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 128-bit 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 <http://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 division by zero 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 <math-underflow.h>
64
65static const _Float128
66 invsqrtpi = L(5.6418958354775628694807945156077258584405E-1),
67 two = 2,
68 one = 1,
69 zero = 0;
70
71
72_Float128
73__ieee754_jnl (int n, _Float128 x)
74{
75 uint32_t se;
76 int32_t i, ix, sgn;
77 _Float128 a, b, temp, di, ret;
78 _Float128 z, w;
79 ieee854_long_double_shape_type u;
80
81
82 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
83 * Thus, J(-n,x) = J(n,-x)
84 */
85
86 u.value = x;
87 se = u.parts32.w0;
88 ix = se & 0x7fffffff;
89
90 /* if J(n,NaN) is NaN */
91 if (ix >= 0x7fff0000)
92 {
93 if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
94 return x + x;
95 }
96
97 if (n < 0)
98 {
99 n = -n;
100 x = -x;
101 se ^= 0x80000000;
102 }
103 if (n == 0)
104 return (__ieee754_j0l (x));
105 if (n == 1)
106 return (__ieee754_j1l (x));
107 sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */
108 x = fabsl (x);
109
110 {
111 SET_RESTORE_ROUNDL (FE_TONEAREST);
112 if (x == 0 || ix >= 0x7fff0000) /* if x is 0 or inf */
113 return sgn == 1 ? -zero : zero;
114 else if ((_Float128) n <= x)
115 {
116 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
117 if (ix >= 0x412D0000)
118 { /* x > 2**302 */
119
120 /* ??? Could use an expansion for large x here. */
121
122 /* (x >> n**2)
123 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
124 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
125 * Let s=sin(x), c=cos(x),
126 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
127 *
128 * n sin(xn)*sqt2 cos(xn)*sqt2
129 * ----------------------------------
130 * 0 s-c c+s
131 * 1 -s-c -c+s
132 * 2 -s+c -c-s
133 * 3 s+c c-s
134 */
135 _Float128 s;
136 _Float128 c;
137 __sincosl (x, &s, &c);
138 switch (n & 3)
139 {
140 case 0:
141 temp = c + s;
142 break;
143 case 1:
144 temp = -c + s;
145 break;
146 case 2:
147 temp = -c - s;
148 break;
149 case 3:
150 temp = c - s;
151 break;
152 }
153 b = invsqrtpi * temp / sqrtl (x);
154 }
155 else
156 {
157 a = __ieee754_j0l (x);
158 b = __ieee754_j1l (x);
159 for (i = 1; i < n; i++)
160 {
161 temp = b;
162 b = b * ((_Float128) (i + i) / x) - a; /* avoid underflow */
163 a = temp;
164 }
165 }
166 }
167 else
168 {
169 if (ix < 0x3fc60000)
170 { /* x < 2**-57 */
171 /* x is tiny, return the first Taylor expansion of J(n,x)
172 * J(n,x) = 1/n!*(x/2)^n - ...
173 */
174 if (n >= 400) /* underflow, result < 10^-4952 */
175 b = zero;
176 else
177 {
178 temp = x * 0.5;
179 b = temp;
180 for (a = one, i = 2; i <= n; i++)
181 {
182 a *= (_Float128) i; /* a = n! */
183 b *= temp; /* b = (x/2)^n */
184 }
185 b = b / a;
186 }
187 }
188 else
189 {
190 /* use backward recurrence */
191 /* x x^2 x^2
192 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
193 * 2n - 2(n+1) - 2(n+2)
194 *
195 * 1 1 1
196 * (for large x) = ---- ------ ------ .....
197 * 2n 2(n+1) 2(n+2)
198 * -- - ------ - ------ -
199 * x x x
200 *
201 * Let w = 2n/x and h=2/x, then the above quotient
202 * is equal to the continued fraction:
203 * 1
204 * = -----------------------
205 * 1
206 * w - -----------------
207 * 1
208 * w+h - ---------
209 * w+2h - ...
210 *
211 * To determine how many terms needed, let
212 * Q(0) = w, Q(1) = w(w+h) - 1,
213 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
214 * When Q(k) > 1e4 good for single
215 * When Q(k) > 1e9 good for double
216 * When Q(k) > 1e17 good for quadruple
217 */
218 /* determine k */
219 _Float128 t, v;
220 _Float128 q0, q1, h, tmp;
221 int32_t k, m;
222 w = (n + n) / (_Float128) x;
223 h = 2 / (_Float128) x;
224 q0 = w;
225 z = w + h;
226 q1 = w * z - 1;
227 k = 1;
228 while (q1 < L(1.0e17))
229 {
230 k += 1;
231 z += h;
232 tmp = z * q1 - q0;
233 q0 = q1;
234 q1 = tmp;
235 }
236 m = n + n;
237 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
238 t = one / (i / x - t);
239 a = t;
240 b = one;
241 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
242 * Hence, if n*(log(2n/x)) > ...
