1 /*-
2  * Copyright (c) 1992, 1993
3  *	The Regents of the University of California.  All rights reserved.
4  *
5  * Redistribution and use in source and binary forms, with or without
6  * modification, are permitted provided that the following conditions
7  * are met:
8  * 1. Redistributions of source code must retain the above copyright
9  *    notice, this list of conditions and the following disclaimer.
10  * 2. Redistributions in binary form must reproduce the above copyright
11  *    notice, this list of conditions and the following disclaimer in the
12  *    documentation and/or other materials provided with the distribution.
13  * 3. All advertising materials mentioning features or use of this software
14  *    must display the following acknowledgement:
15  *	This product includes software developed by the University of
16  *	California, Berkeley and its contributors.
17  * 4. Neither the name of the University nor the names of its contributors
18  *    may be used to endorse or promote products derived from this software
19  *    without specific prior written permission.
20  *
21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31  * SUCH DAMAGE.
32  */
33 
34 /* @(#)gamma.c	8.1 (Berkeley) 6/4/93 */
35 #include <sys/cdefs.h>
36 __FBSDID("$FreeBSD$");
37 
38 /*
39  * This code by P. McIlroy, Oct 1992;
40  *
41  * The financial support of UUNET Communications Services is greatfully
42  * acknowledged.
43  */
44 
45 #include <math.h>
46 #include "mathimpl.h"
47 
48 /* METHOD:
49  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
50  * 	At negative integers, return NaN and raise invalid.
51  *
52  * x < 6.5:
53  *	Use argument reduction G(x+1) = xG(x) to reach the
54  *	range [1.066124,2.066124].  Use a rational
55  *	approximation centered at the minimum (x0+1) to
56  *	ensure monotonicity.
57  *
58  * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
59  *	adjusted for equal-ripples:
60  *
61  *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
62  *
63  *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
64  *	avoid premature round-off.
65  *
66  * Special values:
67  *	-Inf:			return NaN and raise invalid;
68  *	negative integer:	return NaN and raise invalid;
69  *	other x ~< 177.79:	return +-0 and raise underflow;
70  *	+-0:			return +-Inf and raise divide-by-zero;
71  *	finite x ~> 171.63:	return +Inf and raise overflow;
72  *	+Inf:			return +Inf;
73  *	NaN: 			return NaN.
74  *
75  * Accuracy: tgamma(x) is accurate to within
76  *	x > 0:  error provably < 0.9ulp.
77  *	Maximum observed in 1,000,000 trials was .87ulp.
78  *	x < 0:
79  *	Maximum observed error < 4ulp in 1,000,000 trials.
80  */
81 
82 static double neg_gam(double);
83 static double small_gam(double);
84 static double smaller_gam(double);
85 static struct Double large_gam(double);
86 static struct Double ratfun_gam(double, double);
87 
88 /*
89  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
90  * [1.066.., 2.066..] accurate to 4.25e-19.
91  */
92 #define LEFT -.3955078125	/* left boundary for rat. approx */
93 #define x0 .461632144968362356785	/* xmin - 1 */
94 
95 #define a0_hi 0.88560319441088874992
96 #define a0_lo -.00000000000000004996427036469019695
97 #define P0	 6.21389571821820863029017800727e-01
98 #define P1	 2.65757198651533466104979197553e-01
99 #define P2	 5.53859446429917461063308081748e-03
100 #define P3	 1.38456698304096573887145282811e-03
101 #define P4	 2.40659950032711365819348969808e-03
102 #define Q0	 1.45019531250000000000000000000e+00
103 #define Q1	 1.06258521948016171343454061571e+00
104 #define Q2	-2.07474561943859936441469926649e-01
105 #define Q3	-1.46734131782005422506287573015e-01
106 #define Q4	 3.07878176156175520361557573779e-02
107 #define Q5	 5.12449347980666221336054633184e-03
108 #define Q6	-1.76012741431666995019222898833e-03
109 #define Q7	 9.35021023573788935372153030556e-05
110 #define Q8	 6.13275507472443958924745652239e-06
111 /*
112  * Constants for large x approximation (x in [6, Inf])
113  * (Accurate to 2.8*10^-19 absolute)
114  */
115 #define lns2pi_hi 0.418945312500000
116 #define lns2pi_lo -.000006779295327258219670263595
117 #define Pa0	 8.33333333333333148296162562474e-02
118 #define Pa1	-2.77777777774548123579378966497e-03
119 #define Pa2	 7.93650778754435631476282786423e-04
120 #define Pa3	-5.95235082566672847950717262222e-04
121 #define Pa4	 8.41428560346653702135821806252e-04
122 #define Pa5	-1.89773526463879200348872089421e-03
123 #define Pa6	 5.69394463439411649408050664078e-03
124 #define Pa7	-1.44705562421428915453880392761e-02
125 
126 static const double zero = 0., one = 1.0, tiny = 1e-300;
127 
128 double
tgamma(x)129 tgamma(x)
130 	double x;
131 {
132 	struct Double u;
133 
134 	if (x >= 6) {
135 		if(x > 171.63)
136 			return (x / zero);
137 		u = large_gam(x);
138 		return(__exp__D(u.a, u.b));
139 	} else if (x >= 1.0 + LEFT + x0)
140 		return (small_gam(x));
141 	else if (x > 1.e-17)
142 		return (smaller_gam(x));
143 	else if (x > -1.e-17) {
144 		if (x != 0.0)
145 			u.a = one - tiny;	/* raise inexact */
146 		return (one/x);
147 	} else if (!finite(x))
148 		return (x - x);		/* x is NaN or -Inf */
149 	else
150 		return (neg_gam(x));
151 }
152 /*
153  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
154  */
155 static struct Double
large_gam(x)156 large_gam(x)
157 	double x;
158 {
159 	double z, p;
160 	struct Double t, u, v;
161 
162 	z = one/(x*x);
163 	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
164 	p = p/x;
165 
166 	u = __log__D(x);
167 	u.a -= one;
168 	v.a = (x -= .5);
169 	TRUNC(v.a);
170 	v.b = x - v.a;
171 	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
172 	t.b = v.b*u.a + x*u.b;
173 	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
174 	t.b += lns2pi_lo; t.b += p;
175 	u.a = lns2pi_hi + t.b; u.a += t.a;
176 	u.b = t.a - u.a;
177 	u.b += lns2pi_hi; u.b += t.b;
178 	return (u);
179 }
180 /*
181  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
182  * It also has correct monotonicity.
