1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* Modifications for long double are
13 Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
14 and are incorporated herein by permission of the author. The author
15 reserves the right to distribute this material elsewhere under different
16 copying permissions. These modifications are distributed here under
17 the following terms:
18
19 This library is free software; you can redistribute it and/or
20 modify it under the terms of the GNU Lesser General Public
21 License as published by the Free Software Foundation; either
22 version 2.1 of the License, or (at your option) any later version.
23
24 This library is distributed in the hope that it will be useful,
25 but WITHOUT ANY WARRANTY; without even the implied warranty of
26 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
27 Lesser General Public License for more details.
28
29 You should have received a copy of the GNU Lesser General Public
30 License along with this library; if not, see
31 <https://www.gnu.org/licenses/>. */
32
33 /*
34 * __ieee754_jn(n, x), __ieee754_yn(n, x)
35 * floating point Bessel's function of the 1st and 2nd kind
36 * of order n
37 *
38 * Special cases:
39 * y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
40 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
41 * Note 2. About jn(n,x), yn(n,x)
42 * For n=0, j0(x) is called,
43 * for n=1, j1(x) is called,
44 * for n<x, forward recursion us used starting
45 * from values of j0(x) and j1(x).
46 * for n>x, a continued fraction approximation to
47 * j(n,x)/j(n-1,x) is evaluated and then backward
48 * recursion is used starting from a supposed value
49 * for j(n,x). The resulting value of j(0,x) is
50 * compared with the actual value to correct the
51 * supposed value of j(n,x).
52 *
53 * yn(n,x) is similar in all respects, except
54 * that forward recursion is used for all
55 * values of n>1.
56 *
57 */
58
59 #include <errno.h>
60 #include <float.h>
61 #include <math.h>
62 #include <math_private.h>
63 #include <fenv_private.h>
64 #include <math-underflow.h>
65 #include <libm-alias-finite.h>
66
67 static const long double
68 invsqrtpi = 5.64189583547756286948079e-1L, two = 2.0e0L, one = 1.0e0L;
69
70 static const long double zero = 0.0L;
71
72 long double
73 __ieee754_jnl (int n, long double x)
74 {
75 uint32_t se, i0, i1;
76 int32_t i, ix, sgn;
77 long double a, b, temp, di, ret;
78 long double z, w;
79
80 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
81 * Thus, J(-n,x) = J(n,-x)
82 */
83
84 GET_LDOUBLE_WORDS (se, i0, i1, x);
85 ix = se & 0x7fff;
86
87 /* if J(n,NaN) is NaN */
88 if (__glibc_unlikely ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0)))
89 return x + x;
90 if (n < 0)
91 {
92 n = -n;
93 x = -x;
94 se ^= 0x8000;
95 }
96 if (n == 0)
97 return (__ieee754_j0l (x));
98 if (n == 1)
99 return (__ieee754_j1l (x));
100 sgn = (n & 1) & (se >> 15); /* even n -- 0, odd n -- sign(x) */
101 x = fabsl (x);
102 {
103 SET_RESTORE_ROUNDL (FE_TONEAREST);
104 if (__glibc_unlikely ((ix | i0 | i1) == 0 || ix >= 0x7fff))
105 /* if x is 0 or inf */
106 return sgn == 1 ? -zero : zero;
107 else if ((long double) n <= x)
108 {
109 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
110 if (ix >= 0x412D)
111 { /* x > 2**302 */
112
113 /* ??? This might be a futile gesture.
