comm-central/third_party/rust/minimal-lexical/src/bellerophon.rs

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//! An implementation of Clinger's Bellerophon algorithm.
//!
//! This is a moderate path algorithm that uses an extended-precision
//! float, represented in 80 bits, by calculating the bits of slop
//! and determining if those bits could prevent unambiguous rounding.
//!
//! This algorithm requires less static storage than the Lemire algorithm,
//! and has decent performance, and is therefore used when non-decimal,
//! non-power-of-two strings need to be parsed. Clinger's algorithm
//! is described in depth in "How to Read Floating Point Numbers Accurately.",
//!
//! This implementation is loosely based off the Golang implementation,
//! This code is therefore subject to a 3-clause BSD license.
#![cfg(feature = "compact")]
#![doc(hidden)]
use crate::extended_float::ExtendedFloat;
use crate::mask::{lower_n_halfway, lower_n_mask};
use crate::num::Float;
use crate::number::Number;
use crate::rounding::{round, round_nearest_tie_even};
use crate::table::BASE10_POWERS;
// ALGORITHM
// ---------
/// Core implementation of the Bellerophon algorithm.
///
/// Create an extended-precision float, scale it to the proper radix power,
/// calculate the bits of slop, and return the representation. The value
/// will always be guaranteed to be within 1 bit, rounded-down, of the real
/// value. If a negative exponent is returned, this represents we were
/// unable to unambiguously round the significant digits.
///
/// This has been modified to return a biased, rather than unbiased exponent.
pub fn bellerophon<F: Float>(num: &Number) -> ExtendedFloat {
let fp_zero = ExtendedFloat {
mant: 0,
exp: 0,
};
let fp_inf = ExtendedFloat {
mant: 0,
exp: F::INFINITE_POWER,
};
// Early short-circuit, in case of literal 0 or infinity.
// This allows us to avoid narrow casts causing numeric overflow,
// and is a quick check for any radix.
if num.mantissa == 0 || num.exponent <= -0x1000 {
return fp_zero;
} else if num.exponent >= 0x1000 {
return fp_inf;
}
// Calculate our indexes for our extended-precision multiplication.
// This narrowing cast is safe, since exponent must be in a valid range.
let exponent = num.exponent as i32 + BASE10_POWERS.bias;
let small_index = exponent % BASE10_POWERS.step;
let large_index = exponent / BASE10_POWERS.step;
if exponent < 0 {
// Guaranteed underflow (assign 0).
return fp_zero;
}
if large_index as usize >= BASE10_POWERS.large.len() {
// Overflow (assign infinity)
return fp_inf;
}
// Within the valid exponent range, multiply by the large and small
// exponents and return the resulting value.
// Track errors to as a factor of unit in last-precision.
let mut errors: u32 = 0;
if num.many_digits {
errors += error_halfscale();
}
// Multiply by the small power.
// Check if we can directly multiply by an integer, if not,
// use extended-precision multiplication.
let mut fp = ExtendedFloat {
mant: num.mantissa,
exp: 0,
};
match fp.mant.overflowing_mul(BASE10_POWERS.get_small_int(small_index as usize)) {
// Overflow, multiplication unsuccessful, go slow path.
(_, true) => {
normalize(&mut fp);
fp = mul(&fp, &BASE10_POWERS.get_small(small_index as usize));
errors += error_halfscale();
},
// No overflow, multiplication successful.
(mant, false) => {
fp.mant = mant;
normalize(&mut fp);
},
}
// Multiply by the large power.
fp = mul(&fp, &BASE10_POWERS.get_large(large_index as usize));
if errors > 0 {
errors += 1;
}
errors += error_halfscale();
// Normalize the floating point (and the errors).
let shift = normalize(&mut fp);
errors <<= shift;
fp.exp += F::EXPONENT_BIAS;
// Check for literal overflow, even with halfway cases.
if -fp.exp + 1 > 65 {
return fp_zero;
}
// Too many errors accumulated, return an error.
if !error_is_accurate::<F>(errors, &fp) {
// Bias the exponent so we know it's invalid.
fp.exp += F::INVALID_FP;
return fp;
}
// Check if we have a literal 0 or overflow here.
