowed.
The sign of @var{op} is ignored, just the absolute value is used.

When an application is allocating space itself the required size can be
determined with a calculation like the following.  Since @code{mpz_sizeinbase}
always returns at least 1, @code{count} here will be at least one, which
avoids any portability problems with @code{malloc(0)}, though if @code{z} is
zero no space at all is actually needed.

@example
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);
@end example
@end deftypefun

@need 2000
@node Miscellaneous Integer Functions,  , Integer Import and Export, Integer Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous integer functions
@cindex Integer miscellaneous functions

@deftypefun int mpz_fits_ulong_p (mpz_t @var{op})
@deftypefunx int mpz_fits_slong_p (mpz_t @var{op})
@deftypefunx int mpz_fits_uint_p (mpz_t @var{op})
@deftypefunx int mpz_fits_sint_p (mpz_t @var{op})
@deftypefunx int mpz_fits_ushort_p (mpz_t @var{op})
@deftypefunx int mpz_fits_sshort_p (mpz_t @var{op})
Return non-zero iff the value of @var{op} fits in an @code{unsigned long int},
@code{signed long int}, @code{unsigned int}, @code{signed int}, @code{unsigned
short int}, or @code{signed short int}, respectively.  Otherwise, return zero.
@end deftypefun

@deftypefn Macro int mpz_odd_p (mpz_t @var{op})
@deftypefnx Macro int mpz_even_p (mpz_t @var{op})
Determine whether @var{op} is odd or even, respectively.  Return non-zero if
yes, zero if no.  These macros evaluate their argument more than once.
@end deftypefn

@deftypefun size_t mpz_size (mpz_t @var{op})
Return the size of @var{op} measured in number of limbs.  If @var{op} is zero,
the returned value will be zero.
@c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.)
@end deftypefun

@deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base})
Return the size of @var{op} measured in number of digits in base @var{base}.
The base may vary from 2 to 36.  The sign of @var{op} is ignored, just the
absolute value is used.  The result will be exact or 1 too big.  If @var{base}
is a power of 2, the result will always be exact.  If @var{op} is zero the
return value is always 1.

This function is useful in order to allocate the right amount of space before
converting @var{op} to a string.  The right amount of allocation is normally
two more than the value returned by @code{mpz_sizeinbase} (one extra for a
minus sign and one for the null-terminator).
@end deftypefun

@node Rational Number Functions, Floating-point Functions, Integer Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Rational Number Functions
@cindex Rational number functions

This chapter describes the GMP functions for performing arithmetic on rational

Rational numbers are stored in objects of type @code{mpq_t}.

All rational arithmetic functions assume operands have a canonical form, and
canonicalize their result.  The canonical from means that the denominator and
the numerator have no common factors, and that the denominator is positive.
Zero has the unique representation 0/1.

Pure assignment functions do not canonicalize the assigned variable.  It is
the responsibility of the user to canonicalize the assigned variable before
any arithmetic operations are performed on that variable.

@deftypefun void mpq_canonicalize (mpq_t @var{op})
Remove any factors that are common to the numerator and denominator of
@var{op}, and make the denominator positive.
@end deftypefun

* Initializing Rationals::
* Rational Conversions::
* Rational Arithmetic::
* Comparing Rationals::
* Applying Integer Functions::
* I/O of Rationals::

@node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Rational init and assign

@deftypefun void mpq_init (mpq_t @var{dest_rational})
Initialize @var{dest_rational} and set it to 0/1.  Each variable should
normally only be initialized once, or at least cleared out (using the function
@code{mpq_clear}) between each initialization.
@end deftypefun

@deftypefun void mpq_clear (mpq_t @var{rational_number})
Free the space occupied by @var{rational_number}.  Make sure to call this
function for all @code{mpq_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpq_set (mpq_t @var{rop}, mpq_t @var{op})
@deftypefunx void mpq_set_z (mpq_t @var{rop}, mpz_t @var{op})
Assign @var{rop} from @var{op}.
@end deftypefun

@deftypefun void mpq_set_ui (mpq_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2})
@deftypefunx void mpq_set_si (mpq_t @var{rop}, signed long int @var{op1}, unsigned long int @var{op2})
Set the value of @var{rop} to @var{op1}/@var{op2}.  Note that if @var{op1} and
@var{op2} have common factors, @var{rop} has to be passed to
@code{mpq_canonicalize} before any operations are performed on @var{rop}.
@end deftypefun

@deftypefun int mpq_set_str (mpq_t @var{rop}, char *@var{str}, int @var{base})
Set @var{rop} from a null-terminated string @var{str} in the given @var{base}.

The string can be an integer like 41'' or a fraction like 41/152''.  The
fraction must be in canonical form (@pxref{Rational Number Functions}), or if
not then @code{mpq_canonicalize} must be called.

The numerator and optional denominator are parsed the same as in
@code{mpz_set_str} (@pxref{Assigning Integers}).  White space is allowed in
the string, and is simply ignored.  The @var{base} can vary from 2 to 36, or
if @var{base} is 0 then the leading characters are used: @code{0x} for hex,
@code{0} for octal, or decimal otherwise.  Note that this is done separately
for the numerator and denominator, so for instance @code{0xEF/100} is 239/100,
whereas @code{0xEF/0x100} is 239/256.

The return value is 0 if the entire string is a valid number, or @minus{}1 if
not.
@end deftypefun

@deftypefun void mpq_swap (mpq_t @var{rop1}, mpq_t @var{rop2})
Swap the values @var{rop1} and @var{rop2} efficiently.
@end deftypefun

@need 2000
@node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Rational conversion functions
@cindex Conversion functions

@deftypefun double mpq_get_d (mpq_t @var{op})
Convert @var{op} to a @code{double}.
@end deftypefun

@deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op})
@deftypefunx void mpq_set_f (mpq_t @var{rop}, mpf_t @var{op})
Set @var{rop} to the value of @var{op}, without rounding.
@end deftypefun

@deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, mpq_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  The base may vary
from 2 to 36.  The string will be of the form @samp{num/den}, or if the
denominator is 1 then just @samp{num}.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of storage large
enough for the result, that being

@example
mpz_sizeinbase (mpq_numref(@var{op}), @var{base})
+ mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3
@end example

The three extra bytes are for a possible minus sign, possible slash, and the
null-terminator.

A pointer to the result string is returned, being either the allocated block,
or the given @var{str}.
@end deftypefun

@node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Rational arithmetic functions
@cindex Arithmetic functions

@end deftypefun

@deftypefun void mpq_sub (mpq_t @var{difference}, mpq_t @var{minuend}, mpq_t @var{subtrahend})
Set @var{difference} to @var{minuend} @minus{} @var{subtrahend}.
@end deftypefun

@deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand})
Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}.
@end deftypefun

@deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor})
@cindex Division functions
Set @var{quotient} to @var{dividend}/@var{divisor}.
@end deftypefun

@deftypefun void mpq_div_2exp (mpq_t @var{rop}, mpq_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand})
Set @var{negated_operand} to @minus{}@var{operand}.
@end deftypefun

@deftypefun void mpq_abs (mpq_t @var{rop}, mpq_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number})
Set @var{inverted_number} to 1/@var{number}.  If the new denominator is
zero, this routine will divide by zero.
@end deftypefun

@node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Rational comparison functions
@cindex Comparison functions

@deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.

To determine if two rationals are equal, @code{mpq_equal} is faster than
@code{mpq_cmp}.
@end deftypefun

@deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2})
@deftypefnx Macro int mpq_cmp_si (mpq_t @var{op1}, long int @var{num2}, unsigned long int @var{den2})
Compare @var{op1} and @var{num2}/@var{den2}.  Return a positive value if
@math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} =
@var{num2}/@var{den2}}, and a negative value if @math{@var{op1} <
@var{num2}/@var{den2}}.

@var{num2} and @var{den2} are allowed to have common factors.

These functions are implemented as a macros and evaluate their arguments
multiple times.
@end deftypefn

@deftypefn Macro int mpq_sgn (mpq_t @var{op})
@cindex Sign tests
@cindex Rational sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its
arguments multiple times.
@end deftypefn

@deftypefun int mpq_equal (mpq_t @var{op1}, mpq_t @var{op2})
Return non-zero if @var{op1} and @var{op2} are equal, zero if they are
non-equal.  Although @code{mpq_cmp} can be used for the same purpose, this
function is much faster.
@end deftypefun

@node Applying Integer Functions, I/O of Rationals, Comparing Rationals, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Applying Integer Functions to Rationals
@cindex Rational numerator and denominator
@cindex Numerator and denominator

The set of @code{mpq} functions is quite small.  In particular, there are few
functions for either input or output.  The following functions give direct

Note that if an assignment to the numerator and/or denominator could take an
@code{mpq_t} out of the canonical form described at the start of this chapter
(@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be
called before any other @code{mpq} functions are applied to that @code{mpq_t}.

@deftypefn Macro mpz_t mpq_numref (mpq_t @var{op})
@deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op})
Return a reference to the numerator and denominator of @var{op}, respectively.
The @code{mpz} functions can be used on the result of these macros.
@end deftypefn

@deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational})
@deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational})
@deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator})
@deftypefunx void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator})
Get or set the numerator or denominator of a rational.  These functions are
equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or
@code{mpq_denref}.  Direct use of @code{mpq_numref} or @code{mpq_denref} is
@end deftypefun

@need 2000
@node I/O of Rationals,  , Applying Integer Functions, Rational Number Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Rational input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

When using any of these functions, it's a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

Passing a @code{NULL} pointer for a @var{stream} argument to any of these
functions will make them read from @code{stdin} and write to @code{stdout},
respectively.

@deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op})
Output @var{op} on stdio stream @var{stream}, as a string of digits in base
@var{base}.  The base may vary from 2 to 36.  Output is in the form
@samp{num/den} or if the denominator is 1 then just @samp{num}.

Return the number of bytes written, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string of digits from @var{stream} and convert them to a rational in
the number of characters read (including white space), or 0 if a rational

The input can be a fraction like @samp{17/63} or just an integer like
@samp{123}.  Reading stops at the first character not in this form, and white
space is not permitted within the string.  If the input might not be in
canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational
Number Functions}).

The @var{base} can be between 2 and 36, or can be 0 in which case the leading
characters of the string determine the base, @samp{0x} or @samp{0X} for
are examined separately for the numerator and denominator of a fraction, so
for instance @samp{0x10/11} is 16/11, whereas @samp{0x10/0x11} is 16/17.
@end deftypefun

@node Floating-point Functions, Low-level Functions, Rational Number Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions
@cindex User-defined precision
@cindex Precision of floats

GMP floating point numbers are stored in objects of type @code{mpf_t} and
functions operating on them have an @code{mpf_} prefix.

The mantissa of each float has a user-selectable precision, limited only by
available memory.  Each variable has its own precision, and that can be
increased or decreased at any time.

The exponent of each float is a fixed precision, one machine word on most
systems.  In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
@math{2^@W{-68719476768}} to @math{2^@W{68719476736}}, or on a 64-bit system
this will be greater.  Note however @code{mpf_get_str} can only return an
exponent which fits an @code{mp_exp_t} and currently @code{mpf_set_str}
doesn't accept exponents bigger than a @code{long}.

Each variable keeps a size for the mantissa data actually in use.  This means
that if a float is exactly represented in only a few bits then only those bits
will be used in a calculation, even if the selected precision is high.

All calculations are performed to the precision of the destination variable.
Each function is defined to calculate with infinite precision'' followed by
a truncation to the destination precision, but of course the work done is only
what's needed to determine a result under that definition.

The precision selected for a variable is a minimum value, GMP may increase it
a little to facilitate efficient calculation.  Currently this means rounding
up to a whole limb, and then sometimes having a further partial limb,
depending on the high limb of the mantissa.  But applications shouldn't be
concerned by such details.

The mantissa in stored in binary, as might be imagined from the fact
precisions are expressed in bits.  One consequence of this is that decimal
fractions like @math{0.1} cannot be represented exactly.  The same is true of
plain IEEE @code{double} floats.  This makes both highly unsuitable for
calculations involving money or other values that should be exact decimal
fractions.  (Suitably scaled integers, or perhaps rationals, are better
choices.)

@code{mpf} functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the exponent or
results will be unpredictable.  This might change in a future release.

Note that the @code{mpf} functions are @emph{not} intended as a smooth
extension to IEEE P754 arithmetic.  In particular results obtained on one
computer often differ from the results on a computer with a different word
size.

* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::

@node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Initialization Functions
@cindex Float initialization functions
@cindex Initialization functions

@deftypefun void mpf_set_default_prec (unsigned long int @var{prec})
Set the default precision to be @strong{at least} @var{prec} bits.  All
subsequent calls to @code{mpf_init} will use this precision, but previously
initialized variables are unaffected.
@end deftypefun

@deftypefun {unsigned long int} mpf_get_default_prec (void)
Return the default default precision actually used.
@end deftypefun

An @code{mpf_t} object must be initialized before storing the first value in
it.  The functions @code{mpf_init} and @code{mpf_init2} are used for that
purpose.

@deftypefun void mpf_init (mpf_t @var{x})
Initialize @var{x} to 0.  Normally, a variable should be initialized once only
or at least be cleared, using @code{mpf_clear}, between initializations.  The
precision of @var{x} is undefined unless a default precision has already been
established by a call to @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun void mpf_init2 (mpf_t @var{x}, unsigned long int @var{prec})
Initialize @var{x} to 0 and set its precision to be @strong{at least}
@var{prec} bits.  Normally, a variable should be initialized once only or at
least be cleared, using @code{mpf_clear}, between initializations.
@end deftypefun

@deftypefun void mpf_clear (mpf_t @var{x})
Free the space occupied by @var{x}.  Make sure to call this function for all
@code{mpf_t} variables when you are done with them.
@end deftypefun

@need 2000
Here is an example on how to initialize floating-point variables:
@example
@{
mpf_t x, y;
mpf_init (x);           /* use default precision */
mpf_init2 (y, 256);     /* precision @emph{at least} 256 bits */
@dots{}
/* Unless the program is about to exit, do ... */
mpf_clear (x);
mpf_clear (y);
@}
@end example

The following three functions are useful for changing the precision during a
calculation.  A typical use would be for adjusting the precision gradually in
iterative algorithms like Newton-Raphson, making the computation precision
closely match the actual accurate part of the numbers.

@deftypefun {unsigned long int} mpf_get_prec (mpf_t @var{op})
Return the current precision of @var{op}, in bits.
@end deftypefun

@deftypefun void mpf_set_prec (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits.  The
value in @var{rop} will be truncated to the new precision.

This function requires a call to @code{realloc}, and so should not be used in
a tight loop.
@end deftypefun

@deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, unsigned long int @var{prec})
Set the precision of @var{rop} to be @strong{at least} @var{prec} bits,
without changing the memory allocated.

@var{prec} must be no more than the allocated precision for @var{rop}, that
being the precision when @var{rop} was initialized, or in the most recent
@code{mpf_set_prec}.

The value in @var{rop} is unchanged, and in particular if it had a higher
precision than @var{prec} it will retain that higher precision.  New values
written to @var{rop} will use the new @var{prec}.

Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another
@code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original
allocated precision.  Failing to do so will have unpredictable results.

@code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the
original allocated precision.  After @code{mpf_set_prec_raw} it reflects the
@var{prec} value set.

@code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at
different precisions during a calculation, perhaps to gradually increase
precision in an iteration, or just to use various different precisions for
different purposes during a calculation.
@end deftypefun

@need 2000
@node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Assignment Functions
@cindex Float assignment functions
@cindex Assignment functions

These functions assign new values to already initialized floats
(@pxref{Initializing Floats}).

@deftypefun void mpf_set (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_set_d (mpf_t @var{rop}, double @var{op})
@deftypefunx void mpf_set_z (mpf_t @var{rop}, mpz_t @var{op})
@deftypefunx void mpf_set_q (mpf_t @var{rop}, mpq_t @var{op})
Set the value of @var{rop} from @var{op}.
@end deftypefun

@deftypefun int mpf_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
Set the value of @var{rop} from the string in @var{str}.  The string is of the
form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}.
@samp{M} is the mantissa and @samp{N} is the exponent.  The mantissa is always
in the specified base.  The exponent is either in the specified base or, if
@var{base} is negative, in decimal.  The decimal point expected is taken from
the current locale, on systems providing @code{localeconv}.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

Unlike the corresponding @code{mpz} function, the base will not be determined
from the leading characters of the string if @var{base} is 0.  This is so that
numbers like @samp{0.23} are not interpreted as octal.

White space is allowed in the string, and is simply ignored.  [This is not
really true; white-space is ignored in the beginning of the string and within
the mantissa, but not in other places, such as after a minus sign or in the
exponent.  We are considering changing the definition of this function, making
it fail when there is any white-space in the input, since that makes a lot of
to accept @nicode{"3 14"} as meaning 314 as it does now?]

This function returns 0 if the entire string is a valid number in base
@var{base}.  Otherwise it returns @minus{}1.
@end deftypefun

@deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2})
Swap @var{rop1} and @var{rop2} efficiently.  Both the values and the
precisions of the two variables are swapped.
@end deftypefun

@node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Combined Initialization and Assignment Functions
@cindex Initialization and assignment functions
@cindex Float init and assign functions

For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there.  These functions'
names have the form @code{mpf_init_set@dots{}}

Once the float has been initialized by any of the @code{mpf_init_set@dots{}}
functions, it can be used as the source or destination operand for the ordinary
float functions.  Don't use an initialize-and-set function on a variable

@deftypefun void mpf_init_set (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_init_set_ui (mpf_t @var{rop}, unsigned long int @var{op})
@deftypefunx void mpf_init_set_si (mpf_t @var{rop}, signed long int @var{op})
@deftypefunx void mpf_init_set_d (mpf_t @var{rop}, double @var{op})
Initialize @var{rop} and set its value from @var{op}.

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun

@deftypefun int mpf_init_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base})
Initialize @var{rop} and set its value from the string in @var{str}.  See
@code{mpf_set_str} above for details on the assignment operation.

Note that @var{rop} is initialized even if an error occurs.  (I.e., you have to
call @code{mpf_clear} for it.)

The precision of @var{rop} will be taken from the active default precision, as
set by @code{mpf_set_default_prec}.
@end deftypefun

@node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Float conversion functions
@cindex Conversion functions

@deftypefun double mpf_get_d (mpf_t @var{op})
Convert @var{op} to a @code{double}.
@end deftypefun

@deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, mpf_t @var{op})
Find @var{d} and @var{exp} such that @m{@var{d}\times 2^{exp}, @var{d} times 2
raised to @var{exp}}, with @math{0.5@le{}@GMPabs{@var{d}}<1}, is a good
approximation to @var{op}.  This is similar to the standard C function
@code{frexp}.
@end deftypefun

@deftypefun long mpf_get_si (mpf_t @var{op})
@deftypefunx {unsigned long} mpf_get_ui (mpf_t @var{op})
Convert @var{op} to a @code{long} or @code{unsigned long}, truncating any
fraction part.  If @var{op} is too big for the return type, the result is
undefined.

(@pxref{Miscellaneous Float Functions}).
@end deftypefun

@deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
Convert @var{op} to a string of digits in base @var{base}.  @var{base} can be
2 to 36.  Up to @var{n_digits} digits will be generated.  Trailing zeros are
not returned.  No more digits than can be accurately represented by @var{op}
are ever generated.  If @var{n_digits} is 0 then that accurate maximum number
of digits are generated.

If @var{str} is @code{NULL}, the result string is allocated using the current
allocation function (@pxref{Custom Allocation}).  The block will be
@code{strlen(str)+1} bytes, that being exactly enough for the string and
null-terminator.

If @var{str} is not @code{NULL}, it should point to a block of
@math{@var{n_digits} + 2} bytes, that being enough for the mantissa, a
possible minus sign, and a null-terminator.  When @var{n_digits} is 0 to get
all significant digits, an application won't be able to know the space
required, and @var{str} should be @code{NULL} in that case.

The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit.  The applicable exponent is written through
the @var{expptr} pointer.  For example, the number 3.1416 would be returned as
string @nicode{"31416"} and exponent 1.

When @var{op} is zero, an empty string is produced and the exponent returned
is 0.

A pointer to the result string is returned, being either the allocated block
or the given @var{str}.
@end deftypefun

@node Float Arithmetic, Float Comparison, Converting Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Arithmetic Functions
@cindex Float arithmetic functions
@cindex Arithmetic functions

@deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} + @var{op2}}.
@end deftypefun

@deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_ui_sub (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_sub_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @var{op1} @minus{} @var{op2}.
@end deftypefun

@deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}.
@end deftypefun

Division is undefined if the divisor is zero, and passing a zero divisor to the
divide functions will make these functions intentionally divide by zero.  This
lets the user handle arithmetic exceptions in these functions in the same
manner as other arithmetic exceptions.

@deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2})
@deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Division functions
Set @var{rop} to @var{op1}/@var{op2}.
@end deftypefun

@deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op})
@cindex Root extraction functions
Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}.
@end deftypefun

@deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
@cindex Exponentiation functions
@cindex Powering functions
Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}.
@end deftypefun

@deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to @minus{}@var{op}.
@end deftypefun

@deftypefun void mpf_abs (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to the absolute value of @var{op}.
@end deftypefun

@deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to
@var{op2}}.
@end deftypefun

@deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2})
Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to
@var{op2}}.
@end deftypefun

@node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Float comparison functions
@cindex Comparison functions

@deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2})
@deftypefunx int mpf_cmp_d (mpf_t @var{op1}, double @var{op2})
@deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, unsigned long int @var{op2})
@deftypefunx int mpf_cmp_si (mpf_t @var{op1}, signed long int @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.
@end deftypefun

@deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, unsigned long int op3)
Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
equal, zero otherwise.  I.e., test of @var{op1} and @var{op2} are approximately
equal.

