1/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12/*
13 * jn(n, x), yn(n, x)
14 * floating point Bessel's function of the 1st and 2nd kind
15 * of order n
16 *
17 * Special cases:
18 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
19 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
20 * Note 2. About jn(n,x), yn(n,x)
21 * For n=0, j0(x) is called,
22 * for n=1, j1(x) is called,
23 * for n<=x, forward recursion is used starting
24 * from values of j0(x) and j1(x).
25 * for n>x, a continued fraction approximation to
26 * j(n,x)/j(n-1,x) is evaluated and then backward
27 * recursion is used starting from a supposed value
28 * for j(n,x). The resulting value of j(0,x) is
29 * compared with the actual value to correct the
30 * supposed value of j(n,x).
31 *
32 * yn(n,x) is similar in all respects, except
33 * that forward recursion is used for all
34 * values of n>1.
35 */
36
37use super::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1};
38
39const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
40
41/// Integer order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).
42#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
43pub fn jn(n: i32, mut x: f64) -> f64 {
44 let mut ix: u32;
45 let lx: u32;
46 let nm1: i32;
47 let mut i: i32;
48 let mut sign: bool;
49 let mut a: f64;
50 let mut b: f64;
51 let mut temp: f64;
52
53 ix = get_high_word(x);
54 lx = get_low_word(x);
55 sign = (ix >> 31) != 0;
56 ix &= 0x7fffffff;
57
58 // -lx == !lx + 1
59 if ix | ((lx | (!lx).wrapping_add(1)) >> 31) > 0x7ff00000 {
60 /* nan */
61 return x;
62 }
63
64 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
65 * Thus, J(-n,x) = J(n,-x)
66 */
67 /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
68 if n == 0 {
69 return j0(x);
70 }
71 if n < 0 {
72 nm1 = -(n + 1);
73 x = -x;
74 sign = !sign;
75 } else {
76 nm1 = n - 1;
77 }
78 if nm1 == 0 {
79 return j1(x);
80 }
81
82 sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */
83 x = fabs(x);
84 if (ix | lx) == 0 || ix == 0x7ff00000 {
85 /* if x is 0 or inf */
86 b = 0.0;
87 } else if (nm1 as f64) < x {
88 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
89 if ix >= 0x52d00000 {
90 /* x > 2**302 */
91 /* (x >> n**2)
92 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
93 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
94 * Let s=sin(x), c=cos(x),
95 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
96 *
97 * n sin(xn)*sqt2 cos(xn)*sqt2
98 * ----------------------------------
99 * 0 s-c c+s
100 * 1 -s-c -c+s
101 * 2 -s+c -c-s
102 * 3 s+c c-s
103 */
104 temp = match nm1 & 3 {
105 0 => -cos(x) + sin(x),
106 1 => -cos(x) - sin(x),
107 2 => cos(x) - sin(x),
108 // 3
109 _ => cos(x) + sin(x),
110 };
111 b = INVSQRTPI * temp / sqrt(x);
112 } else {
113 a = j0(x);
114 b = j1(x);
115 i = 0;
116 while i < nm1 {
117 i += 1;
118 temp = b;
119 b = b * (2.0 * (i as f64) / x) - a; /* avoid underflow */
120 a = temp;
121 }
122 }
123 } else if ix < 0x3e100000 {
124 /* x < 2**-29 */
125 /* x is tiny, return the first Taylor expansion of J(n,x)
126 * J(n,x) = 1/n!*(x/2)^n - ...
127 */
128 if nm1 > 32 {
129 /* underflow */
130 b = 0.0;
131 } else {
132 temp = x * 0.5;
133 b = temp;
134 a = 1.0;
135 i = 2;
136 while i <= nm1 + 1 {
137 a *= i as f64; /* a = n! */
138 b *= temp; /* b = (x/2)^n */
139 i += 1;
140 }
141 b = b / a;
142 }
143 } else {
144 /* use backward recurrence */
145 /* x x^2 x^2
146 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
147 * 2n - 2(n+1) - 2(n+2)
148 *
149 * 1 1 1
150 * (for large x) = ---- ------ ------ .....
151 * 2n 2(n+1) 2(n+2)
152 * -- - ------ - ------ -
153 * x x x
154 *
155 * Let w = 2n/x and h=2/x, then the above quotient
156 * is equal to the continued fraction:
157 * 1
158 * = -----------------------
159 * 1
160 * w - -----------------
161 * 1
162 * w+h - ---------
163 * w+2h - ...
