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
64static const _Float128
65 invsqrtpi = L(5.6418958354775628694807945156077258584405E-1),
66 two = 2,
67 one = 1,
68 zero = 0;
69
70
71_Float128
72__ieee754_jnl (int n, _Float128 x)
73{
74 u_int32_t se;
75 int32_t i, ix, sgn;
76 _Float128 a, b, temp, di, ret;
77 _Float128 z, w;
78 ieee854_long_double_shape_type u;
79
80
81 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
82 * Thus, J(-n,x) = J(n,-x)
83 */
84
85 u.value = x;
86 se = u.parts32.w0;
87 ix = se & 0x7fffffff;
88
89 /* if J(n,NaN) is NaN */
90 if (ix >= 0x7fff0000)
91 {
92 if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
93 return x + x;
94 }
95
96 if (n < 0)
97 {
98 n = -n;
99 x = -x;
100 se ^= 0x80000000;
101 }
102 if (n == 0)
103 return (__ieee754_j0l (x));
104 if (n == 1)
105 return (__ieee754_j1l (x));
106 sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */
107 x = fabsl (x);
108
109 {
110 SET_RESTORE_ROUNDL (FE_TONEAREST);
111 if (x == 0 || ix >= 0x7fff0000) /* if x is 0 or inf */
112 return sgn == 1 ? -zero : zero;
113 else if ((_Float128) n <= x)
114 {
115 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
116 if (ix >= 0x412D0000)
117 { /* x > 2**302 */
118
119 /* ??? Could use an expansion for large x here. */
120
121 /* (x >> n**2)
122 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
123 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
124 * Let s=sin(x), c=cos(x),
125 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
126 *
127 * n sin(xn)*sqt2 cos(xn)*sqt2
128 * ----------------------------------
129 * 0 s-c c+s
130 * 1 -s-c -c+s
131 * 2 -s+c -c-s
132 * 3 s+c c-s
133 */
134 _Float128 s;
135 _Float128 c;
136 __sincosl (x, &s, &c);
137 switch (n & 3)
138 {
139 case 0:
140 temp = c + s;
141 break;
142 case 1:
143 temp = -c + s;
144 break;
145 case 2:
146 temp = -c - s;
147 break;
148 case 3:
149 temp = c - s;
150 break;
151 }
152 b = invsqrtpi * temp / __ieee754_sqrtl (x);
153 }
154 else
155 {
156 a = __ieee754_j0l (x);
157 b = __ieee754_j1l (x);
158 for (i = 1; i < n; i++)
159 {
160 temp = b;
161 b = b * ((_Float128) (i + i) / x) - a; /* avoid underflow */
162 a = temp;
163 }
164 }
165 }
166 else
167 {
168 if (ix < 0x3fc60000)
169 { /* x < 2**-57 */
170 /* x is tiny, return the first Taylor expansion of J(n,x)
171 * J(n,x) = 1/n!*(x/2)^n - ...
172 */
173 if (n >= 400) /* underflow, result < 10^-4952 */
174 b = zero;
175 else
176 {
177 temp = x * 0.5;
178 b = temp;
179 for (a = one, i = 2; i <= n; i++)
180 {
181 a *= (_Float128) i; /* a = n! */
182 b *= temp; /* b = (x/2)^n */
183 }
184 b = b / a;
185 }
186 }
187 else
188 {
189 /* use backward recurrence */
190 /* x x^2 x^2
191 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
192 * 2n - 2(n+1) - 2(n+2)
193 *
194 * 1 1 1
195 * (for large x) = ---- ------ ------ .....
196 * 2n 2(n+1) 2(n+2)
197 * -- - ------ - ------ -
198 * x x x
199 *
200 * Let w = 2n/x and h=2/x, then the above quotient
201 * is equal to the continued fraction:
202 * 1
203 * = -----------------------
204 * 1
205 * w - -----------------
206 * 1
207 * w+h - ---------
208 * w+2h - ...
209 *
210 * To determine how many terms needed, let
211 * Q(0) = w, Q(1) = w(w+h) - 1,
212 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
213 * When Q(k) > 1e4 good for single
214 * When Q(k) > 1e9 good for double
215 * When Q(k) > 1e17 good for quadruple
216 */
217 /* determine k */
218 _Float128 t, v;
219 _Float128 q0, q1, h, tmp;
220 int32_t k, m;
221 w = (n + n) / (_Float128) x;
222 h = 2 / (_Float128) x;
223 q0 = w;
224 z = w + h;
225 q1 = w * z - 1;
226 k = 1;
227 while (q1 < L(1.0e17))
228 {
229 k += 1;
230 z += h;
231 tmp = z * q1 - q0;
232 q0 = q1;
233 q1 = tmp;
234 }
235 m = n + n;
236 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
237 t = one / (i / x - t);
238 a = t;
239 b = one;
240 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
241 * Hence, if n*(log(2n/x)) > ...