Caution: Currently only whole limbs are compared, and only in an exact
fashion.  In the future values like 1000 and 0111 may be considered the same
to 3 bits (on the basis that their difference is that small).
@end deftypefun

@deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2})
Compute the relative difference between @var{op1} and @var{op2} and store the
result in @var{rop}.  This is @math{@GMPabs{@var{op1}-@var{op2}}/@var{op1}}.
@end deftypefun

@deftypefn Macro int mpf_sgn (mpf_t @var{op})
@cindex Sign tests
@cindex Float sign tests
Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and
@math{-1} if @math{@var{op} < 0}.

This function is actually implemented as a macro.  It evaluates its arguments
multiple times.
@end deftypefn

@node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Float input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio stream.  Passing a @code{NULL} pointer for a @var{stream} argument to
any of these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

When using any of these functions, it is a good idea to include @file{stdio.h}
before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes
for these functions.

@deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op})
Print @var{op} to @var{stream}, as a string of digits.  Return the number of
bytes written, or if an error occurred, return 0.

The mantissa is prefixed with an @samp{0.} and is in the given @var{base},
which may vary from 2 to 36.  An exponent then printed, separated by an
@samp{e}, or if @var{base} is greater than 10 then by an @samp{@@}.  The
exponent is always in decimal.  The decimal point follows the current locale,
on systems providing @code{localeconv}.

Up to @var{n_digits} will be printed from the mantissa, except that no more
digits than are accurately representable by @var{op} will be printed.
@var{n_digits} can be 0 to select that accurate maximum.
@end deftypefun

@deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base})
Read a string in base @var{base} from @var{stream}, and put the read float in
@var{rop}.  The string is of the form @samp{M@@N} or, if the base is 10 or
less, alternatively @samp{MeN}.  @samp{M} is the mantissa and @samp{N} is the
exponent.  The mantissa is always in the specified base.  The exponent is
either in the specified base or, if @var{base} is negative, in decimal.  The
decimal point expected is taken from the current locale, on systems providing
@code{localeconv}.

The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to
@minus{}2.  Negative values are used to specify that the exponent is in
decimal.

Unlike the corresponding @code{mpz} function, the base will not be determined
from the leading characters of the string if @var{base} is 0.  This is so that
numbers like @samp{0.23} are not interpreted as octal.

Return the number of bytes read, or if an error occurred, return 0.
@end deftypefun

@c @deftypefun void mpf_out_raw (FILE *@var{stream}, mpf_t @var{float})
@c Output @var{float} on stdio stream @var{stream}, in raw binary
@c format.  The float is written in a portable format, with 4 bytes of
@c size information, and that many bytes of limbs.  Both the size and the
@c limbs are written in decreasing significance order.
@c @end deftypefun

@c @deftypefun void mpf_inp_raw (mpf_t @var{float}, FILE *@var{stream})
@c Input from stdio stream @var{stream} in the format written by
@c @code{mpf_out_raw}, and put the result in @var{float}.
@c @end deftypefun

@node Miscellaneous Float Functions,  , I/O of Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous float functions
@cindex Float miscellaneous functions

@deftypefun void mpf_ceil (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_floor (mpf_t @var{rop}, mpf_t @var{op})
@deftypefunx void mpf_trunc (mpf_t @var{rop}, mpf_t @var{op})
Set @var{rop} to @var{op} rounded to an integer.  @code{mpf_ceil} rounds to the
next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc}
to the integer towards zero.
@end deftypefun

@deftypefun int mpf_integer_p (mpf_t @var{op})
Return non-zero if @var{op} is an integer.
@end deftypefun

@deftypefun int mpf_fits_ulong_p (mpf_t @var{op})
@deftypefunx int mpf_fits_slong_p (mpf_t @var{op})
@deftypefunx int mpf_fits_uint_p (mpf_t @var{op})
@deftypefunx int mpf_fits_sint_p (mpf_t @var{op})
@deftypefunx int mpf_fits_ushort_p (mpf_t @var{op})
@deftypefunx int mpf_fits_sshort_p (mpf_t @var{op})
Return non-zero if @var{op} would fit in the respective C data type, when
truncated to an integer.
@end deftypefun

@deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, unsigned long int @var{nbits})
Generate a uniformly distributed random float in @var{rop}, such that @math{0
@le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa.

The variable @var{state} must be initialized by calling one of the
@code{gmp_randinit} functions (@ref{Random State Initialization}) before
invoking this function.
@end deftypefun

@deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{exp})
Generate a random float of at most @var{max_size} limbs, with long strings of
zeros and ones in the binary representation.  The exponent of the number is in
the interval @minus{}@var{exp} to @var{exp}.  This function is useful for
testing functions and algorithms, since this kind of random numbers have proven
to be more likely to trigger corner-case bugs.  Negative random numbers are
generated when @var{max_size} is negative.
@end deftypefun

@c @deftypefun size_t mpf_size (mpf_t @var{op})
@c Return the size of @var{op} measured in number of limbs.  If @var{op} is
@c zero, the returned value will be zero.  (@xref{Nomenclature}, for an
@c explanation of the concept @dfn{limb}.)
@c
@c @strong{This function is obsolete.  It will disappear from future GMP
@c releases.}
@c @end deftypefun

@node Low-level Functions, Random Number Functions, Floating-point Functions, Top
@comment  node-name,  next,  previous,  up
@chapter Low-level Functions
@cindex Low-level functions

This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.

@c 1. Some of these function clobber input operands.
@c

The @code{mpn} functions are designed to be as fast as possible, @strong{not}
to provide a coherent calling interface.  The different functions have somewhat
similar interfaces, but there are variations that make them hard to use.  These
functions do as little as possible apart from the real multiple precision
computation, so that no time is spent on things that not all callers need.

A source operand is specified by a pointer to the least significant limb and a
limb count.  A destination operand is specified by just a pointer.  It is the
responsibility of the caller to ensure that the destination has enough space
for storing the result.

With this way of specifying operands, it is possible to perform computations on
subranges of an argument, and store the result into a subrange of a
destination.

A common requirement for all functions is that each source area needs at least
one limb.  No size argument may be zero.  Unless otherwise stated, in-place
operations are allowed where source and destination are the same, but not where
they only partly overlap.

The @code{mpn} functions are the base for the implementation of the
@code{mpz_}, @code{mpf_}, and @code{mpq_} functions.

This example adds the number beginning at @var{s1p} and the number beginning at
@var{s2p} and writes the sum at @var{destp}.  All areas have @var{n} limbs.

@example
cy = mpn_add_n (destp, s1p, s2p, n)
@end example

@noindent
In the notation used here, a source operand is identified by the pointer to
the least significant limb, and the limb count in braces.  For example,
@{@var{s1p}, @var{s1n}@}.

@deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Add @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n}
least significant limbs of the result to @var{rp}.  Return carry, either 0 or
1.

This is the lowest-level function for addition.  It is the preferred function
for addition, since it is written in assembly for most CPUs.  For addition of
a variable to itself (i.e., @var{s1p} equals @var{s2p}, use @code{mpn_lshift}
with a count of 1 for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Add @{@var{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least
significant limbs of the result to @var{rp}.  Return carry, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return carry,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to @var{s2n}.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Subtract @{@var{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the
@var{n} least significant limbs of the result to @var{rp}.  Return borrow,
either 0 or 1.

This is the lowest-level function for subtraction.  It is the preferred
function for subtraction, since it is written in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Subtract @var{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least
significant limbs of the result to @var{rp}.  Return borrow, either 0 or 1.
@end deftypefun

@deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the
@var{s1n} least significant limbs of the result to @var{rp}.  Return borrow,
either 0 or 1.

This function requires that @var{s1n} is greater than or equal to
@var{s2n}.
@end deftypefun

@deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the
2*@var{n}-limb result to @var{rp}.

The destination has to have space for 2*@var{n} limbs, even if the product's
most significant limb is zero.
@end deftypefun

@deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least
significant limbs of the product to @var{rp}.  Return the most significant
limb of the product.  @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are
allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}.

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written in assembly
for most CPUs.

Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift}
with a count equal to the logarithm of @var{s2limb} instead, for optimal speed.
@end deftypefun

@deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least
significant limbs of the product to @{@var{rp}, @var{n}@} and write the result
to @var{rp}.  Return the most significant limb of the product, plus carry-out

This is a low-level function that is a building block for general
multiplication as well as other operations in GMP.  It is written in assembly
for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb})
Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n}
least significant limbs of the product from @{@var{rp}, @var{n}@} and write the
result to @var{rp}.  Return the most significant limb of the product, minus
borrow-out from the subtraction.

This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.  It is written
in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the
result to @var{rp}.  Return the most significant limb of the result.

The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the
result might be one limb smaller.

This function requires that @var{s1n} is greater than or equal to
@var{s2n}.  The destination must be distinct from both input operands.
@end deftypefun

@deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn})
Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient
at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp},
@var{dn}@}.  The quotient is rounded towards 0.

No overlap is permitted between arguments.  @var{nn} must be greater than or
equal to @var{dn}.  The most significant limb of @var{dp} must be non-zero.
The @var{qxn} operand must be zero.
@comment FIXME: Relax overlap requirements!
@end deftypefun

@deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
performance.]

Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the
quotient at @var{r1p}, with the exception of the most significant limb, which
is returned.  The remainder replaces the dividend at @var{rs2p}; it will be
@var{s3n} limbs long (i.e., as many limbs as the divisor).

In addition to an integer quotient, @var{qxn} fraction limbs are developed, and
stored after the integral limbs.  For most usages, @var{qxn} will be zero.

It is required that @var{rs2n} is greater than or equal to @var{s3n}.  It is
required that the most significant bit of the divisor is set.

If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}.  Aside
from that special case, no overlap between arguments is permitted.

Return the most significant limb of the quotient, either 0 or 1.

The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn}
limbs large.
@end deftypefun

@deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb})
@deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}})
Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at
@var{r1p}.  Return the remainder.

The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in
addition @var{qxn} fraction limbs are developed and written to @{@var{r1p},
@var{qxn}@}.  Either or both @var{s2n} and @var{qxn} can be zero.  For most
usages, @var{qxn} will be zero.

@code{mpn_divmod_1} exists for upward source compatibility and is simply a
macro calling @code{mpn_divrem_1} with a @var{qxn} of 0.

The areas at @var{r1p} and @var{s2p} have to be identical or completely
separate, not partially overlapping.
@end deftypefn

@deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n})
performance.]
@end deftypefun

@deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}})
@deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry})
Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing
the result to @{@var{rp}, @var{n}@}.  If 3 divides exactly, the return value is
zero and the result is the quotient.  If not, the return value is non-zero and
the result won't be anything useful.

@code{mpn_divexact_by3c} takes an initial carry parameter, which can be the
return value from a previous call, so a large calculation can be done piece by
piece from low to high.  @code{mpn_divexact_by3} is simply a macro calling
@code{mpn_divexact_by3c} with a 0 carry parameter.

These routines use a multiply-by-inverse and will be faster than
@code{mpn_divrem_1} on CPUs with fast multiplication but slow division.

The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i},
and return value @math{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
@m{b=2\GMPraise{@code{mp\_bits\_per\_limb}}, b=2^mp_bits_per_limb}.  The
return @math{c} is always 0, 1 or 2, and the initial carry @math{i} must also
be 0, 1 or 2 (these are both borrows really).  When @math{c=0} clearly
@math{q=(a-i)/3}.  When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{}
3} is given by @math{3-c}, because @math{b @equiv{} 1 @bmod{} 3} (when
@code{mp_bits_per_limb} is even, which is always so currently).
@end deftypefn

@deftypefun mp_limb_t mpn_mod_1 (mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder.
@var{s1n} can be zero.
@end deftypefun

@deftypefun mp_limb_t mpn_bdivmod (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}, unsigned long int @var{d})
This function puts the low
@math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of @var{q} =
@{@var{s1p}, @var{s1n}@}/@{@var{s2p}, @var{s2n}@} mod @m{2^d,2^@var{d}} at
@var{rp}, and returns the high @var{d} mod @code{mp_bits_per_limb} bits of
@var{q}.

@{@var{s1p}, @var{s1n}@} - @var{q} * @{@var{s2p}, @var{s2n}@} mod @m{2
\GMPraise{@var{s1n}*@code{mp\_bits\_per\_limb}},
2^(@var{s1n}*@nicode{mp\_bits\_per\_limb})} is placed at @var{s1p}.  Since the
low @math{@GMPfloor{@var{d}/@nicode{mp\_bits\_per\_limb}}} limbs of this
difference are zero, it is possible to overwrite the low limbs at @var{s1p}
with this difference, provided @math{@var{rp} @le{} @var{s1p}}.

This function requires that @math{@var{s1n} * @nicode{mp\_bits\_per\_limb}
@ge{} @var{D}}, and that @{@var{s2p}, @var{s2n}@} is odd.

@strong{This interface is preliminary.  It might change incompatibly in future
revisions.}
@end deftypefun

@deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the left are returned in the
least significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@math{@var{rp} @ge{} @var{sp}}.

This function is written in assembly for most CPUs.
@end deftypefun

@deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count})
Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to
@{@var{rp}, @var{n}@}.  The bits shifted out at the right are returned in the
most significant @var{count} bits of the return value (the rest of the return
value is zero).

@var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1.  The
regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided
@math{@var{rp} @le{} @var{sp}}.

This function is written in assembly for most CPUs.
@end deftypefun

@deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a
positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a
negative value if @math{@var{s1} < @var{s2}}.
@end deftypefun

@deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{s1p},
@var{s1n}@} and @{@var{s2p}, @var{s2n}@}.  The result can be up to @var{s2n}
limbs, the return value is the actual number produced.  Both source operands
are destroyed.

@{@var{s1p}, @var{s1n}@} must have at least as many bits as @{@var{s2p},
@var{s2n}@}.  @{@var{s2p}, @var{s2n}@} must be odd.  Both operands must have
non-zero most significant limbs.  No overlap is permitted between @{@var{s1p},
@var{s1n}@} and @{@var{s2p}, @var{s2n}@}.
@end deftypefun

@deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb})
Return the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
@var{s2limb}.  Both operands must be non-zero.
@end deftypefun

@deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, mp_size_t *@var{r2n}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t *@var{s2p}, mp_size_t @var{s2n})
Calculate the greatest common divisor of @{@var{s1p}, @var{s1n}@} and
@{@var{s2p}, @var{s2n}@}.  Store the gcd at @{@var{r1p}, @var{retval}@} and
the first cofactor at @{@var{r2p}, *@var{r2n}@}, with *@var{r2n} negative if
the cofactor is negative.  @var{r1p} and @var{r2p} should each have room for
@math{@var{s1n}+1} limbs, but the return value and value stored through
@var{r2n} indicate the actual number produced.

@math{@{@var{s1p}, @var{s1n}@} @ge{} @{@var{s2p}, @var{s2n}@}} is required,
and both must be non-zero.  The regions @{@var{s1p}, @math{@var{s1n}+1}@} and
@{@var{s2p}, @math{@var{s2n}+1}@} are destroyed (i.e. the operands plus an
extra limb past the end of each).

The cofactor @var{r1} will satisfy @m{r_2 s_1 + k s_2 = r_1, @var{r2}*@var{s1}
+ @var{k}*@var{s2} = @var{r1}}.  The second cofactor @var{k} is not calculated
but can easily be obtained from @m{(r_1 - r_2 s_1) / s_2, (@var{r1} -
@var{r2}*@var{s1}) / @var{s2}}.
@end deftypefun

@deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n})
Compute the square root of @{@var{sp}, @var{n}@} and put the result at
@{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p},
@var{retval}@}.  @var{r2p} needs space for @var{n} limbs, but the return value
indicates how many are produced.

The most significant limb of @{@var{sp}, @var{n}@} must be non-zero.  The
areas @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must
be completely separate.  The areas @{@var{r2p}, @var{n}@} and @{@var{sp},
@var{n}@} must be either identical or completely separate.

If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this
case the return value is zero or non-zero according to whether the remainder
would have been zero or non-zero.

@code{mpz_perfect_square_p}.
@end deftypefun

@deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n})
Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in
base @var{base}, and return the number of characters produced.  There may be
leading zeros in the string.  The string is not in ASCII; to convert it to
printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on
the base and range.  @var{base} can vary from 2 to 256.

The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be
non-zero.  The input @{@var{s1p}, @var{s1n}@} is clobbered, except when
@var{base} is a power of 2, in which case it's unchanged.

The area at @var{str} has to have space for the largest possible number
represented by a @var{s1n} long limb array, plus one extra character.
@end deftypefun

@deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{rp}, const unsigned char *@var{str}, size_t @var{strsize}, int @var{base})
Convert bytes @{@var{str},@var{strsize}@} in the given @var{base} to limbs at
@var{rp}.

@math{@var{str}[0]} is the most significant byte and
@math{@var{str}[@var{strsize}-1]} is the least significant.  Each byte should
be a value in the range 0 to @math{@var{base}-1}, not an ASCII character.
@var{base} can vary from 2 to 256.

The return value is the number of limbs written to @var{rp}.  If the most
significant input byte is non-zero then the high limb at @var{rp} will be
non-zero, and only that exact number of limbs will be required there.

If the most significant input byte is zero then there may be high zero limbs
written to @var{rp} and included in the return value.

@var{strsize} must be at least 1, and no overlap is permitted between
@{@var{str},@var{strsize}@} and the result at @var{rp}.
@end deftypefun

@deftypefun {unsigned long int} mpn_scan0 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next clear bit.

It is required that there be a clear bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.
@end deftypefun

@deftypefun {unsigned long int} mpn_scan1 (const mp_limb_t *@var{s1p}, unsigned long int @var{bit})
Scan @var{s1p} from bit position @var{bit} for the next set bit.

It is required that there be a set bit within the area at @var{s1p} at or
beyond bit position @var{bit}, so that the function has something to return.
@end deftypefun

@deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
@deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n})
Generate a random number of length @var{r1n} and store it at @var{r1p}.  The
most significant limb is always non-zero.  @code{mpn_random} generates
uniformly distributed limb data, @code{mpn_random2} generates long strings of
zeros and ones in the binary representation.

@code{mpn_random2} is intended for testing the correctness of the @code{mpn}
routines.
@end deftypefun

@deftypefun {unsigned long int} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Count the number of set bits in @{@var{s1p}, @var{n}@}.
@end deftypefun

@deftypefun {unsigned long int} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n})
Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p},
@var{n}@}.
@end deftypefun

@deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n})
Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square.
@end deftypefun

@sp 1
@section Nails
@cindex Nails

@strong{Everything in this section is highly experimental and may disappear or
be subject to incompatible changes in a future version of GMP.}

Nails are an experimental feature whereby a few bits are left unused at the
top of each @code{mp_limb_t}.  This can significantly improve carry handling
on some processors.

All the @code{mpn} functions accepting limb data will expect the nail bits to
be zero on entry, and will return data with the nails similarly all zero.
This applies both to limb vectors and to single limb arguments.

Nails can be enabled by configuring with @samp{--enable-nails}.  By default
the number of bits will be chosen according to what suits the host processor,
but a particular number can be selected with @samp{--enable-nails=N}.

At the mpn level, a nail build is neither source nor binary compatible with a
non-nail build, strictly speaking.  But programs acting on limbs only through
the mpn functions are likely to work equally well with either build, and
judicious use of the definitions below should make any program compatible with
either build, at the source level.

For the higher level routines, meaning @code{mpz} etc, a nail build should be
fully source and binary compatible with a non-nail build.

@defmac GMP_NAIL_BITS
@defmacx GMP_NUMB_BITS
@defmacx GMP_LIMB_BITS
@code{GMP_NAIL_BITS} is the number of nail bits, or 0 when nails are not in
use.  @code{GMP_NUMB_BITS} is the number of data bits in a limb.
@code{GMP_LIMB_BITS} is the total number of bits in an @code{mp_limb_t}.  In
all cases

@example
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
@end example
@end defmac

Bit masks for the nail and number parts of a limb.  @code{GMP_NAIL_MASK} is 0
when nails are not in use.

@code{GMP_NAIL_MASK} is not often needed, since the nail part can be obtained
with @code{x >> GMP_NUMB_BITS}, and that means one less large constant, which
can help various RISC chips.
@end defmac

@defmac GMP_NUMB_MAX
The maximum value that can be stored in the number part of a limb.  This is
the same as @code{GMP_NUMB_MASK}, but can be used for clarity when doing
comparisons rather than bit-wise operations.
@end defmac

The term nails'' comes from finger or toe nails, which are at the ends of a
limb (arm or leg).  numb'' is short for number, but is also how the
developers felt after trying for a long time to come up with sensible names
for these things.

In the future (the distant future most likely) a non-zero nail might be
permitted, giving non-unique representations for numbers in a limb vector.
This would help vector processors since carries would only ever need to
propagate one or two limbs.

