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authorDaniel Baumann <daniel.baumann@progress-linux.org>2024-04-19 00:47:55 +0000
committerDaniel Baumann <daniel.baumann@progress-linux.org>2024-04-19 00:47:55 +0000
commit26a029d407be480d791972afb5975cf62c9360a6 (patch)
treef435a8308119effd964b339f76abb83a57c29483 /third_party/rust/minimal-lexical/src/slow.rs
parentInitial commit. (diff)
downloadfirefox-26a029d407be480d791972afb5975cf62c9360a6.tar.xz
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Adding upstream version 124.0.1.upstream/124.0.1
Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>
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+//! Slow, fallback cases where we cannot unambiguously round a float.
+//!
+//! This occurs when we cannot determine the exact representation using
+//! both the fast path (native) cases nor the Lemire/Bellerophon algorithms,
+//! and therefore must fallback to a slow, arbitrary-precision representation.
+
+#![doc(hidden)]
+
+use crate::bigint::{Bigint, Limb, LIMB_BITS};
+use crate::extended_float::{extended_to_float, ExtendedFloat};
+use crate::num::Float;
+use crate::number::Number;
+use crate::rounding::{round, round_down, round_nearest_tie_even};
+use core::cmp;
+
+// ALGORITHM
+// ---------
+
+/// Parse the significant digits and biased, binary exponent of a float.
+///
+/// This is a fallback algorithm that uses a big-integer representation
+/// of the float, and therefore is considerably slower than faster
+/// approximations. However, it will always determine how to round
+/// the significant digits to the nearest machine float, allowing
+/// use to handle near half-way cases.
+///
+/// Near half-way cases are halfway between two consecutive machine floats.
+/// For example, the float `16777217.0` has a bitwise representation of
+/// `100000000000000000000000 1`. Rounding to a single-precision float,
+/// the trailing `1` is truncated. Using round-nearest, tie-even, any
+/// value above `16777217.0` must be rounded up to `16777218.0`, while
+/// any value before or equal to `16777217.0` must be rounded down
+/// to `16777216.0`. These near-halfway conversions therefore may require
+/// a large number of digits to unambiguously determine how to round.
+#[inline]
+pub fn slow<'a, F, Iter1, Iter2>(
+ num: Number,
+ fp: ExtendedFloat,
+ integer: Iter1,
+ fraction: Iter2,
+) -> ExtendedFloat
+where
+ F: Float,
+ Iter1: Iterator<Item = &'a u8> + Clone,
+ Iter2: Iterator<Item = &'a u8> + Clone,
+{
+ // Ensure our preconditions are valid:
+ // 1. The significant digits are not shifted into place.
+ debug_assert!(fp.mant & (1 << 63) != 0);
+
+ // This assumes the sign bit has already been parsed, and we're
+ // starting with the integer digits, and the float format has been
+ // correctly validated.
+ let sci_exp = scientific_exponent(&num);
+
+ // We have 2 major algorithms we use for this:
+ // 1. An algorithm with a finite number of digits and a positive exponent.
+ // 2. An algorithm with a finite number of digits and a negative exponent.
+ let (bigmant, digits) = parse_mantissa(integer, fraction, F::MAX_DIGITS);
+ let exponent = sci_exp + 1 - digits as i32;
+ if exponent >= 0 {
+ positive_digit_comp::<F>(bigmant, exponent)
+ } else {
+ negative_digit_comp::<F>(bigmant, fp, exponent)
+ }
+}
+
+/// Generate the significant digits with a positive exponent relative to mantissa.
+pub fn positive_digit_comp<F: Float>(mut bigmant: Bigint, exponent: i32) -> ExtendedFloat {
+ // Simple, we just need to multiply by the power of the radix.
+ // Now, we can calculate the mantissa and the exponent from this.
+ // The binary exponent is the binary exponent for the mantissa
+ // shifted to the hidden bit.
+ bigmant.pow(10, exponent as u32).unwrap();
+
+ // Get the exact representation of the float from the big integer.
+ // hi64 checks **all** the remaining bits after the mantissa,
+ // so it will check if **any** truncated digits exist.
