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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,
}
}