243 * single 8.8722839355e+01
244 * double 7.09782712893383973096e+02
245 * long double 1.1356523406294143949491931077970765006170e+04
246 * then recurrent value may overflow and the result is
247 * likely underflow to zero
248 */
249 tmp = n;
250 v = two / x;
251 tmp = tmp * __ieee754_logl (fabsl (v * tmp));
252
253 if (tmp < L(1.1356523406294143949491931077970765006170e+04))
254 {
255 for (i = n - 1, di = (_Float128) (i + i); i > 0; i--)
256 {
257 temp = b;
258 b *= di;
259 b = b / x - a;
260 a = temp;
261 di -= two;
262 }
263 }
264 else
265 {
266 for (i = n - 1, di = (_Float128) (i + i); i > 0; i--)
267 {
268 temp = b;
269 b *= di;
270 b = b / x - a;
271 a = temp;
272 di -= two;
273 /* scale b to avoid spurious overflow */
274 if (b > L(1e100))
275 {
276 a /= b;
277 t /= b;
278 b = one;
279 }
280 }
281 }
282 /* j0() and j1() suffer enormous loss of precision at and
283 * near zero; however, we know that their zero points never
284 * coincide, so just choose the one further away from zero.
285 */
286 z = __ieee754_j0l (x);
287 w = __ieee754_j1l (x);
288 if (fabsl (z) >= fabsl (w))
289 b = (t * z / b);
290 else
291 b = (t * w / a);
292 }
293 }
294 if (sgn == 1)
295 ret = -b;
296 else
297 ret = b;
298 }
299 if (ret == 0)
300 {
301 ret = __copysignl (LDBL_MIN, ret) * LDBL_MIN;
302 __set_errno (ERANGE);
303 }
304 else
305 math_check_force_underflow (ret);
306 return ret;
307}
308strong_alias (__ieee754_jnl, __jnl_finite)
309
310_Float128
311__ieee754_ynl (int n, _Float128 x)
312{
313 uint32_t se;
314 int32_t i, ix;
315 int32_t sign;
316 _Float128 a, b, temp, ret;
317 ieee854_long_double_shape_type u;
318
319 u.value = x;
320 se = u.parts32.w0;
321 ix = se & 0x7fffffff;
322
323 /* if Y(n,NaN) is NaN */
324 if (ix >= 0x7fff0000)
325 {
326 if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
327 return x + x;
328 }
329 if (x <= 0)
330 {
331 if (x == 0)
332 return ((n < 0 && (n & 1) != 0) ? 1 : -1) / L(0.0);
333 if (se & 0x80000000)
334 return zero / (zero * x);
335 }
336 sign = 1;
337 if (n < 0)
338 {
339 n = -n;
340 sign = 1 - ((n & 1) << 1);
341 }
342 if (n == 0)
343 return (__ieee754_y0l (x));
344 {
345 SET_RESTORE_ROUNDL (FE_TONEAREST);
346 if (n == 1)
347 {
348 ret = sign * __ieee754_y1l (x);
349 goto out;
350 }
351 if (ix >= 0x7fff0000)
352 return zero;
353 if (ix >= 0x412D0000)
354 { /* x > 2**302 */
355
356 /* ??? See comment above on the possible futility of this. */
357
358 /* (x >> n**2)
359 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
360 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
361 * Let s=sin(x), c=cos(x),
362 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
363 *
364 * n sin(xn)*sqt2 cos(xn)*sqt2
365 * ----------------------------------
366 * 0 s-c c+s
367 * 1 -s-c -c+s
368 * 2 -s+c -c-s
369 * 3 s+c c-s
370 */
371 _Float128 s;
372 _Float128 c;
373 __sincosl (x, &s, &c);
374 switch (n & 3)
375 {
376 case 0:
377 temp = s - c;
378 break;
379 case 1:
380 temp = -s - c;
381 break;
382 case 2:
383 temp = -s + c;
384 break;
385 case 3:
386 temp = s + c;
387 break;
388 }
389 b = invsqrtpi * temp / sqrtl (x);
390 }
391 else
392 {
393 a = __ieee754_y0l (x);
394 b = __ieee754_y1l (x);
395 /* quit if b is -inf */
396 u.value = b;
397 se = u.parts32.w0 & 0xffff0000;
398 for (i = 1; i < n && se != 0xffff0000; i++)
399 {
400 temp = b;
401 b = ((_Float128) (i + i) / x) * b - a;
402 u.value = b;
403 se = u.parts32.w0 & 0xffff0000;
404 a = temp;
405 }
406 }
407 /* If B is +-Inf, set up errno accordingly. */
408 if (! isfinite (b))
409 __set_errno (ERANGE);
410 if (sign > 0)
411 ret = b;
412 else
413 ret = -b;
414 }
415 out:
416 if (isinf (ret))
417 ret = __copysignl (LDBL_MAX, ret) * LDBL_MAX;
418 return ret;
419}
420strong_alias (__ieee754_ynl, __ynl_finite)
421