libm/math/

j1f.rs

/* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
/*
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

use super::{cosf, fabsf, logf, sinf, sqrtf};

const INVSQRTPI: f32 = 5.6418961287e-01; /* 0x3f106ebb */
const TPI: f32 = 6.3661974669e-01; /* 0x3f22f983 */

fn common(ix: u32, x: f32, y1: bool, sign: bool) -> f32 {
    let z: f64;
    let mut s: f64;
    let c: f64;
    let mut ss: f64;
    let mut cc: f64;

    s = sinf(x) as f64;
    if y1 {
        s = -s;
    }
    c = cosf(x) as f64;
    cc = s - c;
    if ix < 0x7f000000 {
        ss = -s - c;
        z = cosf(2.0 * x) as f64;
        if s * c > 0.0 {
            cc = z / ss;
        } else {
            ss = z / cc;
        }
        if ix < 0x58800000 {
            if y1 {
                ss = -ss;
            }
            cc = (ponef(x) as f64) * cc - (qonef(x) as f64) * ss;
        }
    }
    if sign {
        cc = -cc;
    }
    return (((INVSQRTPI as f64) * cc) / (sqrtf(x) as f64)) as f32;
}

/* R0/S0 on [0,2] */
const R00: f32 = -6.2500000000e-02; /* 0xbd800000 */
const R01: f32 = 1.4070566976e-03; /* 0x3ab86cfd */
const R02: f32 = -1.5995563444e-05; /* 0xb7862e36 */
const R03: f32 = 4.9672799207e-08; /* 0x335557d2 */
const S01: f32 = 1.9153760746e-02; /* 0x3c9ce859 */
const S02: f32 = 1.8594678841e-04; /* 0x3942fab6 */
const S03: f32 = 1.1771846857e-06; /* 0x359dffc2 */
const S04: f32 = 5.0463624390e-09; /* 0x31ad6446 */
const S05: f32 = 1.2354227016e-11; /* 0x2d59567e */

/// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f32).
pub fn j1f(x: f32) -> f32 {
    let mut z: f32;
    let r: f32;
    let s: f32;
    let mut ix: u32;
    let sign: bool;

    ix = x.to_bits();
    sign = (ix >> 31) != 0;
    ix &= 0x7fffffff;
    if ix >= 0x7f800000 {
        return 1.0 / (x * x);
    }
    if ix >= 0x40000000 {
        /* |x| >= 2 */
        return common(ix, fabsf(x), false, sign);
    }
    if ix >= 0x39000000 {
        /* |x| >= 2**-13 */
        z = x * x;
        r = z * (R00 + z * (R01 + z * (R02 + z * R03)));
        s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * (S04 + z * S05))));
        z = 0.5 + r / s;
    } else {
        z = 0.5;
    }
    return z * x;
}

const U0: [f32; 5] = [
    -1.9605709612e-01, /* 0xbe48c331 */
    5.0443872809e-02,  /* 0x3d4e9e3c */
    -1.9125689287e-03, /* 0xbafaaf2a */
    2.3525259166e-05,  /* 0x37c5581c */
    -9.1909917899e-08, /* 0xb3c56003 */
];
const V0: [f32; 5] = [
    1.9916731864e-02, /* 0x3ca3286a */
    2.0255257550e-04, /* 0x3954644b */
    1.3560879779e-06, /* 0x35b602d4 */
    6.2274145840e-09, /* 0x31d5f8eb */
    1.6655924903e-11, /* 0x2d9281cf */
];

/// First order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f32).
pub fn y1f(x: f32) -> f32 {
    let z: f32;
    let u: f32;
    let v: f32;
    let ix: u32;

    ix = x.to_bits();
    if (ix & 0x7fffffff) == 0 {
        return -1.0 / 0.0;
    }
    if (ix >> 31) != 0 {
        return 0.0 / 0.0;
    }
    if ix >= 0x7f800000 {
        return 1.0 / x;
    }
    if ix >= 0x40000000 {
        /* |x| >= 2.0 */
        return common(ix, x, true, false);
    }
    if ix < 0x33000000 {
        /* x < 2**-25 */
        return -TPI / x;
    }
    z = x * x;
    u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
    v = 1.0 + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
    return x * (u / v) + TPI * (j1f(x) * logf(x) - 1.0 / x);
}