@node Random Number Functions, Formatted Output, Low-level Functions, Top
@chapter Random Number Functions
@cindex Random number functions

Sequences of pseudo-random numbers in GMP are generated using a variable of
type @code{gmp_randstate_t}, which holds an algorithm selection and a current
state.  Such a variable must be initialized by a call to one of the
@code{gmp_randinit} functions, and can be seeded with one of the
@code{gmp_randseed} functions.

The functions actually generating random numbers are described in @ref{Integer
Random Numbers}, and @ref{Miscellaneous Float Functions}.

The older style random number functions don't accept a @code{gmp_randstate_t}
parameter but instead share a global variable of that type.  They use a
default algorithm and are currently not seeded (though perhaps that will
change in the future).  The new functions accepting a @code{gmp_randstate_t}
are recommended for applications that care about randomness.

* Random State Initialization::
* Random State Seeding::

@node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions
@section Random State Initialization
@cindex Random number state

@deftypefun void gmp_randinit_default (gmp_randstate_t @var{state})
Initialize @var{state} with a default algorithm.  This will be a compromise
between speed and randomness, and is recommended for applications with no
special requirements.
@end deftypefun

@deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a}, @w{unsigned long @var{c}}, @w{unsigned long @var{m2exp}})
Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X +
@var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}.

The low bits of @math{X} in this algorithm are not very random.  The least
significant bit will have a period no more than 2, and the second bit no more
than 4, etc.  For this reason only the high half of each @math{X} is actually
used.

When a random number of more than @math{@var{m2exp}/2} bits is to be
generated, multiple iterations of the recurrence are used and the results
concatenated.
@end deftypefun

@deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, unsigned long @var{size})
Initialize @var{state} for a linear congruential algorithm as per
@code{gmp_randinit_lc_2exp}.  @var{a}, @var{c} and @var{m2exp} are selected
from a table, chosen so that @var{size} bits (or more) of each @math{X} will
be used, ie. @math{@var{m2exp}/2 @ge{} @var{size}}.

If successful the return value is non-zero.  If @var{size} is bigger than the
table data provides then the return value is zero.  The maximum @var{size}
currently supported is 128.
@end deftypefun

@deftypefun void gmp_randinit (gmp_randstate_t @var{state}, @w{gmp_randalg_t @var{alg}}, ...)
@strong{This function is obsolete.}

Initialize @var{state} with an algorithm selected by @var{alg}.  The only
choice is @code{GMP_RAND_ALG_LC}, which is @code{gmp_randinit_lc_2exp_size}.
A third parameter of type @code{unsigned long} is required, this is the
@var{size} for that function.  @code{GMP_RAND_ALG_DEFAULT} or 0 are the same
as @code{GMP_RAND_ALG_LC}.

@code{gmp_randinit} sets bits in @code{gmp_errno} to indicate an error.
@code{GMP_ERROR_UNSUPPORTED_ARGUMENT} if @var{alg} is unsupported, or
@code{GMP_ERROR_INVALID_ARGUMENT} if the @var{size} parameter is too big.
@end deftypefun

@c  Not yet in the library.
@ignore
@deftypefun void gmp_randinit_lc (gmp_randstate_t @var{state}, mpz_t @var{a}, unsigned long int @var{c}, mpz_t @var{m})
Initialize @var{state} for a linear congruential scheme @m{X = (@var{a}X +
@var{c}) @bmod @var{m}, X = (@var{a}*X + @var{c}) mod 2^@var{m}}.
@end deftypefun
@end ignore

@deftypefun void gmp_randclear (gmp_randstate_t @var{state})
Free all memory occupied by @var{state}.
@end deftypefun

@node Random State Seeding,  , Random State Initialization, Random Number Functions
@section Random State Seeding
@cindex Random number seeding

@deftypefun void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed})
@deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, @w{unsigned long int @var{seed}})
Set an initial seed value into @var{state}.

The size of a seed determines how many different sequences of random numbers
that it's possible to generate.  The quality'' of the seed is the randomness
of a given seed compared to the previous seed used, and this affects the
randomness of separate number sequences.  The method for choosing a seed is
critical if the generated numbers are to be used for important applications,
such as generating cryptographic keys.

Traditionally the system time has been used to seed, but care needs to be
taken with this.  If an application seeds often and the resolution of the
system clock is low, then the same sequence of numbers might be repeated.
Also, the system time is quite easy to guess, so if unpredictability is
required then it should definitely not be the only source for the seed value.
On some systems there's a special device @file{/dev/random} which provides
random data better suited for use as a seed.
@end deftypefun

@node Formatted Output, Formatted Input, Random Number Functions, Top
@chapter Formatted Output
@cindex Formatted output
@cindex @code{printf} formatted output

* Formatted Output Strings::
* Formatted Output Functions::
* C++ Formatted Output::

@node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output
@section Format Strings

@code{gmp_printf} and friends accept format strings similar to the standard C
@code{printf} (@pxref{Formatted Output,,,libc,The GNU C Library Reference
Manual}).  A format specification is of the form

@example
% [flags] [width] [.[precision]] [type] conv
@end example

GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t}
and @code{mpf_t} respectively, and @samp{N} for an @code{mp_limb_t} array.
@samp{Z}, @samp{Q} and @samp{N} behave like integers.  @samp{Q} will print a
@samp{/} and a denominator, if needed.  @samp{F} behaves like a float.  For
example,

@example
mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);

mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);

mpf_t f;
int   n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);

const mp_limb_t *ptr;
mp_size_t       size;
gmp_printf ("limb array %Nx\n", ptr, size);
@end example

For @samp{N} the limbs are expected least significant first, as per the
@code{mpn} functions (@pxref{Low-level Functions}).  A negative size can be
given to print the value as a negative.

All the standard C @code{printf} types behave the same as the C library
@code{printf}, and can be freely intermixed with the GMP extensions.  In the
current implementation the standard parts of the format string are simply
handed to @code{printf} and only the GMP extensions handled directly.

The flags accepted are as follows.  GLIBC style @nisamp{'} is only for the
standard C types (not the GMP types), and only if the C library supports it.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{0} @tab pad with zeros (rather than spaces)
@item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0}
@item @nicode{+} @tab always show a sign
@item (space)    @tab show a space or a @samp{-} sign
@item @nicode{'} @tab group digits, GLIBC style (not GMP types)
@end multitable
@end quotation

The optional width and precision can be given as a number within the format
string, or as a @samp{*} to take an extra parameter of type @code{int}, the
same as the standard @code{printf}.

The standard types accepted are as follows.  @samp{h} and @samp{l} are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the output.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{h}  @tab @nicode{short}
@item @nicode{hh} @tab @nicode{char}
@item @nicode{j}  @tab @nicode{intmax_t} or @nicode{uintmax_t}
@item @nicode{l}  @tab @nicode{long} or @nicode{wchar_t}
@item @nicode{ll} @tab @nicode{long long}
@item @nicode{L}  @tab @nicode{long double}
@item @nicode{t}  @tab @nicode{ptrdiff_t}
@item @nicode{z}  @tab @nicode{size_t}
@end multitable
@end quotation

@noindent
The GMP types are

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{F}  @tab @nicode{mpf_t}, float conversions
@item @nicode{Q}  @tab @nicode{mpq_t}, integer conversions
@item @nicode{N}  @tab @nicode{mp_limb_t} array, integer conversions
@item @nicode{Z}  @tab @nicode{mpz_t}, integer conversions
@end multitable
@end quotation

The conversions accepted are as follows.  @samp{a} and @samp{A} are always
supported for @code{mpf_t} but depend on the C library for standard C float
types.  @samp{m} and @samp{p} depend on the C library.

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @nicode{a} @nicode{A} @tab hex floats, C99 style
@item @nicode{c}            @tab character
@item @nicode{d}            @tab decimal integer
@item @nicode{e} @nicode{E} @tab scientific format float
@item @nicode{f}            @tab fixed point float
@item @nicode{i}            @tab same as @nicode{d}
@item @nicode{g} @nicode{G} @tab fixed or scientific float
@item @nicode{m}            @tab @code{strerror} string, GLIBC style
@item @nicode{n}            @tab store characters written so far
@item @nicode{o}            @tab octal integer
@item @nicode{p}            @tab pointer
@item @nicode{s}            @tab string
@item @nicode{u}            @tab unsigned integer
@item @nicode{x} @nicode{X} @tab hex integer
@end multitable
@end quotation

@samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for
types @samp{Z}, @samp{Q} and @samp{N} they are signed.  @samp{u} is not
meaningful for @samp{Z}, @samp{Q} and @samp{N}.

@samp{n} can be used with any type, even the GMP types.

