1 /* $OpenBSD: e_tgammal.c,v 1.4 2013/11/12 20:35:19 martynas Exp $ */
2
3 /*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19 /* tgammal.c
20 *
21 * Gamma function
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double x, y, tgammal();
28 * extern int signgam;
29 *
30 * y = tgammal( x );
31 *
32 *
33 *
34 * DESCRIPTION:
35 *
36 * Returns gamma function of the argument. The result is
37 * correctly signed, and the sign (+1 or -1) is also
38 * returned in a global (extern) variable named signgam.
39 * This variable is also filled in by the logarithmic gamma
40 * function lgamma().
41 *
42 * Arguments |x| <= 13 are reduced by recurrence and the function
43 * approximated by a rational function of degree 7/8 in the
44 * interval (2,3). Large arguments are handled by Stirling's
45 * formula. Large negative arguments are made positive using
46 * a reflection formula.
47 *
48 *
49 * ACCURACY:
50 *
51 * Relative error:
52 * arithmetic domain # trials peak rms
53 * IEEE -40,+40 10000 3.6e-19 7.9e-20
54 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19
55 *
56 * Accuracy for large arguments is dominated by error in powl().
57 *
58 */
59
60 #include <float.h>
61 #include <math.h>
62
63 #include "math_private.h"
64
65 /*
66 tgamma(x+2) = tgamma(x+2) P(x)/Q(x)
67 0 <= x <= 1
68 Relative error
69 n=7, d=8
70 Peak error = 1.83e-20
71 Relative error spread = 8.4e-23
72 */
73
74 static long double P[8] = {
75 4.212760487471622013093E-5L,
76 4.542931960608009155600E-4L,
77 4.092666828394035500949E-3L,
78 2.385363243461108252554E-2L,
79 1.113062816019361559013E-1L,
80 3.629515436640239168939E-1L,
81 8.378004301573126728826E-1L,
82 1.000000000000000000009E0L,
83 };
84 static long double Q[9] = {
85 -1.397148517476170440917E-5L,
86 2.346584059160635244282E-4L,
87 -1.237799246653152231188E-3L,
88 -7.955933682494738320586E-4L,
89 2.773706565840072979165E-2L,
90 -4.633887671244534213831E-2L,
91 -2.243510905670329164562E-1L,
92 4.150160950588455434583E-1L,
93 9.999999999999999999908E-1L,
94 };
95
96 /*
97 static long double P[] = {
98 -3.01525602666895735709e0L,
99 -3.25157411956062339893e1L,
100 -2.92929976820724030353e2L,
101 -1.70730828800510297666e3L,
102 -7.96667499622741999770e3L,
103 -2.59780216007146401957e4L,
104 -5.99650230220855581642e4L,
105 -7.15743521530849602425e4L
106 };
107 static long double Q[] = {
108 1.00000000000000000000e0L,
109 -1.67955233807178858919e1L,
110 8.85946791747759881659e1L,
111 5.69440799097468430177e1L,
112 -1.98526250512761318471e3L,
113 3.31667508019495079814e3L,
114 1.60577839621734713377e4L,
115 -2.97045081369399940529e4L,
116 -7.15743521530849602412e4L
117 };
118 */
119 #define MAXGAML 1755.455L
120 /*static const long double LOGPI = 1.14472988584940017414L;*/
121
122 /* Stirling's formula for the gamma function
123 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
124 z(x) = x
125 13 <= x <= 1024
126 Relative error
127 n=8, d=0
128 Peak error = 9.44e-21
129 Relative error spread = 8.8e-4
130 */
131
132 static long double STIR[9] = {
133 7.147391378143610789273E-4L,
134 -2.363848809501759061727E-5L,
135 -5.950237554056330156018E-4L,
136 6.989332260623193171870E-5L,
137 7.840334842744753003862E-4L,
138 -2.294719747873185405699E-4L,
139 -2.681327161876304418288E-3L,
140 3.472222222230075327854E-3L,
141 8.333333333333331800504E-2L,
142 };
143
144 #define MAXSTIR 1024.0L
145 static const long double SQTPI = 2.50662827463100050242E0L;
146
147 /* 1/tgamma(x) = z P(z)
148 * z(x) = 1/x
149 * 0 < x < 0.03125
150 * Peak relative error 4.2e-23
151 */
152
153 static long double S[9] = {
154 -1.193945051381510095614E-3L,
155 7.220599478036909672331E-3L,
156 -9.622023360406271645744E-3L,
157 -4.219773360705915470089E-2L,
158 1.665386113720805206758E-1L,
159 -4.200263503403344054473E-2L,
160 -6.558780715202540684668E-1L,
161 5.772156649015328608253E-1L,
162 1.000000000000000000000E0L,
163 };
164
165 /* 1/tgamma(-x) = z P(z)
166 * z(x) = 1/x
167 * 0 < x < 0.03125
168 * Peak relative error 5.16e-23
169 * Relative error spread = 2.5e-24
170 */
171
172 static long double SN[9] = {
173 1.133374167243894382010E-3L,
174 7.220837261893170325704E-3L,
175 9.621911155035976733706E-3L,
176 -4.219773343731191721664E-2L,
177 -1.665386113944413519335E-1L,
178 -4.200263503402112910504E-2L,
179 6.558780715202536547116E-1L,
180 5.772156649015328608727E-1L,
181 -1.000000000000000000000E0L,
182 };
183
184 static const long double PIL = 3.1415926535897932384626L;
185
186 static long double stirf ( long double );
187
188 /* Gamma function computed by Stirling's formula.
