about summary refs log tree commit diff
path: root/sysdeps/ieee754/ldbl-96/lgamma_negl.c
blob: 98db0dea58997cb98bcb63cad4405ab26646601d (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
/* lgammal expanding around zeros.
   Copyright (C) 2015-2019 Free Software Foundation, Inc.
   This file is part of the GNU C Library.

   The GNU C Library is free software; you can redistribute it and/or
   modify it under the terms of the GNU Lesser General Public
   License as published by the Free Software Foundation; either
   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
   License along with the GNU C Library; if not, see
   <https://www.gnu.org/licenses/>.  */

#include <float.h>
#include <math.h>
#include <math_private.h>
#include <fenv_private.h>

static const long double lgamma_zeros[][2] =
  {
    { -0x2.74ff92c01f0d82acp+0L, 0x1.360cea0e5f8ed3ccp-68L },
    { -0x2.bf6821437b201978p+0L, -0x1.95a4b4641eaebf4cp-64L },
    { -0x3.24c1b793cb35efb8p+0L, -0xb.e699ad3d9ba6545p-68L },
    { -0x3.f48e2a8f85fca17p+0L, -0xd.4561291236cc321p-68L },
    { -0x4.0a139e16656030cp+0L, -0x3.9f0b0de18112ac18p-64L },
    { -0x4.fdd5de9bbabf351p+0L, -0xd.0aa4076988501d8p-68L },
    { -0x5.021a95fc2db64328p+0L, -0x2.4c56e595394decc8p-64L },
    { -0x5.ffa4bd647d0357ep+0L, 0x2.b129d342ce12071cp-64L },
    { -0x6.005ac9625f233b6p+0L, -0x7.c2d96d16385cb868p-68L },
    { -0x6.fff2fddae1bbff4p+0L, 0x2.9d949a3dc02de0cp-64L },
    { -0x7.000cff7b7f87adf8p+0L, 0x3.b7d23246787d54d8p-64L },
    { -0x7.fffe5fe05673c3c8p+0L, -0x2.9e82b522b0ca9d3p-64L },
    { -0x8.0001a01459fc9f6p+0L, -0xc.b3cec1cec857667p-68L },
    { -0x8.ffffd1c425e81p+0L, 0x3.79b16a8b6da6181cp-64L },
    { -0x9.00002e3bb47d86dp+0L, -0x6.d843fedc351deb78p-64L },
    { -0x9.fffffb606bdfdcdp+0L, -0x6.2ae77a50547c69dp-68L },
    { -0xa.0000049f93bb992p+0L, -0x7.b45d95e15441e03p-64L },
    { -0xa.ffffff9466e9f1bp+0L, -0x3.6dacd2adbd18d05cp-64L },
    { -0xb.0000006b9915316p+0L, 0x2.69a590015bf1b414p-64L },
    { -0xb.fffffff70893874p+0L, 0x7.821be533c2c36878p-64L },
    { -0xc.00000008f76c773p+0L, -0x1.567c0f0250f38792p-64L },
    { -0xc.ffffffff4f6dcf6p+0L, -0x1.7f97a5ffc757d548p-64L },
    { -0xd.00000000b09230ap+0L, 0x3.f997c22e46fc1c9p-64L },
    { -0xd.fffffffff36345bp+0L, 0x4.61e7b5c1f62ee89p-64L },
    { -0xe.000000000c9cba5p+0L, -0x4.5e94e75ec5718f78p-64L },
    { -0xe.ffffffffff28c06p+0L, -0xc.6604ef30371f89dp-68L },
    { -0xf.0000000000d73fap+0L, 0xc.6642f1bdf07a161p-68L },
    { -0xf.fffffffffff28cp+0L, -0x6.0c6621f512e72e5p-64L },
    { -0x1.000000000000d74p+4L, 0x6.0c6625ebdb406c48p-64L },
    { -0x1.0ffffffffffff356p+4L, -0x9.c47e7a93e1c46a1p-64L },
    { -0x1.1000000000000caap+4L, 0x9.c47e7a97778935ap-64L },
    { -0x1.1fffffffffffff4cp+4L, 0x1.3c31dcbecd2f74d4p-64L },
    { -0x1.20000000000000b4p+4L, -0x1.3c31dcbeca4c3b3p-64L },
    { -0x1.2ffffffffffffff6p+4L, -0x8.5b25cbf5f545ceep-64L },
    { -0x1.300000000000000ap+4L, 0x8.5b25cbf5f547e48p-64L },
    { -0x1.4p+4L, 0x7.950ae90080894298p-64L },
    { -0x1.4p+4L, -0x7.950ae9008089414p-64L },
    { -0x1.5p+4L, 0x5.c6e3bdb73d5c63p-68L },
    { -0x1.5p+4L, -0x5.c6e3bdb73d5c62f8p-68L },
    { -0x1.6p+4L, 0x4.338e5b6dfe14a518p-72L },
    { -0x1.6p+4L, -0x4.338e5b6dfe14a51p-72L },
    { -0x1.7p+4L, 0x2.ec368262c7033b3p-76L },
    { -0x1.7p+4L, -0x2.ec368262c7033b3p-76L },
    { -0x1.8p+4L, 0x1.f2cf01972f577ccap-80L },
    { -0x1.8p+4L, -0x1.f2cf01972f577ccap-80L },
    { -0x1.9p+4L, 0x1.3f3ccdd165fa8d4ep-84L },
    { -0x1.9p+4L, -0x1.3f3ccdd165fa8d4ep-84L },
    { -0x1.ap+4L, 0xc.4742fe35272cd1cp-92L },
    { -0x1.ap+4L, -0xc.4742fe35272cd1cp-92L },
    { -0x1.bp+4L, 0x7.46ac70b733a8c828p-96L },
    { -0x1.bp+4L, -0x7.46ac70b733a8c828p-96L },
    { -0x1.cp+4L, 0x4.2862898d42174ddp-100L },
    { -0x1.cp+4L, -0x4.2862898d42174ddp-100L },
    { -0x1.dp+4L, 0x2.4b3f31686b15af58p-104L },
    { -0x1.dp+4L, -0x2.4b3f31686b15af58p-104L },
    { -0x1.ep+4L, 0x1.3932c5047d60e60cp-108L },
    { -0x1.ep+4L, -0x1.3932c5047d60e60cp-108L },
    { -0x1.fp+4L, 0xa.1a6973c1fade217p-116L },
    { -0x1.fp+4L, -0xa.1a6973c1fade217p-116L },
    { -0x2p+4L, 0x5.0d34b9e0fd6f10b8p-120L },
    { -0x2p+4L, -0x5.0d34b9e0fd6f10b8p-120L },
    { -0x2.1p+4L, 0x2.73024a9ba1aa36a8p-124L },
  };

