/* 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 */ 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 } } 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 } }