compiler_builtins/math/libm_math/

j0f.rs

/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */
/*
 * Conversion to float by Ian Lance Taylor, Cygnus Support, [email protected].
 */
/*
 * ====================================================
 * 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, y0: bool) -> f32 {
    let z: f32;
    let s: f32;
    let mut c: f32;
    let mut ss: f32;
    let mut cc: f32;
    /*
     * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
     * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
     */
    s = sinf(x);
    c = cosf(x);
    if y0 {
        c = -c;
    }
    cc = s + c;
    if ix < 0x7f000000 {
        ss = s - c;
        z = -cosf(2.0 * x);
        if s * c < 0.0 {
            cc = z / ss;
        } else {
            ss = z / cc;
        }
        if ix < 0x58800000 {
            if y0 {
                ss = -ss;
            }
            cc = pzerof(x) * cc - qzerof(x) * ss;
        }
    }
    return INVSQRTPI * cc / sqrtf(x);
}

/* R0/S0 on [0, 2.00] */
const R02: f32 = 1.5625000000e-02; /* 0x3c800000 */
const R03: f32 = -1.8997929874e-04; /* 0xb947352e */
const R04: f32 = 1.8295404516e-06; /* 0x35f58e88 */
const R05: f32 = -4.6183270541e-09; /* 0xb19eaf3c */
const S01: f32 = 1.5619102865e-02; /* 0x3c7fe744 */
const S02: f32 = 1.1692678527e-04; /* 0x38f53697 */
const S03: f32 = 5.1354652442e-07; /* 0x3509daa6 */
const S04: f32 = 1.1661400734e-09; /* 0x30a045e8 */

/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f32).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn j0f(mut x: f32) -> f32 {
    let z: f32;
    let r: f32;
    let s: f32;
    let mut ix: u32;

    ix = x.to_bits();
    ix &= 0x7fffffff;
    if ix >= 0x7f800000 {
        return 1.0 / (x * x);
    }
    x = fabsf(x);

    if ix >= 0x40000000 {
        /* |x| >= 2 */
        /* large ulp error near zeros */
        return common(ix, x, false);
    }
    if ix >= 0x3a000000 {
        /* |x| >= 2**-11 */
        /* up to 4ulp error near 2 */
        z = x * x;
        r = z * (R02 + z * (R03 + z * (R04 + z * R05)));
        s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04)));
        return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s);
    }
    if ix >= 0x21800000 {
        /* |x| >= 2**-60 */
        x = 0.25 * x * x;
    }
    return 1.0 - x;
}

const U00: f32 = -7.3804296553e-02; /* 0xbd9726b5 */
const U01: f32 = 1.7666645348e-01; /* 0x3e34e80d */
const U02: f32 = -1.3818567619e-02; /* 0xbc626746 */
const U03: f32 = 3.4745343146e-04; /* 0x39b62a69 */
const U04: f32 = -3.8140706238e-06; /* 0xb67ff53c */
const U05: f32 = 1.9559013964e-08; /* 0x32a802ba */
const U06: f32 = -3.9820518410e-11; /* 0xae2f21eb */
const V01: f32 = 1.2730483897e-02; /* 0x3c509385 */
const V02: f32 = 7.6006865129e-05; /* 0x389f65e0 */
const V03: f32 = 2.5915085189e-07; /* 0x348b216c */
const V04: f32 = 4.4111031494e-10; /* 0x2ff280c2 */

/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f32).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn y0f(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 */
        /* large ulp error near zeros */
        return common(ix, x, true);
    }
    if ix >= 0x39000000 {
        /* x >= 2**-13 */
        /* large ulp error at x ~= 0.89 */
        z = x * x;
        u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06)))));
        v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04)));
        return u / v + TPI * (j0f(x) * logf(x));
    }
    return U00 + TPI * logf(x);
}

/* The asymptotic expansions of pzero is
 *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 * For x >= 2, We approximate pzero by
 *      pzero(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
 *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 */
const PR8: [f32; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00,  /* 0x00000000 */
    -7.0312500000e-02, /* 0xbd900000 */
    -8.0816707611e+00, /* 0xc1014e86 */
    -2.5706311035e+02, /* 0xc3808814 */
    -2.4852163086e+03, /* 0xc51b5376 */
    -5.2530439453e+03, /* 0xc5a4285a */
];
const PS8: [f32; 5] = [
    1.1653436279e+02, /* 0x42e91198 */
    3.8337448730e+03, /* 0x456f9beb */
    4.0597855469e+04, /* 0x471e95db */
    1.1675296875e+05, /* 0x47e4087c */
    4.7627726562e+04, /* 0x473a0bba */
];
const PR5: [f32; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    -1.1412546255e-11, /* 0xad48c58a */
    -7.0312492549e-02, /* 0xbd8fffff */
    -4.1596107483e+00, /* 0xc0851b88 */
    -6.7674766541e+01, /* 0xc287597b */
    -3.3123129272e+02, /* 0xc3a59d9b */
    -3.4643338013e+02, /* 0xc3ad3779 */
];
const PS5: [f32; 5] = [
    6.0753936768e+01, /* 0x42730408 */
    1.0512523193e+03, /* 0x44836813 */
    5.9789707031e+03, /* 0x45bad7c4 */
    9.6254453125e+03, /* 0x461665c8 */
    2.4060581055e+03, /* 0x451660ee */
];

