libm/math/jn.rs
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
41pub fn jn(n: i32, mut x: f64) -> f64 {
42 let mut ix: u32;
43 let lx: u32;
44 let nm1: i32;
45 let mut i: i32;
46 let mut sign: bool;
47 let mut a: f64;
48 let mut b: f64;
49 let mut temp: f64;
50
51 ix = get_high_word(x);
52 lx = get_low_word(x);
53 sign = (ix >> 31) != 0;
54 ix &= 0x7fffffff;
55
56 // -lx == !lx + 1
57 if (ix | (lx | ((!lx).wrapping_add(1))) >> 31) > 0x7ff00000 {
58 /* nan */
59 return x;
60 }
61
62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
63 * Thus, J(-n,x) = J(n,-x)
64 */
65 /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
66 if n == 0 {
67 return j0(x);
68 }
69 if n < 0 {
70 nm1 = -(n + 1);
71 x = -x;
72 sign = !sign;
73 } else {
74 nm1 = n - 1;
75 }
76 if nm1 == 0 {
77 return j1(x);
78 }
79
80 sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */
81 x = fabs(x);
82 if (ix | lx) == 0 || ix == 0x7ff00000 {
83 /* if x is 0 or inf */
84 b = 0.0;
85 } else if (nm1 as f64) < x {
86 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
87 if ix >= 0x52d00000 {
88 /* x > 2**302 */
89 /* (x >> n**2)
90 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
91 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
92 * Let s=sin(x), c=cos(x),
93 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
94 *
95 * n sin(xn)*sqt2 cos(xn)*sqt2
96 * ----------------------------------
97 * 0 s-c c+s
98 * 1 -s-c -c+s
99 * 2 -s+c -c-s
100 * 3 s+c c-s
101 */
102 temp = match nm1 & 3 {
103 0 => -cos(x) + sin(x),
104 1 => -cos(x) - sin(x),
105 2 => cos(x) - sin(x),
106 3 | _ => cos(x) + sin(x),
107 };
108 b = INVSQRTPI * temp / sqrt(x);
109 } else {
110 a = j0(x);
111 b = j1(x);
112 i = 0;
113 while i < nm1 {
114 i += 1;
115 temp = b;
116 b = b * (2.0 * (i as f64) / x) - a; /* avoid underflow */
117 a = temp;
118 }
119 }
120 } else {
121 if ix < 0x3e100000 {
122 /* x < 2**-29 */
123 /* x is tiny, return the first Taylor expansion of J(n,x)
124 * J(n,x) = 1/n!*(x/2)^n - ...
125 */
126 if nm1 > 32 {
127 /* underflow */
128 b = 0.0;
129 } else {
130 temp = x * 0.5;
131 b = temp;
132 a = 1.0;
133 i = 2;
134 while i <= nm1 + 1 {
135 a *= i as f64; /* a = n! */
136 b *= temp; /* b = (x/2)^n */
137 i += 1;
138 }
139 b = b / a;
140 }
141 } else {
142 /* use backward recurrence */
143 /* x x^2 x^2
144 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
145 * 2n - 2(n+1) - 2(n+2)
146 *
147 * 1 1 1
148 * (for large x) = ---- ------ ------ .....
149 * 2n 2(n+1) 2(n+2)
150 * -- - ------ - ------ -
151 * x x x
152 *
153 * Let w = 2n/x and h=2/x, then the above quotient
154 * is equal to the continued fraction:
155 * 1
156 * = -----------------------
157 * 1
158 * w - -----------------
159 * 1
160 * w+h - ---------
161 * w+2h - ...
