1 /* lgamma expanding around zeros.
2 Copyright (C) 2015-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include <float.h>
20 #include <math.h>
21 #include <math-narrow-eval.h>
22 #include <math_private.h>
23 #include <fenv_private.h>
24
25 static const double lgamma_zeros[][2] =
26 {
27 { -0x2.74ff92c01f0d8p+0, -0x2.abec9f315f1ap-56 },
28 { -0x2.bf6821437b202p+0, 0x6.866a5b4b9be14p-56 },
29 { -0x3.24c1b793cb35ep+0, -0xf.b8be699ad3d98p-56 },
30 { -0x3.f48e2a8f85fcap+0, -0x1.70d4561291237p-56 },
31 { -0x4.0a139e1665604p+0, 0xf.3c60f4f21e7fp-56 },
32 { -0x4.fdd5de9bbabf4p+0, 0xa.ef2f55bf89678p-56 },
33 { -0x5.021a95fc2db64p+0, -0x3.2a4c56e595394p-56 },
34 { -0x5.ffa4bd647d034p+0, -0x1.7dd4ed62cbd32p-52 },
35 { -0x6.005ac9625f234p+0, 0x4.9f83d2692e9c8p-56 },
36 { -0x6.fff2fddae1bcp+0, 0xc.29d949a3dc03p-60 },
37 { -0x7.000cff7b7f87cp+0, 0x1.20bb7d2324678p-52 },
38 { -0x7.fffe5fe05673cp+0, -0x3.ca9e82b522b0cp-56 },
39 { -0x8.0001a01459fc8p+0, -0x1.f60cb3cec1cedp-52 },
40 { -0x8.ffffd1c425e8p+0, -0xf.fc864e9574928p-56 },
41 { -0x9.00002e3bb47d8p+0, -0x6.d6d843fedc35p-56 },
42 { -0x9.fffffb606bep+0, 0x2.32f9d51885afap-52 },
43 { -0xa.0000049f93bb8p+0, -0x1.927b45d95e154p-52 },
44 { -0xa.ffffff9466eap+0, 0xe.4c92532d5243p-56 },
45 { -0xb.0000006b9915p+0, -0x3.15d965a6ffea4p-52 },
46 { -0xb.fffffff708938p+0, -0x7.387de41acc3d4p-56 },
47 { -0xc.00000008f76c8p+0, 0x8.cea983f0fdafp-56 },
48 { -0xc.ffffffff4f6ep+0, 0x3.09e80685a0038p-52 },
49 { -0xd.00000000b092p+0, -0x3.09c06683dd1bap-52 },
50 { -0xd.fffffffff3638p+0, 0x3.a5461e7b5c1f6p-52 },
51 { -0xe.000000000c9c8p+0, -0x3.a545e94e75ec6p-52 },
52 { -0xe.ffffffffff29p+0, 0x3.f9f399fb10cfcp-52 },
53 { -0xf.0000000000d7p+0, -0x3.f9f399bd0e42p-52 },
54 { -0xf.fffffffffff28p+0, -0xc.060c6621f513p-56 },
55 { -0x1.000000000000dp+4, -0x7.3f9f399da1424p-52 },
56 { -0x1.0ffffffffffffp+4, -0x3.569c47e7a93e2p-52 },
57 { -0x1.1000000000001p+4, 0x3.569c47e7a9778p-52 },
58 { -0x1.2p+4, 0xb.413c31dcbecdp-56 },
59 { -0x1.2p+4, -0xb.413c31dcbeca8p-56 },
60 { -0x1.3p+4, 0x9.7a4da340a0ab8p-60 },
61 { -0x1.3p+4, -0x9.7a4da340a0ab8p-60 },
62 { -0x1.4p+4, 0x7.950ae90080894p-64 },
63 { -0x1.4p+4, -0x7.950ae90080894p-64 },
64 { -0x1.5p+4, 0x5.c6e3bdb73d5c8p-68 },
65 { -0x1.5p+4, -0x5.c6e3bdb73d5c8p-68 },
66 { -0x1.6p+4, 0x4.338e5b6dfe14cp-72 },
67 { -0x1.6p+4, -0x4.338e5b6dfe14cp-72 },
68 { -0x1.7p+4, 0x2.ec368262c7034p-76 },
69 { -0x1.7p+4, -0x2.ec368262c7034p-76 },
70 { -0x1.8p+4, 0x1.f2cf01972f578p-80 },
71 { -0x1.8p+4, -0x1.f2cf01972f578p-80 },
72 { -0x1.9p+4, 0x1.3f3ccdd165fa9p-84 },
73 { -0x1.9p+4, -0x1.3f3ccdd165fa9p-84 },
74 { -0x1.ap+4, 0xc.4742fe35272dp-92 },
75 { -0x1.ap+4, -0xc.4742fe35272dp-92 },
76 { -0x1.bp+4, 0x7.46ac70b733a8cp-96 },
77 { -0x1.bp+4, -0x7.46ac70b733a8cp-96 },
78 { -0x1.cp+4, 0x4.2862898d42174p-100 },
79 };
80
81 static const double e_hi = 0x2.b7e151628aed2p+0, e_lo = 0xa.6abf7158809dp-56;
82