+ let (mant, is_truncated) = bigmant.hi64();
+ let exp = bigmant.bit_length() as i32 - 64 + F::EXPONENT_BIAS;
+ let mut fp = ExtendedFloat {
+ mant,
+ exp,
+ };
+
+ // Shift the digits into position and determine if we need to round-up.
+ round::<F, _>(&mut fp, |f, s| {
+ round_nearest_tie_even(f, s, |is_odd, is_halfway, is_above| {
+ is_above || (is_halfway && is_truncated) || (is_odd && is_halfway)
+ });
+ });
+ fp
+}
+
+/// Generate the significant digits with a negative exponent relative to mantissa.
+///
+/// This algorithm is quite simple: we have the significant digits `m1 * b^N1`,
+/// where `m1` is the bigint mantissa, `b` is the radix, and `N1` is the radix
+/// exponent. We then calculate the theoretical representation of `b+h`, which
+/// is `m2 * 2^N2`, where `m2` is the bigint mantissa and `N2` is the binary
+/// exponent. If we had infinite, efficient floating precision, this would be
+/// equal to `m1 / b^-N1` and then compare it to `m2 * 2^N2`.
+///
+/// Since we cannot divide and keep precision, we must multiply the other:
+/// if we want to do `m1 / b^-N1 >= m2 * 2^N2`, we can do
+/// `m1 >= m2 * b^-N1 * 2^N2` Going to the decimal case, we can show and example
+/// and simplify this further: `m1 >= m2 * 2^N2 * 10^-N1`. Since we can remove
+/// a power-of-two, this is `m1 >= m2 * 2^(N2 - N1) * 5^-N1`. Therefore, if
+/// `N2 - N1 > 0`, we need have `m1 >= m2 * 2^(N2 - N1) * 5^-N1`, otherwise,
+/// we have `m1 * 2^(N1 - N2) >= m2 * 5^-N1`, where the resulting exponents
+/// are all positive.
+///
+/// This allows us to compare both floats using integers efficiently
+/// without any loss of precision.
+#[allow(clippy::comparison_chain)]
+pub fn negative_digit_comp<F: Float>(
+ bigmant: Bigint,
+ mut fp: ExtendedFloat,
+ exponent: i32,
+) -> ExtendedFloat {
+ // Ensure our preconditions are valid:
+ // 1. The significant digits are not shifted into place.
+ debug_assert!(fp.mant & (1 << 63) != 0);
+
+ // Get the significant digits and radix exponent for the real digits.
+ let mut real_digits = bigmant;
+ let real_exp = exponent;
+ debug_assert!(real_exp < 0);
+
+ // Round down our extended-precision float and calculate `b`.
+ let mut b = fp;
+ round::<F, _>(&mut b, round_down);
+ let b = extended_to_float::<F>(b);
+
+ // Get the significant digits and the binary exponent for `b+h`.
+ let theor = bh(b);
+ let mut theor_digits = Bigint::from_u64(theor.mant);
+ let theor_exp = theor.exp;
+
+ // We need to scale the real digits and `b+h` digits to be the same
+ // order. We currently have `real_exp`, in `radix`, that needs to be
+ // shifted to `theor_digits` (since it is negative), and `theor_exp`
+ // to either `theor_digits` or `real_digits` as a power of 2 (since it
+ // may be positive or negative). Try to remove as many powers of 2
+ // as possible. All values are relative to `theor_digits`, that is,
+ // reflect the power you need to multiply `theor_digits` by.
+ //
+ // Both are on opposite-sides of equation, can factor out a
+ // power of two.
+ //
+ // Example: 10^-10, 2^-10 -> ( 0, 10, 0)
+ // Example: 10^-10, 2^-15 -> (-5, 10, 0)
+ // Example: 10^-10, 2^-5 -> ( 5, 10, 0)
+ // Example: 10^-10, 2^5 -> (15, 10, 0)
+ let binary_exp = theor_exp - real_exp;
+ let halfradix_exp = -real_exp;
+ if halfradix_exp != 0 {
+ theor_digits.pow(5, halfradix_exp as u32).unwrap();
+ }
+ if binary_exp > 0 {
+ theor_digits.pow(2, binary_exp as u32).unwrap();
+ } else if binary_exp < 0 {
+ real_digits.pow(2, (-binary_exp) as u32).unwrap();
+ }
+
+ // Compare our theoretical and real digits and round nearest, tie even.
