## Artifact b21fd5c1b1effb75ddb410a69c3510e7c9da3699:

- Executable file wiki_references/2017/software/Rust/src_from_GitHub/the_repository_clones/rust/src/libcore/num/dec2flt/algorithm.rs — part of check-in [dee8e3e8ea] at 2017-05-19 18:47:53 on branch trunk — wiki reference upgrade (user: vhost7825ssh) [ancestry] [annotate] [blame]

// Copyright 2015 The Rust Project Developers. See the COPYRIGHT // file at the top-level directory of this distribution and at // http://rust-lang.org/COPYRIGHT. // // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your // option. This file may not be copied, modified, or distributed // except according to those terms. //! The various algorithms from the paper. use cmp::min; use cmp::Ordering::{Less, Equal, Greater}; use num::diy_float::Fp; use num::dec2flt::table; use num::dec2flt::rawfp::{self, Unpacked, RawFloat, fp_to_float, next_float, prev_float}; use num::dec2flt::num::{self, Big}; /// Number of significand bits in Fp const P: u32 = 64; // We simply store the best approximation for *all* exponents, so the variable "h" and the // associated conditions can be omitted. This trades performance for a couple kilobytes of space. fn power_of_ten(e: i16) -> Fp { assert!(e >= table::MIN_E); let i = e - table::MIN_E; let sig = table::POWERS.0[i as usize]; let exp = table::POWERS.1[i as usize]; Fp { f: sig, e: exp } } // In most architectures, floating point operations have an explicit bit size, therefore the // precision of the computation is determined on a per-operation basis. #[cfg(any(not(target_arch="x86"), target_feature="sse2"))] mod fpu_precision { pub fn set_precision<T>() { } } // On x86, the x87 FPU is used for float operations if the SSE/SSE2 extensions are not available. // The x87 FPU operates with 80 bits of precision by default, which means that operations will // round to 80 bits causing double rounding to happen when values are eventually represented as // 32/64 bit float values. To overcome this, the FPU control word can be set so that the // computations are performed in the desired precision. #[cfg(all(target_arch="x86", not(target_feature="sse2")))] mod fpu_precision { use mem::size_of; /// A structure used to preserve the original value of the FPU control word, so that it can be /// restored when the structure is dropped. /// /// The x87 FPU is a 16-bits register whose fields are as follows: /// /// | 12-15 | 10-11 | 8-9 | 6-7 | 5 | 4 | 3 | 2 | 1 | 0 | /// |------:|------:|----:|----:|---:|---:|---:|---:|---:|---:| /// | | RC | PC | | PM | UM | OM | ZM | DM | IM | /// /// The documentation for all of the fields is available in the IA-32 Architectures Software /// Developer's Manual (Volume 1). /// /// The only field which is relevant for the following code is PC, Precision Control. This /// field determines the precision of the operations performed by the FPU. It can be set to: /// - 0b00, single precision i.e. 32-bits /// - 0b10, double precision i.e. 64-bits /// - 0b11, double extended precision i.e. 80-bits (default state) /// The 0b01 value is reserved and should not be used. pub struct FPUControlWord(u16); fn set_cw(cw: u16) { unsafe { asm!("fldcw $0" :: "m" (cw) :: "volatile") } } /// Set the precision field of the FPU to `T` and return a `FPUControlWord` pub fn set_precision<T>() -> FPUControlWord { let cw = 0u16; // Compute the value for the Precision Control field that is appropriate for `T`. let cw_precision = match size_of::<T>() { 4 => 0x0000, // 32 bits 8 => 0x0200, // 64 bits _ => 0x0300, // default, 80 bits }; // Get the original value of the control word to restore it later, when the // `FPUControlWord` structure is dropped unsafe { asm!