tgammal.c raw
1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */
2 /*
3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
4 *
5 * Permission to use, copy, modify, and distribute this software for any
6 * purpose with or without fee is hereby granted, provided that the above
7 * copyright notice and this permission notice appear in all copies.
8 *
9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16 */
17 /*
18 * Gamma function
19 *
20 *
21 * SYNOPSIS:
22 *
23 * long double x, y, tgammal();
24 *
25 * y = tgammal( x );
26 *
27 *
28 * DESCRIPTION:
29 *
30 * Returns gamma function of the argument. The result is
31 * correctly signed.
32 *
33 * Arguments |x| <= 13 are reduced by recurrence and the function
34 * approximated by a rational function of degree 7/8 in the
35 * interval (2,3). Large arguments are handled by Stirling's
36 * formula. Large negative arguments are made positive using
37 * a reflection formula.
38 *
39 *
40 * ACCURACY:
41 *
42 * Relative error:
43 * arithmetic domain # trials peak rms
44 * IEEE -40,+40 10000 3.6e-19 7.9e-20
45 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19
46 *
47 * Accuracy for large arguments is dominated by error in powl().
48 *
49 */
50
51 #include "libm.h"
52
53 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
54 long double tgammal(long double x)
55 {
56 return tgamma(x);
57 }
58 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
59 /*
60 tgamma(x+2) = tgamma(x+2) P(x)/Q(x)
61 0 <= x <= 1
62 Relative error
63 n=7, d=8
64 Peak error = 1.83e-20
65 Relative error spread = 8.4e-23
66 */
67 static const long double P[8] = {
68 4.212760487471622013093E-5L,
69 4.542931960608009155600E-4L,
70 4.092666828394035500949E-3L,
71 2.385363243461108252554E-2L,
72 1.113062816019361559013E-1L,
73 3.629515436640239168939E-1L,
74 8.378004301573126728826E-1L,
75 1.000000000000000000009E0L,
76 };
77 static const long double Q[9] = {
78 -1.397148517476170440917E-5L,
79 2.346584059160635244282E-4L,
80 -1.237799246653152231188E-3L,
81 -7.955933682494738320586E-4L,
82 2.773706565840072979165E-2L,
83 -4.633887671244534213831E-2L,
84 -2.243510905670329164562E-1L,
85 4.150160950588455434583E-1L,
86 9.999999999999999999908E-1L,
87 };
88
89 /*
90 static const long double P[] = {
91 -3.01525602666895735709e0L,
92 -3.25157411956062339893e1L,
93 -2.92929976820724030353e2L,
94 -1.70730828800510297666e3L,
95 -7.96667499622741999770e3L,
96 -2.59780216007146401957e4L,
97 -5.99650230220855581642e4L,
98 -7.15743521530849602425e4L
99 };
100 static const long double Q[] = {
101 1.00000000000000000000e0L,
102 -1.67955233807178858919e1L,
103 8.85946791747759881659e1L,
104 5.69440799097468430177e1L,
105 -1.98526250512761318471e3L,
106 3.31667508019495079814e3L,
107 1.60577839621734713377e4L,
108 -2.97045081369399940529e4L,
109 -7.15743521530849602412e4L
110 };
111 */
112 #define MAXGAML 1755.455L
113 /*static const long double LOGPI = 1.14472988584940017414L;*/
114
115 /* Stirling's formula for the gamma function
116 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
117 z(x) = x
118 13 <= x <= 1024
119 Relative error
120 n=8, d=0
121 Peak error = 9.44e-21
122 Relative error spread = 8.8e-4
123 */
124 static const long double STIR[9] = {
125 7.147391378143610789273E-4L,
126 -2.363848809501759061727E-5L,
127 -5.950237554056330156018E-4L,
128 6.989332260623193171870E-5L,
129 7.840334842744753003862E-4L,
130 -2.294719747873185405699E-4L,