static const long double e_hi = 0x2.b7e151628aed2a6cp+0L;
static const long double e_lo = -0x1.408ea77f630b0c38p-64L;

/* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) in Stirling's
   approximation to lgamma function.  */

static const long double lgamma_coeff[] =
  {
    0x1.5555555555555556p-4L,
    -0xb.60b60b60b60b60bp-12L,
    0x3.4034034034034034p-12L,
    -0x2.7027027027027028p-12L,
    0x3.72a3c5631fe46aep-12L,
    -0x7.daac36664f1f208p-12L,
    0x1.a41a41a41a41a41ap-8L,
    -0x7.90a1b2c3d4e5f708p-8L,
    0x2.dfd2c703c0cfff44p-4L,
    -0x1.6476701181f39edcp+0L,
    0xd.672219167002d3ap+0L,
    -0x9.cd9292e6660d55bp+4L,
    0x8.911a740da740da7p+8L,
    -0x8.d0cc570e255bf5ap+12L,
    0xa.8d1044d3708d1c2p+16L,
    -0xe.8844d8a169abbc4p+20L,
  };

#define NCOEFF (sizeof (lgamma_coeff) / sizeof (lgamma_coeff[0]))

/* Polynomial approximations to (|gamma(x)|-1)(x-n)/(x-x0), where n is
   the integer end-point of the half-integer interval containing x and
   x0 is the zero of lgamma in that half-integer interval.  Each
   polynomial is expressed in terms of x-xm, where xm is the midpoint
   of the interval for which the polynomial applies.  */

static const long double poly_coeff[] =
  {
    /* Interval [-2.125, -2] (polynomial degree 13).  */
    -0x1.0b71c5c54d42eb6cp+0L,
    -0xc.73a1dc05f349517p-4L,
    -0x1.ec841408528b6baep-4L,
    -0xe.37c9da26fc3b492p-4L,
    -0x1.03cd87c5178991ap-4L,
    -0xe.ae9ada65ece2f39p-4L,
    0x9.b1185505edac18dp-8L,
    -0xe.f28c130b54d3cb2p-4L,
    0x2.6ec1666cf44a63bp-4L,
    -0xf.57cb2774193bbd5p-4L,
    0x4.5ae64671a41b1c4p-4L,
    -0xf.f48ea8b5bd3a7cep-4L,
    0x6.7d73788a8d30ef58p-4L,
    -0x1.11e0e4b506bd272ep+0L,
    /* Interval [-2.25, -2.125] (polynomial degree 13).  */
    -0xf.2930890d7d675a8p-4L,
    -0xc.a5cfde054eab5cdp-4L,
    0x3.9c9e0fdebb0676e4p-4L,
    -0x1.02a5ad35605f0d8cp+0L,
    0x9.6e9b1185d0b92edp-4L,
    -0x1.4d8332f3d6a3959p+0L,
    0x1.1c0c8cacd0ced3eap+0L,
    -0x1.c9a6f592a67b1628p+0L,
    0x1.d7e9476f96aa4bd6p+0L,
    -0x2.921cedb488bb3318p+0L,
    0x2.e8b3fd6ca193e4c8p+0L,
    -0x3.cb69d9d6628e4a2p+0L,
    0x4.95f12c73b558638p+0L,
    -0x5.d392d0b97c02ab6p+0L,
    /* Interval [-2.375, -2.25] (polynomial degree 14).  */
    -0xd.7d28d505d618122p-4L,
    -0xe.69649a304098532p-4L,
    0xb.0d74a2827d055c5p-4L,
    -0x1.924b09228531c00ep+0L,
    0x1.d49b12bccee4f888p+0L,
    -0x3.0898bb7dbb21e458p+0L,
    0x4.207a6cad6fa10a2p+0L,
    -0x6.39ee630b46093ad8p+0L,
    0x8.e2e25211a3fb5ccp+0L,
    -0xd.0e85ccd8e79c08p+0L,
    0x1.2e45882bc17f9e16p+4L,
    -0x1.b8b6e841815ff314p+4L,
    0x2.7ff8bf7504fa04dcp+4L,
    -0x3.c192e9c903352974p+4L,
