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
tgamma(x)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
large_gam(x)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
small_gam(x)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
smaller_gam(x)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
ratfun_gam(z,c)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
neg_gam(x)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