/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.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.

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

 */

/*

 * jn(n, x), yn(n, x)

 * floating point Bessel's function of the 1st and 2nd kind

 * of order n

 *

 * Special cases:

 *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;

 *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.

 * Note 2. About jn(n,x), yn(n,x)

 *      For n=0, j0(x) is called,

 *      for n=1, j1(x) is called,

 *      for n<=x, forward recursion is used starting

 *      from values of j0(x) and j1(x).

 *      for n>x, a continued fraction approximation to

 *      j(n,x)/j(n-1,x) is evaluated and then backward

 *      recursion is used starting from a supposed value

 *      for j(n,x). The resulting value of j(0,x) is

 *      compared with the actual value to correct the

 *      supposed value of j(n,x).

 *

 *      yn(n,x) is similar in all respects, except

 *      that forward recursion is used for all

 *      values of n>1.

 */

use super::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1};

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

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

pub fn jn(n: i32, mut x: f64) -> f64 {

    let mut ix: u32;

    let lx: u32;

    let nm1: i32;

    let mut i: i32;

    let mut sign: bool;

    let mut a: f64;

    let mut b: f64;

    let mut temp: f64;

    ix = get_high_word(x);

    lx = get_low_word(x);

    sign = (ix >> 31) != 0;

    ix &= 0x7fffffff;

    // -lx == !lx + 1

    if (ix | (lx | ((!lx).wrapping_add(1))) >> 31) > 0x7ff00000 {

        /* nan */

        return x;

    }

    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)

     * Thus, J(-n,x) = J(n,-x)

     */

    /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */

    if n == 0 {

        return j0(x);

    }

    if n < 0 {

        nm1 = -(n + 1);

        x = -x;

        sign = !sign;

    } else {

        nm1 = n - 1;

    }

    if nm1 == 0 {

        return j1(x);

    }

    sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */

    x = fabs(x);

    if (ix | lx) == 0 || ix == 0x7ff00000 {

        /* if x is 0 or inf */

        b = 0.0;

    } else if (nm1 as f64) < x {

        /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */

        if ix >= 0x52d00000 {

            /* x > 2**302 */

            /* (x >> n**2)

             *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)

             *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)

             *      Let s=sin(x), c=cos(x),

             *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then

             *

             *             n    sin(xn)*sqt2    cos(xn)*sqt2

             *          ----------------------------------

             *             0     s-c             c+s

             *             1    -s-c            -c+s

             *             2    -s+c            -c-s

             *             3     s+c             c-s

             */

            temp = match nm1 & 3 {

                0 => -cos(x) + sin(x),

                1 => -cos(x) - sin(x),

                2 => cos(x) - sin(x),

                3 | _ => cos(x) + sin(x),

            };

            b = INVSQRTPI * temp / sqrt(x);

        } else {

            a = j0(x);

            b = j1(x);

            i = 0;

            while i < nm1 {

                i += 1;

                temp = b;

                b = b * (2.0 * (i as f64) / x) - a; /* avoid underflow */

                a = temp;

            }

        }

    } else {

        if ix < 0x3e100000 {

            /* x < 2**-29 */

            /* x is tiny, return the first Taylor expansion of J(n,x)

             * J(n,x) = 1/n!*(x/2)^n  - ...

             */

            if nm1 > 32 {

                /* underflow */

                b = 0.0;

            } else {

                temp = x * 0.5;

                b = temp;

                a = 1.0;

                i = 2;

                while i <= nm1 + 1 {

                    a *= i as f64; /* a = n! */

                    b *= temp; /* b = (x/2)^n */

                    i += 1;

                }

                b = b / a;

            }

        } else {

            /* use backward recurrence */

            /*                      x      x^2      x^2

             *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....

             *                      2n  - 2(n+1) - 2(n+2)

             *

             *                      1      1        1

             *  (for large x)   =  ----  ------   ------   .....

             *                      2n   2(n+1)   2(n+2)

             *                      -- - ------ - ------ -

             *                       x     x         x

             *

             * Let w = 2n/x and h=2/x, then the above quotient

             * is equal to the continued fraction:

             *                  1

             *      = -----------------------

             *                     1

             *         w - -----------------

             *                        1

             *              w+h - ---------

             *                     w+2h - ...

