pxfm 0.1.28

Fast and accurate math
Documentation 
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/*
 * // Copyright (c) Radzivon Bartoshyk 8/2025. 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::double_double::DoubleDouble;
use crate::exponents::fast_ldexp;
use crate::polyeval::f_polyeval6;

//
// // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0)
// #[inline]
// fn raphson_step(x: f64, a: f64) -> f64 {
//     let h = f_fmla(-a * x, x * x, 1.0);
//     f_fmla(1. / 3. * h, x, x)
// }

// y1 = y0(k1 − c(k2 − k3c), c = x*y0*y0*y0
// k1 = 14/9 , k2 = 7/9 , k3 = 2/9
#[inline(always)]
fn halleys_div_free(x: f64, a: f64) -> f64 {
    const K3: f64 = 2. / 9.;
    const K2: f64 = 7. / 9.;
    const K1: f64 = 14. / 9.;
    let c = a * x * x * x;
    let mut y = f_fmla(-K3, c, K2);
    y = f_fmla(-c, y, K1);
    y * x
}

#[inline(always)]
#[allow(unused)]
fn halleys_div_free_fma(x: f64, a: f64) -> f64 {
    const K3: f64 = 2. / 9.;
    const K2: f64 = 7. / 9.;
    const K1: f64 = 14. / 9.;
    let c = a * x * x * x;
    let mut y = f64::mul_add(-K3, c, K2);
    y = f64::mul_add(-c, y, K1);
    y * x
}

#[inline(always)]
fn rcbrt_gen_impl(a: f64) -> f64 {
    // Decompose a = m * 2^e, with m in [0.5, 1)
    let xu = a.to_bits();
    let exp = ((xu >> 52) & 0x7ff) as i32;
    let mut e = ((xu >> 52) & 0x7ff) as i32;
    let mut mant = xu & ((1u64 << 52) - 1);

    if exp == 0x7ff {
        if a.is_infinite() {
            return if a.is_sign_negative() { -0.0 } else { 0.0 };
        }
        return a + a;
    }
    if exp == 0 && a == 0. {
        return if a.is_sign_negative() {
            f64::NEG_INFINITY
        } else {
            f64::INFINITY
        };
    }

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

    e -= 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_inv_cbrt = x^(-1/3);
    // Q = fpminimax(f_inv_cbrt, 5, [|D...|], d, relative, floating);
    // See ./notes/inv_cbrt.sollya

    let p = f_polyeval6(
        m,
        f64::from_bits(0x3ffc7f365bceaf71),
        f64::from_bits(0xbff90e741fb9c896),
        f64::from_bits(0x3ff3e68b9b2cd237),
        f64::from_bits(0xbfe321c5eb24a185),
        f64::from_bits(0x3fc3fa269b897f69),
        f64::from_bits(0xbf916d6f13849fd1),
    );

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

    // 1; 2^{-1/3}; 2^{-2/3}
    static ESCALE: [u64; 3] = [1.0f64.to_bits(), 0x3fe965fea53d6e3d, 0x3fe428a2f98d728b];

    let z = p * f64::from_bits(ESCALE[rem_scale as usize]);

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

    // One Halley's method step
    // then refine in partial double-double precision with Newton-Raphson iteration
    let y0 = halleys_div_free(z, mm);

    let d2y = DoubleDouble::from_exact_mult(y0, y0);
    let d3y = DoubleDouble::quick_mult_f64(d2y, y0);
    let hb = DoubleDouble::quick_mult_f64(d3y, mm);

    let y: f64;

    #[cfg(any(
        all(
            any(target_arch = "x86", target_arch = "x86_64"),
            target_feature = "fma"
        ),
        target_arch = "aarch64"
    ))]
    {
        // decompose double-double in linear FMA sums
        // r = (1.0 - hb.hi - hb.lo) * y0 = y0 - hb.hi * y0 - hb.lo * y0 = fma(-hb.lo, y0, fma(-hb.hi, y0, y0))
        let r = f_fmla(-hb.lo, y0, f_fmla(hb.hi, -y0, y0));
        // // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0) = y0 + 1/3 * r
        y = f_fmla(1. / 3., r, y0);
    }
    #[cfg(not(any(
        all(
            any(target_arch = "x86", target_arch = "x86_64"),
            target_feature = "fma"
        ),
        target_arch = "aarch64"
    )))]
    {
        let m_hb = DoubleDouble::full_add_f64(-hb, 1.0);
        let r = DoubleDouble::quick_mult_f64(m_hb, y0);
        y = f_fmla(1. / 3., r.to_f64(), y0);
    }
    f64::copysign(fast_ldexp(y, -q), a)
}

