pxfm 0.1.29

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::polyeval::f_estrin_polyeval5;
use crate::sin::{range_reduction_small, sincos_eval};
use crate::sin_helper::sincos_eval_dd;
use crate::sin_table::SIN_K_PI_OVER_128;
use crate::sincos_reduce::LargeArgumentReduction;

#[cold]
#[inline(never)]
fn sinmx_accurate(y: DoubleDouble, sin_k: DoubleDouble, cos_k: DoubleDouble, x: f64) -> f64 {
    let r_sincos = sincos_eval_dd(y);

    // k is an integer and -pi / 256 <= y <= pi / 256.
    // Then sin(x) = sin((k * pi/128 + y)
    //             = sin(y) * cos(k*pi/128) + cos(y) * sin(k*pi/128)

    let sin_k_cos_y = DoubleDouble::quick_mult(r_sincos.v_cos, sin_k);
    let cos_k_sin_y = DoubleDouble::quick_mult(r_sincos.v_sin, cos_k);

    let mut rr = DoubleDouble::full_dd_add(sin_k_cos_y, cos_k_sin_y);
    rr = DoubleDouble::full_add_f64(rr, -x);
    rr.to_f64()
}

#[cold]
fn sinmx_near_zero_hard(x: f64) -> f64 {
    const C: [(u64, u64); 8] = [
        (0xb37137ef120d4bbd, 0xb6db8d4e2aa9f813),
        (0xbc6555555554e720, 0xbfc5555555555555),
        (0x3c01110fff8e3ea0, 0x3f81111111111111),
        (0xbb6314569388b856, 0xbf2a01a01a01a01a),
        (0xbb61f946e615f3cd, 0x3ec71de3a556c723),
        (0x3a8998bc94bd3bf0, 0xbe5ae64567f2d4df),
        (0xba702e73490290eb, 0x3de61245e54b6747),
        (0xba0182df5b1ffd4c, 0xbd6ae4894bb27213),
    ];
    let x2 = DoubleDouble::from_exact_mult(x, x);
    let mut p = DoubleDouble::mul_add(
        x2,
        DoubleDouble::from_bit_pair(C[7]),
        DoubleDouble::from_bit_pair(C[6]),
    );
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[5]));
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[4]));
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[3]));
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[2]));
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[1]));
    p = DoubleDouble::mul_add(x2, p, DoubleDouble::from_bit_pair(C[0]));
    p = DoubleDouble::quick_mult_f64(p, x);
    p.to_f64()
}

/// Computes sin(x) - x
///
/// ULP 0.5
pub fn f_sinmx(x: f64) -> f64 {
    let x_e = (x.to_bits() >> 52) & 0x7ff;
    const E_BIAS: u64 = (1u64 << (11 - 1u64)) - 1u64;

    let y: DoubleDouble;
    let k;

    let mut argument_reduction = LargeArgumentReduction::default();

    // |x| < 2^32 (with FMA) or |x| < 2^23 (w/o FMA)
    if x_e < E_BIAS + 16 {
        if x_e < E_BIAS - 6 {
            // |x| < 2^-6
            if x_e < E_BIAS - 32 {
                // |x| < 2^-32
                // Signed zeros.
                if x == 0.0 {
                    return x;
                }

                // For |x| < 2^-32, taylor series sin(x) - x ~ -x^3/6
                let x2 = x * x;
                let c = f_fmla(
                    x2,
                    f64::from_bits(0x3f81111111111111),
                    f64::from_bits(0xbfc5555555555555),
                ) * x2;
                return c * x;
            }

            // Generated by Sollya:
            // d = [2^-26, pi/16];
            // f_sinmx = (sin(x) - x)/x;
            // Q = fpminimax(f_sinmx, [|0, 2, 4, 6, 8, 10, 12|], [|127, 127, D...|], d);
            let x2 = DoubleDouble::from_exact_mult(x, x);
            let p = f_estrin_polyeval5(
                x2.hi,
                f64::from_bits(0x3f81111111111111),
                f64::from_bits(0xbf2a01a01a019d2f),
                f64::from_bits(0x3ec71de3a5269512),
                f64::from_bits(0xbe5ae642b76ba0f5),
                f64::from_bits(0x3de6035da3c7eaed),
            );
            let mut c = DoubleDouble::mul_f64_add(
                x2,
                p,
                DoubleDouble::from_bit_pair((0xbc655542976eb2af, 0xbfc5555555555555)),
            );
            c = DoubleDouble::mul_add(
                x2,
                c,
                DoubleDouble::from_bit_pair((0x34b215c35dc9e9be, 0xb832bde584573661)),
            );
            c = DoubleDouble::quick_mult_f64(c, x);
            let err = f_fmla(
                x2.hi,
                f64::from_bits(0x3cc0000000000000), // 2^-51
                f64::from_bits(0x3bc0000000000000), // 2^-67
            );
            let ub = c.hi + (c.lo + err);
            let lb = c.hi + (c.lo - err);
            if ub == lb {
                return c.to_f64();
            }
            return sinmx_near_zero_hard(x);
        }
        // // Small range reduction.
        (y, k) = range_reduction_small(x);
    } else {
        // Inf or NaN
        if x_e > 2 * E_BIAS {
            // sin(+-Inf) = NaN
            return x + f64::NAN;
        }

        // Large range reduction.
        (k, y) = argument_reduction.reduce(x);
    }

    let r_sincos = sincos_eval(y);

    // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
    let sk = SIN_K_PI_OVER_128[(k & 255) as usize];
    let ck = SIN_K_PI_OVER_128[((k.wrapping_add(64)) & 255) as usize];

    let sin_k = DoubleDouble::from_bit_pair(sk);
    let cos_k = DoubleDouble::from_bit_pair(ck);

    let sin_k_cos_y = DoubleDouble::quick_mult(r_sincos.v_cos, sin_k);
    let cos_k_sin_y = DoubleDouble::quick_mult(r_sincos.v_sin, cos_k);

    // sin_k_cos_y is always >> cos_k_sin_y
    let mut rr = DoubleDouble::from_exact_add(sin_k_cos_y.hi, cos_k_sin_y.hi);
    rr.lo += sin_k_cos_y.lo + cos_k_sin_y.lo;

    rr = DoubleDouble::from_exact_add(rr.hi, rr.lo);
    rr = DoubleDouble::full_add_f64(rr, -x);

    let rlp = rr.lo + r_sincos.err;
    let rlm = rr.lo - r_sincos.err;

    let r_upper = rr.hi + rlp; // (rr.lo + ERR);
    let r_lower = rr.hi + rlm; // (rr.lo - ERR);

    // Ziv's accuracy test
    if r_upper == r_lower {
        return rr.to_f64();
    }

    sinmx_accurate(y, sin_k, cos_k, x)
}

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

    #[test]
    fn f_sinf_test() {
        assert_eq!(f_sinmx(0.0), 0.0);
        assert_eq!(f_sinmx(1.0), -0.1585290151921035);
        assert_eq!(f_sinmx(0.3), -0.0044797933386604245);
        assert_eq!(f_sinmx(-1.0), 0.1585290151921035);
        assert_eq!(f_sinmx(-0.3), 0.0044797933386604245);
        assert_eq!(f_sinmx(std::f64::consts::PI / 2.), -0.5707963267948966);
        assert!(f_sinmx(f64::INFINITY).is_nan());
        assert!(f_sinmx(f64::NEG_INFINITY).is_nan());
    }
}