/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */

/*

 * ====================================================

 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

 *

 * Developed at SunSoft, a Sun Microsystems, Inc. business.

 * Permission to use, copy, modify, and distribute this

 * software is freely granted, provided that this notice

 * is preserved.

 * ====================================================

 */

/* j0(x), y0(x)

 * Bessel function of the first and second kinds of order zero.

 * Method -- j0(x):

 *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...

 *      2. Reduce x to |x| since j0(x)=j0(-x),  and

 *         for x in (0,2)

 *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;

 *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )

 *         for x in (2,inf)

 *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))

 *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)

 *         as follow:

 *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)

 *                      = 1/sqrt(2) * (cos(x) + sin(x))

 *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)

 *                      = 1/sqrt(2) * (sin(x) - cos(x))

 *         (To avoid cancellation, use

 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))

 *          to compute the worse one.)

 *

 *      3 Special cases

 *              j0(nan)= nan

 *              j0(0) = 1

 *              j0(inf) = 0

 *

 * Method -- y0(x):

 *      1. For x<2.

 *         Since

 *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)

 *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.

 *         We use the following function to approximate y0,

 *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2

 *         where

 *              U(z) = u00 + u01*z + ... + u06*z^6

 *              V(z) = 1  + v01*z + ... + v04*z^4

 *         with absolute approximation error bounded by 2**-72.

 *         Note: For tiny x, U/V = u0 and j0(x)~1, hence

 *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)

 *      2. For x>=2.

 *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))

 *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)

 *         by the method mentioned above.

 *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.

 */

use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt};

const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */

const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */

/* common method when |x|>=2 */

fn common(ix: u32, x: f64, y0: bool) -> f64 {

    let s: f64;

    let mut c: f64;

    let mut ss: f64;

    let mut cc: f64;

    let z: f64;

    /*

     * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4))

     * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4))

     *

     * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2)

     * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2)

     * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))

     */

    s = sin(x);

    c = cos(x);

    if y0 {

        c = -c;

    }

    cc = s + c;

    /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */

    if ix < 0x7fe00000 {

        ss = s - c;

        z = -cos(2.0 * x);

        if s * c < 0.0 {

            cc = z / ss;

        } else {

            ss = z / cc;

        }

        if ix < 0x48000000 {

            if y0 {

                ss = -ss;

            }

            cc = pzero(x) * cc - qzero(x) * ss;

        }

    }

    return INVSQRTPI * cc / sqrt(x);

}

/* R0/S0 on [0, 2.00] */

const R02: f64 = 1.56249999999999947958e-02; /* 0x3F8FFFFF, 0xFFFFFFFD */

const R03: f64 = -1.89979294238854721751e-04; /* 0xBF28E6A5, 0xB61AC6E9 */

const R04: f64 = 1.82954049532700665670e-06; /* 0x3EBEB1D1, 0x0C503919 */

const R05: f64 = -4.61832688532103189199e-09; /* 0xBE33D5E7, 0x73D63FCE */

const S01: f64 = 1.56191029464890010492e-02; /* 0x3F8FFCE8, 0x82C8C2A4 */

const S02: f64 = 1.16926784663337450260e-04; /* 0x3F1EA6D2, 0xDD57DBF4 */

const S03: f64 = 5.13546550207318111446e-07; /* 0x3EA13B54, 0xCE84D5A9 */

const S04: f64 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */

/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).

pub fn j0(mut x: f64) -> f64 {

    let z: f64;

    let r: f64;

    let s: f64;

    let mut ix: u32;

    ix = get_high_word(x);

    ix &= 0x7fffffff;

    /* j0(+-inf)=0, j0(nan)=nan */

    if ix >= 0x7ff00000 {

        return 1.0 / (x * x);

    }

    x = fabs(x);

    if ix >= 0x40000000 {

        /* |x| >= 2 */

        /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */

        return common(ix, x, false);

    }

    /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */

    if ix >= 0x3f200000 {

        /* |x| >= 2**-13 */

        /* up to 4ulp error close to 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);

    }

    /* 1 - x*x/4 */

    /* prevent underflow */

    /* inexact should be raised when x!=0, this is not done correctly */

    if ix >= 0x38000000 {

        /* |x| >= 2**-127 */

        x = 0.25 * x * x;

    }

    return 1.0 - x;

}

const U00: f64 = -7.38042951086872317523e-02; /* 0xBFB2E4D6, 0x99CBD01F */

const U01: f64 = 1.76666452509181115538e-01; /* 0x3FC69D01, 0x9DE9E3FC */

const U02: f64 = -1.38185671945596898896e-02; /* 0xBF8C4CE8, 0xB16CFA97 */

const U03: f64 = 3.47453432093683650238e-04; /* 0x3F36C54D, 0x20B29B6B */

const U04: f64 = -3.81407053724364161125e-06; /* 0xBECFFEA7, 0x73D25CAD */

const U05: f64 = 1.95590137035022920206e-08; /* 0x3E550057, 0x3B4EABD4 */

const U06: f64 = -3.98205194132103398453e-11; /* 0xBDC5E43D, 0x693FB3C8 */

const V01: f64 = 1.27304834834123699328e-02; /* 0x3F8A1270, 0x91C9C71A */

const V02: f64 = 7.60068627350353253702e-05; /* 0x3F13ECBB, 0xF578C6C1 */

const V03: f64 = 2.59150851840457805467e-07; /* 0x3E91642D, 0x7FF202FD */

const V04: f64 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */

/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).

pub fn y0(x: f64) -> f64 {

    let z: f64;

    let u: f64;

    let v: f64;

    let ix: u32;

    let lx: u32;

    ix = get_high_word(x);

    lx = get_low_word(x);