    0x5.8040b75f4ef07f98p+4L,
    /* Interval [-2.5, -2.375] (polynomial degree 15).  */
    -0xb.74ea1bcfff94b2cp-4L,
    -0x1.2a82bd590c375384p+0L,
    0x1.88020f828b968634p+0L,
    -0x3.32279f040eb80fa4p+0L,
    0x5.57ac825175943188p+0L,
    -0x9.c2aedcfe10f129ep+0L,
    0x1.12c132f2df02881ep+4L,
    -0x1.ea94e26c0b6ffa6p+4L,
    0x3.66b4a8bb0290013p+4L,
    -0x6.0cf735e01f5990bp+4L,
    0xa.c10a8db7ae99343p+4L,
    -0x1.31edb212b315feeap+8L,
    0x2.1f478592298b3ebp+8L,
    -0x3.c546da5957ace6ccp+8L,
    0x7.0e3d2a02579ba4bp+8L,
    -0xc.b1ea961c39302f8p+8L,
    /* Interval [-2.625, -2.5] (polynomial degree 16).  */
    -0x3.d10108c27ebafad4p-4L,
    0x1.cd557caff7d2b202p+0L,
    0x3.819b4856d3995034p+0L,
    0x6.8505cbad03dd3bd8p+0L,
    0xb.c1b2e653aa0b924p+0L,
    0x1.50a53a38f05f72d6p+4L,
    0x2.57ae00cbd06efb34p+4L,
    0x4.2b1563077a577e9p+4L,
    0x7.6989ed790138a7f8p+4L,
    0xd.2dd28417b4f8406p+4L,
    0x1.76e1b71f0710803ap+8L,
    0x2.9a7a096254ac032p+8L,
    0x4.a0e6109e2a039788p+8L,
    0x8.37ea17a93c877b2p+8L,
    0xe.9506a641143612bp+8L,
    0x1.b680ed4ea386d52p+12L,
    0x3.28a2130c8de0ae84p+12L,
    /* Interval [-2.75, -2.625] (polynomial degree 15).  */
    -0x6.b5d252a56e8a7548p-4L,
    0x1.28d60383da3ac72p+0L,
    0x1.db6513ada8a6703ap+0L,
    0x2.e217118f9d34aa7cp+0L,
    0x4.450112c5cbd6256p+0L,
    0x6.4af99151e972f92p+0L,
    0x9.2db598b5b183cd6p+0L,
    0xd.62bef9c9adcff6ap+0L,
    0x1.379f290d743d9774p+4L,
    0x1.c58271ff823caa26p+4L,
    0x2.93a871b87a06e73p+4L,
    0x3.bf9db66103d7ec98p+4L,
    0x5.73247c111fbf197p+4L,
    0x7.ec8b9973ba27d008p+4L,
    0xb.eca5f9619b39c03p+4L,
    0x1.18f2e46411c78b1cp+8L,
    /* Interval [-2.875, -2.75] (polynomial degree 14).  */
    -0x8.a41b1e4f36ff88ep-4L,
    0xc.da87d3b69dc0f34p-4L,
    0x1.1474ad5c36158ad2p+0L,
    0x1.761ecb90c5553996p+0L,
    0x1.d279bff9ae234f8p+0L,
    0x2.4e5d0055a16c5414p+0L,
    0x2.d57545a783902f8cp+0L,
    0x3.8514eec263aa9f98p+0L,
    0x4.5235e338245f6fe8p+0L,
    0x5.562b1ef200b256c8p+0L,
    0x6.8ec9782b93bd565p+0L,
    0x8.14baf4836483508p+0L,
    0x9.efaf35dc712ea79p+0L,
    0xc.8431f6a226507a9p+0L,
    0xf.80358289a768401p+0L,
    /* Interval [-3, -2.875] (polynomial degree 13).  */
    -0xa.046d667e468f3e4p-4L,
    0x9.70b88dcc006c216p-4L,
    0xa.a8a39421c86ce9p-4L,
    0xd.2f4d1363f321e89p-4L,
    0xd.ca9aa1a3ab2f438p-4L,
    0xf.cf09c31f05d02cbp-4L,
    0x1.04b133a195686a38p+0L,
    0x1.22b54799d0072024p+0L,
    0x1.2c5802b869a36ae8p+0L,
    0x1.4aadf23055d7105ep+0L,
    0x1.5794078dd45c55d6p+0L,
    0x1.7759069da18bcf0ap+0L,
    0x1.8e672cefa4623f34p+0L,
    0x1.b2acfa32c17145e6p+0L,
  };