114 If x exceeds X_TLOSS anyway, the wrapper function
115 will set the result to zero. */
116
117 /* (x >> n**2)
118 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
119 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
120 * Let s=sin(x), c=cos(x),
121 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
122 *
123 * n sin(xn)*sqt2 cos(xn)*sqt2
124 * ----------------------------------
125 * 0 s-c c+s
126 * 1 -s-c -c+s
127 * 2 -s+c -c-s
128 * 3 s+c c-s
129 */
130 long double s;
131 long double c;
132 __sincosl (x, &s, &c);
133 switch (n & 3)
134 {
135 case 0:
136 temp = c + s;
137 break;
138 case 1:
139 temp = -c + s;
140 break;
141 case 2:
142 temp = -c - s;
143 break;
144 case 3:
145 temp = c - s;
146 break;
147 default:
148 __builtin_unreachable ();
149 }
150 b = invsqrtpi * temp / sqrtl (x);
151 }
152 else
153 {
154 a = __ieee754_j0l (x);
155 b = __ieee754_j1l (x);
156 for (i = 1; i < n; i++)
157 {
158 temp = b;
159 b = b * ((long double) (i + i) / x) - a; /* avoid underflow */
160 a = temp;
161 }
162 }
163 }
164 else
165 {
166 if (ix < 0x3fde)
167 { /* x < 2**-33 */
168 /* x is tiny, return the first Taylor expansion of J(n,x)
169 * J(n,x) = 1/n!*(x/2)^n - ...
170 */
171 if (n >= 400) /* underflow, result < 10^-4952 */
172 b = zero;
173 else
174 {
175 temp = x * 0.5;
176 b = temp;
177 for (a = one, i = 2; i <= n; i++)
178 {
179 a *= (long double) i; /* a = n! */
180 b *= temp; /* b = (x/2)^n */
181 }
182 b = b / a;
183 }
184 }
185 else
186 {
187 /* use backward recurrence */
188 /* x x^2 x^2
189 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
190 * 2n - 2(n+1) - 2(n+2)
191 *
192 * 1 1 1
193 * (for large x) = ---- ------ ------ .....
194 * 2n 2(n+1) 2(n+2)
195 * -- - ------ - ------ -
196 * x x x
197 *
198 * Let w = 2n/x and h=2/x, then the above quotient
199 * is equal to the continued fraction:
200 * 1
201 * = -----------------------
202 * 1
203 * w - -----------------
204 * 1
205 * w+h - ---------
206 * w+2h - ...
207 *
208 * To determine how many terms needed, let
209 * Q(0) = w, Q(1) = w(w+h) - 1,
210 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
211 * When Q(k) > 1e4 good for single
212 * When Q(k) > 1e9 good for double
213 * When Q(k) > 1e17 good for quadruple
214 */
215 /* determine k */
216 long double t, v;
217 long double q0, q1, h, tmp;
218 int32_t k, m;
219 w = (n + n) / (long double) x;
220 h = 2.0L / (long double) x;
221 q0 = w;
222 z = w + h;
223 q1 = w * z - 1.0L;
224 k = 1;
225 while (q1 < 1.0e11L)
226 {
227 k += 1;
228 z += h;
229 tmp = z * q1 - q0;
230 q0 = q1;
231 q1 = tmp;
232 }
233 m = n + n;
234 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
235 t = one / (i / x - t);
236 a = t;
237 b = one;
238 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
239 * Hence, if n*(log(2n/x)) > ...
240 * single 8.8722839355e+01
241 * double 7.09782712893383973096e+02
242 * long double 1.1356523406294143949491931077970765006170e+04
243 * then recurrent value may overflow and the result is
244 * likely underflow to zero
245 */
246 tmp = n;
247 v = two / x;
248 tmp = tmp * __ieee754_logl (fabsl (v * tmp));
249
250 if (tmp < 1.1356523406294143949491931077970765006170e+04L)
251 {
252 for (i = n - 1, di = (long double) (i + i); i > 0; i--)
253 {
254 temp = b;
255 b *= di;
256 b = b / x - a;
257 a = temp;
258 di -= two;
259 }
260 }
261 else
262 {
263 for (i = n - 1, di = (long double) (i + i); i > 0; i--)
264 {
265 temp = b;
266 b *= di;
267 b = b / x - a;
268 a = temp;
269 di -= two;
270 /* scale b to avoid spurious overflow */
271 if (b > 1e100L)
272 {
273 a /= b;
274 t /= b;
275 b = one;
276 }
277 }
278 }
279 /* j0() and j1() suffer enormous loss of precision at and
280 * near zero; however, we know that their zero points never
281 * coincide, so just choose the one further away from zero.
