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/* SPDX-License-Identifier: MIT OR Apache-2.0 */
use crate::support::{CastFrom, CastInto, Float, HInt, Int, MinInt, NarrowingDiv};
#[inline]
pub fn fmod<F: Float>(x: F, y: F) -> F
where
F::Int: HInt,
<F::Int as HInt>::D: NarrowingDiv,
{
let _1 = F::Int::ONE;
let sx = x.to_bits() & F::SIGN_MASK;
let ux = x.to_bits() & !F::SIGN_MASK;
let uy = y.to_bits() & !F::SIGN_MASK;
// Cases that return NaN:
// NaN % _
// Inf % _
// _ % NaN
// _ % 0
let x_nan_or_inf = ux & F::EXP_MASK == F::EXP_MASK;
let y_nan_or_zero = uy.wrapping_sub(_1) & F::EXP_MASK == F::EXP_MASK;
if x_nan_or_inf | y_nan_or_zero {
return (x * y) / (x * y);
}
if ux < uy {
// |x| < |y|
return x;
}
let (num, ex) = into_sig_exp::<F>(ux);
let (div, ey) = into_sig_exp::<F>(uy);
// To compute `(num << ex) % (div << ey)`, first
// evaluate `rem = (num << (ex - ey)) % div` ...
let rem = reduction::<F>(num, ex - ey, div);
// ... so the result will be `rem << ey`
if rem.is_zero() {
// Return zero with the sign of `x`
return F::from_bits(sx);
};
// We would shift `rem` up by `ey`, but have to stop at `F::SIG_BITS`
let shift = ey.min(F::SIG_BITS - rem.ilog2());
// Anything past that is added to the exponent field
let bits = (rem << shift) + (F::Int::cast_from(ey - shift) << F::SIG_BITS);
F::from_bits(sx + bits)
}
/// Given the bits of a finite float, return a tuple of
/// - the mantissa with the implicit bit (0 if subnormal, 1 otherwise)
/// - the additional exponent past 1, (0 for subnormal, 0 or more otherwise)
fn into_sig_exp<F: Float>(mut bits: F::Int) -> (F::Int, u32) {
bits &= !F::SIGN_MASK;
// Subtract 1 from the exponent, clamping at 0
let sat = bits.checked_sub(F::IMPLICIT_BIT).unwrap_or(F::Int::ZERO);
(
bits - (sat & F::EXP_MASK),
u32::cast_from(sat >> F::SIG_BITS),
)
}
/// Compute the remainder `(x * 2.pow(e)) % y` without overflow.
fn reduction<F>(mut x: F::Int, e: u32, y: F::Int) -> F::Int
where
F: Float,
F::Int: HInt,
<<F as Float>::Int as HInt>::D: NarrowingDiv,
{
// `f16` only has 5 exponent bits, so even `f16::MAX = 65504.0` is only
// a 40-bit integer multiple of the smallest subnormal.
if F::BITS == 16 {
debug_assert!(F::EXP_MAX - F::EXP_MIN == 29);
debug_assert!(e <= 29);
let u: u16 = x.cast();
let v: u16 = y.cast();
let u = (u as u64) << e;
let v = v as u64;
return F::Int::cast_from((u % v) as u16);
}
// Ensure `x < 2y` for later steps
if x >= (y << 1) {
// This case is only reached with subnormal divisors,
// but it might be better to just normalize all significands
// to make this unnecessary. The further calls could potentially
// benefit from assuming a specific fixed leading bit position.
x %= y;
}
// The simple implementation seems to be fastest for a short reduction
// at this size. The limit here was chosen empirically on an Intel Nehalem.
// Less old CPUs that have faster `u64 * u64 -> u128` might not benefit,
// and 32-bit systems or architectures without hardware multipliers might
// want to do this in more cases.
if F::BITS == 64 && e < 32 {
// Assumes `x < 2y`
for _ in 0..e {
x = x.checked_sub(y).unwrap_or(x);
x <<= 1;
}
return x.checked_sub(y).unwrap_or(x);
}
// Fast path for short reductions
if e < F::BITS {
let w = x.widen() << e;
if let Some((_, r)) = w.checked_narrowing_div_rem(y) {
return r;
}
}
// Assumes `x < 2y`
crate::support::linear_mul_reduction(x, e, y)
}