/* For x >= 8, the asymptotic expansions of pone is
 *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 * We approximate pone by
 *      pone(x) = 1 + (R/S)
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 *        S = 1 + ps0*s^2 + ... + ps4*s^10
 * and
 *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 */

const PR8: [f32; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00, /* 0x00000000 */
    1.1718750000e-01, /* 0x3df00000 */
    1.3239480972e+01, /* 0x4153d4ea */
    4.1205184937e+02, /* 0x43ce06a3 */
    3.8747453613e+03, /* 0x45722bed */
    7.9144794922e+03, /* 0x45f753d6 */
];
const PS8: [f32; 5] = [
    1.1420736694e+02, /* 0x42e46a2c */
    3.6509309082e+03, /* 0x45642ee5 */
    3.6956207031e+04, /* 0x47105c35 */
    9.7602796875e+04, /* 0x47bea166 */
    3.0804271484e+04, /* 0x46f0a88b */
];

const PR5: [f32; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    1.3199052094e-11, /* 0x2d68333f */
    1.1718749255e-01, /* 0x3defffff */
    6.8027510643e+00, /* 0x40d9b023 */
    1.0830818176e+02, /* 0x42d89dca */
    5.1763616943e+02, /* 0x440168b7 */
    5.2871520996e+02, /* 0x44042dc6 */
];
const PS5: [f32; 5] = [
    5.9280597687e+01, /* 0x426d1f55 */
    9.9140142822e+02, /* 0x4477d9b1 */
    5.3532670898e+03, /* 0x45a74a23 */
    7.8446904297e+03, /* 0x45f52586 */
    1.5040468750e+03, /* 0x44bc0180 */
];

const PR3: [f32; 6] = [
    3.0250391081e-09, /* 0x314fe10d */
    1.1718686670e-01, /* 0x3defffab */
    3.9329774380e+00, /* 0x407bb5e7 */
    3.5119403839e+01, /* 0x420c7a45 */
    9.1055007935e+01, /* 0x42b61c2a */
    4.8559066772e+01, /* 0x42423c7c */
];
const PS3: [f32; 5] = [
    3.4791309357e+01, /* 0x420b2a4d */
    3.3676245117e+02, /* 0x43a86198 */
    1.0468714600e+03, /* 0x4482dbe3 */
    8.9081134033e+02, /* 0x445eb3ed */
    1.0378793335e+02, /* 0x42cf936c */
];

const PR2: [f32; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    1.0771083225e-07, /* 0x33e74ea8 */
    1.1717621982e-01, /* 0x3deffa16 */
    2.3685150146e+00, /* 0x401795c0 */
    1.2242610931e+01, /* 0x4143e1bc */
    1.7693971634e+01, /* 0x418d8d41 */
    5.0735230446e+00, /* 0x40a25a4d */
];
const PS2: [f32; 5] = [
    2.1436485291e+01, /* 0x41ab7dec */
    1.2529022980e+02, /* 0x42fa9499 */
    2.3227647400e+02, /* 0x436846c7 */
    1.1767937469e+02, /* 0x42eb5bd7 */
    8.3646392822e+00, /* 0x4105d590 */
];

fn ponef(x: f32) -> f32 {
    let p: &[f32; 6];
    let q: &[f32; 5];
    let z: f32;
    let r: f32;
    let s: f32;
    let mut ix: u32;

    ix = x.to_bits();
    ix &= 0x7fffffff;
    if ix >= 0x41000000 {
        p = &PR8;
        q = &PS8;
    } else if ix >= 0x409173eb {
        p = &PR5;
        q = &PS5;
    } else if ix >= 0x4036d917 {
        p = &PR3;
        q = &PS3;
    } else
    /*ix >= 0x40000000*/
    {
        p = &PR2;
        q = &PS2;
    }
    z = 1.0 / (x * x);
    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
    return 1.0 + r / s;
}

