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/* 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 }
}