183  */
184 static double
small_gam(x)185 small_gam(x)
186 	double x;
187 {
188 	double y, ym1, t;
189 	struct Double yy, r;
190 	y = x - one;
191 	ym1 = y - one;
192 	if (y <= 1.0 + (LEFT + x0)) {
193 		yy = ratfun_gam(y - x0, 0);
194 		return (yy.a + yy.b);
195 	}
196 	r.a = y;
197 	TRUNC(r.a);
198 	yy.a = r.a - one;
199 	y = ym1;
200 	yy.b = r.b = y - yy.a;
201 	/* Argument reduction: G(x+1) = x*G(x) */
202 	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
203 		t = r.a*yy.a;
204 		r.b = r.a*yy.b + y*r.b;
205 		r.a = t;
206 		TRUNC(r.a);
207 		r.b += (t - r.a);
208 	}
209 	/* Return r*tgamma(y). */
210 	yy = ratfun_gam(y - x0, 0);
211 	y = r.b*(yy.a + yy.b) + r.a*yy.b;
212 	y += yy.a*r.a;
213 	return (y);
214 }
215 /*
216  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
217  */
218 static double
smaller_gam(x)219 smaller_gam(x)
220 	double x;
221 {
222 	double t, d;
223 	struct Double r, xx;
224 	if (x < x0 + LEFT) {
225 		t = x, TRUNC(t);
226 		d = (t+x)*(x-t);
227 		t *= t;
228 		xx.a = (t + x), TRUNC(xx.a);
229 		xx.b = x - xx.a; xx.b += t; xx.b += d;
230 		t = (one-x0); t += x;
231 		d = (one-x0); d -= t; d += x;
232 		x = xx.a + xx.b;
233 	} else {
234 		xx.a =  x, TRUNC(xx.a);
235 		xx.b = x - xx.a;
236 		t = x - x0;
237 		d = (-x0 -t); d += x;
238 	}
239 	r = ratfun_gam(t, d);
240 	d = r.a/x, TRUNC(d);
241 	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
242 	return (d + r.a/x);
243 }
244 /*
245  * returns (z+c)^2 * P(z)/Q(z) + a0
246  */
247 static struct Double
ratfun_gam(z,c)248 ratfun_gam(z, c)
249 	double z, c;
250 {
251 	double p, q;
252 	struct Double r, t;
253 
254 	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
255 	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
256 
257 	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
258 	p = p/q;
259 	t.a = z, TRUNC(t.a);		/* t ~= z + c */
260 	t.b = (z - t.a) + c;
261 	t.b *= (t.a + z);
262 	q = (t.a *= t.a);		/* t = (z+c)^2 */
263 	TRUNC(t.a);
264 	t.b += (q - t.a);
265 	r.a = p, TRUNC(r.a);		/* r = P/Q */
266 	r.b = p - r.a;
267 	t.b = t.b*p + t.a*r.b + a0_lo;
268 	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
269 	r.a = t.a + a0_hi, TRUNC(r.a);
270 	r.b = ((a0_hi-r.a) + t.a) + t.b;
271 	return (r);			/* r = a0 + t */
272 }
273 
274 static double
neg_gam(x)275 neg_gam(x)
276 	double x;
277 {
278 	int sgn = 1;
279 	struct Double lg, lsine;
280 	double y, z;
281 
282 	y = ceil(x);
283 	if (y == x)		/* Negative integer. */
284 		return ((x - x) / zero);
285 	z = y - x;
286 	if (z > 0.5)
287 		z = one - z;
288 	y = 0.5 * y;
289 	if (y == ceil(y))
290 		sgn = -1;
291 	if (z < .25)
292 		z = sin(M_PI*z);
293 	else
294 		z = cos(M_PI*(0.5-z));
295 	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
296 	if (x < -170) {
297 		if (x < -190)
298 			return ((double)sgn*tiny*tiny);
299 		y = one - x;		/* exact: 128 < |x| < 255 */
300 		lg = large_gam(y);
301 		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
302 		lg.a -= lsine.a;		/* exact (opposite signs) */
303 		lg.b -= lsine.b;
304 		y = -(lg.a + lg.b);
305 		z = (y + lg.a) + lg.b;
306 		y = __exp__D(y, z);
307 		if (sgn < 0) y = -y;
308 		return (y);
309 	}
310 	y = one-x;
311 	if (one-y == x)
312 		y = tgamma(y);
313 	else		/* 1-x is inexact */
314 		y = -x*tgamma(-x);
315 	if (sgn < 0) y = -y;
316 	return (M_PI / (y*z));
317 }
318