1 /* $NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $ */
2
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. All advertising materials mentioning features or use of this software
16 * must display the following acknowledgement:
17 * This product includes software developed by the University of
18 * California, Berkeley and its contributors.
19 * 4. Neither the name of the University nor the names of its contributors
20 * may be used to endorse or promote products derived from this software
21 * without specific prior written permission.
22 *
23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
26 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * SUCH DAMAGE.
34 */
35
36 /* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
37 #include <sys/cdefs.h>
38 #if 0
39 __FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_tgamma.c 176449 2008-02-22 02:26:51Z das $");
40 #else
41 __RCSID("$NetBSD: b_tgamma.c,v 1.1 2012/05/05 17:54:14 christos Exp $");
42 #endif
43
44 /*
45 * This code by P. McIlroy, Oct 1992;
46 *
47 * The financial support of UUNET Communications Services is greatfully
48 * acknowledged.
49 */
50
51 #include "math.h"
52 #include "math_private.h"
53
54 /* METHOD:
55 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
56 * At negative integers, return NaN and raise invalid.
57 *
58 * x < 6.5:
59 * Use argument reduction G(x+1) = xG(x) to reach the
60 * range [1.066124,2.066124]. Use a rational
61 * approximation centered at the minimum (x0+1) to
62 * ensure monotonicity.
63 *
64 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
65 * adjusted for equal-ripples:
66 *
67 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
68 *
69 * Keep extra precision in multiplying (x-.5)(log(x)-1), to
70 * avoid premature round-off.
71 *
72 * Special values:
73 * -Inf: return NaN and raise invalid;
74 * negative integer: return NaN and raise invalid;
75 * other x ~< 177.79: return +-0 and raise underflow;
76 * +-0: return +-Inf and raise divide-by-zero;
77 * finite x ~> 171.63: return +Inf and raise overflow;
78 * +Inf: return +Inf;
79 * NaN: return NaN.
80 *
81 * Accuracy: tgamma(x) is accurate to within
82 * x > 0: error provably < 0.9ulp.
83 * Maximum observed in 1,000,000 trials was .87ulp.
84 * x < 0:
85 * Maximum observed error < 4ulp in 1,000,000 trials.
86 */
87
88 static double neg_gam(double);
89 static double small_gam(double);
90 static double smaller_gam(double);
91 static struct Double large_gam(double);
92 static struct Double ratfun_gam(double, double);
93
94 /*
95 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
96 * [1.066.., 2.066..] accurate to 4.25e-19.
97 */
98 #define LEFT -.3955078125 /* left boundary for rat. approx */
99 #define x0 .461632144968362356785 /* xmin - 1 */
100
101 #define a0_hi 0.88560319441088874992
102 #define a0_lo -.00000000000000004996427036469019695
103 #define P0 6.21389571821820863029017800727e-01
104 #define P1 2.65757198651533466104979197553e-01
105 #define P2 5.53859446429917461063308081748e-03
106 #define P3 1.38456698304096573887145282811e-03
107 #define P4 2.40659950032711365819348969808e-03
108 #define Q0 1.45019531250000000000000000000e+00
109 #define Q1 1.06258521948016171343454061571e+00
110 #define Q2 -2.07474561943859936441469926649e-01
111 #define Q3 -1.46734131782005422506287573015e-01
112 #define Q4 3.07878176156175520361557573779e-02
113 #define Q5 5.12449347980666221336054633184e-03
114 #define Q6 -1.76012741431666995019222898833e-03
115 #define Q7 9.35021023573788935372153030556e-05
116 #define Q8 6.13275507472443958924745652239e-06
117 /*
118 * Constants for large x approximation (x in [6, Inf])
119 * (Accurate to 2.8*10^-19 absolute)
120 */
121 #define lns2pi_hi 0.418945312500000
122 #define lns2pi_lo -.000006779295327258219670263595
123 #define Pa0 8.33333333333333148296162562474e-02
124 #define Pa1 -2.77777777774548123579378966497e-03