// If we have an exponent of -63, we can still have a valid shift,
// giving a case where we have too many errors and need to round-up.
if -fp.exp + 1 == 65 {
// Have more than 64 bits below the minimum exponent, must be 0.
return fp_zero;
}
round::<F, _>(&mut fp, |f, s| {
round_nearest_tie_even(f, s, |is_odd, is_halfway, is_above| {
is_above || (is_odd && is_halfway)
});
});
fp
}
// ERRORS
// ------
// Calculate if the errors in calculating the extended-precision float.
//
// Specifically, we want to know if we are close to a halfway representation,
// or halfway between `b` and `b+1`, or `b+h`. The halfway representation
// has the form:
// SEEEEEEEHMMMMMMMMMMMMMMMMMMMMMMM100...
// where:
// S = Sign Bit
// E = Exponent Bits
// H = Hidden Bit
// M = Mantissa Bits
//
// The halfway representation has a bit set 1-after the mantissa digits,
// and no bits set immediately afterward, making it impossible to
// round between `b` and `b+1` with this representation.
/// Get the full error scale.
#[inline(always)]
const fn error_scale() -> u32 {
8
}
/// Get the half error scale.
#[inline(always)]
const fn error_halfscale() -> u32 {
error_scale() / 2
}
/// Determine if the number of errors is tolerable for float precision.
fn error_is_accurate<F: Float>(errors: u32, fp: &ExtendedFloat) -> bool {
// Check we can't have a literal 0 denormal float.
debug_assert!(fp.exp >= -64);
// Determine if extended-precision float is a good approximation.
// If the error has affected too many units, the float will be
// inaccurate, or if the representation is too close to halfway
// that any operations could affect this halfway representation.
// See the documentation for dtoa for more information.
// This is always a valid u32, since `fp.exp >= -64`
// will always be positive and the significand size is {23, 52}.
let mantissa_shift = 64 - F::MANTISSA_SIZE - 1;
// The unbiased exponent checks is `unbiased_exp <= F::MANTISSA_SIZE
// - F::EXPONENT_BIAS -64 + 1`, or `biased_exp <= F::MANTISSA_SIZE - 63`,
// or `biased_exp <= mantissa_shift`.
let extrabits = match fp.exp <= -mantissa_shift {
// Denormal, since shifting to the hidden bit still has a negative exponent.
// The unbiased check calculation for bits is `1 - F::EXPONENT_BIAS - unbiased_exp`,
// or `1 - biased_exp`.
true => 1 - fp.exp,
false => 64 - F::MANTISSA_SIZE - 1,
};
// Our logic is as follows: we want to determine if the actual
// mantissa and the errors during calculation differ significantly
// from the rounding point. The rounding point for round-nearest
// is the halfway point, IE, this when the truncated bits start
// with b1000..., while the rounding point for the round-toward
// is when the truncated bits are equal to 0.
// To do so, we can check whether the rounding point +/- the error
// are >/< the actual lower n bits.
//
// For whether we need to use signed or unsigned types for this
// analysis, see this example, using u8 rather than u64 to simplify
// things.
//
// # Comparisons
// cmp1 = (halfway - errors) < extra
// cmp1 = extra < (halfway + errors)
//
// # Large Extrabits, Low Errors
//
// extrabits = 8
// halfway = 0b10000000
// extra = 0b10000010
// errors = 0b00000100
// halfway - errors = 0b01111100
// halfway + errors = 0b10000100
//
// Unsigned:
// halfway - errors = 124
// halfway + errors = 132
// extra = 130
// cmp1 = true
// cmp2 = true
// Signed:
// halfway - errors = 124
// halfway + errors = -124
// extra = -126
// cmp1 = false
// cmp2 = true
//
// # Conclusion
//
// Since errors will always be small, and since we want to detect
// if the representation is accurate, we need to use an **unsigned**
// type for comparisons.
let maskbits = extrabits as u64;
let errors = errors as u64;
// Round-to-nearest, need to use the halfway point.
if extrabits > 64 {
// Underflow, we have a shift larger than the mantissa.
// Representation is valid **only** if the value is close enough
// overflow to the next bit within errors. If it overflows,
// the representation is **not** valid.
!fp.mant.overflowing_add(errors).1
} else {
let mask = lower_n_mask(maskbits);
let extra = fp.mant & mask;
// Round-to-nearest, need to check if we're close to halfway.
