pic_scale/math/

bessel_order_one.rs

/* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.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.
 * ====================================================
 */

#![allow(clippy::excessive_precision)]

const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
#[inline]
fn get_high_word(x: f64) -> u32 {
    (x.to_bits() >> 32) as u32
}

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

    /*
     * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
     * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
     *
     * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
     * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
     * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
     */
    s = x.sin();
    if y1 {
        s = -s;
    }
    let c = x.cos();
    cc = s - c;
    if ix < 0x7fe00000 {
        /* avoid overflow in 2*x */
        ss = -s - c;
        z = (2.0 * x).cos();
        if s * c > 0.0 {
            cc = z / ss;
        } else {
            ss = z / cc;
        }
        if ix < 0x48000000 {
            if y1 {
                ss = -ss;
            }
            cc = pone(x) * cc - qone(x) * ss;
        }
    }
    if sign {
        cc = -cc;
    }
    INVSQRTPI * cc / x.sqrt()
}

/* R0/S0 on [0,2] */
const R00: f64 = -6.25000000000000000000e-02; /* 0xBFB00000, 0x00000000 */
const R01: f64 = 1.40705666955189706048e-03; /* 0x3F570D9F, 0x98472C61 */
const R02: f64 = -1.59955631084035597520e-05; /* 0xBEF0C5C6, 0xBA169668 */
const R03: f64 = 4.96727999609584448412e-08; /* 0x3E6AAAFA, 0x46CA0BD9 */
const S01: f64 = 1.91537599538363460805e-02; /* 0x3F939D0B, 0x12637E53 */
const S02: f64 = 1.85946785588630915560e-04; /* 0x3F285F56, 0xB9CDF664 */
const S03: f64 = 1.17718464042623683263e-06; /* 0x3EB3BFF8, 0x333F8498 */
const S04: f64 = 5.04636257076217042715e-09; /* 0x3E35AC88, 0xC97DFF2C */
const S05: f64 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */

pub fn j1(x: f64) -> f64 {
    let mut z: f64;
    let r: f64;
    let s: f64;
    let mut ix: u32;

    ix = get_high_word(x);
    let sign = (ix >> 31) != 0;
    ix &= 0x7fffffff;
    if ix >= 0x7ff00000 {
        return 1.0 / (x * x);
    }
    if ix >= 0x40000000 {
        /* |x| >= 2 */
        return common(ix, x.abs(), false, sign);
    }
    if ix >= 0x38000000 {
        /* |x| >= 2**-127 */
        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 = r / s;
    } else {
        /* avoid underflow, raise inexact if x!=0 */
        z = x;
    }
    (0.5 + z) * 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: [f64; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
    1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
    1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
    4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
    3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
    7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
];
const PS8: [f64; 5] = [
    1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
    3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
    3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
    9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
    3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
];

const PR5: [f64; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
    1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
    6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
    1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
    5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
    5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
];
const PS5: [f64; 5] = [
    5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
    9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
    5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
    7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
    1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
];

const PR3: [f64; 6] = [
    3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
    1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
    3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
    3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
    9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
    4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
];
const PS3: [f64; 5] = [
    3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
    3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
    1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
    8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
    1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
];

const PR2: [f64; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
    1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
    2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
    1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
    1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
    5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
];
const PS2: [f64; 5] = [
    2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
    1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
    2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
    1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
    8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
];

fn pone(x: f64) -> f64 {
    let p: &[f64; 6];
    let q: &[f64; 5];
    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;
    }
    let z = 1.0 / (x * x);
    let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
    let s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
    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: [f64; 6] = [
    /* for x in [inf, 8]=1/[0,0.125] */
    0.00000000000000000000e+00,  /* 0x00000000, 0x00000000 */
    -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
    -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
    -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
    -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
    -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
];
const QS8: [f64; 6] = [
    1.61395369700722909556e+02,  /* 0x40642CA6, 0xDE5BCDE5 */
    7.82538599923348465381e+03,  /* 0x40BE9162, 0xD0D88419 */
    1.33875336287249578163e+05,  /* 0x4100579A, 0xB0B75E98 */
    7.19657723683240939863e+05,  /* 0x4125F653, 0x72869C19 */
    6.66601232617776375264e+05,  /* 0x412457D2, 0x7719AD5C */
    -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
];

const QR5: [f64; 6] = [
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
    -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
    -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
    -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
    -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
    -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
];
const QS5: [f64; 6] = [
    8.12765501384335777857e+01,  /* 0x405451B2, 0xFF5A11B2 */
    1.99179873460485964642e+03,  /* 0x409F1F31, 0xE77BF839 */
    1.74684851924908907677e+04,  /* 0x40D10F1F, 0x0D64CE29 */
    4.98514270910352279316e+04,  /* 0x40E8576D, 0xAABAD197 */
    2.79480751638918118260e+04,  /* 0x40DB4B04, 0xCF7C364B */
    -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
];

const QR3: [f64; 6] = [
    -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
    -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
    -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
    -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
    -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
    -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
];
const QS3: [f64; 6] = [
    4.76651550323729509273e+01,  /* 0x4047D523, 0xCCD367E4 */
    6.73865112676699709482e+02,  /* 0x40850EEB, 0xC031EE3E */
    3.38015286679526343505e+03,  /* 0x40AA684E, 0x448E7C9A */
    5.54772909720722782367e+03,  /* 0x40B5ABBA, 0xA61D54A6 */
    1.90311919338810798763e+03,  /* 0x409DBC7A, 0x0DD4DF4B */
    -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
];

const QR2: [f64; 6] = [
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
    -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
    -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
    -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
    -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
    -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
];
const QS2: [f64; 6] = [
    2.95333629060523854548e+01,  /* 0x403D888A, 0x78AE64FF */
    2.52981549982190529136e+02,  /* 0x406F9F68, 0xDB821CBA */
    7.57502834868645436472e+02,  /* 0x4087AC05, 0xCE49A0F7 */
    7.39393205320467245656e+02,  /* 0x40871B25, 0x48D4C029 */
    1.55949003336666123687e+02,  /* 0x40637E5E, 0x3C3ED8D4 */
    -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
];

fn qone(x: f64) -> f64 {
    let p: &[f64; 6];
    let q: &[f64; 6];
    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;
    }
    let z = 1.0 / (x * x);
    let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
    let s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
    (0.375 + r / s) / x
}

#[inline]
pub fn jinc_f64(x: f64) -> f64 {
    if x == 0f64 {
        return 0f64;
    }
    j1(x) / x
}

#[inline]
pub fn jinc_f32(x: f32) -> f32 {
    if x == 0f32 {
        return 0f32;
    }
    (j1(x as f64) / x as f64) as f32
}

pub trait Jinc<T> {
    fn jinc() -> fn(T) -> T;
}

impl Jinc<f64> for f64 {
    fn jinc() -> fn(f64) -> f64 {
        jinc_f64
    }
}

impl Jinc<f32> for f32 {
    fn jinc() -> fn(f32) -> f32 {
        jinc_f32
    }
}