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. */
60struct LDouble {
61	long double a;
62	long double b;
63};
64
65#include "b_logl.c"
66#include "b_expl.c"
67
68static const double zero = 0.;
69static 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 */
89static const union ieee_ext_u
90ln2pi_hiu = LD80C(0xd680000000000000,  -2,  4.18945312500000000000e-01L),
91ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
92#define	ln2pi_hi	(ln2pi_hiu.extu_ld)
93#define	ln2pi_lo	(ln2pi_lou.extu_ld)
94
95static 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
117static struct LDouble
118large_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 */
156static 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
162static const union ieee_ext_u
163P0u = LD80C(0xdb629fb9bbdc1c1d,    -2,  4.28486815855585429733e-01L),
164P1u = LD80C(0xe6f4f9f5641aa6be,    -3,  2.25543885805587730552e-01L),
165P2u = LD80C(0xead1bd99fdaf7cc1,    -6,  2.86644652514293482381e-02L),
166P3u = LD80C(0x9ccc8b25838ab1e0,    -8,  4.78512567772456362048e-03L),
167P4u = LD80C(0x8f0c4383ef9ce72a,    -9,  2.18273781132301146458e-03L),
168P5u = LD80C(0xe732ab2c0a2778da,   -13,  2.20487522485636008928e-04L),
169P6u = LD80C(0xce70b27ca822b297,   -16,  2.46095923774929264284e-05L),
170P7u = LD80C(0xa309e2e16fb63663,   -19,  2.42946473022376182921e-06L),
171P8u = LD80C(0xaf9c110efb2c633d,   -23,  1.63549217667765869987e-07L),
172Q1u = LD80C(0xd4d7422719f48f15,    -1,  8.31409582658993993626e-01L),
173Q2u = LD80C(0xe13138ea404f1268,    -5, -5.49785826915643198508e-02L),
174Q3u = LD80C(0xd1c6cc91989352c0,    -4, -1.02429960435139887683e-01L),
175Q4u = LD80C(0xa7e9435a84445579,    -7,  1.02484853505908820524e-02L),
176Q5u = LD80C(0x83c7c34db89b7bda,    -8,  4.02161632832052872697e-03L),
177Q6u = LD80C(0xbed06bf6e1c14e5b,   -11, -7.27898206351223022157e-04L),
178Q7u = LD80C(0xef05bf841d4504c0,   -18,  7.12342421869453515194e-06L),
179Q8u = 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
198static struct LDouble
199ratfun_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 */
241static const union ieee_ext_u
242  xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
243#define	x0	(xm1u.extu_ld)
244
245static const double
246    left = -0.3955078125;	/* left boundary for rat. approx */
247
248static long double
249small_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 */
283static long double
284smaller_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 */
325static const union ieee_ext_u
326piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
327#define	pi	(piu.extu_ld)
328
329static long double
330neg_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 */
381static const double xmax = 1755.54834290446291689;
382static const double iota = 0x1p-116;
383
384long double
385tgammal(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