162 *
163 * To determine how many terms needed, let
164 * Q(0) = w, Q(1) = w(w+h) - 1,
165 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
166 * When Q(k) > 1e4 good for single
167 * When Q(k) > 1e9 good for double
168 * When Q(k) > 1e17 good for quadruple
169 */
170 /* determine k */
171 let mut t: f64;
172 let mut q0: f64;
173 let mut q1: f64;
174 let mut w: f64;
175 let h: f64;
176 let mut z: f64;
177 let mut tmp: f64;
178 let nf: f64;
179
180 let mut k: i32;
181
182 nf = (nm1 as f64) + 1.0;
183 w = 2.0 * nf / x;
184 h = 2.0 / x;
185 z = w + h;
186 q0 = w;
187 q1 = w * z - 1.0;
188 k = 1;
189 while q1 < 1.0e9 {
190 k += 1;
191 z += h;
192 tmp = z * q1 - q0;
193 q0 = q1;
194 q1 = tmp;
195 }
196 t = 0.0;
197 i = k;
198 while i >= 0 {
199 t = 1.0 / (2.0 * ((i as f64) + nf) / x - t);
200 i -= 1;
201 }
202 a = t;
203 b = 1.0;
204 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
205 * Hence, if n*(log(2n/x)) > ...
206 * single 8.8722839355e+01
207 * double 7.09782712893383973096e+02
208 * long double 1.1356523406294143949491931077970765006170e+04
209 * then recurrent value may overflow and the result is
210 * likely underflow to zero
211 */
212 tmp = nf * log(fabs(w));
213 if tmp < 7.09782712893383973096e+02 {
214 i = nm1;
215 while i > 0 {
216 temp = b;
217 b = b * (2.0 * (i as f64)) / x - a;
218 a = temp;
219 i -= 1;
220 }
221 } else {
222 i = nm1;
223 while i > 0 {
224 temp = b;
225 b = b * (2.0 * (i as f64)) / x - a;
226 a = temp;
227 /* scale b to avoid spurious overflow */
228 let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500
229 if b > x1p500 {
230 a /= b;
231 t /= b;
232 b = 1.0;
233 }
234 i -= 1;
235 }
236 }
237 z = j0(x);
238 w = j1(x);
239 if fabs(z) >= fabs(w) {
240 b = t * z / b;
241 } else {
242 b = t * w / a;
243 }
244 }
245 }
246
247 if sign {
248 -b
249 } else {
250 b
251 }
252}
253
254pub fn yn(n: i32, x: f64) -> f64 {
255 let mut ix: u32;
256 let lx: u32;
257 let mut ib: u32;
258 let nm1: i32;
259 let mut sign: bool;
260 let mut i: i32;
261 let mut a: f64;
262 let mut b: f64;
263 let mut temp: f64;
264
265 ix = get_high_word(x);
266 lx = get_low_word(x);
267 sign = (ix >> 31) != 0;
268 ix &= 0x7fffffff;
269
270 // -lx == !lx + 1
271 if (ix | (lx | ((!lx).wrapping_add(1))) >> 31) > 0x7ff00000 {
272 /* nan */
273 return x;
274 }
275 if sign && (ix | lx) != 0 {
276 /* x < 0 */
277 return 0.0 / 0.0;
278 }
279 if ix == 0x7ff00000 {
280 return 0.0;
281 }
282
283 if n == 0 {
284 return y0(x);
285 }
286 if n < 0 {
287 nm1 = -(n + 1);
288 sign = (n & 1) != 0;
289 } else {
290 nm1 = n - 1;
291 sign = false;
292 }
293 if nm1 == 0 {
294 if sign {
295 return -y1(x);
296 } else {
297 return y1(x);
298 }
299 }
300
301 if ix >= 0x52d00000 {
302 /* x > 2**302 */
303 /* (x >> n**2)
304 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
305 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
306 * Let s=sin(x), c=cos(x),
307 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
308 *
309 * n sin(xn)*sqt2 cos(xn)*sqt2
310 * ----------------------------------
311 * 0 s-c c+s
312 * 1 -s-c -c+s
313 * 2 -s+c -c-s
314 * 3 s+c c-s
315 */
316 temp = match nm1 & 3 {
317 0 => -sin(x) - cos(x),
318 1 => -sin(x) + cos(x),
319 2 => sin(x) + cos(x),
320 3 | _ => 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 {
339 -b
340 } else {
341 b
342 }
343}