189 */
stirf(long double x)190 static long double stirf(long double x)
191 {
192 long double y, w, v;
193
194 w = 1.0L/x;
195 /* For large x, use rational coefficients from the analytical expansion. */
196 if( x > 1024.0L )
197 w = (((((6.97281375836585777429E-5L * w
198 + 7.84039221720066627474E-4L) * w
199 - 2.29472093621399176955E-4L) * w
200 - 2.68132716049382716049E-3L) * w
201 + 3.47222222222222222222E-3L) * w
202 + 8.33333333333333333333E-2L) * w
203 + 1.0L;
204 else
205 w = 1.0L + w * __polevll( w, STIR, 8 );
206 y = expl(x);
207 if( x > MAXSTIR )
208 { /* Avoid overflow in pow() */
209 v = powl( x, 0.5L * x - 0.25L );
210 y = v * (v / y);
211 }
212 else
213 {
214 y = powl( x, x - 0.5L ) / y;
215 }
216 y = SQTPI * y * w;
217 return( y );
218 }
219
220 long double
tgammal(long double x)221 tgammal(long double x)
222 {
223 long double p, q, z;
224 int i;
225
226 signgam = 1;
227 if( isnan(x) )
228 return(NAN);
229 if(x == INFINITY)
230 return(INFINITY);
231 if(x == -INFINITY)
232 return(x - x);
233 if( x == 0.0L )
234 return( 1.0L / x );
235 q = fabsl(x);
236
237 if( q > 13.0L )
238 {
239 if( q > MAXGAML )
240 goto goverf;
241 if( x < 0.0L )
242 {
243 p = floorl(q);
244 if( p == q )
245 return (x - x) / (x - x);
246 i = p;
247 if( (i & 1) == 0 )
248 signgam = -1;
249 z = q - p;
250 if( z > 0.5L )
251 {
252 p += 1.0L;
253 z = q - p;
254 }
255 z = q * sinl( PIL * z );
256 z = fabsl(z) * stirf(q);
257 if( z <= PIL/LDBL_MAX )
258 {
259 goverf:
260 return( signgam * INFINITY);
261 }
262 z = PIL/z;
263 }
264 else
265 {
266 z = stirf(x);
267 }
268 return( signgam * z );
269 }
270
271 z = 1.0L;
272 while( x >= 3.0L )
273 {
274 x -= 1.0L;
275 z *= x;
276 }
277
278 while( x < -0.03125L )
279 {
280 z /= x;
281 x += 1.0L;
282 }
283
284 if( x <= 0.03125L )
285 goto small;
286
287 while( x < 2.0L )
288 {
289 z /= x;
290 x += 1.0L;
291 }
292
293 if( x == 2.0L )
294 return(z);
295
296 x -= 2.0L;
297 p = __polevll( x, P, 7 );
298 q = __polevll( x, Q, 8 );
299 z = z * p / q;
300 if( z < 0 )
301 signgam = -1;
302 return z;
303
304 small:
305 if( x == 0.0L )
306 return (x - x) / (x - x);
307 else
308 {
309 if( x < 0.0L )
310 {
311 x = -x;
312 q = z / (x * __polevll( x, SN, 8 ));
313 signgam = -1;
314 }
315 else
316 q = z / (x * __polevll( x, S, 8 ));
317 }
318 return q;
319 }
320