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 | |
37 | use super::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1}; |
38 | |
39 | const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ |
40 | |
41 | pub 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 | |
254 | pub 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 | } |
344 | |