Other types or conversions that might be accepted by the C library
@code{printf} cannot be used through @code{gmp_printf}, this includes for
instance extensions registered with GLIBC @code{register_printf_function}.
Also currently there's no support for POSIX @samp{$} style numbered arguments (perhaps this will be added in the future). The precision field has it's usual meaning for integer @samp{Z} and float @samp{F} types, but is currently undefined for @samp{Q} and should not be used with that. @code{mpf_t} conversions only ever generate as many digits as can be accurately represented by the operand, the same as @code{mpf_get_str} does. Zeros will be used if necessary to pad to the requested precision. This happens even for an @samp{f} conversion of an @code{mpf_t} which is an integer, for instance @math{2^@W{1024}} in an @code{mpf_t} of 128 bits precision will only produce about 40 digits, then pad with zeros to the decimal point. An empty precision field like @samp{%.Fe} or @samp{%.Ff} can be used to specifically request just the significant digits. The decimal point character (or string) is taken from the current locale settings on systems which provide @code{localeconv} (@pxref{Locales,,Locales and Internationalization,libc,The GNU C Library Reference Manual}). The C library will normally do the same for standard float output. The format string is only interpreted as plain @code{char}s, multibyte characters are not recognised. Perhaps this will change in the future. @node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output @section Functions Each of the following functions is similar to the corresponding C library function. The basic @code{printf} forms take a variable argument list. The @code{vprintf} forms take an argument pointer, see @ref{Variadic Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3 va_start}. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return @math{-1} to indicate a write error. All the functions can return @math{-1} if the C library @code{printf} variant in use returns @math{-1}, but this shouldn't normally occur. @deftypefun int gmp_printf (const char *@var{fmt}, ...) @deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap}) Print to the standard output @code{stdout}. Return the number of characters written, or @math{-1} if an error occurred. @end deftypefun @deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, ...) @deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap}) Print to the stream @var{fp}. Return the number of characters written, or @math{-1} if an error occurred. @end deftypefun @deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, ...) @deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in @var{buf}. Return the number of characters written, excluding the terminating null. No overlap is permitted between the space at @var{buf} and the string @var{fmt}. These functions are not recommended, since there's no protection against exceeding the space available at @var{buf}. @end deftypefun @deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, ...) @deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in @var{buf}. No more than @var{size} bytes will be written. To get the full output, @var{size} must be enough for the string and null-terminator. The return value is the total number of characters which ought to have been produced, excluding the terminating null. If @math{@var{retval} @ge{} @var{size}} then the actual output has been truncated to the first @math{@var{size}-1} characters, and a null appended. No overlap is permitted between the region @{@var{buf},@var{size}@} and the @var{fmt} string. Notice the return value is in ISO C99 @code{snprintf} style. This is so even if the C library @code{vsnprintf} is the older GLIBC 2.0.x style. @end deftypefun @deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, ...) @deftypefunx int gmp_vasprintf (char *@var{pp}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in a block of memory obtained from the current memory allocation function (@pxref{Custom Allocation}). The block will be the size of the string and null-terminator. Put the address of the block in *@var{pp}. Return the number of characters produced, excluding the null-terminator. Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return @math{-1} if there's no more memory available, it lets the current allocation function handle that. @end deftypefun @deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, ...) @deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap}) Append to the current obstack object, in the same style as @code{obstack_printf}. Return the number of characters written. A null-terminator is not written. @var{fmt} cannot be within the current obstack object, since the object might move as it grows. These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see @ref{Obstacks,,,libc,The GNU C Library Reference Manual}. @end deftypefun @node C++ Formatted Output, , Formatted Output Functions, Formatted Output @section C++ Formatted Output @cindex C++ @code{ostream} output @cindex @code{ostream} output The following functions are provided in @file{libgmpxx}, which is built if C++ support is enabled (@pxref{Build Options}). Prototypes are available from @code{<gmp.h>}. @deftypefun ostream& operator<< (ostream& @var{stream}, mpz_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. In hex or octal, @var{op} is printed as a signed number, the same as for decimal. This is unlike the standard @code{operator<<} routines on @code{int} etc, which instead give twos complement. @end deftypefun @deftypefun ostream& operator<< (ostream& @var{stream}, mpq_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. Output will be a fraction like @samp{5/9}, or if the denominator is 1 then just a plain integer like @samp{123}. In hex or octal, @var{op} is printed as a signed value, the same as for decimal. If @code{ios::showbase} is set then a base indicator is shown on both the numerator and denominator (if the denominator is required). @end deftypefun @deftypefun ostream& operator<< (ostream& @var{stream}, mpf_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. The decimal point follows the current locale, on systems providing @code{localeconv}. Hex and octal are supported, unlike the standard @code{operator<<} on @code{double}. The mantissa will be in hex or octal, the exponent will be in decimal. For hex the exponent delimiter is an @samp{@@}. This is as per @code{mpf_out_str}. @code{ios::showbase} is supported, and will put a base on the mantissa, for example hex @samp{0x1.8} or @samp{0x0.8}, or octal @samp{01.4} or @samp{00.4}. This last form is slightly strange, but at least differentiates itself from decimal. @end deftypefun These operators mean that GMP types can be printed in the usual C++ way, for example, @example mpz_t z; int n; ... cout << "iteration " << n << " value " << z << "\n"; @end example But note that @code{ostream} output (and @code{istream} input, @pxref{C++ Formatted Input}) is the only overloading available and using for instance @code{+} with an @code{mpz_t} will have unpredictable results. @node Formatted Input, C++ Class Interface, Formatted Output, Top @chapter Formatted Input @cindex Formatted input @cindex @code{scanf} formatted input @menu * Formatted Input Strings:: * Formatted Input Functions:: * C++ Formatted Input:: @end menu @node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input @section Formatted Input Strings @code{gmp_scanf} and friends accept format strings similar to the standard C @code{scanf} (@pxref{Formatted Input,,,libc,The GNU C Library Reference Manual}). A format specification is of the form @example % [flags] [width] [type] conv @end example GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t} and @code{mpf_t} respectively. @samp{Z} and @samp{Q} behave like integers. @samp{Q} will read a @samp{/} and a denominator, if present. @samp{F} behaves like a float. GMP variables don't require an @code{&} when passed to @code{gmp_scanf}, since they're already call-by-reference''. For example, @example /* to read say "a(5) = 1234" */ int n; mpz_t z; gmp_scanf ("a(%d) = %Zd\n", &n, z); mpq_t q1, q2; gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2); /* to read say "topleft (1.55,-2.66)" */ mpf_t x, y; char buf[32]; gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y); @end example All the standard C @code{scanf} types behave the same as in the C library @code{scanf}, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to @code{scanf} and only the GMP extensions handled directly. The flags accepted are as follows. @samp{a} and @samp{'} will depend on support from the C library, and @samp{'} cannot be used with GMP types. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{*} @tab read but don't store @item @nicode{a} @tab allocate a buffer (string conversions) @item @nicode{'} @tab group digits, GLIBC style (not GMP types) @end multitable @end quotation The standard types accepted are as follows. @samp{h} and @samp{l} are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{h} @tab @nicode{short} @item @nicode{hh} @tab @nicode{char} @item @nicode{j} @tab @nicode{intmax_t} or @nicode{uintmax_t} @item @nicode{l} @tab @nicode{long int}, @nicode{double} or @nicode{wchar_t} @item @nicode{ll} @tab @nicode{long long} @item @nicode{L} @tab @nicode{long double} @item @nicode{q} @tab @nicode{quad_t} or @nicode{u_quad_t} @item @nicode{t} @tab @nicode{ptrdiff_t} @item @nicode{z} @tab @nicode{size_t} @end multitable @end quotation @noindent The GMP types are @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{F} @tab @nicode{mpf_t}, float conversions @item @nicode{Q} @tab @nicode{mpq_t}, integer conversions @item @nicode{Z} @tab @nicode{mpz_t}, integer conversions @end multitable @end quotation The conversions accepted are as follows. @samp{p} and @samp{[} will depend on support from the C library, the rest are standard. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{c} @tab character or characters @item @nicode{d} @tab decimal integer @item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G} @tab float @item @nicode{i} @tab integer with base indicator @item @nicode{n} @tab characters read so far @item @nicode{o} @tab octal integer @item @nicode{p} @tab pointer @item @nicode{s} @tab string of non-whitespace characters @item @nicode{u} @tab decimal integer @item @nicode{x} @nicode{X} @tab hex integer @item @nicode{[} @tab string of characters in a set @end multitable @end quotation @samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all read either fixed point or scientific format, and either @samp{e} or @samp{E} for the exponent in scientific format. @samp{x} and @samp{X} are identical, both accept both upper and lower case hexadecimal. @samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative values. For the standard C types these are described as unsigned'' conversions, but that merely affects certain overflow handling, negatives are still allowed (see @code{strtoul}, @ref{Parsing of Integers,,,libc,The GNU C Library Reference Manual}). For GMP types there are no overflows, and @samp{d} and @samp{u} are identical. @samp{Q} type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then @code{mpq_canonicalize} must be called before using it in any calculations (@pxref{Rational Number Functions}). @samp{Qi} will read a base specification separately for the numerator and denominator. For example @samp{0x10/11} would be 16/11, whereas @samp{0x10/0x11} would be 16/17. @samp{n} can be used with any of the types above, even the GMP types. @samp{*} to suppress assignment is allowed, though the field would then do nothing at all. Other conversions or types that might be accepted by the C library @code{scanf} cannot be used through @code{gmp_scanf}. Whitespace is read and discarded before a field, except for @samp{c} and @samp{[} conversions. For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide @code{localeconv} (@pxref{Locales,,Locales and Internationalization,libc,The GNU C Library Reference Manual}). The C library will normally do the same for standard float input. The format string is only interpreted as plain @code{char}s, multibyte characters are not recognised. Perhaps this will change in the future. @node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input @section Formatted Input Functions Each of the following functions is similar to the corresponding C library function. The plain @code{scanf} forms take a variable argument list. The @code{vscanf} forms take an argument pointer, see @ref{Variadic Functions,,,libc,The GNU C Library Reference Manual}, or @samp{man 3 va_start}. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. No overlap is permitted between the @var{fmt} string and any of the results produced. @deftypefun int gmp_scanf (const char *@var{fmt}, ...) @deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap}) Read from the standard input @code{stdin}. @end deftypefun @deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, ...) @deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap}) Read from the stream @var{fp}. @end deftypefun @deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, ...) @deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap}) Read from a null-terminated string @var{s}. @end deftypefun The return value from each of these functions is the same as the standard C99 @code{scanf}, namely the number of fields successfully parsed and stored. @samp{%n} fields and fields read but suppressed by @samp{*} don't count towards the return value. If end of file or file error, or end of string, is reached when a match is required, and when no previous non-suppressed fields have matched, then the return value is EOF instead of 0. A match is required for a literal character in the format string or a field other than @samp{%n}. Whitespace in the format string is only an optional match and won't induce an EOF in this fashion. Leading whitespace read and discarded for a field doesn't count as a match. @node C++ Formatted Input, , Formatted Input Functions, Formatted Input @section C++ Formatted Input @cindex C++ @code{istream} input @cindex @code{istream} input The following functions are provided in @file{libgmpxx}, which is built only if C++ support is enabled (@pxref{Build Options}). Prototypes are available from @code{<gmp.h>}. @deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop}) Read @var{rop} from @var{stream}, using its @code{ios} formatting settings. @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop}) Read @var{rop} from @var{stream}, using its @code{ios} formatting settings. An integer like @samp{123} will be read, or a fraction like @samp{5/9}. If the fraction is not in canonical form then @code{mpq_canonicalize} must be called (@pxref{Rational Number Functions}). @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop}) Read @var{rop} from @var{stream}, using its @code{ios} formatting settings. Hex or octal floats are not supported, but might be in the future. @end deftypefun These operators mean that GMP types can be read in the usual C++ way, for example, @example mpz_t z; ... cin >> z; @end example But note that @code{istream} input (and @code{ostream} output, @pxref{C++ Formatted Output}) is the only overloading available and using for instance @code{+} with an @code{mpz_t} will have unpredictable results. @node C++ Class Interface, BSD Compatible Functions, Formatted Input, Top @chapter C++ Class Interface @cindex C++ Interface This chapter describes the C++ class based interface to GMP. All GMP C language types and functions can be used in C++ programs, since @file{gmp.h} has @code{extern "C"} qualifiers, but the class interface offers overloaded functions and operators which may be more convenient. Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. For GCC this means version 2.91 or later. @strong{Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.} @menu * C++ Interface General:: * C++ Interface Integers:: * C++ Interface Rationals:: * C++ Interface Floats:: * C++ Interface MPFR:: * C++ Interface Random Numbers:: * C++ Interface Limitations:: @end menu @node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface @section C++ Interface General @noindent All the C++ classes and functions are available with @cindex gmpxx.h @example #include <gmpxx.h> @end example Programs should be linked with the @file{libgmpxx} and @file{libgmp} libraries. For example, @example g++ mycxxprog.cc -lgmpxx -lgmp @end example @noindent The classes defined are @deftp Class mpz_class @deftpx Class mpq_class @deftpx Class mpf_class @end deftp The standard operators and various standard functions are overloaded to allow arithmetic with these classes. For example, @example int main (void) @{ mpz_class a, b, c; a = 1234; b = "-5678"; c = a+b; cout << "sum is " << c << "\n"; cout << "absolute value is " << abs(c) << "\n"; return 0; @} @end example An important feature of the implementation is that an expression like @code{a=b+c} results in a single call to the corresponding @code{mpz_add}, without using a temporary for the @code{b+c} part. Expressions which by their nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries though. The classes can be freely intermixed in expressions, as can the classes and the standard types @code{long}, @code{unsigned long} and @code{double}. Smaller types like @code{int} or @code{float} can also be intermixed, since C++ will promote them. Note that @code{bool} is not accepted directly, but must be explicitly cast to an @code{int} first. This is because C++ will automatically convert any pointer to a @code{bool}, so if GMP accepted @code{bool} it would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible. Conversions back from the classes to standard C++ types aren't done automatically, instead member functions like @code{get_si} are provided (see the following sections for details). Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions, @deftypefun mpz_t mpz_class::get_mpz_t () @deftypefunx mpq_t mpq_class::get_mpq_t () @deftypefunx mpf_t mpf_class::get_mpf_t () @end deftypefun These can be used to call a C function which doesn't have a C++ class interface. For example to set @code{a} to the GCD of @code{b} and @code{c}, @example mpz_class a, b, c; ... mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t()); @end example In the other direction, a class can be initialized from the corresponding GMP C type, or assigned to if an explicit constructor is used. In both cases this makes a copy of the value, it doesn't create any sort of association. For example, @example mpz_t z; // ... init and calculate z ... mpz_class x(z); mpz_class y; y = mpz_class (z); @end example There are no namespace setups in @file{gmpxx.h}, all types and functions are simply put into the global namespace. This is what @file{gmp.h} has done in the past, and continues to do for compatibility. The extras provided by @file{gmpxx.h} follow GMP naming conventions and are unlikely to clash with anything. @node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface @section C++ Interface Integers @deftypefun void mpz_class::mpz_class (type @var{n}) Construct an @code{mpz_class}. All the standard C++ types may be used, except @code{long long} and @code{long double}, and all the GMP C++ classes can be used. Any necessary conversion follows the corresponding C function, for example @code{double} follows @code{mpz_set_d} (@pxref{Assigning Integers}). @end deftypefun @deftypefun void mpz_class::mpz_class (mpz_t @var{z}) Construct an @code{mpz_class} from an @code{mpz_t}. The value in @var{z} is copied into the new @code{mpz_class}, there won't be any permanent association between it and @var{z}. @end deftypefun @deftypefun void mpz_class::mpz_class (const char *@var{s}) @deftypefunx void mpz_class::mpz_class (const char *@var{s}, int base) @deftypefunx void mpz_class::mpz_class (const string& @var{s}) @deftypefunx void mpz_class::mpz_class (const string& @var{s}, int base) Construct an @code{mpz_class} converted from a string using @code{mpz_set_str}, (@pxref{Assigning Integers}). If the @var{base} is not given then 0 is used. @end deftypefun @deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d}) @deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d}) Divisions involving @code{mpz_class} round towards zero, as per the @code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}). This corresponds to the rounding used for plain @code{int} calculations on most machines. The @code{mpz_fdiv...} or @code{mpz_cdiv...} functions can always be called directly if desired. For example, @example mpz_class q, a, d; ... mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t()); @end example @end deftypefun @deftypefun mpz_class abs (mpz_class @var{op1}) @deftypefunx int cmp (mpz_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpz_class @var{op2}) @deftypefunx double mpz_class::get_d (void) @deftypefunx long mpz_class::get_si (void) @deftypefunx {unsigned long} mpz_class::get_ui (void) @maybepagebreak @deftypefunx bool mpz_class::fits_sint_p (void) @deftypefunx bool mpz_class::fits_slong_p (void) @deftypefunx bool mpz_class::fits_sshort_p (void) @maybepagebreak @deftypefunx bool mpz_class::fits_uint_p (void) @deftypefunx bool mpz_class::fits_ulong_p (void) @deftypefunx bool mpz_class::fits_ushort_p (void) @maybepagebreak @deftypefunx int sgn (mpz_class @var{op}) @deftypefunx mpz_class sqrt (mpz_class @var{op}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. @end deftypefun @sp 1 Overloaded operators for combinations of @code{mpz_class} and @code{double} are provided for completeness, but it should be noted that if the given @code{double} is not an integer then the way any rounding is done is currently unspecified. The rounding might take place at the start, in the middle, or at the end of the operation, and it might change in the future. Conversions between @code{mpz_class} and @code{double}, however, are defined to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}. And comparisons are always made exactly, as per @code{mpz_cmp_d}. @node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface @section C++ Interface Rationals In all the following constructors, if a fraction is given then it should be in canonical form, or if not then @code{mpq_class::canonicalize} called. @deftypefun void mpq_class::mpq_class (type @var{op}) @deftypefunx void mpq_class::mpq_class (integer @var{num}, integer @var{den}) Construct an @code{mpq_class}. The initial value can be a single value of any type, or a pair of integers (@code{mpz_class} or standard C++ integer types) representing a fraction, except that @code{long long} and @code{long double} are not supported. For example, @example mpq_class q (99); mpq_class q (1.75); mpq_class q (1, 3); @end example @end deftypefun @deftypefun void mpq_class::mpq_class (mpq_t @var{q}) Construct an @code{mpq_class} from an @code{mpq_t}. The value in @var{q} is copied into the new @code{mpq_class}, there won't be any permanent association between it and @var{q}. @end deftypefun @deftypefun void mpq_class::mpq_class (const char *@var{s}) @deftypefunx void mpq_class::mpq_class (const char *@var{s}, int base) @deftypefunx void mpq_class::mpq_class (const string& @var{s}) @deftypefunx void mpq_class::mpq_class (const string& @var{s}, int base) Construct an @code{mpq_class} converted from a string using @code{mpq_set_str}, (@pxref{Initializing Rationals}). If the @var{base} is not given then 0 is used. @end deftypefun @deftypefun void mpq_class::canonicalize () Put an @code{mpq_class} into canonical form, as per @ref{Rational Number Functions}. All arithmetic operators require their operands in canonical form, and will return results in canonical form. @end deftypefun @deftypefun mpq_class abs (mpq_class @var{op}) @deftypefunx int cmp (mpq_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpq_class @var{op2}) @maybepagebreak @deftypefunx double mpq_class::get_d (void) @deftypefunx int sgn (mpq_class @var{op}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. @end deftypefun @deftypefun {mpz_class&} mpq_class::get_num () @deftypefunx {mpz_class&} mpq_class::get_den () Get a reference to an @code{mpz_class} which is the numerator or denominator of an @code{mpq_class}. This can be used both for read and write access. If the object returned is modified, it modifies the original @code{mpq_class}. If direct manipulation might produce a non-canonical value, then @code{mpq_class::canonicalize} must be called before further operations. @end deftypefun @deftypefun mpz_t mpq_class::get_num_mpz_t () @deftypefunx mpz_t mpq_class::get_den_mpz_t () Get a reference to the underlying @code{mpz_t} numerator or denominator of an @code{mpq_class}. This can be passed to C functions expecting an @code{mpz_t}. Any modifications made to the @code{mpz_t} will modify the original @code{mpq_class}. If direct manipulation might produce a non-canonical value, then @code{mpq_class::canonicalize} must be called before further operations. @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop}); Read @var{rop} from @var{stream}, using its @code{ios} formatting settings, the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}). If the @var{rop} read might not be in canonical form then @code{mpq_class::canonicalize} must be called. @end deftypefun @node C++ Interface Floats, C++ Interface MPFR, C++ Interface Rationals, C++ Class Interface @section C++ Interface Floats When an expression requires the use of temporary intermediate @code{mpf_class} values, like @code{f=g*h+x*y}, those temporaries will have the same precision as the destination @code{f}. Explicit constructors can be used if this doesn't suit. @deftypefun {} mpf_class::mpf_class (type @var{op}) @deftypefunx {} mpf_class::mpf_class (type @var{op}, unsigned long @var{prec}) Construct an @code{mpf_class}. Any standard C++ type can be used, except @code{long long} and @code{long double}, and any of the GMP C++ classes can be used. If @var{prec} is given, the initial precision is that value, in bits. If @var{prec} is not given, then the initial precision is determined by the type of @var{op} given. An @code{mpz_class}, @code{mpq_class}, string, or C++ builtin type will give the default @code{mpf} precision (@pxref{Initializing Floats}). An @code{mpf_class} or expression will give the precision of that value. The precision of a binary expression is the higher of the two operands. @example mpf_class f(1.5); // default precision mpf_class f(1.5, 500); // 500 bits (at least) mpf_class f(x); // precision of x mpf_class f(abs(x)); // precision of x mpf_class f(-g, 1000); // 1000 bits (at least) mpf_class f(x+y); // greater of precisions of x and y @end example @end deftypefun @deftypefun mpf_class abs (mpf_class @var{op}) @deftypefunx mpf_class ceil (mpf_class @var{op}) @deftypefunx int cmp (mpf_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpf_class @var{op2}) @maybepagebreak @deftypefunx mpf_class floor (mpf_class @var{op}) @deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2}) @deftypefunx double mpf_class::get_d (void) @deftypefunx long mpf_class::get_si (void) @deftypefunx {unsigned long} mpf_class::get_ui (void) @maybepagebreak @deftypefunx bool mpf_class::fits_sint_p (void) @deftypefunx bool mpf_class::fits_slong_p (void) @deftypefunx bool mpf_class::fits_sshort_p (void) @maybepagebreak @deftypefunx bool mpf_class::fits_uint_p (void) @deftypefunx bool mpf_class::fits_ulong_p (void) @deftypefunx bool mpf_class::fits_ushort_p (void) @maybepagebreak @deftypefunx int sgn (mpf_class @var{op}) @deftypefunx mpf_class sqrt (mpf_class @var{op}) @deftypefunx mpf_class trunc (mpf_class @var{op}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. The accuracy provided by @code{hypot} is not currently guaranteed. @end deftypefun @deftypefun {unsigned long int} mpf_class::get_prec () @deftypefunx void mpf_class::set_prec (unsigned long @var{prec}) @deftypefunx void mpf_class::set_prec_raw (unsigned long @var{prec}) Get or set the current precision of an @code{mpf_class}. The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing Floats}) apply to @code{mpf_class::set_prec_raw}. Note in particular that the @code{mpf_class} must be restored to it's allocated precision before being destroyed. This must be done by application code, there's no automatic mechanism for it. @end deftypefun @node C++ Interface MPFR, C++ Interface Random Numbers, C++ Interface Floats, C++ Class Interface @section C++ Interface MPFR The C++ class interface to MPFR is provided if MPFR is enabled (@pxref{Build Options}). This interface must be regarded as preliminary and possibly subject to incompatible changes in the future, since MPFR itself is preliminary. All definitions can be obtained with @cindex mpfrxx.h @example #include <mpfrxx.h> @end example @noindent This defines @deftp Class mpfr_class @end deftp @noindent which behaves similarly to @code{mpf_class} (@pxref{C++ Interface Floats}). @node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface MPFR, C++ Class Interface @section C++ Interface Random Numbers @deftp Class gmp_randclass The C++ class interface to the GMP random number functions uses @code{gmp_randclass} to hold an algorithm selection and current state, as per @code{gmp_randstate_t}. @end deftp @deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, ...), ...) Construct a @code{gmp_randclass}, using a call to the given @var{randinit} function (@pxref{Random State Initialization}). The arguments expected are the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}. For example, @example gmp_randclass r1 (gmp_randinit_default); gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32); gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp); @end example @code{gmp_randinit_lc_2exp_size} can fail if the size requested is too big, the behaviour of @code{gmp_randclass::gmp_randclass} is undefined in this case (perhaps this will change in the future). @end deftypefun @deftypefun {} gmp_randclass::gmp_randclass (gmp_randalg_t @var{alg}, ...) Construct a @code{gmp_randclass} using the same parameters as @code{gmp_randinit} (@pxref{Random State Initialization}). This function is obsolete and the above @var{randinit} style should be preferred. @end deftypefun @deftypefun void gmp_randclass::seed (unsigned long int @var{s}) @deftypefunx void gmp_randclass::seed (mpz_class @var{s}) Seed a random number generator. See @pxref{Random Number Functions}, for how to choose a good seed. @end deftypefun @deftypefun mpz_class gmp_randclass::get_z_bits (unsigned long @var{bits}) @deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits}) Generate a random integer with a specified number of bits. @end deftypefun @deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n}) Generate a random integer in the range 0 to @math{@var{n}-1} inclusive. @end deftypefun @deftypefun mpf_class gmp_randclass::get_f () @deftypefunx mpf_class gmp_randclass::get_f (unsigned long @var{prec}) Generate a random float @var{f} in the range @math{0 <= @var{f} < 1}. @var{f} will be to @var{prec} bits precision, or if @var{prec} is not given then to the precision of the destination. For example, @example gmp_randclass r; ... mpf_class f (0, 512); // 512 bits precision f = r.get_f(); // random number, 512 bits @end example @end deftypefun @node C++ Interface Limitations, , C++ Interface Random Numbers, C++ Class Interface @section C++ Interface Limitations @table @asis @item @code{mpq_class} and Templated Reading A generic piece of template code probably won't know that @code{mpq_class} requires a @code{canonicalize} call if inputs read with @code{operator>>} might be non-canonical. This can lead to incorrect results. @code{operator>>} behaves as it does for reasons of efficiency. A canonicalize can be quite time consuming on large operands, and is best avoided if it's not necessary. But this potential difficulty reduces the usefulness of @code{mpq_class}. Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in the future, maybe a preprocessor define, a global flag, or an @code{ios} flag pressed into service. Or maybe, at the risk of inconsistency, the @code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t} @code{operator>>} not doing so, for use on those occasions when that's acceptable. Send feedback or alternate ideas to @email{bug-gmp@@gnu.org}. @item Subclassing Subclassing the GMP C++ classes works, but is not currently recommended. Expressions involving subclasses resolve correctly (or seem to), but in normal C++ fashion the subclass doesn't inherit constructors and assignments. There's many of those in the GMP classes, and a good way to reestablish them in a subclass is not yet provided. @item Templated Expressions A subtle difficulty exists when using expressions together with application-defined template functions. Consider the following, with @code{T} intended to be some numeric type, @example template <class T> T fun (const T &, const T &); @end example @noindent When used with, say, plain @code{mpz_class} variables, it works fine: @code{T} is resolved as @code{mpz_class}. @example mpz_class f(1), g(2); fun (f, g); // Good @end example @noindent But when one of the arguments is an expression, it doesn't work. @example mpz_class f(1), g(2), h(3); fun (f, g+h); // Bad @end example This is because @code{g+h} ends up being a certain expression template type internal to @code{gmpxx.h}, which the C++ template resolution rules are unable to automatically convert to @code{mpz_class}. The workaround is simply to add an explicit cast. @example mpz_class f(1), g(2), h(3); fun (f, mpz_class(g+h)); // Good @end example Similarly, within @code{fun} it may be necessary to cast an expression to type @code{T} when calling a templated @code{fun2}. @example template <class T> void fun (T f, T g) @{ fun2 (f, f+g); // Bad @} template <class T> void fun (T f, T g) @{ fun2 (f, T(f+g)); // Good @} @end example @end table @node BSD Compatible Functions, Custom Allocation, C++ Class Interface, Top @comment node-name, next, previous, up @chapter Berkeley MP Compatible Functions @cindex Berkeley MP compatible functions @cindex BSD MP compatible functions These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U*ix systems. The @samp{--enable-mpbsd} option must be used when building GNU MP to make these available (@pxref{Installing GMP}). The original Berkeley MP library has a usage restriction: you cannot use the same variable as both source and destination in a single function call. The compatible functions in GNU MP do not share this restriction---inputs and outputs may overlap. It is not recommended that new programs are written using these functions. Apart from the incomplete set of functions, the interface for initializing @code{MINT} objects is more error prone, and the @code{pow} function collides with @code{pow} in @file{libm.a}. @cindex @file{mp.h} Include the header @file{mp.h} to get the definition of the necessary types and functions. If you are on a BSD derived system, make sure to include GNU @file{mp.h} if you are going to link the GNU @file{libmp.a} to your program. This means that you probably need to give the @samp{-I<dir>} option to the compiler, where @samp{<dir>} is the directory where you have GNU @file{mp.h}. @deftypefun {MINT *} itom (signed short int @var{initial_value}) Allocate an integer consisting of a @code{MINT} object and dynamic limb space. Initialize the integer to @var{initial_value}. Return a pointer to the @code{MINT} object. @end deftypefun @deftypefun {MINT *} xtom (char *@var{initial_value}) Allocate an integer consisting of a @code{MINT} object and dynamic limb space. Initialize the integer from @var{initial_value}, a hexadecimal, null-terminated C string. Return a pointer to the @code{MINT} object. @end deftypefun @deftypefun void move (MINT *@var{src}, MINT *@var{dest}) Set @var{dest} to @var{src} by copying. Both variables must be previously initialized. @end deftypefun @deftypefun void madd (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination}) Add @var{src_1} and @var{src_2} and put the sum in @var{destination}. @end deftypefun @deftypefun void msub (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination}) Subtract @var{src_2} from @var{src_1} and put the difference in @var{destination}. @end deftypefun @deftypefun void mult (MINT *@var{src_1}, MINT *@var{src_2}, MINT *@var{destination}) Multiply @var{src_1} and @var{src_2} and put the product in @var{destination}. @end deftypefun @deftypefun void mdiv (MINT *@var{dividend}, MINT *@var{divisor}, MINT *@var{quotient}, MINT *@var{remainder}) @deftypefunx void sdiv (MINT *@var{dividend}, signed short int @var{divisor}, MINT *@var{quotient}, signed short int *@var{remainder}) Set @var{quotient} to @var{dividend}/@var{divisor}, and @var{remainder} to @var{dividend} mod @var{divisor}. The quotient is rounded towards zero; the remainder has the same sign as the dividend unless it is zero. Some implementations of these functions work differently---or not at all---for negative arguments. @end deftypefun @deftypefun void msqrt (MINT *@var{op}, MINT *@var{root}, MINT *@var{remainder}) Set @var{root} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part of the square root of @var{op}}, like @code{mpz_sqrt}. Set @var{remainder} to @m{(@var{op} - @var{root}^2), @var{op}@minus{}@var{root}*@var{root}}, i.e. zero if @var{op} is a perfect square. If @var{root} and @var{remainder} are the same variable, the results are undefined. @end deftypefun @deftypefun void pow (MINT *@var{base}, MINT *@var{exp}, MINT *@var{mod}, MINT *@var{dest}) Set @var{dest} to (@var{base} raised to @var{exp}) modulo @var{mod}. @end deftypefun @deftypefun void rpow (MINT *@var{base}, signed short int @var{exp}, MINT *@var{dest}) Set @var{dest} to @var{base} raised to @var{exp}. @end deftypefun @deftypefun void gcd (MINT *@var{op1}, MINT *@var{op2}, MINT *@var{res}) Set @var{res} to the greatest common divisor of @var{op1} and @var{op2}. @end deftypefun @deftypefun int mcmp (MINT *@var{op1}, MINT *@var{op2}) Compare @var{op1} and @var{op2}. Return a positive value if @var{op1} > @var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} < @var{op2}. @end deftypefun @deftypefun void min (MINT *@var{dest}) Input a decimal string from @code{stdin}, and put the read integer in @var{dest}. SPC and TAB are allowed in the number string, and are ignored. @end deftypefun @deftypefun void mout (MINT *@var{src}) Output @var{src} to @code{stdout}, as a decimal string. Also output a newline. @end deftypefun @deftypefun {char *} mtox (MINT *@var{op}) Convert @var{op} to a hexadecimal string, and return a pointer to the string. The returned string is allocated using the default memory allocation function, @code{malloc} by default. It will be @code{strlen(str)+1} bytes, that being exactly enough for the string and null-terminator. @end deftypefun @deftypefun void mfree (MINT *@var{op}) De-allocate, the space used by @var{op}. @strong{This function should only be passed a value returned by @code{itom} or @code{xtom}.} @end deftypefun @node Custom Allocation, Language Bindings, BSD Compatible Functions, Top @comment node-name, next, previous, up @chapter Custom Allocation @cindex Custom allocation @cindex Memory allocation @cindex Allocation of memory By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory allocation, and if they fail GMP prints a message to the standard error output and terminates the program. Alternate functions can be specified to allocate memory in a different way or to have a different error action on running out of memory. This feature is available in the Berkeley compatibility library (@pxref{BSD Compatible Functions}) as well as the main GMP library. @deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t)) Replace the current allocation functions from the arguments. If an argument is @code{NULL}, the corresponding default function is used. These functions will be used for all memory allocation done by GMP, apart from temporary space from @code{alloca} if that function is available and GMP is configured to use it (@pxref{Build Options}). @strong{Be sure to call @code{mp_set_memory_functions} only when there are no active GMP objects allocated using the previous memory functions! Usually that means calling it before any other GMP function.} @end deftypefun The functions supplied should fit the following declarations: @deftypefun {void *} allocate_function (size_t @var{alloc_size}) Return a pointer to newly allocated space with at least @var{alloc_size} bytes. @end deftypefun @deftypefun {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size}) Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be @var{new_size} bytes. The block may be moved if necessary or if desired, and in that case the smaller of @var{old_size} and @var{new_size} bytes must be copied to the new location. The return value is a pointer to the resized block, that being the new location if moved or just @var{ptr} if not. @var{ptr} is never @code{NULL}, it's always a previously allocated block. @var{new_size} may be bigger or smaller than @var{old_size}. @end deftypefun @deftypefun void deallocate_function (void *@var{ptr}, size_t @var{size}) De-allocate the space pointed to by @var{ptr}. @var{ptr} is never @code{NULL}, it's always a previously allocated block of @var{size} bytes. @end deftypefun A @dfn{byte} here means the unit used by the @code{sizeof} operator. The @var{old_size} parameters to @var{reallocate_function} and @var{deallocate_function} are passed for convenience, but of course can be ignored if not needed. The default functions using @code{malloc} and friends for instance don't use them. No error return is allowed from any of these functions, if they return then they must have performed the specified operation. In particular note that @var{allocate_function} or @var{reallocate_function} mustn't return @code{NULL}. Getting a different fatal error action is a good use for custom allocation functions, for example giving a graphical dialog rather than the default print to @code{stderr}. How much is possible when genuinely out of memory is another question though. There's currently no defined way for the allocation functions to recover from an error such as out of memory, they must terminate program execution. A @code{longjmp} or throwing a C++ exception will have undefined results. This may change in the future. GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make. Since the default GMP allocation uses @code{malloc} and friends, those functions will be linked in even if the first thing a program does is an @code{mp_set_memory_functions}. It's necessary to change the GMP sources if this is a problem. @node Language Bindings, Algorithms, Custom Allocation, Top @chapter Language Bindings The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency. @c GNUstep Base Library @uref{http://www.gnustep.org} (version 0.9.1) is @c intending to use GMP for its NSDecimal class, which would be an Objective @c C binding for GMP. Has some configure stuff ready, but no code. @c @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces @c in tex, just to separate the URL from the preceding text a bit. @iftex @macro spaceuref {U} @ @ @uref{\U\} @end macro @end iftex @ifnottex @macro spaceuref {U} @uref{\U\} @end macro @end ifnottex @sp 1 @table @asis @item C++ @itemize @bullet @item GMP C++ class interface, @pxref{C++ Class Interface} @* Straightforward interface, expression templates to eliminate temporaries. @item ALP @spaceuref{http://www.inria.fr/saga/logiciels/ALP} @* Linear algebra and polynomials using templates. @item Arithmos @spaceuref{http://win-www.uia.ac.be/u/cant/arithmos} @* Rationals with infinities and square roots. @item CLN @spaceuref{http://clisp.cons.org/~haible/packages-cln.html} @* High level classes for arithmetic. @item LiDIA @spaceuref{http://www.informatik.tu-darmstadt.de/TI/LiDIA} @* A C++ library for computational number theory. @item Linbox @spaceuref{http://www.linalg.org} @* Sparse vectors and matrices. @item NTL @spaceuref{http://www.shoup.net/ntl} @* A C++ number theory library. @end itemize @item Fortran @itemize @bullet @item Omni F77 @spaceuref{http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html} @* Arbitrary precision floats. @end itemize @item Haskell @itemize @bullet @item Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc} @end itemize @item Java @itemize @bullet @item Kaffe @spaceuref{http://www.kaffe.org} @item Kissme @spaceuref{http://kissme.sourceforge.net} @end itemize @item Lisp @itemize @bullet @item GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html} @* In the process of switching to GMP for bignums. @item Librep @spaceuref{http://librep.sourceforge.net} @end itemize @item M4 @itemize @bullet @item GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu} @* Optionally provides an arbitrary precision @code{mpeval}. @end itemize @item ML @itemize @bullet @item MLton compiler @spaceuref{http://www.mlton.org} @end itemize @item Oz @itemize @bullet @item Mozart @spaceuref{http://www.mozart-oz.org} @end itemize @item Pascal @itemize @bullet @item GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de} @* GMP unit. @end itemize @item Perl @itemize @bullet @item GMP module, see @file{demos/perl} in the GMP sources. @item Math::GMP @spaceuref{http://www.cpan.org} @* Compatible with Math::BigInt, but not as many functions as the GMP module above. @item Math::BigInt::GMP @spaceuref{http://www.cpan.org} @* Plug Math::GMP into normal Math::BigInt operations. @end itemize @need 1000 @item Pike @itemize @bullet @item mpz module in the standard distribution, @uref{http://pike.idonex.com} @end itemize @need 500 @item Prolog @itemize @bullet @item SWI Prolog @spaceuref{http://www.swi.psy.uva.nl/projects/SWI-Prolog} @* Arbitrary precision floats. @end itemize @item Python @itemize @bullet @item mpz module in the standard distribution, @uref{http://www.python.org} @item GMPY @uref{http://gmpy.sourceforge.net} @end itemize @item Scheme @itemize @bullet @item RScheme @spaceuref{http://www.rscheme.org} @item STklos @spaceuref{http://kaolin.unice.fr/STklos} @end itemize @item Smalltalk @itemize @bullet @item GNU Smalltalk @spaceuref{http://www.smalltalk.org/versions/GNUSmalltalk.html} @end itemize @item Other @itemize @bullet @item DrGenius @spaceuref{http://drgenius.seul.org} @* Geometry system and mathematical programming language. @item GiNaC @spaceuref{http://www.ginac.de} @* C++ computer algebra using CLN. @item Maxima @uref{http://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma computer algebra using GCL. @item Q @spaceuref{http://www.musikwissenschaft.uni-mainz.de/~ag/q} @* Equational programming system. @item Regina @spaceuref{http://regina.sourceforge.net} @* Topological calculator. @item Yacas @spaceuref{http://www.xs4all.nl/~apinkus/yacas.html} @* Yet another computer algebra system. @end itemize @end table @node Algorithms, Internals, Language Bindings, Top @chapter Algorithms @cindex Algorithms This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms. Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions. @menu * Multiplication Algorithms:: * Division Algorithms:: * Greatest Common Divisor Algorithms:: * Powering Algorithms:: * Root Extraction Algorithms:: * Radix Conversion Algorithms:: * Other Algorithms:: * Assembler Coding:: @end menu @node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms @section Multiplication @cindex Multiplication algorithms N@cross{}N limb multiplications and squares are done using one of four algorithms, as the size N increases. @quotation @multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item Algorithm @tab Threshold @item Basecase @tab (none) @item Karatsuba @tab @code{MUL_KARATSUBA_THRESHOLD} @item Toom-3 @tab @code{MUL_TOOM3_THRESHOLD} @item FFT @tab @code{MUL_FFT_THRESHOLD} @end multitable @end quotation Similarly for squaring, with the @code{SQR} thresholds. Note though that the FFT is only used if GMP is configured with @samp{--enable-fft}, @pxref{Build Options}. N@cross{}M multiplications of operands with different sizes above @code{MUL_KARATSUBA_THRESHOLD} are currently done by splitting into M@cross{}M pieces. The Karatsuba and Toom-3 routines then operate only on equal size operands. This is not very efficient, and is slated for improvement in the future. @menu * Basecase Multiplication:: * Karatsuba Multiplication:: * Toom-Cook 3-Way Multiplication:: * FFT Multiplication:: * Other Multiplication:: @end menu @node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms @subsection Basecase Multiplication Basecase N@cross{}M multiplication is a straightforward rectangular set of cross-products, the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method. This is an @m{O(NM),O(N*M)} algorithm. See Knuth section 4.3.1 algorithm M (@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code. Assembler implementations of @code{mpn_mul_basecase} are essentially the same as the generic C code, but have all the usual assembler tricks and obscurities introduced for speed. A square can be done in roughly half the time of a multiply, by using the fact that the cross products above and below the diagonal are the same. A triangle of products below the diagonal is formed, doubled (left shift by one bit), and then the products on the diagonal added. This can be seen in @file{mpn/generic/sqr_basecase.c}. Again the assembler implementations take essentially the same approach. @tex \def\GMPline#1#2#3#4#5#6{% \hbox {% \vrule height 2.5ex depth 1ex \hbox to 2em {\hfil{#2}\hfil}% \vrule \hbox to 2em {\hfil{#3}\hfil}% \vrule \hbox to 2em {\hfil{#4}\hfil}% \vrule \hbox to 2em {\hfil{#5}\hfil}% \vrule \hbox to 2em {\hfil{#6}\hfil}% \vrule}} \GMPdisplay{ \hbox{% \vbox{% \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}% \vfill}% \vbox{% \hbox{% \hbox to 2em {\hfil u0\hfil}% \hbox to 2em {\hfil u1\hfil}% \hbox to 2em {\hfil u2\hfil}% \hbox to 2em {\hfil u3\hfil}% \hbox to 2em {\hfil u4\hfil}}% \vskip 0.7ex \hrule \GMPline{u0}{d}{}{}{}{}% \hrule \GMPline{u1}{}{d}{}{}{}% \hrule \GMPline{u2}{}{}{d}{}{}% \hrule \GMPline{u3}{}{}{}{d}{}% \hrule \GMPline{u4}{}{}{}{}{d}% \hrule}}} @end tex @ifnottex @example @group u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+ @end group @end example @end ifnottex In practice squaring isn't a full 2@cross{} faster than multiplying, it's usually around 1.5@cross{}. Less than 1.5@cross{} probably indicates @code{mpn_sqr_basecase} wants improving on that CPU. On some CPUs @code{mpn_mul_basecase} can be faster than the generic C @code{mpn_sqr_basecase}. @code{SQR_BASECASE_THRESHOLD} is the size at which to use @code{mpn_sqr_basecase}, this will be zero if that routine should be used always. @node Karatsuba Multiplication, Toom-Cook 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms @subsection Karatsuba Multiplication The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here. The inputs @math{x} and @math{y} are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd). @tex % GMPboxwidth used for all the multiplication pictures \global\newdimen\GMPboxwidth \global\GMPboxwidth=5em % GMPboxdepth and GMPboxheight are also used for the float pictures \global\newdimen\GMPboxdepth \global\GMPboxdepth=1ex \global\newdimen\GMPboxheight \global\GMPboxheight=2ex \gdef\GMPvrule{\vrule height \GMPboxheight depth \GMPboxdepth} \def\GMPbox#1#2{% \vbox {% \hrule \hbox to 2\GMPboxwidth{% \GMPvrule \hfil$#1$\hfil \vrule \hfil$#2$\hfil \vrule}% \hrule}} \GMPdisplay{% \vbox{% \hbox to 2\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPbox{x_1}{x_0} \vskip 0.5ex \GMPbox{y_1}{y_0} }} @end tex @ifnottex @example @group high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+ @end group @end example @end ifnottex Let @math{b} be the power of 2 where the split occurs, ie.@: if @ms{x,0} is @math{k} limbs (@ms{y,0} the same) then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}. With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the following holds, @display @m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0, x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0} @end display This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs, whereas a basecase multiply of N@cross{}N limbs is equivalent to four multiplies of (N/2)@cross{}(N/2). The factors @math{(b^2+b)} etc represent the positions where the three products must be added. @tex \def\GMPboxA#1#2{% \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}% \vrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}% \vrule} \hrule}} \def\GMPboxB#1#2{% \hbox{% \raise \GMPboxdepth \hbox to \GMPboxwidth {\hfil #1\hskip 0.5em}% \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}% \vrule}% \hrule}}} \GMPdisplay{% \vbox{% \hbox to 4\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPboxA{x_1y_1}{x_0y_0} \vskip 0.5ex \GMPboxB{$+$}{x_1y_1} \vskip 0.5ex \GMPboxB{$+$}{x_0y_0} \vskip 0.5ex \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)} }} @end tex @ifnottex @example @group high low +--------+--------+ +--------+--------+ | x1*y1 | | x0*y0 | +--------+--------+ +--------+--------+ +--------+--------+ add | x1*y1 | +--------+--------+ +--------+--------+ add | x0*y0 | +--------+--------+ +--------+--------+ sub | (x1-x0)*(y1-y0) | +--------+--------+ @end group @end example @end ifnottex The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1), high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb additions, rather than @m{6k,6*k}, but in GMP extra function call overheads outweigh the saving. Squaring is similar to multiplying, but with @math{x=y} the formula reduces to an equivalent with three squares, @display @m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2, x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2} @end display The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now always positive. A similar formula for both multiplying and squaring can be constructed with a middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}. But those sums can exceed @math{k} limbs, leading to more carry handling and additions than the form above. Karatsuba multiplication is asymptotically an @math{O(N^@W{1.585})} algorithm, the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies each 1/2 the size of the inputs. This is a big improvement over the basecase multiply at @math{O(N^2)} and the advantage soon overcomes the extra additions Karatsuba performs. @code{MUL_KARATSUBA_THRESHOLD} can be as little as 10 limbs. The @code{SQR} threshold is usually about twice the @code{MUL}. The basecase algorithm will take a time of the form @m{M(N) = aN^2 + bN + c, M(N) = a*N^2 + b*N + c} and the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N + e}. Clearly per-crossproduct speedups in the basecase code reduce @math{a} and decrease the threshold, but linear style speedups reducing @math{b} will actually increase the threshold. The latter can be seen for instance when adding an optimized @code{mpn_sqr_diagonal} to @code{mpn_sqr_basecase}. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest. @node Toom-Cook 3-Way Multiplication, FFT Multiplication, Karatsuba Multiplication, Multiplication Algorithms @subsection Toom-Cook 3-Way Multiplication The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A. The 3-way form used in GMP is described here. The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the others). @tex \def\GMPbox#1#2#3{% \vbox{% \hrule \vfil \hbox to 3\GMPboxwidth {% \GMPvrule \hfil$#1$\hfil \vrule \hfil$#2$\hfil \vrule \hfil$#3$\hfil \vrule}% \vfil \hrule }} \GMPdisplay{% \vbox{% \hbox to 3\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPbox{x_2}{x_1}{x_0} \vskip 0.5ex \GMPbox{y_2}{y_1}{y_0} \vskip 0.5ex }} @end tex @ifnottex @example @group high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+ @end group @end example @end ifnottex @noindent These parts are treated as the coefficients of two polynomials @display @group @m{X(t) = x_2t^2 + x_1t + x_0, X(t) = x2*t^2 + x1*t + x0} @m{Y(t) = y_2t^2 + y_1t + y_0, Y(t) = y2*t^2 + y1*t + y0} @end group @end display Again let @math{b} equal the power of 2 which is the size of the @ms{x,0}, @ms{x,1}, @ms{y,0} and @ms{y,1} pieces, ie.@: if they're @math{k} limbs each then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}. With this @math{x=X(b)} and @math{y=Y(b)}. Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients are @display @m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0, W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0} @end display @noindent The @m{w_i,w[i]} are going to be determined, and when they are they'll give the final result using @math{w=W(b)}, since @m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}. The coefficients will be roughly @math{b^2} each, and the final @math{W(b)} will be an addition like, @tex \def\GMPbox#1#2{% \moveright #1\GMPboxwidth \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil$#2$\hfil}% \vrule}% \hrule }} \GMPdisplay{% \vbox{% \hbox to 6\GMPboxwidth {high \hfil low}% \vskip 0.7ex \GMPbox{0}{w_4} \vskip 0.5ex \GMPbox{1}{w_3} \vskip 0.5ex \GMPbox{2}{w_2} \vskip 0.5ex \GMPbox{3}{w_1} \vskip 0.5ex \GMPbox{4}{w_1} }} @end tex @ifnottex @example @group high low +-------+-------+ | w4 | +-------+-------+ +--------+-------+ | w3 | +--------+-------+ +--------+-------+ | w2 | +--------+-------+ +--------+-------+ | w1 | +--------+-------+ +-------+-------+ | w0 | +-------+-------+ @end group @end example @end ifnottex The @m{w_i,w[i]} coefficients could be formed by a simple set of cross products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2}, @m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all nine @m{x_iy_j,x[i]*y[j]} for @math{i,j=0,1,2}, and would be equivalent merely to a basecase multiply. Instead the following approach is used. @math{X(t)} and @math{Y(t)} are evaluated and multiplied at 5 points, giving values of @math{W(t)} at those points. The points used can be chosen in various ways, but in GMP the following are used @quotation @multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item Point @tab Value @item @math{t=0} @tab @m{x_0y_0,x0*y0}, which gives @ms{w,0} immediately @item @math{t=2} @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0)*(4*y2+2*y1+y0)} @item @math{t=1} @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0)*(y2+y1+y0)} @item @m{t={1\over2},t=1/2} @tab @m{(x_2+2x_1+4x_0)(y_2+2y_1+4y_0),(x2+2*x1+4*x0)*(y2+2*y1+4*y0)} @item @m{t=\infty,t=inf} @tab @m{x_2y_2,x2*y2}, which gives @ms{w,4} immediately @end multitable @end quotation At @m{t={1\over2},t=1/2} the value calculated is actually @m{16X({1\over2})Y({1\over2}), 16*X(1/2)*Y(1/2)}, giving a value for @m{16W({1\over2}),16*W(1/2)}, and this is always an integer. At @m{t=\infty,t=inf} the value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much easier to think of as simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately (much like @m{x_0y_0,x0*y0} at @math{t=0} gives @ms{w,0} immediately). Now each of the points substituted into @m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+@dots{}+w0} gives a linear combination of the @m{w_i,w[i]} coefficients, and the value of those combinations has just been calculated. @tex \GMPdisplay{%$\matrix{%
W(0)           & = &       &   &      &   &      &   &      &   &   w_0 \cr
16W({1\over2}) & = &   w_4 & + & 2w_3 & + & 4w_2 & + & 8w_1 & + & 16w_0 \cr
W(1)           & = &   w_4 & + &  w_3 & + &  w_2 & + &  w_1 & + &   w_0 \cr
W(2)           & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + &   w_0 \cr
W(\infty)      & = &   w_4 \cr
