1/* lgammaf expanding around zeros.
2 Copyright (C) 2015-2018 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 <http://www.gnu.org/licenses/>. */
18
19#include <float.h>
20#include <math.h>
21#include <math_private.h>
22
23static const float lgamma_zeros[][2] =
24 {
25 { -0x2.74ff94p+0f, 0x1.3fe0f2p-24f },
26 { -0x2.bf682p+0f, -0x1.437b2p-24f },
27 { -0x3.24c1b8p+0f, 0x6.c34cap-28f },
28 { -0x3.f48e2cp+0f, 0x1.707a04p-24f },
29 { -0x4.0a13ap+0f, 0x1.e99aap-24f },
30 { -0x4.fdd5ep+0f, 0x1.64454p-24f },
31 { -0x5.021a98p+0f, 0x2.03d248p-24f },
32 { -0x5.ffa4cp+0f, 0x2.9b82fcp-24f },
33 { -0x6.005ac8p+0f, -0x1.625f24p-24f },
34 { -0x6.fff3p+0f, 0x2.251e44p-24f },
35 { -0x7.000dp+0f, 0x8.48078p-28f },
36 { -0x7.fffe6p+0f, 0x1.fa98c4p-28f },
37 { -0x8.0001ap+0f, -0x1.459fcap-28f },
38 { -0x8.ffffdp+0f, -0x1.c425e8p-24f },
39 { -0x9.00003p+0f, 0x1.c44b82p-24f },
40 { -0xap+0f, 0x4.9f942p-24f },
41 { -0xap+0f, -0x4.9f93b8p-24f },
42 { -0xbp+0f, 0x6.b9916p-28f },
43 { -0xbp+0f, -0x6.b9915p-28f },
44 { -0xcp+0f, 0x8.f76c8p-32f },
45 { -0xcp+0f, -0x8.f76c7p-32f },
46 { -0xdp+0f, 0xb.09231p-36f },
47 { -0xdp+0f, -0xb.09231p-36f },
48 { -0xep+0f, 0xc.9cba5p-40f },
49 { -0xep+0f, -0xc.9cba5p-40f },
50 { -0xfp+0f, 0xd.73f9fp-44f },
51 };
52
53static const float e_hi = 0x2.b7e15p+0f, e_lo = 0x1.628aeep-24f;
54
55/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
56 approximation to lgamma function. */
57
58static const float lgamma_coeff[] =
59 {
60 0x1.555556p-4f,
61 -0xb.60b61p-12f,
62 0x3.403404p-12f,
63 };
64
65#define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))
66
67/* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
68 the integer end-point of the half-integer interval containing x and
69 x0 is the zero of lgamma in that half-integer interval. Each
70 polynomial is expressed in terms of x-xm, where xm is the midpoint
71 of the interval for which the polynomial applies. */
72
73static const float poly_coeff[] =
74 {
75 /* Interval [-2.125, -2] (polynomial degree 5). */
76 -0x1.0b71c6p+0f,
77 -0xc.73a1ep-4f,
78 -0x1.ec8462p-4f,
79 -0xe.37b93p-4f,
80 -0x1.02ed36p-4f,
81 -0xe.cbe26p-4f,
82 /* Interval [-2.25, -2.125] (polynomial degree 5). */
83 -0xf.29309p-4f,
84 -0xc.a5cfep-4f,
85 0x3.9c93fcp-4f,
86 -0x1.02a2fp+0f,
87 0x9.896bep-4f,
88 -0x1.519704p+0f,
89 /* Interval [-2.375, -2.25] (polynomial degree 5). */
90 -0xd.7d28dp-4f,
91 -0xe.6964cp-4f,
92 0xb.0d4f1p-4f,
93 -0x1.9240aep+0f,
94 0x1.dadabap+0f,
95 -0x3.1778c4p+0f,
96 /* Interval [-2.5, -2.375] (polynomial degree 6). */
97 -0xb.74ea2p-4f,
98 -0x1.2a82cp+0f,
99 0x1.880234p+0f,
100 -0x3.320c4p+0f,
101 0x5.572a38p+0f,
102 -0x9.f92bap+0f,
103 0x1.1c347ep+4f,
104 /* Interval [-2.625, -2.5] (polynomial degree 6). */
105 -0x3.d10108p-4f,
106 0x1.cd5584p+0f,
107 0x3.819c24p+0f,
108 0x6.84cbb8p+0f,
109 0xb.bf269p+0f,
110 0x1.57fb12p+4f,
111 0x2.7b9854p+4f,
112 /* Interval [-2.75, -2.625] (polynomial degree 6). */
113 -0x6.b5d25p-4f,
114 0x1.28d604p+0f,
115 0x1.db6526p+0f,
116 0x2.e20b38p+0f,
117 0x4.44c378p+0f,
118 0x6.62a08p+0f,
119 0x9.6db3ap+0f,
120 /* Interval [-2.875, -2.75] (polynomial degree 5). */
121 -0x8.a41b2p-4f,
122 0xc.da87fp-4f,
123 0x1.147312p+0f,
124 0x1.7617dap+0f,
125 0x1.d6c13p+0f,
126 0x2.57a358p+0f,
127 /* Interval [-3, -2.875] (polynomial degree 5). */
128 -0xa.046d6p-4f,
129 0x9.70b89p-4f,
130 0xa.a89a6p-4f,
131 0xd.2f2d8p-4f,
132 0xd.e32b4p-4f,
133 0xf.fb741p-4f,
134 };
135
136static const size_t poly_deg[] =
137 {
138 5,
139 5,
140 5,
141 6,
142 6,
143 6,
144 5,
145 5,
146 };
147
148static const size_t poly_end[] =
149 {
150 5,
151 11,
152 17,
153 24,
154 31,
155 38,
156 44,
157 50,
158 };
159
160/* Compute sin (pi * X) for -0.25 <= X <= 0.5. */
161
162static float
163lg_sinpi (float x)
164{
165 if (x <= 0.25f)