+ let ord = real_digits.data.cmp(&theor_digits.data);
+ round::<F, _>(&mut fp, |f, s| {
+ round_nearest_tie_even(f, s, |is_odd, _, _| {
+ // Can ignore `is_halfway` and `is_above`, since those were
+ // calculates using less significant digits.
+ match ord {
+ cmp::Ordering::Greater => true,
+ cmp::Ordering::Less => false,
+ cmp::Ordering::Equal if is_odd => true,
+ cmp::Ordering::Equal => false,
+ }
+ });
+ });
+ fp
+}
+
+/// Add a digit to the temporary value.
+macro_rules! add_digit {
+ ($c:ident, $value:ident, $counter:ident, $count:ident) => {{
+ let digit = $c - b'0';
+ $value *= 10 as Limb;
+ $value += digit as Limb;
+
+ // Increment our counters.
+ $counter += 1;
+ $count += 1;
+ }};
+}
+
+/// Add a temporary value to our mantissa.
+macro_rules! add_temporary {
+ // Multiply by the small power and add the native value.
+ (@mul $result:ident, $power:expr, $value:expr) => {
+ $result.data.mul_small($power).unwrap();
+ $result.data.add_small($value).unwrap();
+ };
+
+ // # Safety
+ //
+ // Safe is `counter <= step`, or smaller than the table size.
+ ($format:ident, $result:ident, $counter:ident, $value:ident) => {
+ if $counter != 0 {
+ // SAFETY: safe, since `counter <= step`, or smaller than the table size.
+ let small_power = unsafe { f64::int_pow_fast_path($counter, 10) };
+ add_temporary!(@mul $result, small_power as Limb, $value);
+ $counter = 0;
+ $value = 0;
+ }
+ };
+
+ // Add a temporary where we won't read the counter results internally.
+ //
+ // # Safety
+ //
+ // Safe is `counter <= step`, or smaller than the table size.
+ (@end $format:ident, $result:ident, $counter:ident, $value:ident) => {
+ if $counter != 0 {
+ // SAFETY: safe, since `counter <= step`, or smaller than the table size.
+ let small_power = unsafe { f64::int_pow_fast_path($counter, 10) };
+ add_temporary!(@mul $result, small_power as Limb, $value);
+ }
+ };
+
+ // Add the maximum native value.
+ (@max $format:ident, $result:ident, $counter:ident, $value:ident, $max:ident) => {
+ add_temporary!(@mul $result, $max, $value);
+ $counter = 0;
+ $value = 0;
+ };
+}
+
+/// Round-up a truncated value.
+macro_rules! round_up_truncated {
+ ($format:ident, $result:ident, $count:ident) => {{
+ // Need to round-up.
+ // Can't just add 1, since this can accidentally round-up
+ // values to a halfway point, which can cause invalid results.
+ add_temporary!(@mul $result, 10, 1);
+ $count += 1;
+ }};
+}
+
+/// Check and round-up the fraction if any non-zero digits exist.
+macro_rules! round_up_nonzero {
+ ($format:ident, $iter:expr, $result:ident, $count:ident) => {{
+ for &digit in $iter {
+ if digit != b'0' {
+ round_up_truncated!($format, $result, $count);
+ return ($result, $count);
+ }
+ }
+ }};
+}
+
+/// Parse the full mantissa into a big integer.
+///
+/// Returns the parsed mantissa and the number of digits in the mantissa.
+/// The max digits is the maximum number of digits plus one.
+pub fn parse_mantissa<'a, Iter1, Iter2>(
+ mut integer: Iter1,
+ mut fraction: Iter2,
+ max_digits: usize,
+) -> (Bigint, usize)
+where
+ Iter1: Iterator<Item = &'a u8> + Clone,
+ Iter2: Iterator<Item = &'a u8> + Clone,
+{
+ // Iteratively process all the data in the mantissa.
