xref: /freebsd/lib/msun/bsdsrc/b_tgamma.c (revision c66ec88fed842fbaad62c30d510644ceb7bd2d71)
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 /* @(#)gamma.c	8.1 (Berkeley) 6/4/93 */
33 #include <sys/cdefs.h>
34 __FBSDID("$FreeBSD$");
35 
36 /*
37  * This code by P. McIlroy, Oct 1992;
38  *
39  * The financial support of UUNET Communications Services is greatfully
40  * acknowledged.
41  */
42 
43 #include <math.h>
44 #include "mathimpl.h"
45 
46 /* METHOD:
47  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
48  * 	At negative integers, return NaN and raise invalid.
49  *
50  * x < 6.5:
51  *	Use argument reduction G(x+1) = xG(x) to reach the
52  *	range [1.066124,2.066124].  Use a rational
53  *	approximation centered at the minimum (x0+1) to
54  *	ensure monotonicity.
55  *
56  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
57  *	adjusted for equal-ripples:
58  *
59  *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
60  *
61  *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
62  *	avoid premature round-off.
63  *
64  * Special values:
65  *	-Inf:			return NaN and raise invalid;
66  *	negative integer:	return NaN and raise invalid;
67  *	other x ~< 177.79:	return +-0 and raise underflow;
68  *	+-0:			return +-Inf and raise divide-by-zero;
69  *	finite x ~> 171.63:	return +Inf and raise overflow;
70  *	+Inf:			return +Inf;
71  *	NaN: 			return NaN.
72  *
73  * Accuracy: tgamma(x) is accurate to within
74  *	x > 0:  error provably < 0.9ulp.
75  *	Maximum observed in 1,000,000 trials was .87ulp.
76  *	x < 0:
77  *	Maximum observed error < 4ulp in 1,000,000 trials.
78  */
79 
80 static double neg_gam(double);
81 static double small_gam(double);
82 static double smaller_gam(double);
83 static struct Double large_gam(double);
84 static struct Double ratfun_gam(double, double);
85 
86 /*
87  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
88  * [1.066.., 2.066..] accurate to 4.25e-19.
89  */
90 #define LEFT -.3955078125	/* left boundary for rat. approx */
91 #define x0 .461632144968362356785	/* xmin - 1 */
92 
93 #define a0_hi 0.88560319441088874992
94 #define a0_lo -.00000000000000004996427036469019695
95 #define P0	 6.21389571821820863029017800727e-01
96 #define P1	 2.65757198651533466104979197553e-01
97 #define P2	 5.53859446429917461063308081748e-03
98 #define P3	 1.38456698304096573887145282811e-03
99 #define P4	 2.40659950032711365819348969808e-03
100 #define Q0	 1.45019531250000000000000000000e+00
101 #define Q1	 1.06258521948016171343454061571e+00
102 #define Q2	-2.07474561943859936441469926649e-01
103 #define Q3	-1.46734131782005422506287573015e-01
104 #define Q4	 3.07878176156175520361557573779e-02
105 #define Q5	 5.12449347980666221336054633184e-03
106 #define Q6	-1.76012741431666995019222898833e-03
107 #define Q7	 9.35021023573788935372153030556e-05
108 #define Q8	 6.13275507472443958924745652239e-06
109 /*
110  * Constants for large x approximation (x in [6, Inf])
111  * (Accurate to 2.8*10^-19 absolute)
112  */
113 #define lns2pi_hi 0.418945312500000
114 #define lns2pi_lo -.000006779295327258219670263595
115 #define Pa0	 8.33333333333333148296162562474e-02
116 #define Pa1	-2.77777777774548123579378966497e-03
117 #define Pa2	 7.93650778754435631476282786423e-04
118 #define Pa3	-5.95235082566672847950717262222e-04
119 #define Pa4	 8.41428560346653702135821806252e-04
120 #define Pa5	-1.89773526463879200348872089421e-03
121 #define Pa6	 5.69394463439411649408050664078e-03
122 #define Pa7	-1.44705562421428915453880392761e-02
123 
124 static const double zero = 0., one = 1.0, tiny = 1e-300;
125 
126 double
127 tgamma(x)
128 	double x;
129 {
130 	struct Double u;
131 
132 	if (x >= 6) {
133 		if(x > 171.63)
134 			return (x / zero);
135 		u = large_gam(x);
136 		return(__exp__D(u.a, u.b));
137 	} else if (x >= 1.0 + LEFT + x0)
138 		return (small_gam(x));
139 	else if (x > 1.e-17)
140 		return (smaller_gam(x));
141 	else if (x > -1.e-17) {
142 		if (x != 0.0)
143 			u.a = one - tiny;	/* raise inexact */
144 		return (one/x);
145 	} else if (!finite(x))
146 		return (x - x);		/* x is NaN or -Inf */
147 	else
148 		return (neg_gam(x));
149 }
150 /*
151  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
152  */
153 static struct Double
154 large_gam(x)
155 	double x;
156 {
157 	double z, p;
158 	struct Double t, u, v;
159 
160 	z = one/(x*x);
161 	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
162 	p = p/x;
163 
164 	u = __log__D(x);
165 	u.a -= one;
166 	v.a = (x -= .5);
167 	TRUNC(v.a);
168 	v.b = x - v.a;
169 	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
170 	t.b = v.b*u.a + x*u.b;
171 	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
172 	t.b += lns2pi_lo; t.b += p;
173 	u.a = lns2pi_hi + t.b; u.a += t.a;
174 	u.b = t.a - u.a;
175 	u.b += lns2pi_hi; u.b += t.b;
176 	return (u);
177 }
178 /*
179  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
180  * It also has correct monotonicity.