("fnstcw $0" : "=*m" (&cw) ::: "volatile") } // Set the control word to the desired precision. This is achieved by masking away the old // precision (bits 8 and 9, 0x300) and replacing it with the precision flag computed above. set_cw((cw & 0xFCFF) | cw_precision); FPUControlWord(cw) } impl Drop for FPUControlWord { fn drop(&mut self) { set_cw(self.0) } } } /// The fast path of Bellerophon using machine-sized integers and floats. /// /// This is extracted into a separate function so that it can be attempted before constructing /// a bignum. pub fn fast_path<T: RawFloat>(integral: &[u8], fractional: &[u8], e: i64) -> Option<T> { let num_digits = integral.len() + fractional.len(); // log_10(f64::MAX_SIG) ~ 15.95. We compare the exact value to MAX_SIG near the end, // this is just a quick, cheap rejection (and also frees the rest of the code from // worrying about underflow). if num_digits > 16 { return None; } if e.abs() >= T::CEIL_LOG5_OF_MAX_SIG as i64 { return None; } let f = num::from_str_unchecked(integral.iter().chain(fractional.iter())); if f > T::MAX_SIG { return None; } // The fast path crucially depends on arithmetic being rounded to the correct number of bits // without any intermediate rounding. On x86 (without SSE or SSE2) this requires the precision // of the x87 FPU stack to be changed so that it directly rounds to 64/32 bit. // The `set_precision` function takes care of setting the precision on architectures which // require setting it by changing the global state (like the control word of the x87 FPU). let _cw = fpu_precision::set_precision::<T>(); // The case e < 0 cannot be folded into the other branch. Negative powers result in // a repeating fractional part in binary, which are rounded, which causes real // (and occasionally quite significant!) errors in the final result. if e >= 0 { Some(T::from_int(f) * T::short_fast_pow10(e as usize)) } else { Some(T::from_int(f) / T::short_fast_pow10(e.abs() as usize)) } } /// Algorithm Bellerophon is trivial code justified by non-trivial numeric analysis. /// /// It rounds ``f`` to a float with 64 bit significand and multiplies it by the best approximation /// of `10^e` (in the same floating point format). This is often enough to get the correct result. /// However, when the result is close to halfway between two adjacent (ordinary) floats, the /// compound rounding error from multiplying two approximation means the result may be off by a /// few bits. When this happens, the iterative Algorithm R fixes things up. /// /// The hand-wavy "close to halfway" is made precise by the numeric analysis in the paper. /// In the words of Clinger: /// /// > Slop, expressed in units of the least significant bit, is an inclusive bound for the error /// > accumulated during the floating point calculation of the approximation to f * 10^e. (Slop is /// > not a bound for the true error, but bounds the difference between the approximation z and /// > the best possible approximation that uses p bits of significand.) pub fn bellerophon<T: RawFloat>(f: &Big, e: i16) -> T { let slop; if f <= &Big::from_u64(T::MAX_SIG) { // The cases abs(e) < log5(2^N) are in fast_path() slop = if e >= 0 { 0 } else { 3 }; } else { slop = if e >= 0 { 1 } else { 4 }; } let z = rawfp::big_to_fp(f).mul(&power_of_ten(e)).normalize(); let exp_p_n = 1 << (P - T::SIG_BITS as u32); let lowbits: i64 = (z.f % exp_p_n) as i64; // Is the slop large enough to make a difference when // rounding to n bits? if (lowbits - exp_p_n as i64 / 2).abs() <= slop { algorithm_r(f, e, fp_to_float(z)) } else { fp_to_float(z) } } /// An iterative algorithm that improves a floating point approximation of `f * 10^e`. /// /// Each iteration gets one unit in the last place closer, which of course takes terribly long to /// converge if `z0` is even mildly off. Luckily, when used as fallback for Bellerophon, the /// starting approximation is off by at most one ULP. fn algorithm_r<T: RawFloat>(f: &Big, e: i16, z0: T) -> T { let mut z = z0; loop { let raw = z.unpack(); let (m, k) = (raw.sig, raw.k); let mut x = f.clone(); let mut y = Big::from_u64(m); // Find positive integers `x`, `y` such that `x / y` is exactly `(f * 10^e) / (m * 2^k)`. // This not only avoids