+ // We do this via small, intermediate values which once we reach
+ // the maximum number of digits we can process without overflow,
+ // we add the temporary to the big integer.
+ let mut counter: usize = 0;
+ let mut count: usize = 0;
+ let mut value: Limb = 0;
+ let mut result = Bigint::new();
+
+ // Now use our pre-computed small powers iteratively.
+ // This is calculated as `⌊log(2^BITS - 1, 10)⌋`.
+ let step: usize = if LIMB_BITS == 32 {
+ 9
+ } else {
+ 19
+ };
+ let max_native = (10 as Limb).pow(step as u32);
+
+ // Process the integer digits.
+ 'integer: loop {
+ // Parse a digit at a time, until we reach step.
+ while counter < step && count < max_digits {
+ if let Some(&c) = integer.next() {
+ add_digit!(c, value, counter, count);
+ } else {
+ break 'integer;
+ }
+ }
+
+ // Check if we've exhausted our max digits.
+ if count == max_digits {
+ // Need to check if we're truncated, and round-up accordingly.
+ // SAFETY: safe since `counter <= step`.
+ add_temporary!(@end format, result, counter, value);
+ round_up_nonzero!(format, integer, result, count);
+ round_up_nonzero!(format, fraction, result, count);
+ return (result, count);
+ } else {
+ // Add our temporary from the loop.
+ // SAFETY: safe since `counter <= step`.
+ add_temporary!(@max format, result, counter, value, max_native);
+ }
+ }
+
+ // Skip leading fraction zeros.
+ // Required to get an accurate count.
+ if count == 0 {
+ for &c in &mut fraction {
+ if c != b'0' {
+ add_digit!(c, value, counter, count);
+ break;
+ }
+ }
+ }
+
+ // Process the fraction digits.
+ 'fraction: loop {
+ // Parse a digit at a time, until we reach step.
+ while counter < step && count < max_digits {
+ if let Some(&c) = fraction.next() {
+ add_digit!(c, value, counter, count);
+ } else {
+ break 'fraction;
+ }
+ }
+
+ // Check if we've exhausted our max digits.
+ if count == max_digits {
+ // SAFETY: safe since `counter <= step`.
+ add_temporary!(@end format, result, counter, value);
+ round_up_nonzero!(format, fraction, result, count);
+ return (result, count);
+ } else {
+ // Add our temporary from the loop.
+ // SAFETY: safe since `counter <= step`.
+ add_temporary!(@max format, result, counter, value, max_native);
+ }
+ }
+
+ // We will always have a remainder, as long as we entered the loop
+ // once, or counter % step is 0.
+ // SAFETY: safe since `counter <= step`.
+ add_temporary!(@end format, result, counter, value);
+
+ (result, count)
+}
+
+// SCALING
+// -------
+
+/// Calculate the scientific exponent from a `Number` value.
+/// Any other attempts would require slowdowns for faster algorithms.
+#[inline]
+pub fn scientific_exponent(num: &Number) -> i32 {
+ // Use power reduction to make this faster.
+ let mut mantissa = num.mantissa;
+ let mut exponent = num.exponent;
+ while mantissa >= 10000 {
+ mantissa /= 10000;
+ exponent += 4;
+ }
+ while mantissa >= 100 {
+ mantissa /= 100;
+ exponent += 2;
+ }
+ while mantissa >= 10 {
+ mantissa /= 10;
+ exponent += 1;
+ }
+ exponent as i32
+}
+
+/// Calculate `b` from a a representation of `b` as a float.
+#[inline]
+pub fn b<F: Float>(float: F) -> ExtendedFloat {
+ ExtendedFloat {
+ mant: float.mantissa(),
+ exp: float.exponent(),
+ }
+}
+
+/// Calculate `b+h` from a a representation of `b` as a float.
+#[inline]
+pub fn bh<F: Float>(float: F) -> ExtendedFloat {
+ let fp = b(float);
+ ExtendedFloat {
+ mant: (fp.mant << 1) + 1,
+ exp: fp.exp - 1,
+ }
+}