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Accurate decimal-to-float parsing routines.

Browse source 
This commit primarily adds implementations of the algorithms from William
Clinger's paper "How to Read Floating Point Numbers Accurately". It also
includes a lot of infrastructure necessary for those algorithms, and some
unit tests.

Since these algorithms reject a few (extreme) inputs that were previously
accepted, this could be seen as a [breaking-change]
This commit is contained in:
Robin Kruppe 2015-07-26 17:50:29 +02:00
parent b7e39a1c2d
commit ba792a4baa
13 changed files with 2787 additions and 15 deletions
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353
src/libcore/num/dec2flt/algorithm.rs Normal file
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// 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 num::flt2dec::strategy::grisu::Fp;
use prelude::v1::*;
use cmp::min;
use cmp::Ordering::{Less, Equal, Greater};
use super::table;
use super::rawfp::{self, Unpacked, RawFloat, fp_to_float, next_float, prev_float};
use super::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 space (11 KiB versus... 5 KiB or so?)
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 }
}
/// 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;
}
let e = e as i16; // Can't overflow because e.abs() <= LOG5_OF_EXP_N
// 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 occasioally quite significant!) errors in the final result.
// The case `e == 0`, however, is unnecessary for correctness. It's just measurably faster.
if e == 0 {
Some(T::from_int(f))
} else if e > 0 {
Some(T::from_int(f) * fp_to_float(power_of_ten(e)))
} else {
Some(T::from_int(f) / fp_to_float(power_of_ten(-e)))
}
}
/// 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 adjecent (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 = f64::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 remainer
// 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)
}
}