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 #include <sys/cdefs.h>
45
46 /*
47 * See bsdsrc/t_tgamma.c for implementation details.
48 */
49
50 #include <float.h>
51
52 #if LDBL_MAX_EXP != 0x4000
53 #error "Unsupported long double format"
54 #endif
55
56 #include "math.h"
57 #include "math_private.h"
58
59 /* Used in b_log.c and below. */
60 struct LDouble {
61 long double a;
62 long double b;
63 };
64
65 #include "b_logl.c"
66 #include "b_expl.c"
67
68 static const double zero = 0.;
69 static const volatile double tiny = 1e-300;
70 /*
71 * x >= 6
72 *
73 * Use the asymptotic approximation (Stirling's formula) adjusted for
74 * equal-ripples:
75 *
76 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
77 *
78 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
79 * premature round-off.
80 *
81 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
82 */
83
84 /*
85 * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
86 * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
87 * variables are clearly misnamed.
88 */
89 static const union ieee_ext_u
90 ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
91 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
92 #define ln2pi_hi (ln2pi_hiu.extu_ld)
93 #define ln2pi_lo (ln2pi_lou.extu_ld)
94
95 static const union ieee_ext_u
96 Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
97 Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
98 Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
99 Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
100 Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
101 Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
102 Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
103 Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
104 Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
105 Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
106 #define Pa0 (Pa0u.extu_ld)
107 #define Pa1 (Pa1u.extu_ld)
108 #define Pa2 (Pa2u.extu_ld)
109 #define Pa3 (Pa3u.extu_ld)
110 #define Pa4 (Pa4u.extu_ld)
111 #define Pa5 (Pa5u.extu_ld)
112 #define Pa6 (Pa6u.extu_ld)
113 #define Pa7 (Pa7u.extu_ld)
114 #define Pa8 (Pa8u.extu_ld)
115 #define Pa9 (Pa9u.extu_ld)
116
117 static struct LDouble
large_gam(long double x)118 large_gam(long double x)
119 {
120 long double p, z, thi, tlo, xhi, xlo;
121 struct LDouble u;
122
123 z = 1 / (x * x);
124 p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
125 z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
126 p = p / x;
127
128 u = __log__LD(x);
129 u.a -= 1;
130
131 /* Split (x - 0.5) in high and low parts. */
132 x -= 0.5L;
133 xhi = (float)x;
134 xlo = x - xhi;
135
136 /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
137 thi = xhi * u.a;
138 tlo = xlo * u.a + x * u.b;
139
140 /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
141 tlo += ln2pi_lo;
142 tlo += p;
143 u.a = ln2pi_hi + tlo;
144 u.a += thi;
145 u.b = thi - u.a;
146 u.b += ln2pi_hi;
147 u.b += tlo;
148 return (u);
149 }
150 /*
151 * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
152 * [1.066.., 2.066..] accurate to 4.25e-19.
153 *
154 * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
155 */
156 static const union ieee_ext_u
157 a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
158 a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
159 #define a0_hi (a0_hiu.extu_ld)
160 #define a0_lo (a0_lou.extu_ld)
161
162 static const union ieee_ext_u
163 P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
164 P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
165 P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
166 P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
167 P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
168 P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
169 P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
170 P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
171 P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
172 Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
173 Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
174 Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
175 Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
176 Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
177 Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
178 Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
179 Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
180 #define P0 (P0u.extu_ld)
181 #define P1 (P1u.extu_ld)
182 #define P2 (P2u.extu_ld)
183 #define P3 (P3u.extu_ld)
184 #define P4 (P4u.extu_ld)
185 #define P5 (P5u.extu_ld)
186 #define P6 (P6u.extu_ld)
187 #define P7 (P7u.extu_ld)
188 #define P8 (P8u.extu_ld)
189 #define Q1 (Q1u.extu_ld)
190 #define Q2 (Q2u.extu_ld)
191 #define Q3 (Q3u.extu_ld)
192 #define Q4 (Q4u.extu_ld)
193 #define Q5 (Q5u.extu_ld)
194 #define Q6 (Q6u.extu_ld)
195 #define Q7 (Q7u.extu_ld)
196 #define Q8 (Q8u.extu_ld)
197
198 static struct LDouble
ratfun_gam(long double z,long double c)199 ratfun_gam(long double z, long double c)
200 {
201 long double p, q, thi, tlo;
202 struct LDouble r;
203
204 q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
205 z * (Q6 + z * (Q7 + z * Q8)))))));
206 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
207 z * (P6 + z * (P7 + z * P8)))))));
208 p = p / q;
209
210 /* Split z into high and low parts. */
211 thi = (float)z;
212 tlo = (z - thi) + c;
213 tlo *= (thi + z);
214
215 /* Split (z+c)^2 into high and low parts. */
216 thi *= thi;
217 q = thi;
218 thi = (float)thi;
219 tlo += (q - thi);
220
221 /* Split p/q into high and low parts. */
222 r.a = (float)p;
223 r.b = p - r.a;
224
225 tlo = tlo * p + thi * r.b + a0_lo;
226 thi *= r.a; /* t = (z+c)^2*(P/Q) */
227 r.a = (float)(thi + a0_hi);
228 r.b = ((a0_hi - r.a) + thi) + tlo;
229 return (r); /* r = a0 + t */
230 }
231 /*
232 * x < 6
233 *
234 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
235 * 2.066124]. Use a rational approximation centered at the minimum
236 * (x0+1) to ensure monotonicity.