// IE, b10100 | 100000, where `|` signifies the truncation point.
let halfway = lower_n_halfway(maskbits);
let cmp1 = halfway.wrapping_sub(errors) < extra;
let cmp2 = extra < halfway.wrapping_add(errors);
// If both comparisons are true, we have significant rounding error,
// and the value cannot be exactly represented. Otherwise, the
// representation is valid.
!(cmp1 && cmp2)
}
}
// MATH
// ----
/// Normalize float-point number.
///
/// Shift the mantissa so the number of leading zeros is 0, or the value
/// itself is 0.
///
/// Get the number of bytes shifted.
pub fn normalize(fp: &mut ExtendedFloat) -> i32 {
// Note:
// Using the ctlz intrinsic via leading_zeros is way faster (~10x)
// than shifting 1-bit at a time, via while loop, and also way
// faster (~2x) than an unrolled loop that checks at 32, 16, 4,
// 2, and 1 bit.
//
// Using a modulus of pow2 (which will get optimized to a bitwise
// and with 0x3F or faster) is slightly slower than an if/then,
// however, removing the if/then will likely optimize more branched
// code as it removes conditional logic.
// Calculate the number of leading zeros, and then zero-out
// any overflowing bits, to avoid shl overflow when self.mant == 0.
if fp.mant != 0 {
let shift = fp.mant.leading_zeros() as i32;
fp.mant <<= shift;
fp.exp -= shift;
shift
} else {
0
}
}
/// Multiply two normalized extended-precision floats, as if by `a*b`.
///
/// The precision is maximal when the numbers are normalized, however,
/// decent precision will occur as long as both values have high bits
/// set. The result is not normalized.
///
/// Algorithm:
/// 1. Non-signed multiplication of mantissas (requires 2x as many bits as input).
/// 2. Normalization of the result (not done here).
/// 3. Addition of exponents.
pub fn mul(x: &ExtendedFloat, y: &ExtendedFloat) -> ExtendedFloat {
// Logic check, values must be decently normalized prior to multiplication.
debug_assert!(x.mant >> 32 != 0);
debug_assert!(y.mant >> 32 != 0);
// Extract high-and-low masks.
// Mask is u32::MAX for older Rustc versions.
const LOMASK: u64 = 0xffff_ffff;
let x1 = x.mant >> 32;
let x0 = x.mant & LOMASK;
let y1 = y.mant >> 32;
let y0 = y.mant & LOMASK;
// Get our products
let x1_y0 = x1 * y0;
let x0_y1 = x0 * y1;
let x0_y0 = x0 * y0;
let x1_y1 = x1 * y1;
let mut tmp = (x1_y0 & LOMASK) + (x0_y1 & LOMASK) + (x0_y0 >> 32);
// round up
tmp += 1 << (32 - 1);
ExtendedFloat {
mant: x1_y1 + (x1_y0 >> 32) + (x0_y1 >> 32) + (tmp >> 32),
exp: x.exp + y.exp + 64,
}
}
// POWERS
// ------
/// Precalculated powers of base N for the Bellerophon algorithm.
pub struct BellerophonPowers {
// Pre-calculated small powers.
pub small: &'static [u64],
// Pre-calculated large powers.
pub large: &'static [u64],
/// Pre-calculated small powers as 64-bit integers
pub small_int: &'static [u64],
// Step between large powers and number of small powers.
pub step: i32,
// Exponent bias for the large powers.
pub bias: i32,
/// ceil(log2(radix)) scaled as a multiplier.
pub log2: i64,
/// Bitshift for the log2 multiplier.
pub log2_shift: i32,
}
/// Allow indexing of values without bounds checking
impl BellerophonPowers {
#[inline]
pub fn get_small(&self, index: usize) -> ExtendedFloat {
let mant = self.small[index];
let exp = (1 - 64) + ((self.log2 * index as i64) >> self.log2_shift);
ExtendedFloat {
mant,
exp: exp as i32,
}
}
#[inline]
pub fn get_large(&self, index: usize) -> ExtendedFloat {
let mant = self.large[index];
let biased_e = index as i64 * self.step as i64 - self.bias as i64;
let exp = (1 - 64) + ((self.log2 * biased_e) >> self.log2_shift);
ExtendedFloat {
mant,
exp: exp as i32,
}
}
#[inline]
pub fn get_small_int(&self, index: usize) -> u64 {
self.small_int[index]
}
}