242 * single 8.8722839355e+01
243 * double 7.09782712893383973096e+02
244 * long double 1.1356523406294143949491931077970765006170e+04
245 * then recurrent value may overflow and the result is
246 * likely underflow to zero
247 */
248 tmp = n;
249 v = two / x;
250 tmp = tmp * __ieee754_logl (fabsl (v * tmp));
251
252 if (tmp < L(1.1356523406294143949491931077970765006170e+04))
253 {
254 for (i = n - 1, di = (_Float128) (i + i); i > 0; i--)
255 {
256 temp = b;
257 b *= di;
258 b = b / x - a;
259 a = temp;
260 di -= two;
261 }
262 }
263 else
264 {
265 for (i = n - 1, di = (_Float128) (i + i); i > 0; i--)
266 {
267 temp = b;
268 b *= di;
269 b = b / x - a;
270 a = temp;
271 di -= two;
272 /* scale b to avoid spurious overflow */
273 if (b > L(1e100))
274 {
275 a /= b;
276 t /= b;
277 b = one;
278 }
279 }
280 }
281 /* j0() and j1() suffer enormous loss of precision at and
282 * near zero; however, we know that their zero points never
283 * coincide, so just choose the one further away from zero.
284 */
285 z = __ieee754_j0l (x);
286 w = __ieee754_j1l (x);
287 if (fabsl (z) >= fabsl (w))
288 b = (t * z / b);
289 else
290 b = (t * w / a);
291 }
292 }
293 if (sgn == 1)
294 ret = -b;
295 else
296 ret = b;
297 }
298 if (ret == 0)
299 {
300 ret = __copysignl (LDBL_MIN, ret) * LDBL_MIN;
301 __set_errno (ERANGE);
302 }
303 else
304 math_check_force_underflow (ret);
305 return ret;
306}
307strong_alias (__ieee754_jnl, __jnl_finite)
308
309_Float128
310__ieee754_ynl (int n, _Float128 x)
311{
312 u_int32_t se;
313 int32_t i, ix;
314 int32_t sign;
315 _Float128 a, b, temp, ret;
316 ieee854_long_double_shape_type u;
317
318 u.value = x;
319 se = u.parts32.w0;
320 ix = se & 0x7fffffff;
321
322 /* if Y(n,NaN) is NaN */
323 if (ix >= 0x7fff0000)
324 {
325 if ((u.parts32.w0 & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3)
326 return x + x;
327 }
328 if (x <= 0)
329 {
330 if (x == 0)
331 return ((n < 0 && (n & 1) != 0) ? 1 : -1) / L(0.0);
332 if (se & 0x80000000)
333 return zero / (zero * x);
334 }
335 sign = 1;
336 if (n < 0)
337 {
338 n = -n;
339 sign = 1 - ((n & 1) << 1);
340 }
341 if (n == 0)
342 return (__ieee754_y0l (x));
343 {
344 SET_RESTORE_ROUNDL (FE_TONEAREST);
345 if (n == 1)
346 {
347 ret = sign * __ieee754_y1l (x);
348 goto out;
349 }
350 if (ix >= 0x7fff0000)
351 return zero;
352 if (ix >= 0x412D0000)
353 { /* x > 2**302 */
354
355 /* ??? See comment above on the possible futility of this. */
356
357 /* (x >> n**2)
358 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
359 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
360 * Let s=sin(x), c=cos(x),
361 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
362 *
363 * n sin(xn)*sqt2 cos(xn)*sqt2
364 * ----------------------------------
365 * 0 s-c c+s
366 * 1 -s-c -c+s
367 * 2 -s+c -c-s
368 * 3 s+c c-s
369 */
370 _Float128 s;
371 _Float128 c;
372 __sincosl (x, &s, &c);
373 switch (n & 3)
374 {
375 case 0:
376 temp = s - c;
377 break;
378 case 1:
379 temp = -s - c;
380 break;
381 case 2:
382 temp = -s + c;
383 break;
384 case 3:
385 temp = s + c;
386 break;
387 }
388 b = invsqrtpi * temp / __ieee754_sqrtl (x);
389 }
390 else
391 {
392 a = __ieee754_y0l (x);
393 b = __ieee754_y1l (x);
394 /* quit if b is -inf */
395 u.value = b;
396 se = u.parts32.w0 & 0xffff0000;
397 for (i = 1; i < n && se != 0xffff0000; i++)
398 {
399 temp = b;
400 b = ((_Float128) (i + i) / x) * b - a;
401 u.value = b;
402 se = u.parts32.w0 & 0xffff0000;
403 a = temp;
404 }
405 }
406 /* If B is +-Inf, set up errno accordingly. */
407 if (! isfinite (b))
408 __set_errno (ERANGE);
409 if (sign > 0)
410 ret = b;
411 else
412 ret = -b;
413 }
414 out:
415 if (isinf (ret))
416 ret = __copysignl (LDBL_MAX, ret) * LDBL_MAX;
417 return ret;
418}
419strong_alias (__ieee754_ynl, __ynl_finite)
420