125 #define Pa2 7.93650778754435631476282786423e-04
126 #define Pa3 -5.95235082566672847950717262222e-04
127 #define Pa4 8.41428560346653702135821806252e-04
128 #define Pa5 -1.89773526463879200348872089421e-03
129 #define Pa6 5.69394463439411649408050664078e-03
130 #define Pa7 -1.44705562421428915453880392761e-02
131
132 static const double zero = 0., one = 1.0, tiny = 1e-300;
133
134 double
tgamma(double x)135 tgamma(double x)
136 {
137 struct Double u;
138
139 if (x >= 6) {
140 if(x > 171.63)
141 return (x / zero);
142 u = large_gam(x);
143 return(__exp__D(u.a, u.b));
144 } else if (x >= 1.0 + LEFT + x0)
145 return (small_gam(x));
146 else if (x > 1.e-17)
147 return (smaller_gam(x));
148 else if (x > -1.e-17) {
149 if (x != 0.0)
150 u.a = one - tiny; /* raise inexact */
151 return (one/x);
152 } else if (!finite(x))
153 return (x - x); /* x is NaN or -Inf */
154 else
155 return (neg_gam(x));
156 }
157 /*
158 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
159 */
160 static struct Double
large_gam(double x)161 large_gam(double x)
162 {
163 double z, p;
164 struct Double t, u, v;
165
166 z = one/(x*x);
167 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
168 p = p/x;
169
170 u = __log__D(x);
171 u.a -= one;
172 v.a = (x -= .5);
173 TRUNC(v.a);
174 v.b = x - v.a;
175 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
176 t.b = v.b*u.a + x*u.b;
177 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
178 t.b += lns2pi_lo; t.b += p;
179 u.a = lns2pi_hi + t.b; u.a += t.a;
180 u.b = t.a - u.a;
181 u.b += lns2pi_hi; u.b += t.b;
182 return (u);
183 }
184 /*
185 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
186 * It also has correct monotonicity.
187 */
188 static double
small_gam(double x)189 small_gam(double x)
190 {
191 double y, ym1, t;
192 struct Double yy, r;
193 y = x - one;
194 ym1 = y - one;
195 if (y <= 1.0 + (LEFT + x0)) {
196 yy = ratfun_gam(y - x0, 0);
197 return (yy.a + yy.b);
198 }
199 r.a = y;
200 TRUNC(r.a);
201 yy.a = r.a - one;
202 y = ym1;
203 yy.b = r.b = y - yy.a;
204 /* Argument reduction: G(x+1) = x*G(x) */
205 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
206 t = r.a*yy.a;
207 r.b = r.a*yy.b + y*r.b;
208 r.a = t;
209 TRUNC(r.a);
210 r.b += (t - r.a);
211 }
212 /* Return r*tgamma(y). */
213 yy = ratfun_gam(y - x0, 0);
214 y = r.b*(yy.a + yy.b) + r.a*yy.b;
215 y += yy.a*r.a;
216 return (y);
217 }
218 /*
219 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
220 */
221 static double
smaller_gam(double x)222 smaller_gam(double x)
223 {
224 double t, d;
225 struct Double r, xx;
226 if (x < x0 + LEFT) {
227 t = x, TRUNC(t);
228 d = (t+x)*(x-t);
229 t *= t;
230 xx.a = (t + x), TRUNC(xx.a);
231 xx.b = x - xx.a; xx.b += t; xx.b += d;
232 t = (one-x0); t += x;
233 d = (one-x0); d -= t; d += x;
234 x = xx.a + xx.b;
235 } else {
236 xx.a = x, TRUNC(xx.a);
237 xx.b = x - xx.a;
238 t = x - x0;
239 d = (-x0 -t); d += x;
240 }
241 r = ratfun_gam(t, d);
242 d = r.a/x, TRUNC(d);
243 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
244 return (d + r.a/x);
245 }
246 /*
247 * returns (z+c)^2 * P(z)/Q(z) + a0
248 */
249 static struct Double
ratfun_gam(double z,double c)250 ratfun_gam(double z, double c)
251 {
252 double p, q;
253 struct Double r, t;
254
255 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
256 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
257
258 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
259 p = p/q;
260 t.a = z, TRUNC(t.a); /* t ~= z + c */
261 t.b = (z - t.a) + c;
262 t.b *= (t.a + z);
263 q = (t.a *= t.a); /* t = (z+c)^2 */
264 TRUNC(t.a);
265 t.b += (q - t.a);
266 r.a = p, TRUNC(r.a); /* r = P/Q */
267 r.b = p - r.a;
268 t.b = t.b*p + t.a*r.b + a0_lo;
269 t.a *= r.a; /* t = (z+c)^2*(P/Q) */
270 r.a = t.a + a0_hi, TRUNC(r.a);
271 r.b = ((a0_hi-r.a) + t.a) + t.b;
272 return (r); /* r = a0 + t */
273 }
274
275 static double
neg_gam(double x)276 neg_gam(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