83 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
84 approximation to lgamma function. */
85
86 static const double lgamma_coeff[] =
87 {
88 0x1.5555555555555p-4,
89 -0xb.60b60b60b60b8p-12,
90 0x3.4034034034034p-12,
91 -0x2.7027027027028p-12,
92 0x3.72a3c5631fe46p-12,
93 -0x7.daac36664f1f4p-12,
94 0x1.a41a41a41a41ap-8,
95 -0x7.90a1b2c3d4e6p-8,
96 0x2.dfd2c703c0dp-4,
97 -0x1.6476701181f3ap+0,
98 0xd.672219167003p+0,
99 -0x9.cd9292e6660d8p+4,
100 };
101
102 #define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
103
104 /* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
105 the integer end-point of the half-integer interval containing x and
106 x0 is the zero of lgamma in that half-integer interval. Each
107 polynomial is expressed in terms of x-xm, where xm is the midpoint
108 of the interval for which the polynomial applies. */
109
110 static const double poly_coeff[] =
111 {
112 /* Interval [-2.125, -2] (polynomial degree 10). */
113 -0x1.0b71c5c54d42fp+0,
114 -0xc.73a1dc05f3758p-4,
115 -0x1.ec84140851911p-4,
116 -0xe.37c9da23847e8p-4,
117 -0x1.03cd87cdc0ac6p-4,
118 -0xe.ae9aedce12eep-4,
119 0x9.b11a1780cfd48p-8,
120 -0xe.f25fc460bdebp-4,
121 0x2.6e984c61ca912p-4,
122 -0xf.83fea1c6d35p-4,
123 0x4.760c8c8909758p-4,
124 /* Interval [-2.25, -2.125] (polynomial degree 11). */
125 -0xf.2930890d7d678p-4,
126 -0xc.a5cfde054eaa8p-4,
127 0x3.9c9e0fdebd99cp-4,
128 -0x1.02a5ad35619d9p+0,
129 0x9.6e9b1167c164p-4,
130 -0x1.4d8332eba090ap+0,
131 0x1.1c0c94b1b2b6p+0,
132 -0x1.c9a70d138c74ep+0,
133 0x1.d7d9cf1d4c196p+0,
134 -0x2.91fbf4cd6abacp+0,
135 0x2.f6751f74b8ff8p+0,
136 -0x3.e1bb7b09e3e76p+0,
137 /* Interval [-2.375, -2.25] (polynomial degree 12). */
138 -0xd.7d28d505d618p-4,
139 -0xe.69649a3040958p-4,
140 0xb.0d74a2827cd6p-4,
141 -0x1.924b09228a86ep+0,
142 0x1.d49b12bcf6175p+0,
143 -0x3.0898bb530d314p+0,
144 0x4.207a6be8fda4cp+0,
145 -0x6.39eef56d4e9p+0,
146 0x8.e2e42acbccec8p+0,
147 -0xd.0d91c1e596a68p+0,
148 0x1.2e20d7099c585p+4,
149 -0x1.c4eb6691b4ca9p+4,
150 0x2.96a1a11fd85fep+4,
151 /* Interval [-2.5, -2.375] (polynomial degree 13). */
152 -0xb.74ea1bcfff948p-4,
153 -0x1.2a82bd590c376p+0,
154 0x1.88020f828b81p+0,
155 -0x3.32279f040d7aep+0,
156 0x5.57ac8252ce868p+0,
157 -0x9.c2aedd093125p+0,
158 0x1.12c132716e94cp+4,
159 -0x1.ea94dfa5c0a6dp+4,
160 0x3.66b61abfe858cp+4,
161 -0x6.0cfceb62a26e4p+4,
162 0xa.beeba09403bd8p+4,
163 -0x1.3188d9b1b288cp+8,
164 0x2.37f774dd14c44p+8,
165 -0x3.fdf0a64cd7136p+8,
166 /* Interval [-2.625, -2.5] (polynomial degree 13). */
167 -0x3.d10108c27ebbp-4,
168 0x1.cd557caff7d2fp+0,
169 0x3.819b4856d36cep+0,
170 0x6.8505cbacfc42p+0,
171 0xb.c1b2e6567a4dp+0,
172 0x1.50a53a3ce6c73p+4,
173 0x2.57adffbb1ec0cp+4,
174 0x4.2b15549cf400cp+4,
175 0x7.698cfd82b3e18p+4,
176 0xd.2decde217755p+4,
177 0x1.7699a624d07b9p+8,
178 0x2.98ecf617abbfcp+8,
179 0x4.d5244d44d60b4p+8,
180 0x8.e962bf7395988p+8,
181 /* Interval [-2.75, -2.625] (polynomial degree 12). */