164 *
165 * To determine how many terms needed, let
166 * Q(0) = w, Q(1) = w(w+h) - 1,
167 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
168 * When Q(k) > 1e4 good for single
169 * When Q(k) > 1e9 good for double
170 * When Q(k) > 1e17 good for quadruple
171 */
172 /* determine k */
173 let mut t: f64;
174 let mut q0: f64;
175 let mut q1: f64;
176 let mut w: f64;
177 let h: f64;
178 let mut z: f64;
179 let mut tmp: f64;
180 let nf: f64;
181
182 let mut k: i32;
183
184 nf = (nm1 as f64) + 1.0;
185 w = 2.0 * nf / x;
186 h = 2.0 / x;
187 z = w + h;
188 q0 = w;
189 q1 = w * z - 1.0;
190 k = 1;
191 while q1 < 1.0e9 {
192 k += 1;
193 z += h;
194 tmp = z * q1 - q0;
195 q0 = q1;
196 q1 = tmp;
197 }
198 t = 0.0;
199 i = k;
200 while i >= 0 {
201 t = 1.0 / (2.0 * ((i as f64) + nf) / x - t);
202 i -= 1;
203 }
204 a = t;
205 b = 1.0;
206 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
207 * Hence, if n*(log(2n/x)) > ...
208 * single 8.8722839355e+01
209 * double 7.09782712893383973096e+02
210 * long double 1.1356523406294143949491931077970765006170e+04
211 * then recurrent value may overflow and the result is
212 * likely underflow to zero
213 */
214 tmp = nf * log(fabs(w));
215 if tmp < 7.09782712893383973096e+02 {
216 i = nm1;
217 while i > 0 {
218 temp = b;
219 b = b * (2.0 * (i as f64)) / x - a;
220 a = temp;
221 i -= 1;
222 }
223 } else {
224 i = nm1;
225 while i > 0 {
226 temp = b;
227 b = b * (2.0 * (i as f64)) / x - a;
228 a = temp;
229 /* scale b to avoid spurious overflow */
230 let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500
231 if b > x1p500 {
232 a /= b;
233 t /= b;
234 b = 1.0;
235 }
236 i -= 1;
237 }
238 }
239 z = j0(x);
240 w = j1(x);
241 if fabs(z) >= fabs(w) {
242 b = t * z / b;
243 } else {
244 b = t * w / a;
245 }
246 }
247
248 if sign { -b } else { b }
249}
250
251/// Integer order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).
252#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
253pub fn yn(n: i32, x: f64) -> f64 {
254 let mut ix: u32;
255 let lx: u32;
256 let mut ib: u32;
257 let nm1: i32;
258 let mut sign: bool;
259 let mut i: i32;
260 let mut a: f64;
261 let mut b: f64;
262 let mut temp: f64;
263
264 ix = get_high_word(x);
265 lx = get_low_word(x);
266 sign = (ix >> 31) != 0;
267 ix &= 0x7fffffff;
268
269 // -lx == !lx + 1
270 if ix | ((lx | (!lx).wrapping_add(1)) >> 31) > 0x7ff00000 {
271 /* nan */
272 return x;
273 }
274 if sign && (ix | lx) != 0 {
275 /* x < 0 */
276 return 0.0 / 0.0;
277 }
278 if ix == 0x7ff00000 {
279 return 0.0;
280 }
281
282 if n == 0 {
283 return y0(x);
284 }
285 if n < 0 {
286 nm1 = -(n + 1);
287 sign = (n & 1) != 0;
288 } else {
289 nm1 = n - 1;
290 sign = false;
291 }
292 if nm1 == 0 {
293 if sign {
294 return -y1(x);
295 } else {
296 return y1(x);
297 }
298 }
299
300 if ix >= 0x52d00000 {
301 /* x > 2**302 */
302 /* (x >> n**2)
303 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
304 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
305 * Let s=sin(x), c=cos(x),
306 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
307 *
308 * n sin(xn)*sqt2 cos(xn)*sqt2
309 * ----------------------------------
310 * 0 s-c c+s
311 * 1 -s-c -c+s
312 * 2 -s+c -c-s
313 * 3 s+c c-s
314 */
315 temp = match nm1 & 3 {
316 0 => -sin(x) - cos(x),
317 1 => -sin(x) + cos(x),
318 2 => sin(x) + cos(x),
319 // 3
320 _ => sin(x) - cos(x),
321 };
322 b = INVSQRTPI * temp / sqrt(x);
323 } else {
324 a = y0(x);
325 b = y1(x);
326 /* quit if b is -inf */
327 ib = get_high_word(b);
328 i = 0;
329 while i < nm1 && ib != 0xfff00000 {
330 i += 1;
331 temp = b;
332 b = (2.0 * (i as f64) / x) * b - a;
333 ib = get_high_word(b);
334 a = temp;
335 }
336 }
337
338 if sign { -b } else { b }
339}
340