    /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */

    if ((ix << 1) | lx) == 0 {

        return -1.0 / 0.0;

    }

    if (ix >> 31) != 0 {

        return 0.0 / 0.0;

    }

    if ix >= 0x7ff00000 {

        return 1.0 / x;

    }

    if ix >= 0x40000000 {

        /* x >= 2 */

        /* large ulp errors near zeros: 3.958, 7.086,.. */

        return common(ix, x, true);

    }

    /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */

    if ix >= 0x3e400000 {

        /* x >= 2**-27 */

        /* large ulp error near the first zero, 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 * (j0(x) * log(x));

    }

    return U00 + TPI * log(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: [f64; 6] = [

    /* for x in [inf, 8]=1/[0,0.125] */

    0.00000000000000000000e+00,  /* 0x00000000, 0x00000000 */

    -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */

    -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */

    -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */

    -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */

    -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */

];

const PS8: [f64; 5] = [

    1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */

    3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */

    4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */

    1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */

    4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */

];

const PR5: [f64; 6] = [

    /* for x in [8,4.5454]=1/[0.125,0.22001] */

    -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */

    -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */

    -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */

    -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */

    -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */

    -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */

];

const PS5: [f64; 5] = [

    6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */

    1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */

    5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */

    9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */

    2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */

];

const PR3: [f64; 6] = [

    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

    -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */

    -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */

    -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */

    -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */

    -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */

    -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */

];

const PS3: [f64; 5] = [

    3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */

    3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */

    1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */

    1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */

    1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */

];

const PR2: [f64; 6] = [

    /* for x in [2.8570,2]=1/[0.3499,0.5] */

    -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */

    -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */

    -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */

    -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */

    -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */

    -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */

];

const PS2: [f64; 5] = [

    2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */

    1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */

    2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */

    1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */

    1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */

];

fn pzero(x: f64) -> f64 {

    let p: &[f64; 6];

    let q: &[f64; 5];

    let z: f64;

    let r: f64;

    let s: f64;

    let mut ix: u32;

    ix = get_high_word(x);

    ix &= 0x7fffffff;

    if ix >= 0x40200000 {

        p = &PR8;

        q = &PS8;

    } else if ix >= 0x40122E8B {

        p = &PR5;

        q = &PS5;

    } else if ix >= 0x4006DB6D {

        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: [f64; 6] = [

    /* for x in [inf, 8]=1/[0,0.125] */

    0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */

    7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */

    1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */

    5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */

    8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */

    3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */

];

const QS8: [f64; 6] = [

    1.63776026895689824414e+02,  /* 0x406478D5, 0x365B39BC */

    8.09834494656449805916e+03,  /* 0x40BFA258, 0x4E6B0563 */

    1.42538291419120476348e+05,  /* 0x41016652, 0x54D38C3F */

    8.03309257119514397345e+05,  /* 0x412883DA, 0x83A52B43 */

    8.40501579819060512818e+05,  /* 0x4129A66B, 0x28DE0B3D */

    -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */

];

const QR5: [f64; 6] = [

    /* for x in [8,4.5454]=1/[0.125,0.22001] */

    1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */

    7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */

    5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */

    1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */

    1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */

    1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */

];

const QS5: [f64; 6] = [

    8.27766102236537761883e+01,  /* 0x4054B1B3, 0xFB5E1543 */

    2.07781416421392987104e+03,  /* 0x40A03BA0, 0xDA21C0CE */

    1.88472887785718085070e+04,  /* 0x40D267D2, 0x7B591E6D */

    5.67511122894947329769e+04,  /* 0x40EBB5E3, 0x97E02372 */

    3.59767538425114471465e+04,  /* 0x40E19118, 0x1F7A54A0 */

    -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */

];

const QR3: [f64; 6] = [

    /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

    4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */

    7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */

    3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */

    4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */

    1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */

    1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */

];

const QS3: [f64; 6] = [

    4.87588729724587182091e+01,  /* 0x40486122, 0xBFE343A6 */

    7.09689221056606015736e+02,  /* 0x40862D83, 0x86544EB3 */

    3.70414822620111362994e+03,  /* 0x40ACF04B, 0xE44DFC63 */

    6.46042516752568917582e+03,  /* 0x40B93C6C, 0xD7C76A28 */

    2.51633368920368957333e+03,  /* 0x40A3A8AA, 0xD94FB1C0 */

    -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */

];

const QR2: [f64; 6] = [

    /* for x in [2.8570,2]=1/[0.3499,0.5] */

    1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */

    7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */

    1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */

    1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */

    3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */

    1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */

];

const QS2: [f64; 6] = [

    3.03655848355219184498e+01,  /* 0x403E5D96, 0xF7C07AED */

    2.69348118608049844624e+02,  /* 0x4070D591, 0xE4D14B40 */

    8.44783757595320139444e+02,  /* 0x408A6645, 0x22B3BF22 */

    8.82935845112488550512e+02,  /* 0x408B977C, 0x9C5CC214 */

    2.12666388511798828631e+02,  /* 0x406A9553, 0x0E001365 */

    -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */

];

fn qzero(x: f64) -> f64 {

    let p: &[f64; 6];

    let q: &[f64; 6];

    let s: f64;

    let r: f64;

    let z: f64;

    let mut ix: u32;

    ix = get_high_word(x);

    ix &= 0x7fffffff;

    if ix >= 0x40200000 {

        p = &QR8;

        q = &QS8;

    } else if ix >= 0x40122E8B {

        p = &QR5;

        q = &QS5;

    } else if ix >= 0x4006DB6D {

        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;

}