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, contained 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 * The algorithm remains, but the code has been re-arranged to facilitate
42 * porting to other precisions.
43 */
44
45 #include <float.h>
46
47 #include "math.h"
48 #include "math_private.h"
49
50 /* Used in b_log.c and below. */
51 struct Double {
52 double a;
53 double b;
54 };
55
56 #include "b_log.c"
57 #include "b_exp.c"
58
59 /*
60 * The range is broken into several subranges. Each is handled by its
61 * helper functions.
62 *
63 * x >= 6.0: large_gam(x)
64 * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0.
65 * xleft > x > iota: smaller_gam(x) where iota = 1e-17.
66 * iota > x > -itoa: Handle x near 0.
67 * -iota > x : neg_gam
68 *
69 * Special values:
70 * -Inf: return NaN and raise invalid;
71 * negative integer: return NaN and raise invalid;
72 * other x ~< 177.79: return +-0 and raise underflow;
73 * +-0: return +-Inf and raise divide-by-zero;
74 * finite x ~> 171.63: return +Inf and raise overflow;
75 * +Inf: return +Inf;
76 * NaN: return NaN.
77 *
78 * Accuracy: tgamma(x) is accurate to within
79 * x > 0: error provably < 0.9ulp.
80 * Maximum observed in 1,000,000 trials was .87ulp.
81 * x < 0:
82 * Maximum observed error < 4ulp in 1,000,000 trials.
83 */
84
85 /*
86 * Constants for large x approximation (x in [6, Inf])
87 * (Accurate to 2.8*10^-19 absolute)
88 */
89
90 static const double zero = 0.;
91 static const volatile double tiny = 1e-300;
92 /*
93 * x >= 6
94 *
95 * Use the asymptotic approximation (Stirling's formula) adjusted fof
96 * equal-ripples:
97 *
98 * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
99 *
100 * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
101 * premature round-off.
102 *
103 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
104 */
105 static const double
106 ln2pi_hi = 0.41894531250000000,
107 ln2pi_lo = -6.7792953272582197e-6,
108 Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
109 Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */
110 Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */
111 Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */
112 Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */
113 Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */
114 Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */
115 Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */
116
117 static struct Double
large_gam(double x)118 large_gam(double x)
119 {
120 double p, z, thi, tlo, xhi, xlo;
121 struct Double 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))))));
126 p = p / x;
127
128 u = __log__D(x);
129 u.a -= 1;
130
131 /* Split (x - 0.5) in high and low parts. */
132 x -= 0.5;
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 double
157 #if 0
158 a0_hi = 8.8560319441088875e-1,
159 a0_lo = -4.9964270364690197e-17,
160 #else
161 a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */
162 a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */
163 #endif
164 P0 = 6.2138957182182086e-1,
165 P1 = 2.6575719865153347e-1,
166 P2 = 5.5385944642991746e-3,
167 P3 = 1.3845669830409657e-3,
168 P4 = 2.4065995003271137e-3,
169 Q0 = 1.4501953125000000e+0,
170 Q1 = 1.0625852194801617e+0,
171 Q2 = -2.0747456194385994e-1,
172 Q3 = -1.4673413178200542e-1,
173 Q4 = 3.0787817615617552e-2,
174 Q5 = 5.1244934798066622e-3,
175 Q6 = -1.7601274143166700e-3,
176 Q7 = 9.3502102357378894e-5,
177 Q8 = 6.1327550747244396e-6;
178
179 static struct Double
ratfun_gam(double z,double c)180 ratfun_gam(double z, double c)
181 {
182 double p, q, thi, tlo;
183 struct Double r;
184
185 q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
186 z * (Q6 + z * (Q7 + z * Q8)))))));
187 p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4)));
188 p = p / q;
189
190 /* Split z into high and low parts. */
191 thi = (float)z;
192 tlo = (z - thi) + c;
193 tlo *= (thi + z);
194
195 /* Split (z+c)^2 into high and low parts. */
196 thi *= thi;
197 q = thi;
198 thi = (float)thi;
199 tlo += (q - thi);
200
201 /* Split p/q into high and low parts. */
202 r.a = (float)p;
203 r.b = p - r.a;
204
205 tlo = tlo * p + thi * r.b + a0_lo;
206 thi *= r.a; /* t = (z+c)^2*(P/Q) */
207 r.a = (float)(thi + a0_hi);
208 r.b = ((a0_hi - r.a) + thi) + tlo;
209 return (r); /* r = a0 + t */
210 }
211 /*
212 * x < 6
213 *
214 * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
215 * 2.066124]. Use a rational approximation centered at the minimum
216 * (x0+1) to ensure monotonicity.
217 *
218 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
219 * It also has correct monotonicity.