237 *
238 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
239 * It also has correct monotonicity.
240 */
241 static const union ieee_ext_u
242 xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
243 #define x0 (xm1u.extu_ld)
244
245 static const double
246 left = -0.3955078125; /* left boundary for rat. approx */
247
248 static long double
small_gam(long double x)249 small_gam(long double x)
250 {
251 long double t, y, ym1;
252 struct LDouble yy, r;
253
254 y = x - 1;
255
256 if (y <= 1 + (left + x0)) {
257 yy = ratfun_gam(y - x0, 0);
258 return (yy.a + yy.b);
259 }
260
261 r.a = (float)y;
262 yy.a = r.a - 1;
263 y = y - 1 ;
264 r.b = yy.b = y - yy.a;
265
266 /* Argument reduction: G(x+1) = x*G(x) */
267 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
268 t = r.a * yy.a;
269 r.b = r.a * yy.b + y * r.b;
270 r.a = (float)t;
271 r.b += (t - r.a);
272 }
273
274 /* Return r*tgamma(y). */
275 yy = ratfun_gam(y - x0, 0);
276 y = r.b * (yy.a + yy.b) + r.a * yy.b;
277 y += yy.a * r.a;
278 return (y);
279 }
280 /*
281 * Good on (0, 1+x0+left]. Accurate to 1 ulp.
282 */
283 static long double
smaller_gam(long double x)284 smaller_gam(long double x)
285 {
286 long double d, t, xhi, xlo;
287 struct LDouble r;
288
289 if (x < x0 + left) {
290 t = (float)x;
291 d = (t + x) * (x - t);
292 t *= t;
293 xhi = (float)(t + x);
294 xlo = x - xhi;
295 xlo += t;
296 xlo += d;
297 t = 1 - x0;
298 t += x;
299 d = 1 - x0;
300 d -= t;
301 d += x;
302 x = xhi + xlo;
303 } else {
304 xhi = (float)x;
305 xlo = x - xhi;
306 t = x - x0;
307 d = - x0 - t;
308 d += x;
309 }
310
311 r = ratfun_gam(t, d);
312 d = (float)(r.a / x);
313 r.a -= d * xhi;
314 r.a -= d * xlo;
315 r.a += r.b;
316
317 return (d + r.a / x);
318 }
319 /*
320 * x < 0
321 *
322 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
323 * At negative integers, return NaN and raise invalid.
324 */
325 static const union ieee_ext_u
326 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
327 #define pi (piu.extu_ld)
328
329 static long double
neg_gam(long double x)330 neg_gam(long double x)
331 {
332 int sgn = 1;
333 long double y, z;
334
335 y = ceill(x);
336 if (y == x) /* Negative integer. */
337 return ((x - x) / zero);
338
339 z = y - x;
340 if (z > 0.5)
341 z = 1 - z;
342
343 y = y / 2;
344 if (y == ceill(y))
345 sgn = -1;
346
347 if (z < 0.25)
348 z = sinpil(z);
349 else
350 z = cospil(0.5 - z);
351
352 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
353 if (x < -1753) {
354
355 if (x < -1760)
356 return (sgn * tiny * tiny);
357 y = expl(lgammal(x) / 2);
358 y *= y;
359 return (sgn < 0 ? -y : y);
360 }
361
362
363 y = 1 - x;
364 if (1 - y == x)
365 y = tgammal(y);
366 else /* 1-x is inexact */
367 y = - x * tgammal(-x);
368
369 if (sgn < 0) y = -y;
370 return (pi / (y * z));
371 }
372 /*
373 * xmax comes from lgamma(xmax) - emax * log(2) = 0.
374 * static const float xmax = 35.040095f
375 * static const double xmax = 171.624376956302725;
376 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
377 * ld128: 1.75554834290446291700388921607020320e+03L,
378 *
379 * iota is a sloppy threshold to isolate x = 0.
380 */
381 static const double xmax = 1755.54834290446291689;
382 static const double iota = 0x1p-116;
383
384 long double
tgammal(long double x)385 tgammal(long double x)
386 {
387 struct LDouble u;
388
389 ENTERI();
390
391 if (x >= 6) {
392 if (x > xmax)
393 RETURNI(x / zero);
394 u = large_gam(x);
395 RETURNI(__exp__LD(u.a, u.b));
396 }
397
398 if (x >= 1 + left + x0)
399 RETURNI(small_gam(x));
400
401 if (x > iota)
402 RETURNI(smaller_gam(x));
403
404 if (x > -iota) {
405 if (x != 0)
406 u.a = 1 - tiny; /* raise inexact */
407 RETURNI(1 / x);
408 }
409
410 if (!isfinite(x))
411 RETURNI(x - x); /* x is NaN or -Inf */
412
413 RETURNI(neg_gam(x));
414 }
415