pxfm 0.1.28

Fast and accurate math
Documentation 
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/*
 * // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
 * //
 * // Redistribution and use in source and binary forms, with or without modification,
 * // are permitted provided that the following conditions are met:
 * //
 * // 1.  Redistributions of source code must retain the above copyright notice, this
 * // list of conditions and the following disclaimer.
 * //
 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
 * // this list of conditions and the following disclaimer in the documentation
 * // and/or other materials provided with the distribution.
 * //
 * // 3.  Neither the name of the copyright holder nor the names of its
 * // contributors may be used to endorse or promote products derived from
 * // this software without specific prior written permission.
 * //
 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */
use crate::common::f_fmla;
use crate::cube_roots::cbrtf::halley_refine_d;
use crate::double_double::DoubleDouble;
use crate::exponents::fast_ldexp;

pub(crate) trait CbrtBackend {
    fn fma(&self, x: f64, y: f64, z: f64) -> f64;
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64;
    fn halley(&self, x: f64, a: f64) -> f64;
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble;
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble;
}

pub(crate) struct GenericCbrtBackend {}

impl CbrtBackend for GenericCbrtBackend {
    #[inline(always)]
    fn fma(&self, x: f64, y: f64, z: f64) -> f64 {
        f_fmla(x, y, z)
    }
    #[inline(always)]
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64 {
        use crate::polyeval::f_polyeval4;
        f_polyeval4(x, a0, a1, a2, a3)
    }
    #[inline(always)]
    fn halley(&self, x: f64, a: f64) -> f64 {
        halley_refine_d(x, a)
    }
    #[inline(always)]
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble {
        DoubleDouble::from_exact_mult(x, y)
    }
    #[inline(always)]
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble {
        DoubleDouble::quick_mult_f64(x, y)
    }
}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
pub(crate) struct FmaCbrtBackend {}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
impl CbrtBackend for FmaCbrtBackend {
    #[inline(always)]
    fn fma(&self, x: f64, y: f64, z: f64) -> f64 {
        f64::mul_add(x, y, z)
    }
    #[inline(always)]
    fn polyeval4(&self, x: f64, a0: f64, a1: f64, a2: f64, a3: f64) -> f64 {
        use crate::polyeval::d_polyeval4;
        d_polyeval4(x, a0, a1, a2, a3)
    }
    #[inline(always)]
    fn halley(&self, x: f64, a: f64) -> f64 {
        use crate::cube_roots::cbrtf::halley_refine_d_fma;
        halley_refine_d_fma(x, a)
    }
    #[inline(always)]
    fn exact_mul(&self, x: f64, y: f64) -> DoubleDouble {
        DoubleDouble::from_exact_mult_fma(x, y)
    }
    #[inline(always)]
    fn quick_mult_f64(&self, x: DoubleDouble, y: f64) -> DoubleDouble {
        DoubleDouble::quick_mult_f64_fma(x, y)
    }
}

#[inline(always)]
fn cbrt_gen_impl<B: CbrtBackend>(x: f64, backend: B) -> f64 {
    // 1; 2^{1/3}; 2^{2/3}
    static ESCALE: [f64; 3] = [
        1.0,
        f64::from_bits(0x3ff428a2f98d728b),
        f64::from_bits(0x3ff965fea53d6e3d),
    ];

    let bits = x.to_bits();
    let mut exp = ((bits >> 52) & 0x7ff) as i32;
    let mut mant = bits & ((1u64 << 52) - 1);

    if exp == 0x7ff || x == 0.0 {
        return x + x;
    }

    // Normalize subnormal
    if exp == 0 && x != 0.0 {
        let norm = x * f64::from_bits(0x4350000000000000); // * 2^54
        let norm_bits = norm.to_bits();
        mant = norm_bits & ((1u64 << 52) - 1);
        exp = ((norm_bits >> 52) & 0x7ff) as i32 - 54;
    }

    exp -= 1023;

    mant |= 0x3ff << 52;
    let m = f64::from_bits(mant);