182 -0x6.b5d252a56e8a8p-4,
183 0x1.28d60383da3a6p+0,
184 0x1.db6513ada89bep+0,
185 0x2.e217118fa8c02p+0,
186 0x4.450112c651348p+0,
187 0x6.4af990f589b8cp+0,
188 0x9.2db5963d7a238p+0,
189 0xd.62c03647da19p+0,
190 0x1.379f81f6416afp+4,
191 0x1.c5618b4fdb96p+4,
192 0x2.9342d0af2ac4ep+4,
193 0x3.d9cdf56d2b186p+4,
194 0x5.ab9f91d5a27a4p+4,
195 /* Interval [-2.875, -2.75] (polynomial degree 11). */
196 -0x8.a41b1e4f36ff8p-4,
197 0xc.da87d3b69dbe8p-4,
198 0x1.1474ad5c36709p+0,
199 0x1.761ecb90c8c5cp+0,
200 0x1.d279bff588826p+0,
201 0x2.4e5d003fb36a8p+0,
202 0x2.d575575566842p+0,
203 0x3.85152b0d17756p+0,
204 0x4.5213d921ca13p+0,
205 0x5.55da7dfcf69c4p+0,
206 0x6.acef729b9404p+0,
207 0x8.483cc21dd0668p+0,
208 /* Interval [-3, -2.875] (polynomial degree 11). */
209 -0xa.046d667e468f8p-4,
210 0x9.70b88dcc006cp-4,
211 0xa.a8a39421c94dp-4,
212 0xd.2f4d1363f98ep-4,
213 0xd.ca9aa19975b7p-4,
214 0xf.cf09c2f54404p-4,
215 0x1.04b1365a9adfcp+0,
216 0x1.22b54ef213798p+0,
217 0x1.2c52c25206bf5p+0,
218 0x1.4aa3d798aace4p+0,
219 0x1.5c3f278b504e3p+0,
220 0x1.7e08292cc347bp+0,
221 };
222
223 static const size_t poly_deg[] =
224 {
225 10,
226 11,
227 12,
228 13,
229 13,
230 12,
231 11,
232 11,
233 };
234
235 static const size_t poly_end[] =
236 {
237 10,
238 22,
239 35,
240 49,
241 63,
242 76,
243 88,
244 100,
245 };
246
247 /* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
248
249 static double
250 lg_sinpi (double x)
251 {
252 if (x <= 0.25)
253 return __sin (M_PI * x);
254 else
255 return __cos (M_PI * (0.5 - x));
256 }
257
258 /* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
259
260 static double
261 lg_cospi (double x)
262 {
263 if (x <= 0.25)
264 return __cos (M_PI * x);
265 else
266 return __sin (M_PI * (0.5 - x));
267 }
268
269 /* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
270
271 static double
272 lg_cotpi (double x)
273 {
274 return lg_cospi (x) / lg_sinpi (x);
275 }
276
277 /* Compute lgamma of a negative argument -28 < X < -2, setting
278 *SIGNGAMP accordingly. */
279
280 double
281 __lgamma_neg (double x, int *signgamp)
282 {
283 /* Determine the half-integer region X lies in, handle exact
284 integers and determine the sign of the result. */
285 int i = floor (-2 * x);
286 if ((i & 1) == 0 && i == -2 * x)
287 return 1.0 / 0.0;
288 double xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
289 i -= 4;
290 *signgamp = ((i & 2) == 0 ? -1 : 1);
291
292 SET_RESTORE_ROUND (FE_TONEAREST);
293
294 /* Expand around the zero X0 = X0_HI + X0_LO. */
295 double x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
296 double xdiff = x - x0_hi - x0_lo;
297
298 /* For arguments in the range -3 to -2, use polynomial
299 approximations to an adjusted version of the gamma function. */
300 if (i < 2)
301 {
302 int j = floor (-8 * x) - 16;
303 double xm = (-33 - 2 * j) * 0.0625;
304 double x_adj = x - xm;
305 size_t deg = poly_deg[j];