dealing with the signs of `e` and `k`, we also eliminate the // power of two common to `10^e` and `2^k` to make the numbers smaller. make_ratio(&mut x, &mut y, e, k); let m_digits = [(m & 0xFF_FF_FF_FF) as u32, (m >> 32) as u32]; // This is written a bit awkwardly because our bignums don't support // negative numbers, so we use the absolute value + sign information. // The multiplication with m_digits can't overflow. If `x` or `y` are large enough that // we need to worry about overflow, then they are also large enough that `make_ratio` has // reduced the fraction by a factor of 2^64 or more. let (d2, d_negative) = if x >= y { // Don't need x any more, save a clone(). x.sub(&y).mul_pow2(1).mul_digits(&m_digits); (x, false) } else { // Still need y - make a copy. let mut y = y.clone(); y.sub(&x).mul_pow2(1).mul_digits(&m_digits); (y, true) }; if d2 < y { let mut d2_double = d2; d2_double.mul_pow2(1); if m == T::MIN_SIG && d_negative && d2_double > y { z = prev_float(z); } else { return z; } } else if d2 == y { if m % 2 == 0 { if m == T::MIN_SIG && d_negative { z = prev_float(z); } else { return z; } } else if d_negative { z = prev_float(z); } else { z = next_float(z); } } else if d_negative { z = prev_float(z); } else { z = next_float(z); } } } /// Given `x = f` and `y = m` where `f` represent input decimal digits as usual and `m` is the /// significand of a floating point approximation, make the ratio `x / y` equal to /// `(f * 10^e) / (m * 2^k)`, possibly reduced by a power of two both have in common. fn make_ratio(x: &mut Big, y: &mut Big, e: i16, k: i16) { let (e_abs, k_abs) = (e.abs() as usize, k.abs() as usize); if e >= 0 { if k >= 0 { // x = f * 10^e, y = m * 2^k, except that we reduce the fraction by some power of two. let common = min(e_abs, k_abs); x.mul_pow5(e_abs).mul_pow2(e_abs - common); y.mul_pow2(k_abs - common); } else { // x = f * 10^e * 2^abs(k), y = m // This can't overflow because it requires positive `e` and negative `k`, which can // only happen for values extremely close to 1, which means that `e` and `k` will be // comparatively tiny. x.mul_pow5(e_abs).mul_pow2(e_abs + k_abs); } } else { if k >= 0 { // x = f, y = m * 10^abs(e) * 2^k // This can't overflow either, see above. y.mul_pow5(e_abs).mul_pow2(k_abs + e_abs); } else { // x = f * 2^abs(k), y = m * 10^abs(e), again reducing by a common power of two. let common = min(e_abs, k_abs); x.mul_pow2(k_abs - common); y.mul_pow5(e_abs).mul_pow2(e_abs - common); } } } /// Conceptually, Algorithm M is the simplest way to convert a decimal to a float. /// /// We form a ratio that is equal to `f * 10^e`, then throwing in powers of two until it gives /// a valid float significand. The binary exponent `k` is the number of times we multiplied /// numerator or denominator by two, i.e., at all times `f * 10^e` equals `(u / v) * 2^k`. /// When we have found out significand, we only need to round by inspecting the remainder of the /// division, which is done in helper functions further below. /// /// This algorithm is super slow, even with the optimization described in `quick_start()`. /// However, it's the simplest of the algorithms to adapt for overflow, underflow, and subnormal /// results. This implementation takes over when Bellerophon and Algorithm R are overwhelmed. /// Detecting underflow and overflow is easy: The ratio still isn't an in-range significand, /// yet the minimum/maximum exponent has been reached. In the case of overflow, we simply return /// infinity. /// /// Handling underflow and subnormals is trickier. One big problem is that, with the minimum /// exponent, the ratio might still be too large for a significand. See underflow() for details. pub fn algorithm_m<T: RawFloat>(f: &Big, e: i16) -> T { let mut u; let mut v; let e_abs = e.abs() as usize; let mut k = 0; if e < 0 { u = f.clone(); v = Big::from_small(1); v.mul_pow5(e_abs).mul_pow2(e_abs); } else { // FIXME possible optimization: generalize