166 return __sinf ((float) M_PI * x);
167 else
168 return __cosf ((float) M_PI * (0.5f - x));
169}
170
171/* Compute cos (pi * X) for -0.25 <= X <= 0.5. */
172
173static float
174lg_cospi (float x)
175{
176 if (x <= 0.25f)
177 return __cosf ((float) M_PI * x);
178 else
179 return __sinf ((float) M_PI * (0.5f - x));
180}
181
182/* Compute cot (pi * X) for -0.25 <= X <= 0.5. */
183
184static float
185lg_cotpi (float x)
186{
187 return lg_cospi (x) / lg_sinpi (x);
188}
189
190/* Compute lgamma of a negative argument -15 < X < -2, setting
191 *SIGNGAMP accordingly. */
192
193float
194__lgamma_negf (float x, int *signgamp)
195{
196 /* Determine the half-integer region X lies in, handle exact
197 integers and determine the sign of the result. */
198 int i = __floorf (-2 * x);
199 if ((i & 1) == 0 && i == -2 * x)
200 return 1.0f / 0.0f;
201 float xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
202 i -= 4;
203 *signgamp = ((i & 2) == 0 ? -1 : 1);
204
205 SET_RESTORE_ROUNDF (FE_TONEAREST);
206
207 /* Expand around the zero X0 = X0_HI + X0_LO. */
208 float x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
209 float xdiff = x - x0_hi - x0_lo;
210
211 /* For arguments in the range -3 to -2, use polynomial
212 approximations to an adjusted version of the gamma function. */
213 if (i < 2)
214 {
215 int j = __floorf (-8 * x) - 16;
216 float xm = (-33 - 2 * j) * 0.0625f;
217 float x_adj = x - xm;
218 size_t deg = poly_deg[j];
219 size_t end = poly_end[j];
220 float g = poly_coeff[end];
221 for (size_t j = 1; j <= deg; j++)
222 g = g * x_adj + poly_coeff[end - j];
223 return __log1pf (g * xdiff / (x - xn));
224 }
225
226 /* The result we want is log (sinpi (X0) / sinpi (X))
227 + log (gamma (1 - X0) / gamma (1 - X)). */
228 float x_idiff = fabsf (xn - x), x0_idiff = fabsf (xn - x0_hi - x0_lo);
229 float log_sinpi_ratio;
230 if (x0_idiff < x_idiff * 0.5f)
231 /* Use log not log1p to avoid inaccuracy from log1p of arguments
232 close to -1. */
233 log_sinpi_ratio = __ieee754_logf (lg_sinpi (x0_idiff)
234 / lg_sinpi (x_idiff));
235 else
236 {
237 /* Use log1p not log to avoid inaccuracy from log of arguments
238 close to 1. X0DIFF2 has positive sign if X0 is further from
239 XN than X is from XN, negative sign otherwise. */
240 float x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5f;
241 float sx0d2 = lg_sinpi (x0diff2);
242 float cx0d2 = lg_cospi (x0diff2);
243 log_sinpi_ratio = __log1pf (2 * sx0d2
244 * (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
245 }
246
247 float log_gamma_ratio;
248 float y0 = math_narrow_eval (1 - x0_hi);
249 float y0_eps = -x0_hi + (1 - y0) - x0_lo;
250 float y = math_narrow_eval (1 - x);
251 float y_eps = -x + (1 - y);
252 /* We now wish to compute LOG_GAMMA_RATIO
253 = log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)). XDIFF
254 accurately approximates the difference Y0 + Y0_EPS - Y -
255 Y_EPS. Use Stirling's approximation. */
256 float log_gamma_high
257 = (xdiff * __log1pf ((y0 - e_hi - e_lo + y0_eps) / e_hi)
258 + (y - 0.5f + y_eps) * __log1pf (xdiff / y));
259 /* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)). */
260 float y0r = 1 / y0, yr = 1 / y;
261 float y0r2 = y0r * y0r, yr2 = yr * yr;
262 float rdiff = -xdiff / (y * y0);
263 float bterm[NCOEFF];
264 float dlast = rdiff, elast = rdiff * yr * (yr + y0r);
265 bterm[0] = dlast * lgamma_coeff[0];
266 for (size_t j = 1; j < NCOEFF; j++)
267 {
268 float dnext = dlast * y0r2 + elast;
269 float enext = elast * yr2;
270 bterm[j] = dnext * lgamma_coeff[j];
271 dlast = dnext;
272 elast = enext;
273 }
274 float log_gamma_low = 0;
275 for (size_t j = 0; j < NCOEFF; j++)
276 log_gamma_low += bterm[NCOEFF - 1 - j];
277 log_gamma_ratio = log_gamma_high + log_gamma_low;
278
279 return log_sinpi_ratio + log_gamma_ratio;
280}
281