306 size_t end = poly_end[j];
307 double g = poly_coeff[end];
308 for (size_t j = 1; j <= deg; j++)
309 g = g * x_adj + poly_coeff[end - j];
310 return __log1p (g * xdiff / (x - xn));
311 }
312
313 /* The result we want is log (sinpi (X0) / sinpi (X))
314 + log (gamma (1 - X0) / gamma (1 - X)). */
315 double x_idiff = fabs (xn - x), x0_idiff = fabs (xn - x0_hi - x0_lo);
316 double log_sinpi_ratio;
317 if (x0_idiff < x_idiff * 0.5)
318 /* Use log not log1p to avoid inaccuracy from log1p of arguments
319 close to -1. */
320 log_sinpi_ratio = __ieee754_log (lg_sinpi (x0_idiff)
321 / lg_sinpi (x_idiff));
322 else
323 {
324 /* Use log1p not log to avoid inaccuracy from log of arguments
325 close to 1. X0DIFF2 has positive sign if X0 is further from
326 XN than X is from XN, negative sign otherwise. */
327 double x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5;
328 double sx0d2 = lg_sinpi (x0diff2);
329 double cx0d2 = lg_cospi (x0diff2);
330 log_sinpi_ratio = __log1p (2 * sx0d2
331 * (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
332 }
333
334 double log_gamma_ratio;
335 double y0 = math_narrow_eval (1 - x0_hi);
336 double y0_eps = -x0_hi + (1 - y0) - x0_lo;
337 double y = math_narrow_eval (1 - x);
338 double y_eps = -x + (1 - y);
339 /* We now wish to compute LOG_GAMMA_RATIO
340 = log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
341 accurately approximates the difference Y0 + Y0_EPS - Y -
342 Y_EPS. Use Stirling's approximation. First, we may need to
343 adjust into the range where Stirling's approximation is
344 sufficiently accurate. */
345 double log_gamma_adj = 0;
346 if (i < 6)
347 {
348 int n_up = (7 - i) / 2;
349 double ny0, ny0_eps, ny, ny_eps;
350 ny0 = math_narrow_eval (y0 + n_up);
351 ny0_eps = y0 - (ny0 - n_up) + y0_eps;
352 y0 = ny0;
353 y0_eps = ny0_eps;
354 ny = math_narrow_eval (y + n_up);
355 ny_eps = y - (ny - n_up) + y_eps;
356 y = ny;
357 y_eps = ny_eps;
358 double prodm1 = __lgamma_product (xdiff, y - n_up, y_eps, n_up);
359 log_gamma_adj = -__log1p (prodm1);
360 }
361 double log_gamma_high
362 = (xdiff * __log1p ((y0 - e_hi - e_lo + y0_eps) / e_hi)
363 + (y - 0.5 + y_eps) * __log1p (xdiff / y) + log_gamma_adj);
364 /* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
365 double y0r = 1 / y0, yr = 1 / y;
366 double y0r2 = y0r * y0r, yr2 = yr * yr;
367 double rdiff = -xdiff / (y * y0);
368 double bterm[NCOEFF];
369 double dlast = rdiff, elast = rdiff * yr * (yr + y0r);
370 bterm[0] = dlast * lgamma_coeff[0];
371 for (size_t j = 1; j < NCOEFF; j++)
372 {
373 double dnext = dlast * y0r2 + elast;
374 double enext = elast * yr2;
375 bterm[j] = dnext * lgamma_coeff[j];
376 dlast = dnext;
377 elast = enext;
378 }
379 double log_gamma_low = 0;
380 for (size_t j = 0; j < NCOEFF; j++)
381 log_gamma_low += bterm[NCOEFF - 1 - j];
382 log_gamma_ratio = log_gamma_high + log_gamma_low;
383
384 return log_sinpi_ratio + log_gamma_ratio;
385 }