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. */
51struct 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
90static const double zero = 0.;
91static 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 */
105static 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
117static struct Double
118large_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 */
156static 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
179static struct Double
180ratfun_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 */
221static const double
222    left = -0.3955078125,	/* left boundary for rat. approx */
223    x0 = 4.6163214496836236e-1;	/* xmin - 1 */
224
225static double
226small_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 */
259static double
260smaller_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 */
301static double
302neg_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 */
361static const double xmax = 171.624376956302725;
362static const double iota = 0x1p-56;
363
364double
365tgamma(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