static const size_t poly_deg[] =
  {
    13,
    13,
    14,
    15,
    16,
    15,
    14,
    13,
  };

static const size_t poly_end[] =
  {
    13,
    27,
    42,
    58,
    75,
    91,
    106,
    120,
  };

/* Compute sin (pi * X) for -0.25 <= X <= 0.5.  */

static long double
lg_sinpi (long double x)
{
  if (x <= 0.25L)
    return __sinl (M_PIl * x);
  else
    return __cosl (M_PIl * (0.5L - x));
}

/* Compute cos (pi * X) for -0.25 <= X <= 0.5.  */

static long double
lg_cospi (long double x)
{
  if (x <= 0.25L)
    return __cosl (M_PIl * x);
  else
    return __sinl (M_PIl * (0.5L - x));
}

/* Compute cot (pi * X) for -0.25 <= X <= 0.5.  */

static long double
lg_cotpi (long double x)
{
  return lg_cospi (x) / lg_sinpi (x);
}

/* Compute lgamma of a negative argument -33 < X < -2, setting
   *SIGNGAMP accordingly.  */

long double
__lgamma_negl (long double x, int *signgamp)
{
  /* Determine the half-integer region X lies in, handle exact
     integers and determine the sign of the result.  */
  int i = floorl (-2 * x);
  if ((i & 1) == 0 && i == -2 * x)
    return 1.0L / 0.0L;
  long double xn = ((i & 1) == 0 ? -i / 2 : (-i - 1) / 2);
  i -= 4;
  *signgamp = ((i & 2) == 0 ? -1 : 1);

