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/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.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.
* ====================================================
*/
use super::{fabsf, j0f, j1f, logf, y0f, y1f};
/// Integer order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f32).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn jnf(n: i32, mut x: f32) -> f32 {
let mut ix: u32;
let mut nm1: i32;
let mut sign: bool;
let mut i: i32;
let mut a: f32;
let mut b: f32;
let mut temp: f32;
ix = x.to_bits();
sign = (ix >> 31) != 0;
ix &= 0x7fffffff;
if ix > 0x7f800000 {
/* nan */
return x;
}
/* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
if n == 0 {
return j0f(x);
}
if n < 0 {
nm1 = -(n + 1);
x = -x;
sign = !sign;
} else {
nm1 = n - 1;
}
if nm1 == 0 {
return j1f(x);
}
sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */
x = fabsf(x);
if ix == 0 || ix == 0x7f800000 {
/* if x is 0 or inf */
b = 0.0;
} else if (nm1 as f32) < x {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
a = j0f(x);
b = j1f(x);
i = 0;
while i < nm1 {
i += 1;
temp = b;
b = b * (2.0 * (i as f32) / x) - a;
a = temp;
}
} else if ix < 0x35800000 {
/* x < 2**-20 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
if nm1 > 8 {
/* underflow */
nm1 = 8;
}
temp = 0.5 * x;
b = temp;
a = 1.0;
i = 2;
while i <= nm1 + 1 {
a *= i as f32; /* 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: f32;
let mut q0: f32;
let mut q1: f32;
let mut w: f32;
let h: f32;
let mut z: f32;
let mut tmp: f32;
let nf: f32;
let mut k: i32;
nf = (nm1 as f32) + 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 < 1.0e4 {
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 f32) + 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 * logf(fabsf(w));
if tmp < 88.721679688 {
i = nm1;
while i > 0 {
temp = b;
b = 2.0 * (i as f32) * b / x - a;
a = temp;
i -= 1;
}
} else {
i = nm1;
while i > 0 {
temp = b;
b = 2.0 * (i as f32) * b / x - a;
a = temp;
/* scale b to avoid spurious overflow */
let x1p60 = f32::from_bits(0x5d800000); // 0x1p60 == 2^60
if b > x1p60 {
a /= b;
t /= b;
b = 1.0;
}
i -= 1;
}
}
z = j0f(x);
w = j1f(x);
if fabsf(z) >= fabsf(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 (f32).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn ynf(n: i32, x: f32) -> f32 {
let mut ix: u32;
let mut ib: u32;
let nm1: i32;
let mut sign: bool;
let mut i: i32;
let mut a: f32;
let mut b: f32;
let mut temp: f32;
ix = x.to_bits();
sign = (ix >> 31) != 0;
ix &= 0x7fffffff;
if ix > 0x7f800000 {
/* nan */
return x;
}
if sign && ix != 0 {
/* x < 0 */
return 0.0 / 0.0;
}
if ix == 0x7f800000 {
return 0.0;
}
if n == 0 {
return y0f(x);
}
if n < 0 {
nm1 = -(n + 1);
sign = (n & 1) != 0;
} else {
nm1 = n - 1;
sign = false;
}
if nm1 == 0 {
if sign {
return -y1f(x);
} else {
return y1f(x);
}
}
a = y0f(x);
b = y1f(x);
/* quit if b is -inf */
ib = b.to_bits();
i = 0;
while i < nm1 && ib != 0xff800000 {
i += 1;
temp = b;
b = (2.0 * (i as f32) / x) * b - a;
ib = b.to_bits();
a = temp;
}
if sign { -b } else { b }
}