             *

             * To determine how many terms needed, let

             * Q(0) = w, Q(1) = w(w+h) - 1,

             * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),

             * When Q(k) > 1e4      good for single

             * When Q(k) > 1e9      good for double

             * When Q(k) > 1e17     good for quadruple

             */

            /* determine k */

            let mut t: f64;

            let mut q0: f64;

            let mut q1: f64;

            let mut w: f64;

            let h: f64;

            let mut z: f64;

            let mut tmp: f64;

            let nf: f64;

            let mut k: i32;

            nf = (nm1 as f64) + 1.0;

            w = 2.0 * nf / x;

            h = 2.0 / x;

            z = w + h;

            q0 = w;

            q1 = w * z - 1.0;

            k = 1;

            while q1 < 1.0e9 {

                k += 1;

                z += h;

                tmp = z * q1 - q0;

                q0 = q1;

                q1 = tmp;

            }

            t = 0.0;

            i = k;

            while i >= 0 {

                t = 1.0 / (2.0 * ((i as f64) + nf) / x - t);

                i -= 1;

            }

            a = t;

            b = 1.0;

            /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)

             *  Hence, if n*(log(2n/x)) > ...

             *  single 8.8722839355e+01

             *  double 7.09782712893383973096e+02

             *  long double 1.1356523406294143949491931077970765006170e+04

             *  then recurrent value may overflow and the result is

             *  likely underflow to zero

             */

            tmp = nf * log(fabs(w));

            if tmp < 7.09782712893383973096e+02 {

                i = nm1;

                while i > 0 {

                    temp = b;

                    b = b * (2.0 * (i as f64)) / x - a;

                    a = temp;

                    i -= 1;

                }

            } else {

                i = nm1;

                while i > 0 {

                    temp = b;

                    b = b * (2.0 * (i as f64)) / x - a;

                    a = temp;

                    /* scale b to avoid spurious overflow */

                    let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500

                    if b > x1p500 {

                        a /= b;

                        t /= b;

                        b = 1.0;

                    }

                    i -= 1;

                }

            }

            z = j0(x);

            w = j1(x);

            if fabs(z) >= fabs(w) {

                b = t * z / b;

            } else {

                b = t * w / a;

            }

        }

    }

    if sign { -b } else { b }

}

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

pub fn yn(n: i32, x: f64) -> f64 {

    let mut ix: u32;

    let lx: u32;

    let mut ib: u32;

    let nm1: i32;

    let mut sign: bool;

    let mut i: i32;

    let mut a: f64;

    let mut b: f64;

    let mut temp: f64;

    ix = get_high_word(x);

    lx = get_low_word(x);

    sign = (ix >> 31) != 0;

    ix &= 0x7fffffff;

    // -lx == !lx + 1

    if (ix | (lx | ((!lx).wrapping_add(1))) >> 31) > 0x7ff00000 {

        /* nan */

        return x;

    }

    if sign && (ix | lx) != 0 {

        /* x < 0 */

        return 0.0 / 0.0;

    }

    if ix == 0x7ff00000 {

        return 0.0;

    }

    if n == 0 {

        return y0(x);

    }

    if n < 0 {

        nm1 = -(n + 1);

        sign = (n & 1) != 0;

    } else {

        nm1 = n - 1;

        sign = false;

    }

    if nm1 == 0 {

        if sign {

            return -y1(x);

        } else {

            return y1(x);

        }

    }

    if ix >= 0x52d00000 {

        /* x > 2**302 */

        /* (x >> n**2)

         *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)

         *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)

         *      Let s=sin(x), c=cos(x),

         *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then

         *

         *             n    sin(xn)*sqt2    cos(xn)*sqt2

         *          ----------------------------------

         *             0     s-c             c+s

         *             1    -s-c            -c+s

         *             2    -s+c            -c-s

         *             3     s+c             c-s

         */

        temp = match nm1 & 3 {

            0 => -sin(x) - cos(x),

            1 => -sin(x) + cos(x),

            2 => sin(x) + cos(x),

            3 | _ => sin(x) - cos(x),

        };

        b = INVSQRTPI * temp / sqrt(x);

    } else {

        a = y0(x);

        b = y1(x);

        /* quit if b is -inf */

        ib = get_high_word(b);

        i = 0;

        while i < nm1 && ib != 0xfff00000 {

            i += 1;

            temp = b;

            b = (2.0 * (i as f64) / x) * b - a;

            ib = get_high_word(b);

            a = temp;

        }

    }

    if sign { -b } else { b }

}