big_to_fp so that we can do the equivalent of // fp_to_float(big_to_fp(u)) here, only without the double rounding. u = f.clone(); u.mul_pow5(e_abs).mul_pow2(e_abs); v = Big::from_small(1); } quick_start::<T>(&mut u, &mut v, &mut k); let mut rem = Big::from_small(0); let mut x = Big::from_small(0); let min_sig = Big::from_u64(T::MIN_SIG); let max_sig = Big::from_u64(T::MAX_SIG); loop { u.div_rem(&v, &mut x, &mut rem); if k == T::MIN_EXP_INT { // We have to stop at the minimum exponent, if we wait until `k < T::MIN_EXP_INT`, // then we'd be off by a factor of two. Unfortunately this means we have to special- // case normal numbers with the minimum exponent. // FIXME find a more elegant formulation, but run the `tiny-pow10` test to make sure // that it's actually correct! if x >= min_sig && x <= max_sig { break; } return underflow(x, v, rem); } if k > T::MAX_EXP_INT { return T::INFINITY; } if x < min_sig { u.mul_pow2(1); k -= 1; } else if x > max_sig { v.mul_pow2(1); k += 1; } else { break; } } let q = num::to_u64(&x); let z = rawfp::encode_normal(Unpacked::new(q, k)); round_by_remainder(v, rem, q, z) } /// Skip over most AlgorithmM iterations by checking the bit length. fn quick_start<T: RawFloat>(u: &mut Big, v: &mut Big, k: &mut i16) { // The bit length is an estimate of the base two logarithm, and log(u / v) = log(u) - log(v). // The estimate is off by at most 1, but always an under-estimate, so the error on log(u) // and log(v) are of the same sign and cancel out (if both are large). Therefore the error // for log(u / v) is at most one as well. // The target ratio is one where u/v is in an in-range significand. Thus our termination // condition is log2(u / v) being the significand bits, plus/minus one. // FIXME Looking at the second bit could improve the estimate and avoid some more divisions. let target_ratio = T::SIG_BITS as i16; let log2_u = u.bit_length() as i16; let log2_v = v.bit_length() as i16; let mut u_shift: i16 = 0; let mut v_shift: i16 = 0; assert!(*k == 0); loop { if *k == T::MIN_EXP_INT { // Underflow or subnormal. Leave it to the main function. break; } if *k == T::MAX_EXP_INT { // Overflow. Leave it to the main function. break; } let log2_ratio = (log2_u + u_shift) - (log2_v + v_shift); if log2_ratio < target_ratio - 1 { u_shift += 1; *k -= 1; } else if log2_ratio > target_ratio + 1 { v_shift += 1; *k += 1; } else { break; } } u.mul_pow2(u_shift as usize); v.mul_pow2(v_shift as usize); } fn underflow<T: RawFloat>(x: Big, v: Big, rem: Big) -> T { if x < Big::from_u64(T::MIN_SIG) { let q = num::to_u64(&x); let z = rawfp::encode_subnormal(q); return round_by_remainder(v, rem, q, z); } // Ratio isn't an in-range significand with the minimum exponent, so we need to round off // excess bits and adjust the exponent accordingly. The real value now looks like this: // // x lsb // /--------------\/ // 1010101010101010.10101010101010 * 2^k // \-----/\-------/ \------------/ // q trunc. (represented by rem) // // Therefore, when the rounded-off bits are != 0.5 ULP, they decide the rounding // on their own. When they are equal and the remainder is non-zero, the value still // needs to be rounded up. Only when the rounded off bits are 1/2 and the remainder // is zero, we have a half-to-even situation. let bits = x.bit_length(); let lsb = bits - T::SIG_BITS as usize; let q = num::get_bits(&x, lsb, bits); let k = T::MIN_EXP_INT + lsb as i16; let z = rawfp::encode_normal(Unpacked::new(q, k)); let q_even = q % 2 == 0; match num::compare_with_half_ulp(&x, lsb) { Greater => next_float(z), Less => z, Equal if rem.is_zero() && q_even => z, Equal => next_float(z), } } /// Ordinary round-to-even, obfuscated by having to round based on the remainder of a division. fn round_by_remainder<T: RawFloat>(v: Big, r: Big, q: u64, z: T) -> T { let mut v_minus_r = v; v_minus_r.sub(&r); if r < v_minus_r { z } else if r > v_minus_r { next_float(z) } else if q % 2 == 0 { z } else { next_float(z) } }