282 */
283 z = __ieee754_j0l (x);
284 w = __ieee754_j1l (x);
285 if (fabsl (z) >= fabsl (w))
286 b = (t * z / b);
287 else
288 b = (t * w / a);
289 }
290 }
291 if (sgn == 1)
292 ret = -b;
293 else
294 ret = b;
295 }
296 if (ret == 0)
297 {
298 ret = copysignl (LDBL_MIN, ret) * LDBL_MIN;
299 __set_errno (ERANGE);
300 }
301 else
302 math_check_force_underflow (ret);
303 return ret;
304 }
305 libm_alias_finite (__ieee754_jnl, __jnl)
306
307 long double
308 __ieee754_ynl (int n, long double x)
309 {
310 uint32_t se, i0, i1;
311 int32_t i, ix;
312 int32_t sign;
313 long double a, b, temp, ret;
314
315
316 GET_LDOUBLE_WORDS (se, i0, i1, x);
317 ix = se & 0x7fff;
318 /* if Y(n,NaN) is NaN */
319 if (__builtin_expect ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0), 0))
320 return x + x;
321 if (__builtin_expect ((ix | i0 | i1) == 0, 0))
322 /* -inf or inf and divide-by-zero exception. */
323 return ((n < 0 && (n & 1) != 0) ? 1.0L : -1.0L) / 0.0L;
324 if (__builtin_expect (se & 0x8000, 0))
325 return zero / (zero * x);
326 sign = 1;
327 if (n < 0)
328 {
329 n = -n;
330 sign = 1 - ((n & 1) << 1);
331 }
332 if (n == 0)
333 return (__ieee754_y0l (x));
334 {
335 SET_RESTORE_ROUNDL (FE_TONEAREST);
336 if (n == 1)
337 {
338 ret = sign * __ieee754_y1l (x);
339 goto out;
340 }
341 if (__glibc_unlikely (ix == 0x7fff))
342 return zero;
343 if (ix >= 0x412D)
344 { /* x > 2**302 */
345
346 /* ??? See comment above on the possible futility of this. */
347
348 /* (x >> n**2)
349 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
350 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
351 * Let s=sin(x), c=cos(x),
352 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
353 *
354 * n sin(xn)*sqt2 cos(xn)*sqt2
355 * ----------------------------------
356 * 0 s-c c+s
357 * 1 -s-c -c+s
358 * 2 -s+c -c-s
359 * 3 s+c c-s
360 */
361 long double s;
362 long double c;
363 __sincosl (x, &s, &c);
364 switch (n & 3)
365 {
366 case 0:
367 temp = s - c;
368 break;
369 case 1:
370 temp = -s - c;
371 break;
372 case 2:
373 temp = -s + c;
374 break;
375 case 3:
376 temp = s + c;
377 break;
378 default:
379 __builtin_unreachable ();
380 }
381 b = invsqrtpi * temp / sqrtl (x);
382 }
383 else
384 {
385 a = __ieee754_y0l (x);
386 b = __ieee754_y1l (x);
387 /* quit if b is -inf */
388 GET_LDOUBLE_WORDS (se, i0, i1, b);
389 /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */
390 for (i = 1; i < n && se != 0xffffffff; i++)
391 {
392 temp = b;
393 b = ((long double) (i + i) / x) * b - a;
394 GET_LDOUBLE_WORDS (se, i0, i1, b);
395 a = temp;
396 }
397 }
398 /* If B is +-Inf, set up errno accordingly. */
399 if (! isfinite (b))
400 __set_errno (ERANGE);
401 if (sign > 0)
402 ret = b;
403 else
404 ret = -b;
405 }
406 out:
407 if (isinf (ret))
408 ret = copysignl (LDBL_MAX, ret) * LDBL_MAX;
409 return ret;
410 }
411 libm_alias_finite (__ieee754_ynl, __ynl)