  SET_RESTORE_ROUNDL (FE_TONEAREST);

  /* Expand around the zero X0 = X0_HI + X0_LO.  */
  long double x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
  long double xdiff = x - x0_hi - x0_lo;

  /* For arguments in the range -3 to -2, use polynomial
     approximations to an adjusted version of the gamma function.  */
  if (i < 2)
    {
      int j = floorl (-8 * x) - 16;
      long double xm = (-33 - 2 * j) * 0.0625L;
      long double x_adj = x - xm;
      size_t deg = poly_deg[j];
      size_t end = poly_end[j];
      long double g = poly_coeff[end];
      for (size_t j = 1; j <= deg; j++)
	g = g * x_adj + poly_coeff[end - j];
      return __log1pl (g * xdiff / (x - xn));
    }

  /* The result we want is log (sinpi (X0) / sinpi (X))
     + log (gamma (1 - X0) / gamma (1 - X)).  */
  long double x_idiff = fabsl (xn - x), x0_idiff = fabsl (xn - x0_hi - x0_lo);
  long double log_sinpi_ratio;
  if (x0_idiff < x_idiff * 0.5L)
    /* Use log not log1p to avoid inaccuracy from log1p of arguments
       close to -1.  */
    log_sinpi_ratio = __ieee754_logl (lg_sinpi (x0_idiff)
				      / lg_sinpi (x_idiff));
  else
    {
      /* Use log1p not log to avoid inaccuracy from log of arguments
	 close to 1.  X0DIFF2 has positive sign if X0 is further from
	 XN than X is from XN, negative sign otherwise.  */
      long double x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5L;
      long double sx0d2 = lg_sinpi (x0diff2);
      long double cx0d2 = lg_cospi (x0diff2);
      log_sinpi_ratio = __log1pl (2 * sx0d2
				  * (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
    }

  long double log_gamma_ratio;
  long double y0 = 1 - x0_hi;
  long double y0_eps = -x0_hi + (1 - y0) - x0_lo;
  long double y = 1 - x;
  long double y_eps = -x + (1 - y);
  /* We now wish to compute LOG_GAMMA_RATIO
     = log (gamma (Y0 + Y0_EPS) / gamma (Y + Y_EPS)).  XDIFF
     accurately approximates the difference Y0 + Y0_EPS - Y -
     Y_EPS.  Use Stirling's approximation.  First, we may need to
     adjust into the range where Stirling's approximation is
     sufficiently accurate.  */
  long double log_gamma_adj = 0;
  if (i < 8)
    {
      int n_up = (9 - i) / 2;
      long double ny0, ny0_eps, ny, ny_eps;
      ny0 = y0 + n_up;
      ny0_eps = y0 - (ny0 - n_up) + y0_eps;
      y0 = ny0;
      y0_eps = ny0_eps;
      ny = y + n_up;
      ny_eps = y - (ny - n_up) + y_eps;
      y = ny;
      y_eps = ny_eps;
      long double prodm1 = __lgamma_productl (xdiff, y - n_up, y_eps, n_up);
      log_gamma_adj = -__log1pl (prodm1);
    }
  long double log_gamma_high
    = (xdiff * __log1pl ((y0 - e_hi - e_lo + y0_eps) / e_hi)
       + (y - 0.5L + y_eps) * __log1pl (xdiff / y) + log_gamma_adj);
  /* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)).  */
  long double y0r = 1 / y0, yr = 1 / y;
  long double y0r2 = y0r * y0r, yr2 = yr * yr;
  long double rdiff = -xdiff / (y * y0);
  long double bterm[NCOEFF];
  long double dlast = rdiff, elast = rdiff * yr * (yr + y0r);
  bterm[0] = dlast * lgamma_coeff[0];
  for (size_t j = 1; j < NCOEFF; j++)
    {
      long double dnext = dlast * y0r2 + elast;
      long double enext = elast * yr2;
      bterm[j] = dnext * lgamma_coeff[j];
      dlast = dnext;
      elast = enext;
    }
  long double log_gamma_low = 0;
  for (size_t j = 0; j < NCOEFF; j++)
    log_gamma_low += bterm[NCOEFF - 1 - j];
  log_gamma_ratio = log_gamma_high + log_gamma_low;

  return log_sinpi_ratio + log_gamma_ratio;
}