diff options
Diffstat (limited to 'library/compiler-builtins/libm/src/math/jn.rs')
| -rw-r--r-- | library/compiler-builtins/libm/src/math/jn.rs | 339 |
1 files changed, 339 insertions, 0 deletions
diff --git a/library/compiler-builtins/libm/src/math/jn.rs b/library/compiler-builtins/libm/src/math/jn.rs new file mode 100644 index 00000000000..31f8d9c5382 --- /dev/null +++ b/library/compiler-builtins/libm/src/math/jn.rs @@ -0,0 +1,339 @@ +/* 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). +#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] +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). +#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] +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 } +} |