220 */
221 static const double
222 left = -0.3955078125, /* left boundary for rat. approx */
223 x0 = 4.6163214496836236e-1; /* xmin - 1 */
224
225 static double
small_gam(double x)226 small_gam(double x)
227 {
228 double t, y, ym1;
229 struct Double yy, r;
230
231 y = x - 1;
232 if (y <= 1 + (left + x0)) {
233 yy = ratfun_gam(y - x0, 0);
234 return (yy.a + yy.b);
235 }
236
237 r.a = (float)y;
238 yy.a = r.a - 1;
239 y = y - 1 ;
240 r.b = yy.b = y - yy.a;
241
242 /* Argument reduction: G(x+1) = x*G(x) */
243 for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
244 t = r.a * yy.a;
245 r.b = r.a * yy.b + y * r.b;
246 r.a = (float)t;
247 r.b += (t - r.a);
248 }
249
250 /* Return r*tgamma(y). */
251 yy = ratfun_gam(y - x0, 0);
252 y = r.b * (yy.a + yy.b) + r.a * yy.b;
253 y += yy.a * r.a;
254 return (y);
255 }
256 /*
257 * Good on (0, 1+x0+left]. Accurate to 1 ulp.
258 */
259 static double
smaller_gam(double x)260 smaller_gam(double x)
261 {
262 double d, rhi, rlo, t, xhi, xlo;
263 struct Double r;
264
265 if (x < x0 + left) {
266 t = (float)x;
267 d = (t + x) * (x - t);
268 t *= t;
269 xhi = (float)(t + x);
270 xlo = x - xhi;
271 xlo += t;
272 xlo += d;
273 t = 1 - x0;
274 t += x;
275 d = 1 - x0;
276 d -= t;
277 d += x;
278 x = xhi + xlo;
279 } else {
280 xhi = (float)x;
281 xlo = x - xhi;
282 t = x - x0;
283 d = - x0 - t;
284 d += x;
285 }
286
287 r = ratfun_gam(t, d);
288 d = (float)(r.a / x);
289 r.a -= d * xhi;
290 r.a -= d * xlo;
291 r.a += r.b;
292
293 return (d + r.a / x);
294 }
295 /*
296 * x < 0
297 *
298 * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
299 * At negative integers, return NaN and raise invalid.
300 */
301 static double
neg_gam(double x)302 neg_gam(double x)
303 {
304 int sgn = 1;
305 struct Double lg, lsine;
306 double y, z;
307
308 y = ceil(x);
309 if (y == x) /* Negative integer. */
310 return ((x - x) / zero);
311
312 z = y - x;
313 if (z > 0.5)
314 z = 1 - z;
315
316 y = y / 2;
317 if (y == ceil(y))
318 sgn = -1;
319
320 if (z < 0.25)
321 z = sinpi(z);
322 else
323 z = cospi(0.5 - z);
324
325 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
326 if (x < -170) {
327
328 if (x < -190)
329 return (sgn * tiny * tiny);
330
331 y = 1 - x; /* exact: 128 < |x| < 255 */
332 lg = large_gam(y);
333 lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */
334 lg.a -= lsine.a; /* exact (opposite signs) */
335 lg.b -= lsine.b;
336 y = -(lg.a + lg.b);
337 z = (y + lg.a) + lg.b;
338 y = __exp__D(y, z);
339 if (sgn < 0) y = -y;
340 return (y);
341 }
342
343 y = 1 - x;
344 if (1 - y == x)
345 y = tgamma(y);
346 else /* 1-x is inexact */
347 y = - x * tgamma(-x);
348
349 if (sgn < 0) y = -y;
350 return (M_PI / (y * z));
351 }
352 /*
353 * xmax comes from lgamma(xmax) - emax * log(2) = 0.
354 * static const float xmax = 35.040095f
355 * static const double xmax = 171.624376956302725;
356 * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
357 * ld128: 1.75554834290446291700388921607020320e+03L,
358 *
359 * iota is a sloppy threshold to isolate x = 0.
360 */
361 static const double xmax = 171.624376956302725;
362 static const double iota = 0x1p-56;
363
364 double
tgamma(double x)365 tgamma(double x)
366 {
367 struct Double u;
368
369 if (x >= 6) {
370 if (x > xmax)
371 return (x / zero);
372 u = large_gam(x);
373 return (__exp__D(u.a, u.b));
374 }
375
376 if (x >= 1 + left + x0)
377 return (small_gam(x));
378
379 if (x > iota)
380 return (smaller_gam(x));
381
382 if (x > -iota) {
383 if (x != 0.)
384 u.a = 1 - tiny; /* raise inexact */
385 return (1 / x);
386 }
387
388 if (!isfinite(x))
389 return (x - x); /* x is NaN or -Inf */
390
391 return (neg_gam(x));
392 }
393
394 #if (LDBL_MANT_DIG == 53)
395 __weak_reference(tgamma, tgammal);
396 #endif
397