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
|
---|