131 -2.681327161876304418288E-3L,
132 3.472222222230075327854E-3L,
133 8.333333333333331800504E-2L,
134 };
135
136 #define MAXSTIR 1024.0L
137 static const long double SQTPI = 2.50662827463100050242E0L;
138
139 /* 1/tgamma(x) = z P(z)
140 * z(x) = 1/x
141 * 0 < x < 0.03125
142 * Peak relative error 4.2e-23
143 */
144 static const long double S[9] = {
145 -1.193945051381510095614E-3L,
146 7.220599478036909672331E-3L,
147 -9.622023360406271645744E-3L,
148 -4.219773360705915470089E-2L,
149 1.665386113720805206758E-1L,
150 -4.200263503403344054473E-2L,
151 -6.558780715202540684668E-1L,
152 5.772156649015328608253E-1L,
153 1.000000000000000000000E0L,
154 };
155
156 /* 1/tgamma(-x) = z P(z)
157 * z(x) = 1/x
158 * 0 < x < 0.03125
159 * Peak relative error 5.16e-23
160 * Relative error spread = 2.5e-24
161 */
162 static const long double SN[9] = {
163 1.133374167243894382010E-3L,
164 7.220837261893170325704E-3L,
165 9.621911155035976733706E-3L,
166 -4.219773343731191721664E-2L,
167 -1.665386113944413519335E-1L,
168 -4.200263503402112910504E-2L,
169 6.558780715202536547116E-1L,
170 5.772156649015328608727E-1L,
171 -1.000000000000000000000E0L,
172 };
173
174 static const long double PIL = 3.1415926535897932384626L;
175
176 /* Gamma function computed by Stirling's formula.
177 */
178 static long double stirf(long double x)
179 {
180 long double y, w, v;
181
182 w = 1.0/x;
183 /* For large x, use rational coefficients from the analytical expansion. */
184 if (x > 1024.0)
185 w = (((((6.97281375836585777429E-5L * w
186 + 7.84039221720066627474E-4L) * w
187 - 2.29472093621399176955E-4L) * w
188 - 2.68132716049382716049E-3L) * w
189 + 3.47222222222222222222E-3L) * w
190 + 8.33333333333333333333E-2L) * w
191 + 1.0;
192 else
193 w = 1.0 + w * __polevll(w, STIR, 8);
194 y = expl(x);
195 if (x > MAXSTIR) { /* Avoid overflow in pow() */
196 v = powl(x, 0.5L * x - 0.25L);
197 y = v * (v / y);
198 } else {
199 y = powl(x, x - 0.5L) / y;
200 }
201 y = SQTPI * y * w;
202 return y;
203 }
204
205 long double tgammal(long double x)
206 {
207 long double p, q, z;
208
209 if (!isfinite(x))
210 return x + INFINITY;
211
212 q = fabsl(x);
213 if (q > 13.0) {
214 if (x < 0.0) {
215 p = floorl(q);
216 z = q - p;
217 if (z == 0)
218 return 0 / z;
219 if (q > MAXGAML) {
220 z = 0;
221 } else {
222 if (z > 0.5) {
223 p += 1.0;
224 z = q - p;
225 }
226 z = q * sinl(PIL * z);
227 z = fabsl(z) * stirf(q);
228 z = PIL/z;
229 }
230 if (0.5 * p == floorl(q * 0.5))
231 z = -z;
232 } else if (x > MAXGAML) {
233 z = x * 0x1p16383L;
234 } else {
235 z = stirf(x);
236 }
237 return z;
238 }
239
240 z = 1.0;
241 while (x >= 3.0) {
242 x -= 1.0;
243 z *= x;
244 }
245 while (x < -0.03125L) {
246 z /= x;
247 x += 1.0;
248 }
249 if (x <= 0.03125L)
250 goto small;
251 while (x < 2.0) {
252 z /= x;
253 x += 1.0;
254 }
255 if (x == 2.0)
256 return z;
257
258 x -= 2.0;
259 p = __polevll(x, P, 7);
260 q = __polevll(x, Q, 8);
261 z = z * p / q;
262 return z;
263
264 small:
265 /* z==1 if x was originally +-0 */
266 if (x == 0 && z != 1)
267 return x / x;
268 if (x < 0.0) {
269 x = -x;
270 q = z / (x * __polevll(x, SN, 8));
271 } else
272 q = z / (x * __polevll(x, S, 8));
273 return q;
274 }
275 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
276 // TODO: broken implementation to make things compile
277 long double tgammal(long double x)
278 {
279 return tgamma(x);
280 }
281 #endif
282