181  */
182 static double
183 small_gam(x)
184 	double x;
185 {
186 	double y, ym1, t;
187 	struct Double yy, r;
188 	y = x - one;
189 	ym1 = y - one;
190 	if (y <= 1.0 + (LEFT + x0)) {
191 		yy = ratfun_gam(y - x0, 0);
192 		return (yy.a + yy.b);
193 	}
194 	r.a = y;
195 	TRUNC(r.a);
196 	yy.a = r.a - one;
197 	y = ym1;
198 	yy.b = r.b = y - yy.a;
199 	/* Argument reduction: G(x+1) = x*G(x) */
200 	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
201 		t = r.a*yy.a;
202 		r.b = r.a*yy.b + y*r.b;
203 		r.a = t;
204 		TRUNC(r.a);
205 		r.b += (t - r.a);
206 	}
207 	/* Return r*tgamma(y). */
208 	yy = ratfun_gam(y - x0, 0);
209 	y = r.b*(yy.a + yy.b) + r.a*yy.b;
210 	y += yy.a*r.a;
211 	return (y);
212 }
213 /*
214  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
215  */
216 static double
217 smaller_gam(x)
218 	double x;
219 {
220 	double t, d;
221 	struct Double r, xx;
222 	if (x < x0 + LEFT) {
223 		t = x, TRUNC(t);
224 		d = (t+x)*(x-t);
225 		t *= t;
226 		xx.a = (t + x), TRUNC(xx.a);
227 		xx.b = x - xx.a; xx.b += t; xx.b += d;
228 		t = (one-x0); t += x;
229 		d = (one-x0); d -= t; d += x;
230 		x = xx.a + xx.b;
231 	} else {
232 		xx.a =  x, TRUNC(xx.a);
233 		xx.b = x - xx.a;
234 		t = x - x0;
235 		d = (-x0 -t); d += x;
236 	}
237 	r = ratfun_gam(t, d);
238 	d = r.a/x, TRUNC(d);
239 	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
240 	return (d + r.a/x);
241 }
242 /*
243  * returns (z+c)^2 * P(z)/Q(z) + a0
244  */
245 static struct Double
246 ratfun_gam(z, c)
247 	double z, c;
248 {
249 	double p, q;
250 	struct Double r, t;
251 
252 	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
253 	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
254 
255 	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
256 	p = p/q;
257 	t.a = z, TRUNC(t.a);		/* t ~= z + c */
258 	t.b = (z - t.a) + c;
259 	t.b *= (t.a + z);
260 	q = (t.a *= t.a);		/* t = (z+c)^2 */
261 	TRUNC(t.a);
262 	t.b += (q - t.a);
263 	r.a = p, TRUNC(r.a);		/* r = P/Q */
264 	r.b = p - r.a;
265 	t.b = t.b*p + t.a*r.b + a0_lo;
266 	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
267 	r.a = t.a + a0_hi, TRUNC(r.a);
268 	r.b = ((a0_hi-r.a) + t.a) + t.b;
269 	return (r);			/* r = a0 + t */
270 }
271 
272 static double
273 neg_gam(x)
274 	double x;
275 {
276 	int sgn = 1;
277 	struct Double lg, lsine;
278 	double y, z;
279 
280 	y = ceil(x);
281 	if (y == x)		/* Negative integer. */
282 		return ((x - x) / zero);
283 	z = y - x;
284 	if (z > 0.5)
285 		z = one - z;
286 	y = 0.5 * y;
287 	if (y == ceil(y))
288 		sgn = -1;
289 	if (z < .25)
290 		z = sin(M_PI*z);
291 	else
292 		z = cos(M_PI*(0.5-z));
293 	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
294 	if (x < -170) {
295 		if (x < -190)
296 			return ((double)sgn*tiny*tiny);
297 		y = one - x;		/* exact: 128 < |x| < 255 */
298 		lg = large_gam(y);
299 		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
300 		lg.a -= lsine.a;		/* exact (opposite signs) */
301 		lg.b -= lsine.b;
302 		y = -(lg.a + lg.b);
303 		z = (y + lg.a) + lg.b;
304 		y = __exp__D(y, z);
305 		if (sgn < 0) y = -y;
306 		return (y);
307 	}
308 	y = one-x;
309 	if (one-y == x)
310 		y = tgamma(y);
311 	else		/* 1-x is inexact */
312 		y = -x*tgamma(-x);
313 	if (sgn < 0) y = -y;
314 	return (M_PI / (y*z));
315 }
316