}} @end tex @ifnottex @example @group W(0) = w0 16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0 W(1) = w4 + w3 + w2 + w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4 @end group @end example @end ifnottex This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each @m{w_i,w[i]}, by subtracting multiples of one equation from another. In the code the set of five values @math{W(0)},@dots{},@m{W(\infty),W(inf)} will represent those certain linear combinations. By adding or subtracting one from another as necessary, values which are each @m{w_i,w[i]} alone are arrived at. This involves only a few subtractions of small multiples (some of which are powers of 2), and so is fast. A couple of divisions remain by powers of 2 and one division by 3 (or by 6 rather), and that last uses the special @code{mpn_divexact_by3} (@pxref{Exact Division}). In the code the values @ms{w,4}, @ms{w,2} and @ms{w,0} are formed in the destination with pointers @code{E}, @code{C} and @code{A}, and @ms{w,3} and @ms{w,1} in temporary space @code{D} and @code{B} are added to them. There are extra limbs @code{tD}, @code{tC} and @code{tB} at the high end of @ms{w,3}, @ms{w,2} and @ms{w,1} which are handled separately. The final addition then is as follows. @tex \def\GMPboxT#1{% \vbox{% \hrule \hbox {\GMPvrule\hskip 0.4em #1\hskip 0.4em \vrule}% \hrule }} \GMPdisplay{% \vbox{% \hbox to 6\GMPboxwidth {high \hfil low}% \vskip 0.7ex \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil@code{E}\hfil} \vrule \hbox to 2\GMPboxwidth {\hfil@code{C}\hfil} \vrule \hbox to 2\GMPboxwidth {\hfil@code{A}\hfil} \vrule}% \hrule}% \vskip 0.5ex \moveright \GMPboxwidth \vbox{% \hrule \hbox to 4\GMPboxwidth {% \GMPvrule \hfil @code{D}\hfil \vrule \hfil @code{B}\hfil \vrule} \hrule}% \vskip 0.5ex \hbox{% \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tD}}}% \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tC}}}% \hbox to \GMPboxwidth{\hfil \GMPboxT{\code{tB}}}} }} @end tex @ifnottex @example @group high low +-------+-------+-------+-------+-------+-------+ | E | C | A | +-------+-------+-------+-------+-------+-------+ +------+-------++------+-------+ | D || B | +------+-------++------+-------+ -- -- -- |tD| |tC| |tB| -- -- -- @end group @end example @end ifnottex The conversion of @math{W(t)} values to the coefficients is interpolation. A polynomial of degree 4 like @math{W(t)} is uniquely determined by values known at 5 different points. The points can be chosen to make the linear equations come out with a convenient set of steps for isolating the @m{w_i,w[i]}. In @file{mpn/generic/mul_n.c} the @code{interpolate3} routine performs the interpolation. The open-coded one-pass version may be a bit hard to understand, the steps performed can be better seen in the @code{USE_MORE_MPN} version. Squaring follows the same procedure as multiplication, but there's only one @math{X(t)} and it's evaluated at 5 points, and those values squared to give values of @math{W(t)}. The interpolation is then identical, and in fact the same @code{interpolate3} subroutine is used for both squaring and multiplying. Toom-3 is asymptotically @math{O(N^@W{1.465})}, the exponent being @m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the original size. This is an improvement over Karatsuba at @math{O(N^@W{1.585})}, though Toom-Cook does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size. Near the crossover between Toom-3 and Karatsuba there's generally a range of sizes where the difference between the two is small. @code{MUL_TOOM3_THRESHOLD} is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring. A graph of time versus size for the two shows the effect, see @file{tune/README}. At the fairly small sizes where the Toom-3 thresholds occur it's worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there's a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted. The formula given above for the Karatsuba algorithm has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening. An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using a vector to represent the @math{x} and @math{y} splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn't have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done. @node FFT Multiplication, Other Multiplication, Toom-Cook 3-Way Multiplication, Multiplication Algorithms @subsection FFT Multiplication At large to very large sizes a Fermat style FFT multiplication is used, following Sch@"onhage and Strassen (@pxref{References}). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief description of the form used in GMP is given here. The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given @math{N}. A full product @m{xy,x*y} is obtained by choosing @m{N \ge \mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding @math{x} and @math{y} with high zero limbs. The modular product is the native form for the algorithm, so padding to get a full product is unavoidable. The algorithm follows a split, evaluate, pointwise multiply, interpolate and combine similar to that described above for Karatsuba and Toom-3. A @math{k} parameter controls the split, with an FFT-@math{k} splitting into @math{2^k} pieces of @math{M=N/2^k} bits each. @math{N} must be a multiple of @m{2^k\times@code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so the split falls on limb boundaries, avoiding bit shifts in the split and combine stages. The evaluations, pointwise multiplications, and interpolation, are all done modulo @m{2^{N'}+1, 2^N'+1} where @math{N'} is @math{2M+k+3} rounded up to a multiple of @math{2^k} and of @code{mp_bits_per_limb}. The results of interpolation will be the following negacyclic convolution of the input pieces, and the choice of @math{N'} ensures these sums aren't truncated. @tex $$w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j$$ @end tex @ifnottex @example --- \ b w[n] = / (-1) * x[i] * y[j] --- i+j==b*2^k+n b=0,1 @end example @end ifnottex The points used for the evaluation are @math{g^i} for @math{i=0} to @math{2^k-1} where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}. @math{g} is a @m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary cancellations at the interpolation stage, and it's also a power of 2 so the fast fourier transforms used for the evaluation and interpolation do only shifts, adds and negations. The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size @math{N'}. The interpolation is an inverse fast fourier transform. The resulting set of sums of @m{x_iy_j, x[i]*y[j]} are added at appropriate offsets to give the final result. Squaring is the same, but @math{x} is the only input so it's one transform at the evaluate stage and the pointwise multiplies are squares. The interpolation is the same. For a mod @math{2^N+1} product, an FFT-@math{k} is an @m{O(N^{k/(k-1)}), O(N^(k/(k-1)))} algorithm, the exponent representing @math{2^k} recursed modular multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original. Each successive @math{k} is an asymptotic improvement, but overheads mean each is only faster at bigger and bigger sizes. In the code, @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} are the thresholds where each @math{k} is used. Each new @math{k} effectively swaps some multiplying for some shifts, adds and overheads. A mod @math{2^N+1} product can be formed with a normal @math{N@cross{}N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT and Toom-3 etc can be compared directly. A @math{k=4} FFT at @math{O(N^@W{1.333})} can be expected to be the first faster than Toom-3 at @math{O(N^@W{1.465})}. In practice this is what's found, with @code{MUL_FFT_MODF_THRESHOLD} and @code{SQR_FFT_MODF_THRESHOLD} being between 300 and 1000 limbs, depending on the CPU. So far it's been found that only very large FFTs recurse into pointwise multiplies above these sizes. When an FFT is to give a full product, the change of @math{N} to @math{2N} doesn't alter the theoretical complexity for a given @math{k}, but for the purposes of considering where an FFT might be first used it can be assumed that the FFT is recursing into a normal multiply and that on that basis it's doing @math{2^k} recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}. This would mean @math{k=7} at @math{O(N^@W{1.4})} would be the first FFT faster than Toom-3. In practice @code{MUL_FFT_THRESHOLD} and @code{SQR_FFT_THRESHOLD} have been found to be in the @math{k=8} range, somewhere between 3000 and 10000 limbs. The way @math{N} is split into @math{2^k} pieces and then @math{2M+k+3} is rounded up to a multiple of @math{2^k} and @code{mp_bits_per_limb} means that when @math{2^k@ge{}@nicode{mp\_bits\_per\_limb}} the effective @math{N} is a multiple of @m{2^{2k-1},2^(2k-1)} bits. The @math{+k+3} means some values of @math{N} just under such a multiple will be rounded to the next. The complexity calculations above assume that a favourable size is used, meaning one which isn't padded through rounding, and it's also assumed that the extra @math{+k+3} bits are negligible at typical FFT sizes. The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a step-effect into measured speeds. For example @math{k=8} will round @math{N} up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb groups of sizes for which @code{mpn_mul_n} runs at the same speed. Or for @math{k=9} groups of 2048 limbs, @math{k=10} groups of 8192 limbs, etc. In practice it's been found each @math{k} is used at quite small multiples of its size constraint and so the step effect is quite noticeable in a time versus size graph. The threshold determinations currently measure at the mid-points of size steps, but this is sub-optimal since at the start of a new step it can happen that it's better to go back to the previous @math{k} for a while. Something more sophisticated for @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} will be needed. @node Other Multiplication, , FFT Multiplication, Multiplication Algorithms @subsection Other Multiplication The 3-way Toom-Cook algorithm described above (@pxref{Toom-Cook 3-Way Multiplication}) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C. This is not currently used, though it's possible a Toom-4 might fit in between Toom-3 and the FFTs. The notes here are merely for interest. In general a split into @math{r+1} pieces is made, and evaluations and pointwise multiplications done at @m{2r+1,2*r+1} points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an @math{(r+1)}-way algorithm is @m{O(N^{log(2r+1)/log(r+1)}, O(N^(log(2*r+1)/log(r+1)))}. Only the pointwise multiplications count towards big-@math{O} complexity, but the time spent in the evaluate and interpolate stages grows with @math{r} and has a significant practical impact, with the asymptotic advantage of each @math{r} realized only at bigger and bigger sizes. The overheads grow as @m{O(Nr),O(N*r)}, whereas in an @math{r=2^k} FFT they grow only as @m{O(N \log r), O(N*log(r))}. Knuth algorithm C evaluates at points 0,1,2,@dots{},@m{2r,2*r}, but exercise 4 uses @math{-r},@dots{},0,@dots{},@math{r} and the latter saves some small multiplies in the evaluate stage (or rather trades them for additions), and has a further saving of nearly half the interpolate steps. The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately. The divisors at step C7 become @math{j^2} and the multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}. Splitting odd and even parts through positive and negative points can be thought of as using @math{-1} as a square root of unity. If a 4th root of unity was available then a further split and speedup would be possible, but no such root exists for plain integers. Going to complex integers with @m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in cartesian form it takes three real multiplies to do a complex multiply. The existence of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}. Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it's very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP. @node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms @section Division Algorithms @cindex Division algorithms @menu * Single Limb Division:: * Basecase Division:: * Divide and Conquer Division:: * Exact Division:: * Exact Remainder:: * Small Quotient Division:: @end menu @node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms @subsection Single Limb Division N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU. The multiply by inverse follows section 8 of Division by Invariant Integers using Multiplication'' by Granlund and Montgomery (@pxref{References}) and is implemented as @code{udiv_qrnnd_preinv} in @file{gmp-impl.h}. The idea is to have a fixed-point approximation to @math{1/d} (see @code{invert_limb}) and then multiply by the high limb (plus one bit) of the dividend to get a quotient @math{q}. With @math{d} normalized (high bit set), @math{q} is no more than 1 too small. Subtracting @m{qd,q*d} from the dividend gives a remainder, and reveals whether @math{q} or @math{q-1} is correct. The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it's only better on inputs bigger than say 4 or 5 limbs. When a divisor must be normalized, either for the generic C @code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @math{a} is the dividend and @math{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set. The bit shifts for the dividend are usually accomplished on the fly'' meaning by extracting the appropriate bits at each step. Done this way the quotient limbs come out aligned ready to store. When only the remainder is wanted, an alternative is to take the dividend limbs unshifted and calculate @m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k \bmod d2^k, r*2^k mod d*2^k}. This can help on CPUs with poor bit shifts or few registers. The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2@cross{}2 multiply, but the four 1@cross{}1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2@cross{}1 calculating @m{qd,q*d}. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput. A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1@cross{}1 multiply for the inverse effectively becomes two @m{1\over2@cross{}1, (1/2)x1} for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined. @node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms @subsection Basecase Division Basecase N@cross{}M division is like long division done by hand, but in base @m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}. See Knuth section 4.3.1 algorithm D, and @file{mpn/generic/sb_divrem_mn.c}. Briefly stated, while the dividend remains larger than the divisor, a high quotient limb is formed and the N@cross{}1 product @m{qd,q*d} subtracted at the top end of the dividend. With a normalized divisor (most significant bit set), each quotient limb can be formed with a 2@cross{}1 division and a 1@cross{}1 multiplication plus some subtractions. The 2@cross{}1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse (the same as in @ref{Single Limb Division}) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely. With Q=N@minus{}M being the number of quotient limbs, this is an @m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase Q@cross{}M multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs. @node Divide and Conquer Division, Exact Division, Basecase Division, Division Algorithms @subsection Divide and Conquer Division For divisors larger than @code{DIV_DC_THRESHOLD}, division is done by dividing. Or to be precise by a recursive divide and conquer algorithm based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}). The algorithm consists essentially of recognising that a 2N@cross{}N division can be done with the basecase division algorithm (@pxref{Basecase Division}), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (@pxref{Multiplication Algorithms}). The digits'' of the quotient are formed by recursive N@cross{}(N/2) divisions. If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication then the work is about the same as a basecase division, but with more function call overheads and with some subtractions separated from the multiplies. These overheads mean that it's only when N/2 is above @code{MUL_KARATSUBA_THRESHOLD} that divide and conquer is of use. @code{DIV_DC_THRESHOLD} is based on the divisor size N, so it will be somewhere above twice @code{MUL_KARATSUBA_THRESHOLD}, but how much above depends on the CPU. An optimized @code{mpn_mul_basecase} can lower @code{DIV_DC_THRESHOLD} a little by offering a ready-made advantage over repeated @code{mpn_submul_1} calls. Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where @math{M(N)} is the time for an N@cross{}N multiplication done with FFTs. The actual time is a sum over multiplications of the recursed sizes, as can be seen near the end of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}. With higher algorithms the @math{M(N)} term improves and the multiplier tends to @m{\log N, log(N)}. In practice, at moderate to large sizes, a 2N@cross{}N division is about 2 to 4 times slower than an N@cross{}N multiplication. Newton's method used for division is asymptotically @math{O(M(N))} and should therefore be superior to divide and conquer, but it's believed this would only be for large to very large N. @node Exact Division, Exact Remainder, Divide and Conquer Division, Division Algorithms @subsection Exact Division A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean's exact division algorithm uses this knowledge to make some significant optimizations (@pxref{References}). The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from @m{4 \mathord{\times} 7 \bmod 10, 4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7 @equiv{} 1 mod 10}. So @m{8\mathord{\times}543 = 4344,8*543=4344} can be subtracted from the dividend leaving 363810. Notice the low digit has become zero. The procedure is repeated at the second digit, with the next quotient digit 7 (@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting @m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800. And finally at the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7 mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0. So the quotient is 678. Notice however that the multiplies and subtractions don't need to extend past the low three digits of the dividend, since that's enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2N@cross{}N division like this one, only about half the work of a normal basecase division is necessary. For an N@cross{}M exact division producing Q=N@minus{}M quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2@cross{}1 divide and multiply. Secondly, the crossproducts are reduced when @math{Q>M} to @m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q@le{}M} to @m{Q(Q-1)/2, Q*(Q-1)/2}. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number. The modular inverse used is calculated efficiently by @code{modlimb_invert} in @file{gmp-impl.h}. This does four multiplies for a 32-bit limb, or six for a 64-bit limb. @file{tune/modlinv.c} has some alternate implementations that might suit processors better at bit twiddling than multiplying. The sub-quadratic exact division described by Jebelean in Exact Division with Karatsuba Complexity'' is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (@pxref{Divide and Conquer Division}), but operating from low to high. A further possibility not currently implemented is Bidirectional Exact Integer Division'' by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2N@cross{}N division. A special case exact division by 3 exists in @code{mpn_divexact_by3}, supporting Toom-3 multiplication and @code{mpq} canonicalizations. It forms quotient digits with a multiply by the modular inverse of 3 (which is @code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next limb. The multiplications don't need to be on the dependent chain, as long as the effect of the borrows is applied. Only a few optimized assembler implementations currently exist. @node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms @subsection Exact Remainder If the exact division algorithm is done with a full subtraction at each stage and the dividend isn't a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend @math{a}, divisor @math{d}, quotient @math{q}, and @m{b = 2 \GMPraise{@code{mp\_bits\_per\_limb}}, b = 2^mp_bits_per_limb}, then this remainder @math{r} is of the form @tex $$a = qd + r b^n$$ @end tex @ifnottex @example a = q*d + r*b^n @end example @end ifnottex @math{n} represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for @math{q}. @math{r} will be in the range @math{0@le{}r<d} and can be viewed as a remainder, but one shifted up by a factor of @math{b^n}. Carrying out full subtractions at each stage means the same number of cross products must be done as a normal division, but there's still some single limb divisions saved. When @math{d} is a single limb some simplifications arise, providing good speedups on a number of processors. @code{mpn_bdivmod}, @code{mpn_divexact_by3}, @code{mpn_modexact_1_odd} and the @code{redc} function in @code{mpz_powm} differ subtly in how they return @math{r}, leading to some negations in the above formula, but all are essentially the same. Clearly @math{r} is zero when @math{a} is a multiple of @math{d}, and this leads to divisibility or congruence tests which are potentially more efficient than a normal division. The factor of @math{b^n} on @math{r} can be ignored in a GCD when @math{d} is odd, hence the use of @code{mpn_bdivmod} in @code{mpn_gcd}, and the use of @code{mpn_modexact_1_odd} by @code{mpn_gcd_1} and @code{mpz_kronecker_ui} etc (@pxref{Greatest Common Divisor Algorithms}). Montgomery's REDC method for modular multiplications uses operands of the form of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n}) (yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of @math{b^n} in the exact remainder to reach a product in the same form @m{(xy)b^{-n}, (x*y)*b^-n} (@pxref{Modular Powering Algorithm}). Notice that @math{r} generally gives no useful information about the ordinary remainder @math{a @bmod d} since @math{b^n @bmod d} could be anything. If however @math{b^n @equiv{} 1 @bmod d}, then @math{r} is the negative of the ordinary remainder. This occurs whenever @math{d} is a factor of @math{b^n-1}, as for example with 3 in @code{mpn_divexact_by3}. Other such factors include 5, 17 and 257, but no particular use has been found for this. @node Small Quotient Division, , Exact Remainder, Division Algorithms @subsection Small Quotient Division An N@cross{}M division where the number of quotient limbs Q=N@minus{}M is small can be optimized somewhat. An ordinary basecase division normalizes the divisor by shifting it to make the high bit set, shifting the dividend accordingly, and shifting the remainder back down at the end of the calculation. This is wasteful if only a few quotient limbs are to be formed. Instead a division of just the top @m{\rm2Q,2*Q} limbs of the dividend by the top Q limbs of the divisor can be used to form a trial quotient. This requires only those limbs normalized, not the whole of the divisor and dividend. A multiply and subtract then applies the trial quotient to the M@minus{}Q unused limbs of the divisor and N@minus{}Q dividend limbs (which includes Q limbs remaining from the trial quotient division). The starting trial quotient can be 1 or 2 too big, but all cases of 2 too big and most cases of 1 too big are detected by first comparing the most significant limbs that will arise from the subtraction. An addback is done if the quotient still turns out to be 1 too big. This whole procedure is essentially the same as one step of the basecase algorithm done in a Q limb base, though with the trial quotient test done only with the high limbs, not an entire Q limb digit'' product. The correctness of this weaker test can be established by following the argument of Knuth section 4.3.1 exercise 20 but with the @m{v_2 \GMPhat q > b \GMPhat r + u_2, v2*q>b*r+u2} condition appropriately relaxed. @need 1000 @node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms @section Greatest Common Divisor @cindex Greatest common divisor algorithms @menu * Binary GCD:: * Accelerated GCD:: * Extended GCD:: * Jacobi Symbol:: @end menu @node Binary GCD, Accelerated GCD, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms @subsection Binary GCD At small sizes GMP uses an @math{O(N^2)} binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing operands @math{a} and @math{b} using @math{@gcd{}(a,b) = @gcd{}(@min{}(a,b),@abs{}(a-b))}, and also that if @math{a} and @math{b} are first made odd then @math{@abs{}(a-b)} is even and factors of two can be discarded. Variants like letting @math{a-b} become negative and doing a different next step are of interest only as far as they suit particular CPUs, since on small operands it's machine dependent factors that determine performance. The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using @math{a @bmod b} but this has so far been found to be slower everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E. When the implied quotient is large, meaning @math{b} is much smaller than @math{a}, then a division is worthwhile. This is the basis for the initial @math{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter for both N@cross{}1 and 1@cross{}1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them. @node Accelerated GCD, Extended GCD, Binary GCD, Greatest Common Divisor Algorithms @subsection Accelerated GCD For sizes above @code{GCD_ACCEL_THRESHOLD}, GMP uses the Accelerated GCD algorithm described independently by Weber and Jebelean (the latter as the Generalized Binary'' algorithm), @pxref{References}. This algorithm is still @math{O(N^2)}, but is much faster than the binary algorithm since it does fewer multi-precision operations. It consists of alternating the @math{k}-ary reduction by Sorenson, and a dmod'' exact remainder reduction. For operands @math{u} and @math{v} the @math{k}-ary reduction replaces @math{u} with @m{nv-du,n*v-d*u} where @math{n} and @math{d} are single limb values chosen to give two trailing zero limbs on that value, which can be stripped. @math{n} and @math{d} are calculated using an algorithm similar to half of a two limb GCD (see @code{find_a} in @file{mpn/generic/gcd.c}). When @math{u} and @math{v} differ in size by more than a certain number of bits, a dmod is performed to zero out bits at the low end of the larger. It consists of an exact remainder style division applied to an appropriate number of bits (@pxref{Exact Division}, and @pxref{Exact Remainder}). This is faster than a @math{k}-ary reduction but useful only when the operands differ in size. There's a dmod after each @math{k}-ary reduction, and if the dmod leaves the operands still differing in size then it's repeated. The @math{k}-ary reduction step can introduce spurious factors into the GCD calculated, and these are eliminated at the end by taking GCDs with the original inputs @math{@gcd{}(u,@gcd{}(v,g))} using the binary algorithm. Since @math{g} is almost always small this takes very little time. At small sizes the algorithm needs a good implementation of @code{find_a}. At larger sizes it's dominated by @code{mpn_addmul_1} applying @math{n} and @math{d}. @node Extended GCD, Jacobi Symbol, Accelerated GCD, Greatest Common Divisor Algorithms @subsection Extended GCD The extended GCD calculates @math{@gcd{}(a,b)} and also cofactors @math{x} and @math{y} satisfying @m{ax+by=\gcd(a@C{}b), a*x+b*y=gcd(a@C{}b)}. Lehmer's multi-step improvement of the extended Euclidean algorithm is used. See Knuth section 4.5.2 algorithm L, and @file{mpn/generic/gcdext.c}. This is an @math{O(N^2)} algorithm. The multipliers at each step are found using single limb calculations for sizes up to @code{GCDEXT_THRESHOLD}, or double limb calculations above that. The single limb code is faster but doesn't produce full-limb multipliers, hence not making full use of the @code{mpn_addmul_1} calls. When a CPU has a data-dependent multiplier, meaning one which is faster on operands with fewer bits, the extra work in the double-limb calculation might only save some looping overheads, leading to a large @code{GCDEXT_THRESHOLD}. Currently the single limb calculation doesn't optimize for the small quotients that often occur, and this can lead to unusually low values of @code{GCDEXT_THRESHOLD}, depending on the CPU. An analysis of double-limb calculations can be found in A Double-Digit Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}). The code in GMP was developed independently. It should be noted that when a double limb calculation is used, it's used for the whole of that GCD, it doesn't fall back to single limb part way through. This is because as the algorithm proceeds, the inputs @math{a} and @math{b} are reduced, but the cofactors @math{x} and @math{y} grow, so the multipliers at each step are applied to a roughly constant total number of limbs. @node Jacobi Symbol, , Extended GCD, Greatest Common Divisor Algorithms @subsection Jacobi Symbol @code{mpz_jacobi} and @code{mpz_kronecker} are currently implemented with a simple binary algorithm similar to that described for the GCDs (@pxref{Binary GCD}). They're not very fast when both inputs are large. Lehmer's multi-step improvement or a binary based multi-step algorithm is likely to be better. When one operand fits a single limb, and that includes @code{mpz_kronecker_ui} and friends, an initial reduction is done with either @code{mpn_mod_1} or @code{mpn_modexact_1_odd}, followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps. @need 1000 @node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms @section Powering Algorithms @cindex Powering algorithms @menu * Normal Powering Algorithm:: * Modular Powering Algorithm:: @end menu @node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms @subsection Normal Powering Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The left to right'' variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory. @node Modular Powering Algorithm, , Normal Powering Algorithm, Powering Algorithms @subsection Modular Powering Modular powering is implemented using a @math{2^k}-ary sliding window algorithm, as per Handbook of Applied Cryptography'' algorithm 14.85 (@pxref{References}). @math{k} is chosen according to the size of the exponent. Larger exponents use larger values of @math{k}, the choice being made to minimize the average number of multiplications that must supplement the squaring. The modular multiplies and squares use either a simple division or the REDC method by Montgomery (@pxref{References}). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (@pxref{Exact Remainder}). The current REDC has some limitations. It's only @math{O(N^2)} so above @code{POWM_THRESHOLD} division becomes faster and is used. It doesn't attempt to detect small bases, but rather always uses a REDC form, which is usually a full size operand. And lastly it's only applied to odd moduli. @node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms @section Root Extraction Algorithms @cindex Root extraction algorithms @menu * Square Root Algorithm:: * Nth Root Algorithm:: * Perfect Square Algorithm:: * Perfect Power Algorithm:: @end menu @node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms @subsection Square Root Square roots are taken using the Karatsuba Square Root'' algorithm by Paul Zimmermann (@pxref{References}). This is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method. In the Karatsuba multiplication range this is an @m{O({3\over2} M(N/2)),O(1.5*M(N/2))} algorithm, where @math{M(n)} is the time to multiply two numbers of @math{n} limbs. In the FFT multiplication range this grows to a bound of @m{O(6 M(N/2)),O(6*M(N/2))}. In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. The algorithm does all its calculations in integers and the resulting @code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}. The extended precision given by @code{mpf_sqrt_ui} is obtained by padding with zero limbs. @node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms @subsection Nth Root Integer Nth roots are taken using Newton's method with the following iteration, where @math{A} is the input and @math{n} is the root to be taken. @tex $$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$ @end tex @ifnottex @example 1 A a[i+1] = - * ( --------- + (n-1)*a[i] ) n a[i]^(n-1) @end example @end ifnottex The initial approximation @m{a_1,a[1]} is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When @math{n} is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized. @node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms @subsection Perfect Square @code{mpz_perfect_square_p} is able to quickly exclude most non-squares by checking whether the input is a quadratic residue modulo some small integers. The first test is modulo 256 which means simply examining the least significant byte. Only 44 different values occur as the low byte of a square, so 82.8% of non-squares can be immediately excluded. Similar tests modulo primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits then primes up to 53 are used, excluding 99.99%. A single N@cross{}1 remainder using @code{PP} from @file{gmp-impl.h} quickly gives all these remainders. A square root must still be taken for any value that passes the residue tests, to verify it's really a square and not one of the 0.086% (or 0.000156% for 64 bits) non-squares that get through. @xref{Square Root Algorithm}. @node Perfect Power Algorithm, , Perfect Square Algorithm, Root Extraction Algorithms @subsection Perfect Power Detecting perfect powers is required by some factorization algorithms. Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root extractions, though naturally only prime roots need to be considered. (@xref{Nth Root Algorithm}.) If a prime divisor @math{p} with multiplicity @math{e} can be found, then only roots which are divisors of @math{e} need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked. @node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms @section Radix Conversion @cindex Radix conversion algorithms Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation. @menu * Binary to Radix:: * Radix to Binary:: @end menu @node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms @subsection Binary to Radix Conversions from binary to a power-of-2 radix use a simple and fast @math{O(N)} bit extraction algorithm. Conversions from binary to other radices use one of two algorithms. Sizes below @code{GET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method. Repeated divisions by @math{b^n} are made, where @math{b} is the radix and @math{n} is the biggest power that fits in a limb. But instead of simply using the remainder @math{r} from such divisions, an extra divide step is done to give a fractional limb representing @math{r/b^n}. The digits of @math{r} can then be extracted using multiplications by @math{b} rather than divisions. Special case code is provided for decimal, allowing multiplications by 10 to optimize to shifts and adds. Above @code{GET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used. For an input @math{t}, powers @m{b^{n2^i},b^(n*2^i)} of the radix are calculated, until a power between @math{t} and @m{\sqrt{t},sqrt(t)} is reached. @math{t} is then divided by that largest power, giving a quotient which is the digits above that power, and a remainder which is those below. These two parts are in turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than @code{GET_STR_DC_THRESHOLD} limbs, the basecase algorithm described above is used. The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (@pxref{Divide and Conquer Division}), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers @m{b^{n2^i},b^(n*2^i)} must first be overcome. @code{GET_STR_PRECOMPUTE_THRESHOLD} and @code{GET_STR_DC_THRESHOLD} represent the same basic thing, the point where it becomes worth doing a big division to cut the input in half. @code{GET_STR_PRECOMPUTE_THRESHOLD} includes the cost of calculating the radix power required, whereas @code{GET_STR_DC_THRESHOLD} assumes that's already available, which is the case when recursing. Since the base case produces digits from least to most significant but they want to be stored from most to least, it's necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by @code{chars_per_bit_exactly} from @code{mp_bases}, rounding up. The result is either correct or one too big. Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100@dots{} and 99@dots{} is only in the last few bits and the work to identify 99@dots{} might well be almost as much as a full conversion. @code{mpf_get_str} doesn't currently use the algorithm described here, it multiplies or divides by a power of @math{b} to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by @math{b^n} to bring out digits. This is @math{O(N^2)} and is certainly not optimal. The @math{r/b^n} scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power. Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, ie. the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to @m{\sqrt{t},sqrt(t)}, not restricted to a @math{2^i} factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup. @node Radix to Binary, , Binary to Radix, Radix Conversion Algorithms @subsection Radix to Binary Conversions from a power-of-2 radix into binary use a simple and fast @math{O(N)} bitwise concatenation algorithm. Conversions from other radices use one of two algorithms. Sizes below @code{SET_STR_THRESHOLD} use a basic @math{O(N^2)} method. Groups of @math{n} digits are converted to limbs, where @math{n} is the biggest power of the base @math{b} which will fit in a limb, then those groups are accumulated into the result by multiplying by @math{b^n} and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (@pxref{References}). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10. Above @code{SET_STR_THRESHOLD} a sub-quadratic algorithm is used. First groups of @math{n} digits are converted into limbs. Then adjacent limbs are combined into limb pairs with @m{xb^n+y,x*b^n+y}, where @math{x} and @math{y} are the limbs. Adjacent limb pairs are combined into quads similarly with @m{xb^{2n}+y,x*b^(2n)+y}. This continues until a single block remains, that being the result. The advantage of this method is that the multiplications for each @math{x} are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome. @code{SET_STR_THRESHOLD} usually ends up quite big, around 5000 digits, and on some processors much bigger still. @code{SET_STR_THRESHOLD} is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn't used by the base case and so would need some sort of initial calculation or estimate. The main reason @code{SET_STR_THRESHOLD} is so much bigger than the corresponding @code{GET_STR_PRECOMPUTE_THRESHOLD} is that @code{mpn_mul_1} is much faster than @code{mpn_divrem_1} (often by a factor of 10, or more). @need 1000 @node Other Algorithms, Assembler Coding, Radix Conversion Algorithms, Algorithms @section Other Algorithms @menu * Factorial Algorithm:: * Binomial Coefficients Algorithm:: * Fibonacci Numbers Algorithm:: * Lucas Numbers Algorithm:: @end menu @node Factorial Algorithm, Binomial Coefficients Algorithm, Other Algorithms, Other Algorithms @subsection Factorial Factorials @math{n!} are calculated by a simple product from @math{1} to @math{n}, but arranged into certain sub-products. First as many factors as fit in a limb are accumulated, then two of those multiplied to give a 2-limb product. When two 2-limb products are ready they're multiplied to a 4-limb product, and when two 4-limbs are ready they're multiplied to an 8-limb product, etc. A stack of outstanding products is built up, with two of the same size multiplied together when ready. Arranging for multiplications to have operands the same (or nearly the same) size means the Karatsuba and higher multiplication algorithms can be used. And even on sizes below the Karatsuba threshold an N@cross{}N multiply will give a basecase multiply more to work on. An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift. Another possibility would be to determine the prime factorization of the result (which can be done easily), and use a powering method, at each stage squaring then multiplying in those primes with a 1 in their exponent at that point. The advantage would be some multiplies turned into squares. @node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms @subsection Binomial Coefficients Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated by first arranging @math{k @le{} n/2} using @m{\left({n}\atop{k}\right) = \left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then evaluating the following product simply from @math{i=2} to @math{i=k}. @tex $$\left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i}$$ @end tex @ifnottex @example k (n-k+i) C(n,k) = (n-k+1) * prod ------- i=2 i @end example @end ifnottex It's easy to show that each denominator @math{i} will divide the product so far, so the exact division algorithm is used (@pxref{Exact Division}). The numerators @math{n-k+i} and denominators @math{i} are first accumulated into as many fit a limb, to save multi-precision operations, though for @code{mpz_bin_ui} this applies only to the divisors, since @math{n} is an @code{mpz_t} and @math{n-k+i} in general won't fit in a limb at all. An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly. Another possibility, if @math{n} is not too big, would be to determine the prime factorization of the result based on the factorials involved, and power up those primes appropriately. This would help most when @math{k} is near @math{n/2}. @node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms @subsection Fibonacci Numbers The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]} values efficiently. For small @math{n}, a table of single limb values in @code{__gmp_fib_table} is used. On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb up to @m{F_{93},F[93]}. For convenience the table starts at @m{F_{-1},F[-1]}. Beyond the table, values are generated with a binary powering algorithm, calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to low across the bits of @math{n}. The formulas used are @tex \eqalign{ F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr F_{2k-1} &= F_k^2 + F_{k-1}^2 \cr F_{2k} &= F_{2k+1} - F_{2k-1} } @end tex @ifnottex @example F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1] @end example @end ifnottex At each step, @math{k} is the high @math{b} bits of @math{n}. If the next bit of @math{n} is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if it's a 1 then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process repeated until all bits of @math{n} are incorporated. Notice these formulas require just two squares per bit of @math{n}. It'd be possible to handle the first few @math{n} above the single limb table with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} = F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually turns out to be faster for only about 10 or 20 values of @math{n}, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table. Using a table avoids lots of calculations on small numbers, and makes small @math{n} go fast. A bigger table would make more small @math{n} go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm. @code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but @code{mpz_fib_ui} is only interested in @m{F_n,F[n]}. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as @math{n} is odd or even. @tex \eqalign{ F_{2k} &= F_k (F_k + 2F_{k-1}) \cr F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k } @end tex @ifnottex @example F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k @end example @end ifnottex @m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a multiply. For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with @code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done. @node Lucas Numbers Algorithm, , Fibonacci Numbers Algorithm, Other Algorithms @subsection Lucas Numbers @code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas. @tex \eqalign{ L_k &= F_k + 2F_{k-1} \cr L_{k-1} &= 2F_k - F_{k-1} } @end tex @ifnottex @example L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1] @end example @end ifnottex @code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be saved. Trailing zero bits on @math{n} can be handled with a single square each. @tex $$L_{2k} = L_k^2 - 2(-1)^k$$ @end tex @ifnottex @example L[2k] = L[k]^2 - 2*(-1)^k @end example @end ifnottex And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what @code{mpz_fib_ui} does. @tex $$L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k$$ @end tex @ifnottex @example L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k @end example @end ifnottex @node Assembler Coding, , Other Algorithms, Algorithms @section Assembler Coding The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes. At larger sizes algorithm selection becomes more important, but of course speedups in low level routines will still speed up everything proportionally. Carry handling and widening multiplies that are important for GMP can't be easily expressed in C. GCC @code{asm} blocks help a lot and are provided in @file{longlong.h}, but hand coding low level routines invariably offers a speedup over generic C by a factor of anything from 2 to 10. @menu * Assembler Code Organisation:: * Assembler Basics:: * Assembler Carry Propagation:: * Assembler Cache Handling:: * Assembler Floating Point:: * Assembler SIMD Instructions:: * Assembler Software Pipelining:: * Assembler Loop Unrolling:: @end menu @node Assembler Code Organisation, Assembler Basics, Assembler Coding, Assembler Coding @subsection Code Organisation The various @file{mpn} subdirectories contain machine-dependent code, written in C or assembler. The @file{mpn/generic} subdirectory contains default code, used when there's no machine-specific version of a particular file. Each @file{mpn} subdirectory is for an ISA family. Generally 32-bit and 64-bit variants in a family cannot share code and will have separate directories. Within a family further subdirectories may exist for CPU variants. @node Assembler Basics, Assembler Carry Propagation, Assembler Code Organisation, Assembler Coding @subsection Assembler Basics @code{mpn_addmul_1} and @code{mpn_submul_1} are the most important routines for overall GMP performance. All multiplications and divisions come down to repeated calls to these. @code{mpn_add_n}, @code{mpn_sub_n}, @code{mpn_lshift} and @code{mpn_rshift} are next most important. On some CPUs assembler versions of the internal functions @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} give significant speedups, mainly through avoiding function call overheads. They can also potentially make better use of a wide superscalar processor. The restrictions on overlaps between sources and destinations (@pxref{Low-level Functions}) are designed to facilitate a variety of implementations. For example, knowing @code{mpn_add_n} won't have partly overlapping sources and destination means reading can be done far ahead of writing on superscalar processors, and loops can be vectorized on a vector processor, depending on the carry handling. @node Assembler Carry Propagation, Assembler Cache Handling, Assembler Basics, Assembler Coding @subsection Carry Propagation The problem that presents most challenges in GMP is propagating carries from one limb to the next. In functions like @code{mpn_addmul_1} and @code{mpn_add_n}, carries are the only dependencies between limb operations. On processors with carry flags, a straightforward CISC style @code{adc} is generally best. AMD K6 @code{mpn_addmul_1} however is an example of an unusual set of circumstances where a branch works out better. On RISC processors generally an add and compare for overflow is used. This sort of thing can be seen in @file{mpn/generic/aors_n.c}. Some carry propagation schemes require 4 instructions, meaning at least 4 cycles per limb, but other schemes may use just 1 or 2. On wide superscalar processors performance may be completely determined by the number of dependent instructions between carry-in and carry-out for each limb. On vector processors good use can be made of the fact that a carry bit only very rarely propagates more than one limb. When adding a single bit to a limb, there's only a carry out if that limb was @code{0xFF...FF} which on random data will be only 1 in @m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}. @file{mpn/cray/add_n.c} is an example of this, it adds all limbs in parallel, adds one set of carry bits in parallel and then only rarely needs to fall through to a loop propagating further carries. On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code for the RISC style idioms that are necessary to handle carry bits in C. Often conditional jumps are generated where @code{adc} or @code{sbb} forms would be better. And so unfortunately almost any loop involving carry bits needs to be coded in assembler for best results. @node Assembler Cache Handling, Assembler Floating Point, Assembler Carry Propagation, Assembler Coding @subsection Cache Handling GMP aims to perform well both on operands that fit entirely in L1 cache and those which don't. Basic routines like @code{mpn_add_n} or @code{mpn_lshift} are often used on large operands, so L2 and main memory performance is important for them. @code{mpn_mul_1} and @code{mpn_addmul_1} are mostly used for multiply and square basecases, so L1 performance matters most for them, unless assembler versions of @code{mpn_mul_basecase} and @code{mpn_sqr_basecase} exist, in which case the remaining uses are mostly for larger operands. For L2 or main memory operands, memory access times will almost certainly be more than the calculation time. The aim therefore is to maximize memory throughput, by starting a load of the next cache line which processing the contents of the previous one. Clearly this is only possible if the chip has a lock-up free cache or some sort of prefetch instruction. Most current chips have both these features. Prefetching sources combines well with loop unrolling, since a prefetch can be initiated once per unrolled loop (or more than once if the loop covers more than one cache line). On CPUs without write-allocate caches, prefetching destinations will ensure individual stores don't go further down the cache hierarchy, limiting bandwidth. Of course for calculations which are slow anyway, like @code{mpn_divrem_1}, write-throughs might be fine. The distance ahead to prefetch will be determined by memory latency versus throughput. The aim of course is to have data arriving continuously, at peak throughput. Some CPUs have limits on the number of fetches or prefetches in progress. If a special prefetch instruction doesn't exist then a plain load can be used, but in that case care must be taken not to attempt to read past the end of an operand, since that might produce a segmentation violation. Some CPUs or systems have hardware that detects sequential memory accesses and initiates suitable cache movements automatically, making life easy. @node Assembler Floating Point, Assembler SIMD Instructions, Assembler Cache Handling, Assembler Coding @subsection Floating Point Floating point arithmetic is used in GMP for multiplications on CPUs with poor integer multipliers. It's mostly useful for @code{mpn_mul_1}, @code{mpn_addmul_1} and @code{mpn_submul_1} on 64-bit machines, and @code{mpn_mul_basecase} on both 32-bit and 64-bit machines. With IEEE 53-bit double precision floats, integer multiplications producing up to 53 bits will give exact results. Breaking a 64@cross{}64 multiplication into eight 16@cross{}@math{32@rightarrow{}48} bit pieces is convenient. With some care though six 21@cross{}@math{32@rightarrow{}53} bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit. For the @code{mpn_mul_1} family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast conversion of these unsigned 32-bit pieces to floating point is highly machine-dependent. In some cases, reading the data into the integer unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option. Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than @m{2^{53},2^53}, signed and unsigned are the same, but most processors lack unsigned conversions. @sp 2 Here is a diagram showing 16@cross{}32 bit products for an @code{mpn_mul_1} or @code{mpn_addmul_1} with a 64-bit limb. The single limb operand V is split into four 16-bit parts. The multi-limb operand U is split in the loop into two 32-bit parts. @tex \global\newdimen\GMPbits \global\GMPbits=0.18em \def\GMPbox#1#2#3{% \hbox{% \hbox to 128\GMPbits{\hfil \vbox{% \hrule \hbox to 48\GMPbits {\GMPvrule \hfil#2$\hfil \vrule}% \hrule}% \hskip #1\GMPbits}% \raise \GMPboxdepth \hbox{\hskip 2em #3}}} % \GMPdisplay{% \vbox{% \hbox{% \hbox to 128\GMPbits {\hfil \vbox{% \hrule \hbox to 64\GMPbits{% \GMPvrule \hfil$v48$\hfil \vrule \hfil$v32$\hfil \vrule \hfil$v16$\hfil \vrule \hfil$v00$\hfil \vrule} \hrule}}% \raise \GMPboxdepth \hbox{\hskip 2em V Operand}} \vskip 0.5ex \hbox{% \hbox to 128\GMPbits {\hfil \raise \GMPboxdepth \hbox{$\times$\hskip 1.5em}% \vbox{% \hrule \hbox to 64\GMPbits {% \GMPvrule \hfil$u32$\hfil \vrule \hfil$u00$\hfil \vrule}% \hrule}}% \raise \GMPboxdepth \hbox{\hskip 2em U Operand (one limb)}}% \vskip 0.5ex \hbox{\vbox to 2ex{\hrule width 128\GMPbits}}% \GMPbox{0}{u00 \times v00}{$p00$\hskip 1.5em 48-bit products}% \vskip 0.5ex \GMPbox{16}{u00 \times v16}{$p16$} \vskip 0.5ex \GMPbox{32}{u00 \times v32}{$p32$} \vskip 0.5ex \GMPbox{48}{u00 \times v48}{$p48$} \vskip 0.5ex \GMPbox{32}{u32 \times v00}{$r32$} \vskip 0.5ex \GMPbox{48}{u32 \times v16}{$r48$} \vskip 0.5ex \GMPbox{64}{u32 \times v32}{$r64$} \vskip 0.5ex \GMPbox{80}{u32 \times v48}{$r80} }} @end tex @ifnottex @example @group +---+---+---+---+ |v48|v32|v16|v00| V operand +---+---+---+---+ +-------+---+---+ x | u32 | u00 | U operand (one limb) +---------------+ --------------------------------- +-----------+ | u00 x v00 | p00 48-bit products +-----------+ +-----------+ | u00 x v16 | p16 +-----------+ +-----------+ | u00 x v32 | p32 +-----------+ +-----------+ | u00 x v48 | p48 +-----------+ +-----------+ | u32 x v00 | r32 +-----------+ +-----------+ | u32 x v16 | r48 +-----------+ +-----------+ | u32 x v32 | r64 +-----------+ +-----------+ | u32 x v48 | r80 +-----------+ @end group @end example @end ifnottex @math{p32} and @math{r32} can be summed using floating-point addition, and likewise @math{p48} and @math{r48}. @math{p00} and @math{p16} can be summed with @math{r64} and @math{r80} from the previous iteration. For each loop then, four 49-bit quantities are transfered to the integer unit, aligned as follows, @tex % GMPbox here should be 49 bits wide, but use 51 to better show p16+r80' % crossing into the upper 64 bits. \def\GMPbox#1#2#3{% \hbox{% \hbox to 128\GMPbits {% \hfil \vbox{% \hrule \hbox to 51\GMPbits {\GMPvrule \hfil#2$\hfil \vrule}% \hrule}% \hskip #1\GMPbits}% \raise \GMPboxdepth \hbox{\hskip 1.5em$#3$\hfil}% }} \newbox\b \setbox\b\hbox{64 bits}% \newdimen\bw \bw=\wd\b \advance\bw by 2em \newdimen\x \x=128\GMPbits \advance\x by -2\bw \divide\x by4 \GMPdisplay{% \vbox{% \hbox to 128\GMPbits {% \GMPvrule \raise 0.5ex \vbox{\hrule \hbox to \x {}}% \hfil 64 bits\hfil \raise 0.5ex \vbox{\hrule \hbox to \x {}}% \vrule \raise 0.5ex \vbox{\hrule \hbox to \x {}}% \hfil 64 bits\hfil \raise 0.5ex \vbox{\hrule \hbox to \x {}}% \vrule}% \vskip 0.7ex \GMPbox{0}{p00+r64'}{i00} \vskip 0.5ex \GMPbox{16}{p16+r80'}{i16} \vskip 0.5ex \GMPbox{32}{p32+r32}{i32} \vskip 0.5ex \GMPbox{48}{p48+r48}{i48} }} @end tex @ifnottex @example @group |-----64bits----|-----64bits----| +------------+ | p00 + r64' | i00 +------------+ +------------+ | p16 + r80' | i16 +------------+ +------------+ | p32 + r32 | i32 +------------+ +------------+ | p48 + r48 | i48 +------------+ @end group @end example @end ifnottex The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (@math{i48} extends 33 bits into the high half). @node Assembler SIMD Instructions, Assembler Software Pipelining, Assembler Floating Point, Assembler Coding @subsection SIMD Instructions The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP. SIMD multiplications of say four 16@cross{}16 bit multiplies only do as much work as one 32@cross{}32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile. On the 80x86 chips, MMX has so far found a use in @code{mpn_rshift} and @code{mpn_lshift} since it allows 64-bit operations, and is used in a special case for 16-bit multipliers in the P55 @code{mpn_mul_1}. 3DNow and SSE haven't found a use so far. @node Assembler Software Pipelining, Assembler Loop Unrolling, Assembler SIMD Instructions, Assembler Coding @subsection Software Pipelining Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop taking a checksum of an array of limbs might have a load and an add, but the load wouldn't be for that add, rather for the one next time around the loop. Each load then is effectively scheduled back in the previous iteration, allowing latency to be hidden. Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete, and only where a CPU has multiple functional units so that other work can be done while waiting. A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling. Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration. Loop unrolling can help this, with each unrolled block able to use different registers for different values, even if some shuffling is still needed just before going back to the top of the loop. @node Assembler Loop Unrolling, , Assembler Software Pipelining, Assembler Coding @subsection Loop Unrolling Loop unrolling consists of replicating code so that several limbs are processed in each loop. At a minimum this reduces loop overheads by a corresponding factor, but it can also allow better register usage, for example alternately using one register combination and then another. Judicious use of @command{m4} macros can help avoid lots of duplication in the source code. Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and mask. But other multiples can be used too, just by subtracting each @var{n} limbs processed from a counter and waiting for less than @var{n} remaining (or offsetting the counter by @var{n} so it goes negative when there's less than @var{n} remaining). The limbs not a multiple of the unrolling can be handled in various ways, for example @itemize @bullet @item A simple loop at the end (or the start) to process the excess. Care will be wanted that it isn't too much slower than the unrolled part. @item A set of binary tests, for example after an 8-limb unrolling, test for 4 more limbs to process, then a further 2 more or not, and finally 1 more or not. This will probably take more code space than a simple loop. @item A @code{switch} statement, providing separate code for each possible excess, for example an 8-limb unrolling would have separate code for 0 remaining, 1 remaining, etc, up to 7 remaining. This might take a lot of code, but may be the best way to optimize all cases in combination with a deep pipelined loop. @item A computed jump into the middle of the loop, thus making the first iteration handle the excess. This should make times smoothly increase with size, which is attractive, but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining. @end itemize One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end, then delete instructions at the start which have no valid antecedents, and delete instructions at the end whose results are unwanted. Sizes not a multiple of the unrolling can then be handled as desired. @node Internals, Contributors, Algorithms, Top @chapter Internals @strong{This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.} @menu * Integer Internals:: * Rational Internals:: * Float Internals:: * Raw Output Internals:: * C++ Interface Internals:: @end menu @node Integer Internals, Rational Internals, Internals, Internals @section Integer Internals @code{mpz_t} variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows. @table @asis @item @code{_mp_size} The number of limbs, or the negative of that when representing a negative integer. Zero is represented by @code{_mp_size} set to zero, in which case the @code{_mp_d} data is unused. @item @code{_mp_d} A pointer to an array of limbs which is the magnitude. These are stored little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the least significant limb and @code{_mp_d[ABS(_mp_size)-1]} is the most significant. Whenever @code{_mp_size} is non-zero, the most significant limb is non-zero. Currently there's always at least one limb allocated, so for instance @code{mpz_set_ui} never needs to reallocate, and @code{mpz_get_ui} can fetch @code{_mp_d[0]} unconditionally (though its value is then only wanted if @code{_mp_size} is non-zero). @item @code{_mp_alloc} @code{_mp_alloc} is the number of limbs currently allocated at @code{_mp_d}, and naturally @code{_mp_alloc >= ABS(_mp_size)}. When an @code{mpz} routine is about to (or might be about to) increase @code{_mp_size}, it checks @code{_mp_alloc} to see whether there's enough space, and reallocates if not. @code{MPZ_REALLOC} is generally used for this. @end table The various bitwise logical functions like @code{mpz_and} behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn't pretty, but sign and magnitude are best for other routines. Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences). @node Rational Internals, Float Internals, Integer Internals, Internals @section Rational Internals @code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and denominator (@pxref{Integer Internals}). The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1. It's believed that casting out common factors at each stage of a calculation is best in general. A GCD is an @math{O(N^2)} operation so it's better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in @code{mpq_mul} to make only two cross GCDs necessary, not four. This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per @ref{Efficiency}, that's left to applications. The @code{mpq_t} framework might still suit, with @code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and denominator, or of course @code{mpz_t} variables can be used directly. @node Float Internals, Raw Output Internals, Rational Internals, Internals @section Float Internals Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this. @code{mpf_t} floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude. @c FIXME: The arrow heads don't join to the lines exactly. @tex \global\newdimen\GMPboxwidth \GMPboxwidth=5em \global\newdimen\GMPboxheight \GMPboxheight=3ex \def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}} \GMPdisplay{% \vbox{% \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb} \vskip 0.7ex \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}} \hbox { \hbox to 3\GMPboxwidth {% \setbox 0 = \hbox{@code{\_mp\_exp}}% \dimen0=3\GMPboxwidth \advance\dimen0 by -\wd0 \divide\dimen0 by 2 \advance\dimen0 by -1em \setbox1 = \hbox{$\rightarrow$}% \dimen1=\dimen0 \advance\dimen1 by -\wd1 \GMPcentreline{\dimen0}% \hfil \box0% \hfil \GMPcentreline{\dimen1{}}% \box1} \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}} \vskip 0.5ex \vbox {% \hrule \hbox{% \vrule height 2ex depth 1ex \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule} \hrule } \hbox {% \hbox to 0.8 pt {} \hbox to 3\GMPboxwidth {% \hfil$\cdot$} \hbox {$\leftarrow$radix point\hfil}} \hbox to 5\GMPboxwidth{% \setbox 0 = \hbox{@code{\_mp\_size}}% \dimen0 = 5\GMPboxwidth \advance\dimen0 by -\wd0 \divide\dimen0 by 2 \advance\dimen0 by -1em \dimen1 = \dimen0 \setbox1 = \hbox{$\leftarrow$}% \setbox2 = \hbox{$\rightarrow\$}%
\hbox to 0.3 em {}%
\box1
\GMPcentreline{\dimen0}%
\hfil
\box0
\hfil
\GMPcentreline{\dimen1}%
\box2}
}}
@end tex
@ifnottex
@example
most                   least
significant            significant
limb                   limb