const PR3: [f32; 6] = [
    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
    -2.5470459075e-09, /* 0xb12f081b */
    -7.0311963558e-02, /* 0xbd8fffb8 */
    -2.4090321064e+00, /* 0xc01a2d95 */
    -2.1965976715e+01, /* 0xc1afba52 */
    -5.8079170227e+01, /* 0xc2685112 */
    -3.1447946548e+01, /* 0xc1fb9565 */
];
const PS3: [f32; 5] = [
    3.5856033325e+01, /* 0x420f6c94 */
    3.6151397705e+02, /* 0x43b4c1ca */
    1.1936077881e+03, /* 0x44953373 */
    1.1279968262e+03, /* 0x448cffe6 */
    1.7358093262e+02, /* 0x432d94b8 */
];

const PR2: [f32; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    -8.8753431271e-08, /* 0xb3be98b7 */
    -7.0303097367e-02, /* 0xbd8ffb12 */
    -1.4507384300e+00, /* 0xbfb9b1cc */
    -7.6356959343e+00, /* 0xc0f4579f */
    -1.1193166733e+01, /* 0xc1331736 */
    -3.2336456776e+00, /* 0xc04ef40d */
];
const PS2: [f32; 5] = [
    2.2220300674e+01, /* 0x41b1c32d */
    1.3620678711e+02, /* 0x430834f0 */
    2.7047027588e+02, /* 0x43873c32 */
    1.5387539673e+02, /* 0x4319e01a */
    1.4657617569e+01, /* 0x416a859a */
];

fn pzerof(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 qzero is
 *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 * We approximate pzero by
 *      qzero(x) = s*(-1.25 + (R/S))
 * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 *        S = 1 + qS0*s^2 + ... + qS5*s^12
 * and
 *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 */
const QR8: [f32; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00, /* 0x00000000 */
    7.3242187500e-02, /* 0x3d960000 */
    1.1768206596e+01, /* 0x413c4a93 */
    5.5767340088e+02, /* 0x440b6b19 */
    8.8591972656e+03, /* 0x460a6cca */
    3.7014625000e+04, /* 0x471096a0 */
];
const QS8: [f32; 6] = [
    1.6377603149e+02,  /* 0x4323c6aa */
    8.0983447266e+03,  /* 0x45fd12c2 */
    1.4253829688e+05,  /* 0x480b3293 */
    8.0330925000e+05,  /* 0x49441ed4 */
    8.4050156250e+05,  /* 0x494d3359 */
    -3.4389928125e+05, /* 0xc8a7eb69 */
];

const QR5: [f32; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    1.8408595828e-11, /* 0x2da1ec79 */
    7.3242180049e-02, /* 0x3d95ffff */
    5.8356351852e+00, /* 0x40babd86 */
    1.3511157227e+02, /* 0x43071c90 */
    1.0272437744e+03, /* 0x448067cd */
    1.9899779053e+03, /* 0x44f8bf4b */
];
const QS5: [f32; 6] = [
    8.2776611328e+01,  /* 0x42a58da0 */
    2.0778142090e+03,  /* 0x4501dd07 */
    1.8847289062e+04,  /* 0x46933e94 */
    5.6751113281e+04,  /* 0x475daf1d */
    3.5976753906e+04,  /* 0x470c88c1 */
    -5.3543427734e+03, /* 0xc5a752be */
];

const QR3: [f32; 6] = [
    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
    4.3774099900e-09, /* 0x3196681b */
    7.3241114616e-02, /* 0x3d95ff70 */
    3.3442313671e+00, /* 0x405607e3 */
    4.2621845245e+01, /* 0x422a7cc5 */
    1.7080809021e+02, /* 0x432acedf */
    1.6673394775e+02, /* 0x4326bbe4 */
];
const QS3: [f32; 6] = [
    4.8758872986e+01,  /* 0x42430916 */
    7.0968920898e+02,  /* 0x44316c1c */
    3.7041481934e+03,  /* 0x4567825f */
    6.4604252930e+03,  /* 0x45c9e367 */
    2.5163337402e+03,  /* 0x451d4557 */
    -1.4924745178e+02, /* 0xc3153f59 */
];

const QR2: [f32; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    1.5044444979e-07, /* 0x342189db */
    7.3223426938e-02, /* 0x3d95f62a */
    1.9981917143e+00, /* 0x3fffc4bf */
    1.4495602608e+01, /* 0x4167edfd */
    3.1666231155e+01, /* 0x41fd5471 */
    1.6252708435e+01, /* 0x4182058c */
];
const QS2: [f32; 6] = [
    3.0365585327e+01,  /* 0x41f2ecb8 */
    2.6934811401e+02,  /* 0x4386ac8f */
    8.4478375244e+02,  /* 0x44533229 */
    8.8293585205e+02,  /* 0x445cbbe5 */
    2.1266638184e+02,  /* 0x4354aa98 */
    -5.3109550476e+00, /* 0xc0a9f358 */
];

fn qzerof(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 (-0.125 + r / s) / x;
}