#[cfg(any(target_arch = "x86", target_arch = "x86_64"))]
#[target_feature(enable = "avx", enable = "fma")]
unsafe fn rcbrt_fma_impl(a: f64) -> f64 {
    // Decompose a = m * 2^e, with m in [0.5, 1)
    let xu = a.to_bits();
    let exp = ((xu >> 52) & 0x7ff) as i32;
    let mut e = ((xu >> 52) & 0x7ff) as i32;
    let mut mant = xu & ((1u64 << 52) - 1);

    if exp == 0x7ff {
        if a.is_infinite() {
            return if a.is_sign_negative() { -0.0 } else { 0.0 };
        }
        return a + a;
    }
    if exp == 0 && a == 0. {
        return if a.is_sign_negative() {
            f64::NEG_INFINITY
        } else {
            f64::INFINITY
        };
    }

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

    e -= 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_inv_cbrt = x^(-1/3);
    // Q = fpminimax(f_inv_cbrt, 5, [|D...|], d, relative, floating);
    // See ./notes/inv_cbrt.sollya

    use crate::polyeval::d_polyeval6;
    let p = d_polyeval6(
        m,
        f64::from_bits(0x3ffc7f365bceaf71),
        f64::from_bits(0xbff90e741fb9c896),
        f64::from_bits(0x3ff3e68b9b2cd237),
        f64::from_bits(0xbfe321c5eb24a185),
        f64::from_bits(0x3fc3fa269b897f69),
        f64::from_bits(0xbf916d6f13849fd1),
    );

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

    // 1; 2^{-1/3}; 2^{-2/3}
    static ESCALE: [u64; 3] = [1.0f64.to_bits(), 0x3fe965fea53d6e3d, 0x3fe428a2f98d728b];

    let z = p * f64::from_bits(ESCALE[rem_scale as usize]);

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

    // One Halley's method step
    // then refine in partial double-double precision with Newton-Raphson iteration
    let y0 = halleys_div_free_fma(z, mm);

    let d2y = DoubleDouble::from_exact_mult_fma(y0, y0);
    let d3y = DoubleDouble::quick_mult_f64_fma(d2y, y0);
    let hb = DoubleDouble::quick_mult_f64_fma(d3y, mm);

    // decompose double-double in linear FMA sums
    // r = (1.0 - hb.hi - hb.lo) * y0 = y0 - hb.hi * y0 - hb.lo * y0 = fma(-hb.lo, y0, fma(-hb.hi, y0, y0))
    let r = f64::mul_add(-hb.lo, y0, f64::mul_add(hb.hi, -y0, y0));
    // // y1 = y0 + 1/3 * y0 * (1 - a * y0 * y0 * y0) = y0 + 1/3 * r
    let y = f64::mul_add(1. / 3., r, y0);
    f64::copysign(fast_ldexp(y, -q), a)
}

/// Computes 1/cbrt(x)
///
/// ULP 0.5
pub fn f_rcbrt(a: f64) -> f64 {
    #[cfg(not(any(target_arch = "x86", target_arch = "x86_64")))]
    {
        rcbrt_gen_impl(a)
    }
    #[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")
            {
                rcbrt_fma_impl
            } else {
                fn def_rcbrt(x: f64) -> f64 {
                    rcbrt_gen_impl(x)
                }
                def_rcbrt
            }
        });
        unsafe { q(a) }
    }
}

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

    #[test]
    fn test_rcbrt() {
        assert_eq!(f_rcbrt(0.9999999999999717), 1.0000000000000095);
        assert_eq!(f_rcbrt(-68355745214719140000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.),
                   -0.000000000000000000000000000000000000000002445728958868668);
        assert_eq!(f_rcbrt(-96105972807656840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.),
                   -0.0000000000000000000000000000000000000000000000000000000002183148143573148);
        assert_eq!(f_rcbrt(0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000139491540182158),
                   8949883389846071000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.);
        assert_eq!(f_rcbrt(0.00008386280387617153), 22.846001824951983);
        assert_eq!(f_rcbrt(-125.0), -0.2);
        assert_eq!(f_rcbrt(125.0), 0.2);
        assert_eq!(f_rcbrt(1.0), 1.0);
        assert_eq!(f_rcbrt(-1.0), -1.0);
        assert_eq!(f_rcbrt(0.0), f64::INFINITY);
        assert_eq!(f_rcbrt(-27.0), -1. / 3.);
        assert_eq!(
            f_rcbrt(2417851639214765300000000.),
            0.000000007450580596938716
        );
        assert_eq!(f_rcbrt(27.0), 1. / 3.);
        assert_eq!(f_rcbrt(64.0), 0.25);
        assert_eq!(f_rcbrt(-64.0), -0.25);
        assert_eq!(f_rcbrt(f64::NEG_INFINITY), -0.0);
        assert_eq!(f_rcbrt(f64::INFINITY), 0.0);
        assert!(f_rcbrt(f64::NAN).is_nan());
    }
}