_mp_d
|---- _mp_exp --->           |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|

<-------- _mp_size --------->
@sp 1
@end example
@end ifnottex

@noindent
The fields are as follows.

@table @asis
@item @code{_mp_size}
The number of limbs currently in use, or the negative of that when
representing a negative value.  Zero is represented by @code{_mp_size} and
@code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is
unused.  (In the future @code{_mp_exp} might be undefined when representing
zero.)

@item @code{_mp_prec}
The precision of the mantissa, in limbs.  In any calculation the aim is to
produce @code{_mp_prec} limbs of result (the most significant being non-zero).

@item @code{_mp_d}
A pointer to the array of limbs which is the absolute value of the mantissa.
These are stored little endian'' as per the @code{mpn} functions, so
@code{_mp_d[0]} is the least significant limb and
@code{_mp_d[ABS(_mp_size)-1]} the most significant.

The most significant limb is always non-zero, but there are no other
restrictions on its value, in particular the highest 1 bit can be anywhere
within the limb.

@code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being
for convenience (see below).  There are no reallocations during a calculation,
only in a change of precision with @code{mpf_set_prec}.

@item @code{_mp_exp}
The exponent, in limbs, determining the location of the implied radix point.
Zero means the radix point is just above the most significant limb.  Positive
values mean a radix point offset towards the lower limbs and hence a value
@math{@ge{} 1}, as for example in the diagram above.  Negative exponents mean
a radix point further above the highest limb.

Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data
are treated as zero.
@end table

@sp 1
@noindent
The following various points should be noted.

@table @asis
@item Low Zeros
The least significant limbs @code{_mp_d[0]} etc can be zero, though such low
zeros can always be ignored.  Routines likely to produce low zeros check and
avoid them to save time in subsequent calculations, but for most routines
they're quite unlikely and aren't checked.

@item Mantissa Size Range
The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if
the value can be represented in less.  This means low precision values or
small integers stored in a high precision @code{mpf_t} can still be operated
on efficiently.

@code{_mp_size} can also be greater than @code{_mp_prec}.  Firstly a value is
allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d},
and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves
@code{_mp_size} unchanged and so the size can be arbitrarily bigger than
@code{_mp_prec}.

@item Rounding
All rounding is done on limb boundaries.  Calculating @code{_mp_prec} limbs
with the high non-zero will ensure the application requested minimum precision
is obtained.

The use of simple trunc'' rounding towards zero is efficient, since there's
no need to examine extra limbs and increment or decrement.

@item Bit Shifts
Since the exponent is in limbs, there are no bit shifts in basic operations
like @code{mpf_add} and @code{mpf_mul}.  When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.

Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.

@item Use of @code{_mp_prec+1} Limbs
The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just
@code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its
@code{_mp_prec} limbs.  If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
@code{_mp_size} becomes @code{_mp_prec+1}.

Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not
needed for the intended precision, only the @code{_mp_prec} high limbs.  But
zeroing it out or moving the rest down is unnecessary.  Subsequent routines
reading the value will simply take the high limbs they need, and this will be
@code{_mp_prec} if their target has that same precision.  This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.

Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs
if available.  This ensures that a variable which has @code{_mp_size} equal to
@code{_mp_prec+1} will get its full exact value copied.  Strictly speaking
this is unnecessary since only @code{_mp_prec} limbs are needed for the
application's requested precision, but it's considered that an @code{mpf_set}
from one variable into another of the same precision ought to produce an exact
copy.

@item Application Precisions
@code{__GMPF_BITS_TO_PREC} converts an application requested precision to an
@code{_mp_prec}.  The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of @code{_mp_d} is only
non-zero and therefore might contain only one bit.

@code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra
limb from @code{_mp_prec} before converting to bits.  The net effect of
reading back with @code{mpf_get_prec} is simply the precision rounded up to a
multiple of @code{mp_bits_per_limb}.

Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to @code{_mp_d}.  For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
@code{_mp_prec} of 9.  @code{_mp_d} then gets 10 limbs allocated.  Reading
back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and
multiply by 32, giving 256 bits.

Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice
multiple of the limb size.
@end table

@node Raw Output Internals, C++ Interface Internals, Float Internals, Internals
@section Raw Output Internals

@noindent
@code{mpz_out_raw} uses the following format.

@tex
\global\newdimen\GMPboxwidth \GMPboxwidth=5em
\global\newdimen\GMPboxheight \GMPboxheight=3ex
\def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}}
\GMPdisplay{%
\vbox{%
\def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}}
\vbox {%
\hrule
\hbox{%
\vrule height 2.5ex depth 1.5ex
\hbox to \GMPboxwidth {\hfil size\hfil}%
\vrule
\hbox to 3\GMPboxwidth {\hfil data bytes\hfil}%
\vrule}
\hrule}
}}
@end tex
@ifnottex
@example
+------+------------------------+
| size |       data bytes       |
+------+------------------------+
@end example
@end ifnottex

The size is 4 bytes written most significant byte first, being the number of
subsequent data bytes, or the twos complement negative of that when a negative
integer is represented.  The data bytes are the absolute value of the integer,
written most significant byte first.

The most significant data byte is always non-zero, so the output is the same
on all systems, irrespective of limb size.

In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
of the limb size.  @code{mpz_inp_raw} will still accept this, for
compatibility.

The use of big endian'' for both the size and data fields is deliberate, it
makes the data easy to read in a hex dump of a file.  Unfortunately it also
means that the limb data must be reversed when reading or writing, so neither
a big endian nor little endian system can just read and write @code{_mp_d}.

@node C++ Interface Internals,  , Raw Output Internals, Internals
@section C++ Interface Internals

A system of expression templates is used to ensure something like @code{a=b+c}
turns into a simple call to @code{mpz_add} etc.  For @code{mpf_class} and
@code{mpfr_class} the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
@code{f=w*x+y*z}.  These are important features which a naive implementation
cannot provide.

A simplified description of the scheme follows.  The true scheme is
complicated by the fact that expressions have different return types.  For
detailed information, refer to the source code.

To perform an operation, say, addition, we first define a function object''
evaluating it,

@example
struct __gmp_binary_plus
@{
static void eval(mpf_t f, mpf_t g, mpf_t h) @{ mpf_add(f, g, h); @}
@};
@end example

@noindent
And an additive expression'' object,

@example
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
@{
return __gmp_expr
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
@}
@end example

The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<...>>} is used to
encapsulate any possible kind of expression into a single template type.  In
fact even @code{mpf_class} etc are @code{typedef} specializations of
@code{__gmp_expr}.

Next we define assignment of @code{__gmp_expr} to @code{mpf_class}.

@example
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
@{
expr.eval(this->get_mpf_t(), this->precision());
return *this;
@}

template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, unsigned long int precision)
@{
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
@}
@end example

where @code{expr.val1} and @code{expr.val2} are references to the expression's
operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the
@code{__gmp_expr}).

This way, the expression is actually evaluated only at the time of assignment,
when the required precision (that of @code{f}) is known.  Furthermore the
target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly
with @code{f} as the output argument.

Compound expressions are handled by defining operators taking subexpressions
as their arguments, like this:

@example
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
@{
return __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
(expr1, expr2);
@}
@end example

And the corresponding specializations of @code{__gmp_expr::eval}:

@example
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, unsigned long int precision)
@{
// declare two temporaries
mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
@}
@end example

The expression is thus recursively evaluated to any level of complexity and
all subexpressions are evaluated to the precision of @code{f}.

@node Contributors, References, Internals, Top
@comment  node-name,  next,  previous,  up
@appendix Contributors
@cindex Contributors

Torbjorn Granlund wrote the original GMP library and is still developing and
maintaining it.  Several other individuals and organizations have contributed
to GMP in various ways.  Here is a list in chronological order:

Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early
versions of the library.

Richard Stallman contributed to the interface design and revised the first
version of this manual.

Brian Beuning and Doug Lea helped with testing of early versions of the

John Amanatides of York University in Canada contributed the function
@code{mpz_probab_prime_p}.

Paul Zimmermann of Inria sparked the development of GMP 2, with his
comparisons between bignum packages.

Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul)
contributed @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and
@code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2.

Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure.
He has also made valuable suggestions and tested numerous intermediary
releases.

Joachim Hollman was involved in the design of the @code{mpf} interface, and in
the @code{mpz} design revisions for version 2.

Bennet Yee contributed the initial versions of @code{mpz_jacobi} and
@code{mpz_legendre}.

Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and
@file{mpn/m68k/rshift.S} (now in @file{.asm} form).

The development of floating point functions of GNU MP 2, were supported in part
by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial
System SOlving).

GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the
IDA Center for Computing Sciences, USA.

Robert Harley of Inria, France and David Seal of ARM, England, suggested clever
improvements for population count.

Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
multiplication functions for GMP 3.  He also contributed the ARM assembly
code.

Torsten Ekedahl of the Mathematical department of Stockholm University provided
significant inspiration during several phases of the GMP development.  His
mathematical expertise helped improve several algorithms.

Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the
REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square
root.  The ECMNET project Paul is organizing was a driving force behind many
of the optimizations in GMP 3.

Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.

Kent Boortz made the Macintosh port.

Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros,
parameter tuning, speed measuring, the configure system, function inlining,
divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number
functions, printf and scanf functions, perl interface, demo expression parser,
the algorithms chapter in the manual, @file{gmpasm-mode.el}, and various
miscellaneous improvements elsewhere.

Steve Root helped write the optimized alpha 21264 assembly code.

Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++
@code{istream} input routines.

GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde.
Torbjorn's work was partially funded by the IDA Center for Computing Sciences,
USA.

(This list is chronological, not ordered after significance.  If you have
contributed to GMP but are not listed above, please tell @email{tege@@swox.com}

Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system
environment.

@node References, GNU Free Documentation License, Contributors, Top
@comment  node-name,  next,  previous,  up
@appendix References
@cindex References

@c  FIXME: In tex, the @uref's are unhyphenated, which is good for clarity,
@c  but being long words they upset paragraph formatting (the preceding line
@c  can get badly stretched).  Would like an conditional @* style line break
@c  if the uref is too long to fit on the last line of the paragraph, but it's
@c  not clear how to do that.  For now explicit @texlinebreak{}s are used on
@c  paragraphs that come out bad.

@section Books

@itemize @bullet
@item
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM: A Study in
Analytic Number Theory and Computational Complexity'', Wiley, John & Sons,
1998.

@item
Henri Cohen, A Course in Computational Algebraic Number Theory'', Graduate
Texts in Mathematics number 138, Springer-Verlag, 1993.
@texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen}

@item
Donald E. Knuth, The Art of Computer Programming'', volume 2,
Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998.
@texlinebreak{} @uref{http://www-cs-faculty.stanford.edu/~knuth/taocp.html}

@item
John D. Lipson, Elements of Algebra and Algebraic Computing'',
The Benjamin Cummings Publishing Company Inc, 1981.

@item
Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of
Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/}

@item
Richard M. Stallman, Using and Porting GCC'', Free Software Foundation, 1999,
available online @uref{http://www.gnu.org/software/gcc/onlinedocs/}, and in
the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/}
@end itemize

@section Papers

@itemize @bullet
@item
Christoph Burnikel and Joachim Ziegler, Fast Recursive Division'',
Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{}
@uref{http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022}

@item
Torbjorn Granlund and Peter L. Montgomery, Division by Invariant Integers
using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June
1994.  Also available @uref{ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz}
(and .psl.gz).

@item
Peter L. Montgomery, Modular Multiplication Without Trial Division'', in
Mathematics of Computation, volume 44, number 170, April 1985.

@item
Tudor Jebelean,
An algorithm for exact division'',
Journal of Symbolic Computation,
volume 15, 1993, pp. 169-180.
Research report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz}

@item
Tudor Jebelean, Exact Division with Karatsuba Complexity - Extended
Abstract'', RISC-Linz technical report 96-31, @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz}

@item
Tudor Jebelean, Practical Integer Division with Karatsuba Complexity'',
ISSAC 97, pp. 339-341.  Technical report available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz}

@item
Tudor Jebelean, A Generalization of the Binary GCD Algorithm'', ISSAC 93,
pp. 111-116.  Technical report version available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz}

@item
Tudor Jebelean, A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD
of Long Integers'', Journal of Symbolic Computation, volume 19, 1995,
pp. 145-157.  Technical report version also available @texlinebreak{}
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz}

@item
Werner Krandick and Tudor Jebelean, Bidirectional Exact Integer Division'',
Journal of Symbolic Computation, volume 21, 1996, pp. 441-455.  Early
technical report version also available
@uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz}

@item
R. Moenck and A. Borodin, Fast Modular Transforms via Division'',
Proceedings of the 13th Annual IEEE Symposium on Switching and Automata
Theory, October 1972, pp. 90-96.  Reprinted as Fast Modular Transforms'',
Journal of Computer and System Sciences, volume 8, number 3, June 1974,
pp. 366-386.

@item
Arnold Sch@"onhage and Volker Strassen, Schnelle Multiplikation grosser
Zahlen'', Computing 7, 1971, pp. 281-292.

@item
Kenneth Weber, The accelerated integer GCD algorithm'',
ACM Transactions on Mathematical Software,
volume 21, number 1, March 1995, pp. 111-122.

@item
Paul Zimmermann, Karatsuba Square Root'', INRIA Research Report 3805,
November 1999, @uref{http://www.inria.fr/RRRT/RR-3805.html}

@item
Paul Zimmermann, A Proof of GMP Fast Division and Square Root
Implementations'', @texlinebreak{}
@uref{http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz}

@item
Dan Zuras, On Squaring and Multiplying Large Integers'', ARITH-11: IEEE
Symposium on Computer Arithmetic, 1993, pp. 260 to 271.  Reprinted as More
on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers,
volume 43, number 8, August 1994, pp. 899-908.
@end itemize

@node GNU Free Documentation License, Concept Index, References, Top
@include fdl.texi

@node Concept Index, Function Index, GNU Free Documentation License, Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

@node Function Index,  , Concept Index, Top
@comment  node-name,  next,  previous,  up
@unnumbered Function and Type Index
@printindex fn

@bye

@c Local variables:
@c fill-column: 78
@c compile-command: "make gmp.info"
@c End:

`