1 /*-
2 * SPDX-License-Identifier: BSD-3-Clause
3 *
4 * Copyright (c) 1992, 1993
5 * The Regents of the University of California. All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 * 3. Neither the name of the University nor the names of its contributors
16 * may be used to endorse or promote products derived from this software
17 * without specific prior written permission.
18 *
19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
29 * SUCH DAMAGE.
30 */
31
32 /*
33 * The original code, FreeBSD's old svn r93211, contain the following
34 * attribution:
35 *
36 * This code by P. McIlroy, Oct 1992;
37 *
38 * The financial support of UUNET Communications Services is greatfully
39 * acknowledged.
40 *
41 * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
42 */
43
44 /*
45 * See bsdsrc/t_tgamma.c for implementation details.
46 */
47
48 #include <float.h>
49
50 #if LDBL_MAX_EXP != 0x4000
51 #error "Unsupported long double format"
52 #endif
53
54 #ifdef __i386__
55 #include <ieeefp.h>
56 #endif
57
58 #include "fpmath.h"
59 #include "math.h"
60 #include "math_private.h"
61
62 /* Used in b_log.c and below. */
63 struct Double {
64 long double a;
65 long double b;
66 };
67
68 #include "b_logl.c"
69 #include "b_expl.c"
70
71 static const double zero = 0.;
72 static const volatile double tiny = 1e-300;
73 /*
74 * x >= 6
75 *
76 * Use the asymptotic approximation (Stirling's formula) adjusted for
77 * equal-ripples:
78 *
79 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
80 *
81 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
82 * premature round-off.
83 *
84 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
85 */
86
87 /*
88 * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
89 * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
90 * variables are clearly misnamed.
91 */
92 static const union IEEEl2bits
93 ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
94 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
95 #define ln2pi_hi (ln2pi_hiu.e)
96 #define ln2pi_lo (ln2pi_lou.e)
97
98 static const union IEEEl2bits
99 Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
100 Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
101 Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
102 Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
103 Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
104 Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
105 Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
106 Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
107 Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
108 Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
109 #define Pa0 (Pa0u.e)
110 #define Pa1 (Pa1u.e)
111 #define Pa2 (Pa2u.e)
112 #define Pa3 (Pa3u.e)
113 #define Pa4 (Pa4u.e)
114 #define Pa5 (Pa5u.e)
115 #define Pa6 (Pa6u.e)
116 #define Pa7 (Pa7u.e)
117 #define Pa8 (Pa8u.e)
118 #define Pa9 (Pa9u.e)
119
120 static struct Double
large_gam(long double x)121 large_gam(long double x)
122 {
123 long double p, z, thi, tlo, xhi, xlo;
124 long double logx;
125 struct Double u;
126
127 z = 1 / (x * x);
128 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
129 z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
130 p = p / x;
131
132 u = __log__D(x);
133 u.a -= 1;
134
135 /* Split (x - 0.5) in high and low parts. */
136 x -= 0.5L;
137 xhi = (float)x;
138 xlo = x - xhi;
139
140 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
141 thi = xhi * u.a;
142 tlo = xlo * u.a + x * u.b;
143
144 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
145 tlo += ln2pi_lo;
146 tlo += p;
147 u.a = ln2pi_hi + tlo;
148 u.a += thi;
149 u.b = thi - u.a;
150 u.b += ln2pi_hi;
151 u.b += tlo;
152 return (u);
153 }
154 /*
155 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
156 * [1.066.., 2.066..] accurate to 4.25e-19.
157 *
158 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
159 */
160 static const union IEEEl2bits
161 a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
162 a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
163 #define a0_hi (a0_hiu.e)
164 #define a0_lo (a0_lou.e)
165
166 static const union IEEEl2bits
167 P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
168 P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
169 P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
170 P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
171 P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
172 P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
173 P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
174 P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
175 P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
176 Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
177 Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
178 Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
179 Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
180 Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
181 Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
182 Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
183 Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
184 #define P0 (P0u.e)
185 #define P1 (P1u.e)
186 #define P2 (P2u.e)
187 #define P3 (P3u.e)
188 #define P4 (P4u.e)
189 #define P5 (P5u.e)
190 #define P6 (P6u.e)
191 #define P7 (P7u.e)
192 #define P8 (P8u.e)
193 #define Q1 (Q1u.e)
194 #define Q2 (Q2u.e)
195 #define Q3 (Q3u.e)
196 #define Q4 (Q4u.e)
197 #define Q5 (Q5u.e)
198 #define Q6 (Q6u.e)
199 #define Q7 (Q7u.e)
200 #define Q8 (Q8u.e)
201
202 static struct Double
ratfun_gam(long double z,long double c)203 ratfun_gam(long double z, long double c)
204 {
205 long double p, q, thi, tlo;
206 struct Double r;
207
208 q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
209 z * (Q6 + z * (Q7 + z * Q8)))))));
210 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
211 z * (P6 + z * (P7 + z * P8)))))));
212 p = p / q;
213
214 /* Split z into high and low parts. */
215 thi = (float)z;
216 tlo = (z - thi) + c;
217 tlo *= (thi + z);
218
219 /* Split (z+c)^2 into high and low parts. */
220 thi *= thi;
221 q = thi;
222 thi = (float)thi;
223 tlo += (q - thi);
224
225 /* Split p/q into high and low parts. */
226 r.a = (float)p;
227 r.b = p - r.a;
228
229 tlo = tlo * p + thi * r.b + a0_lo;
230 thi *= r.a; /* t = (z+c)^2*(P/Q) */
231 r.a = (float)(thi + a0_hi);
232 r.b = ((a0_hi - r.a) + thi) + tlo;
233 return (r); /* r = a0 + t */
234 }
235 /*
236 * x < 6
237 *
238 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
239 * 2.066124]. Use a rational approximation centered at the minimum
240 * (x0+1) to ensure monotonicity.