    // Polynomial for x^(1/3) on [1.0; 2.0]
    // Generated by Sollya:
    // d = [1.0, 2.0];
    // f_cbrt = x^(1/3);
    // Q = fpminimax(f_cbrt, 4, [|D...|], d, relative, floating);
    // See ./notes/cbrt.sollya

    let p = backend.polyeval4(
        m,
        f64::from_bits(0x3fe1b0babceeaafa),
        f64::from_bits(0x3fe2c9a3e8e06a3c),
        f64::from_bits(0xbfc4dc30afb71885),
        f64::from_bits(0x3f97a8d3e05458e4),
    );

    // split exponent e = 3*q + r with r in {0,1,2}
    // use div_euclid/rem_euclid to get r >= 0
    let q = exp.div_euclid(3);
    let rem_scale = exp.rem_euclid(3);

    let z = p * ESCALE[rem_scale as usize];

    let mm = fast_ldexp(m, rem_scale); // bring mantissa into [1;8]

    let r = 1.0 / mm;

    // One Halley's method step
    // then refine in partial double-double precision with Newton-Raphson iteration
    let y0 = backend.halley(z, mm);
    let d2y = backend.exact_mul(y0, y0);
    let d3y = backend.quick_mult_f64(d2y, y0);
    // Newton-Raphson step
    // h = (x^3 - a) * r
    // y1 = y0 - 1/3 * h * y0
    let h = ((d3y.hi - mm) + d3y.lo) * r;
    // y1 = y0 - 1/3*y0*(h.lo + h.hi) = y0 - 1/3 *y0*h.lo - 1/3 * y0 * h.hi
    let y = backend.fma(-f64::from_bits(0x3fd5555555555555), y0 * h, y0);

    f64::copysign(fast_ldexp(y, q), x)
}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
#[target_feature(enable = "avx", enable = "fma")]
unsafe fn cbrt_fma_impl(x: f64) -> f64 {
    cbrt_gen_impl(x, FmaCbrtBackend {})
}

/// Computes cube root
///
/// Max found ULP 0.5
pub fn f_cbrt(x: f64) -> f64 {
    #[cfg(not(any(target_arch = "x86", target_arch = "x86_64")))]
    {
        cbrt_gen_impl(x, GenericCbrtBackend {})
    }
    #[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
    {
        use std::sync::OnceLock;
        static EXECUTOR: OnceLock<unsafe fn(f64) -> f64> = OnceLock::new();
        let q = EXECUTOR.get_or_init(|| {
            if std::arch::is_x86_feature_detected!("avx")
                && std::arch::is_x86_feature_detected!("fma")
            {
                cbrt_fma_impl
            } else {
                fn def_cbrt(x: f64) -> f64 {
                    cbrt_gen_impl(x, GenericCbrtBackend {})
                }
                def_cbrt
            }
        });
        unsafe { q(x) }
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_cbrt() {
        assert_eq!(f_cbrt(0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005432309223745),
                   0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000017579026781511548);
        assert_eq!(f_cbrt(1.225158611559834), 1.0700336588124544);
        assert_eq!(f_cbrt(0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000139491540182158), 1.1173329935611586e-103);
        assert_eq!(f_cbrt(27.0), 3.0);
        assert_eq!(f_cbrt(64.0), 4.0);
        assert_eq!(f_cbrt(125.0), 5.0);
        assert_eq!(f_cbrt(216.0), 6.0);
        assert_eq!(f_cbrt(343.0), 7.0);
        assert_eq!(f_cbrt(512.0), 8.0);
        assert_eq!(f_cbrt(729.0), 9.0);
        assert_eq!(f_cbrt(-729.0), -9.0);
        assert_eq!(f_cbrt(-512.0), -8.0);
        assert_eq!(f_cbrt(-343.0), -7.0);
        assert_eq!(f_cbrt(-216.0), -6.0);
        assert_eq!(f_cbrt(-125.0), -5.0);
        assert_eq!(f_cbrt(-64.0), -4.0);
        assert_eq!(f_cbrt(-27.0), -3.0);
        assert_eq!(f_cbrt(0.0), 0.0);
        assert_eq!(f_cbrt(f64::INFINITY), f64::INFINITY);
        assert_eq!(f_cbrt(f64::NEG_INFINITY), f64::NEG_INFINITY);
        assert!(f_cbrt(f64::NAN).is_nan());
    }
}