/* For x >= 8, the asymptotic expansions of qone is
 *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 *      qone(x) = s*(0.375 + (R/S))
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 *        S = 1 + qs1*s^2 + ... + qs6*s^12
 * and
 *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 */

const QR8: [f32; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00,  /* 0x00000000 */
    -1.0253906250e-01, /* 0xbdd20000 */
    -1.6271753311e+01, /* 0xc1822c8d */
    -7.5960174561e+02, /* 0xc43de683 */
    -1.1849806641e+04, /* 0xc639273a */
    -4.8438511719e+04, /* 0xc73d3683 */
];
const QS8: [f32; 6] = [
    1.6139537048e+02,  /* 0x43216537 */
    7.8253862305e+03,  /* 0x45f48b17 */
    1.3387534375e+05,  /* 0x4802bcd6 */
    7.1965775000e+05,  /* 0x492fb29c */
    6.6660125000e+05,  /* 0x4922be94 */
    -2.9449025000e+05, /* 0xc88fcb48 */
];

const QR5: [f32; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    -2.0897993405e-11, /* 0xadb7d219 */
    -1.0253904760e-01, /* 0xbdd1fffe */
    -8.0564479828e+00, /* 0xc100e736 */
    -1.8366960144e+02, /* 0xc337ab6b */
    -1.3731937256e+03, /* 0xc4aba633 */
    -2.6124443359e+03, /* 0xc523471c */
];
const QS5: [f32; 6] = [
    8.1276550293e+01,  /* 0x42a28d98 */
    1.9917987061e+03,  /* 0x44f8f98f */
    1.7468484375e+04,  /* 0x468878f8 */
    4.9851425781e+04,  /* 0x4742bb6d */
    2.7948074219e+04,  /* 0x46da5826 */
    -4.7191835938e+03, /* 0xc5937978 */
];

const QR3: [f32; 6] = [
    -5.0783124372e-09, /* 0xb1ae7d4f */
    -1.0253783315e-01, /* 0xbdd1ff5b */
    -4.6101160049e+00, /* 0xc0938612 */
    -5.7847221375e+01, /* 0xc267638e */
    -2.2824453735e+02, /* 0xc3643e9a */
    -2.1921012878e+02, /* 0xc35b35cb */
];
const QS3: [f32; 6] = [
    4.7665153503e+01,  /* 0x423ea91e */
    6.7386511230e+02,  /* 0x4428775e */
    3.3801528320e+03,  /* 0x45534272 */
    5.5477290039e+03,  /* 0x45ad5dd5 */
    1.9031191406e+03,  /* 0x44ede3d0 */
    -1.3520118713e+02, /* 0xc3073381 */
];

const QR2: [f32; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    -1.7838172539e-07, /* 0xb43f8932 */
    -1.0251704603e-01, /* 0xbdd1f475 */
    -2.7522056103e+00, /* 0xc0302423 */
    -1.9663616180e+01, /* 0xc19d4f16 */
    -4.2325313568e+01, /* 0xc2294d1f */
    -2.1371921539e+01, /* 0xc1aaf9b2 */
];
const QS2: [f32; 6] = [
    2.9533363342e+01,  /* 0x41ec4454 */
    2.5298155212e+02,  /* 0x437cfb47 */
    7.5750280762e+02,  /* 0x443d602e */
    7.3939318848e+02,  /* 0x4438d92a */
    1.5594900513e+02,  /* 0x431bf2f2 */
    -4.9594988823e+00, /* 0xc09eb437 */
];

fn qonef(x: f32) -> f32 {
    let p: &[f32; 6];
    let q: &[f32; 6];
    let s: f32;
    let r: f32;
    let z: f32;
    let mut ix: u32;

    ix = x.to_bits();
    ix &= 0x7fffffff;
    if ix >= 0x41000000 {
        p = &QR8;
        q = &QS8;
    } else if ix >= 0x409173eb {
        p = &QR5;
        q = &QS5;
    } else if ix >= 0x4036d917 {
        p = &QR3;
        q = &QS3;
    } else
    /*ix >= 0x40000000*/
    {
        p = &QR2;
        q = &QS2;
    }
    z = 1.0 / (x * x);
    r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
    s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
    return (0.375 + r / s) / x;
}

// PowerPC tests are failing on LLVM 13: https://github.com/rust-lang/rust/issues/88520
#[cfg(not(target_arch = "powerpc64"))]
#[cfg(test)]
mod tests {
    use super::{j1f, y1f};
    #[test]
    fn test_j1f_2488() {
        // 0x401F3E49
        assert_eq!(j1f(2.4881766_f32), 0.49999475_f32);
    }
    #[test]
    fn test_y1f_2002() {
        //allow slightly different result on x87
        let res = y1f(2.0000002_f32);
        if cfg!(all(target_arch = "x86", not(target_feature = "sse2"))) && (res == -0.10703231_f32)
        {
            return;
        }
        assert_eq!(res, -0.10703229_f32);
    }
}