241 *
242 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
243 * It also has correct monotonicity.
244 */
245 static const union IEEEl2bits
246 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
247 #define x0 (xm1u.e)
248
249 static const double
250 left = -0.3955078125; /* left boundary for rat. approx */
251
252 static long double
small_gam(long double x)253 small_gam(long double x)
254 {
255 long double t, y, ym1;
256 struct Double yy, r;
257
258 y = x - 1;
259
260 if (y <= 1 + (left + x0)) {
261 yy = ratfun_gam(y - x0, 0);
262 return (yy.a + yy.b);
263 }
264
265 r.a = (float)y;
266 yy.a = r.a - 1;
267 y = y - 1 ;
268 r.b = yy.b = y - yy.a;
269
270 /* Argument reduction: G(x+1) = x*G(x) */
271 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
272 t = r.a * yy.a;
273 r.b = r.a * yy.b + y * r.b;
274 r.a = (float)t;
275 r.b += (t - r.a);
276 }
277
278 /* Return r*tgamma(y). */
279 yy = ratfun_gam(y - x0, 0);
280 y = r.b * (yy.a + yy.b) + r.a * yy.b;
281 y += yy.a * r.a;
282 return (y);
283 }
284 /*
285 * Good on (0, 1+x0+left]. Accurate to 1 ulp.
286 */
287 static long double
smaller_gam(long double x)288 smaller_gam(long double x)
289 {
290 long double d, rhi, rlo, t, xhi, xlo;
291 struct Double r;
292
293 if (x < x0 + left) {
294 t = (float)x;
295 d = (t + x) * (x - t);
296 t *= t;
297 xhi = (float)(t + x);
298 xlo = x - xhi;
299 xlo += t;
300 xlo += d;
301 t = 1 - x0;
302 t += x;
303 d = 1 - x0;
304 d -= t;
305 d += x;
306 x = xhi + xlo;
307 } else {
308 xhi = (float)x;
309 xlo = x - xhi;
310 t = x - x0;
311 d = - x0 - t;
312 d += x;
313 }
314
315 r = ratfun_gam(t, d);
316 d = (float)(r.a / x);
317 r.a -= d * xhi;
318 r.a -= d * xlo;
319 r.a += r.b;
320
321 return (d + r.a / x);
322 }
323 /*
324 * x < 0
325 *
326 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
327 * At negative integers, return NaN and raise invalid.
328 */
329 static const union IEEEl2bits
330 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
331 #define pi (piu.e)
332
333 static long double
neg_gam(long double x)334 neg_gam(long double x)
335 {
336 int sgn = 1;
337 struct Double lg, lsine;
338 long double y, z;
339
340 y = ceill(x);
341 if (y == x) /* Negative integer. */
342 return ((x - x) / zero);
343
344 z = y - x;
345 if (z > 0.5)
346 z = 1 - z;
347
348 y = y / 2;
349 if (y == ceill(y))
350 sgn = -1;
351
352 if (z < 0.25)
353 z = sinpil(z);
354 else
355 z = cospil(0.5 - z);
356
357 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
358 if (x < -1753) {
359
360 if (x < -1760)
361 return (sgn * tiny * tiny);
362 y = expl(lgammal(x) / 2);
363 y *= y;
364 return (sgn < 0 ? -y : y);
365 }
366
367
368 y = 1 - x;
369 if (1 - y == x)
370 y = tgammal(y);
371 else /* 1-x is inexact */
372 y = - x * tgammal(-x);
373
374 if (sgn < 0) y = -y;
375 return (pi / (y * z));
376 }
377 /*
378 * xmax comes from lgamma(xmax) - emax * log(2) = 0.
379 * static const float xmax = 35.040095f
380 * static const double xmax = 171.624376956302725;
381 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
382 * ld128: 1.75554834290446291700388921607020320e+03L,
383 *
384 * iota is a sloppy threshold to isolate x = 0.
385 */
386 static const double xmax = 1755.54834290446291689;
387 static const double iota = 0x1p-116;
388
389 long double
tgammal(long double x)390 tgammal(long double x)
391 {
392 struct Double u;
393
394 ENTERI();
395
396 if (x >= 6) {
397 if (x > xmax)
398 RETURNI(x / zero);
399 u = large_gam(x);
400 RETURNI(__exp__D(u.a, u.b));
401 }
402
403 if (x >= 1 + left + x0)
404 RETURNI(small_gam(x));
405
406 if (x > iota)
407 RETURNI(smaller_gam(x));
408
409 if (x > -iota) {
410 if (x != 0)
411 u.a = 1 - tiny; /* raise inexact */
412 RETURNI(1 / x);
413 }
414
415 if (!isfinite(x))
416 RETURNI(x - x); /* x is NaN or -Inf */
417
418 RETURNI(neg_gam(x));
419 }
420