about summary refs log tree commit diff
path: root/sysdeps/ieee754/flt-32/lgamma_negf.c
blob: c60f6ca4477cb2e016bce8609ca77ab125ba022c (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
/* lgammaf expanding around zeros.
   Copyright (C) 2015-2018 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
   <http://www.gnu.org/licenses/>.  */

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

static const float lgamma_zeros[][2] =
  {
    { -0x2.74ff94p+0f, 0x1.3fe0f2p-24f },
    { -0x2.bf682p+0f, -0x1.437b2p-24f },
    { -0x3.24c1b8p+0f, 0x6.c34cap-28f },
    { -0x3.f48e2cp+0f, 0x1.707a04p-24f },
    { -0x4.0a13ap+0f, 0x1.e99aap-24f },
    { -0x4.fdd5ep+0f, 0x1.64454p-24f },
    { -0x5.021a98p+0f, 0x2.03d248p-24f },
    { -0x5.ffa4cp+0f, 0x2.9b82fcp-24f },
    { -0x6.005ac8p+0f, -0x1.625f24p-24f },
    { -0x6.fff3p+0f, 0x2.251e44p-24f },
    { -0x7.000dp+0f, 0x8.48078p-28f },
    { -0x7.fffe6p+0f, 0x1.fa98c4p-28f },
    { -0x8.0001ap+0f, -0x1.459fcap-28f },
    { -0x8.ffffdp+0f, -0x1.c425e8p-24f },
    { -0x9.00003p+0f, 0x1.c44b82p-24f },
    { -0xap+0f, 0x4.9f942p-24f },
    { -0xap+0f, -0x4.9f93b8p-24f },
    { -0xbp+0f, 0x6.b9916p-28f },
    { -0xbp+0f, -0x6.b9915p-28f },
    { -0xcp+0f, 0x8.f76c8p-32f },
    { -0xcp+0f, -0x8.f76c7p-32f },
    { -0xdp+0f, 0xb.09231p-36f },
    { -0xdp+0f, -0xb.09231p-36f },
    { -0xep+0f, 0xc.9cba5p-40f },
    { -0xep+0f, -0xc.9cba5p-40f },
    { -0xfp+0f, 0xd.73f9fp-44f },
  };

static const float e_hi = 0x2.b7e15p+0f, e_lo = 0x1.628aeep-24f;

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

static const float lgamma_coeff[] =
  {
    0x1.555556p-4f,
    -0xb.60b61p-12f,
    0x3.403404p-12f,
  };

#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 float poly_coeff[] =
  {
    /* Interval [-2.125, -2] (polynomial degree 5).  */
    -0x1.0b71c6p+0f,
    -0xc.73a1ep-4f,
    -0x1.ec8462p-4f,
    -0xe.37b93p-4f,
    -0x1.02ed36p-4f,
    -0xe.cbe26p-4f,
    /* Interval [-2.25, -2.125] (polynomial degree 5).  */
    -0xf.29309p-4f,
    -0xc.a5cfep-4f,
    0x3.9c93fcp-4f,
    -0x1.02a2fp+0f,
    0x9.896bep-4f,
    -0x1.519704p+0f,
    /* Interval [-2.375, -2.25] (polynomial degree 5).  */
    -0xd.7d28dp-4f,
    -0xe.6964cp-4f,
    0xb.0d4f1p-4f,
    -0x1.9240aep+0f,
    0x1.dadabap+0f,
    -0x3.1778c4p+0f,
    /* Interval [-2.5, -2.375] (polynomial degree 6).  */
    -0xb.74ea2p-4f,
    -0x1.2a82cp+0f,
    0x1.880234p+0f,
    -0x3.320c4p+0f,
    0x5.572a38p+0f,
    -0x9.f92bap+0f,
    0x1.1c347ep+4f,
    /* Interval [-2.625, -2.5] (polynomial degree 6).  */
    -0x3.d10108p-4f,
    0x1.cd5584p+0f,
    0x3.819c24p+0f,
    0x6.84cbb8p+0f,
    0xb.bf269p+0f,
    0x1.57fb12p+4f,
    0x2.7b9854p+4f,
    /* Interval [-2.75, -2.625] (polynomial degree 6).  */
    -0x6.b5d25p-4f,
    0x1.28d604p+0f,
    0x1.db6526p+0f,
    0x2.e20b38p+0f,
    0x4.44c378p+0f,
    0x6.62a08p+0f,
    0x9.6db3ap+0f,
    /* Interval [-2.875, -2.75] (polynomial degree 5).  */
    -0x8.a41b2p-4f,
    0xc.da87fp-4f,
    0x1.147312p+0f,
    0x1.7617dap+0f,
    0x1.d6c13p+0f,
    0x2.57a358p+0f,
    /* Interval [-3, -2.875] (polynomial degree 5).  */
    -0xa.046d6p-4f,
    0x9.70b89p-4f,
    0xa.a89a6p-4f,
    0xd.2f2d8p-4f,
    0xd.e32b4p-4f,
    0xf.fb741p-4f,
  };

static const size_t poly_deg[] =
  {
    5,
    5,
    5,
    6,
    6,
    6,
    5,
    5,
  };

static const size_t poly_end[] =
  {
    5,
    11,
    17,
    24,
    31,
    38,
    44,
    50,
  };

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

static float
lg_sinpi (float x)
{
  if (x <= 0.25f)
    return __sinf ((float) M_PI * x);
  else
    return __cosf ((float) M_PI * (0.5f - x));
}

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

static float
lg_cospi (float x)
{
  if (x <= 0.25f)
    return __cosf ((float) M_PI * x);
  else
    return __sinf ((float) M_PI * (0.5f - x));
}

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

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

/* Compute lgamma of a negative argument -15 < X < -2, setting
   *SIGNGAMP accordingly.  */

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

  SET_RESTORE_ROUNDF (FE_TONEAREST);

  /* Expand around the zero X0 = X0_HI + X0_LO.  */
  float x0_hi = lgamma_zeros[i][0], x0_lo = lgamma_zeros[i][1];
  float 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 = __floorf (-8 * x) - 16;
      float xm = (-33 - 2 * j) * 0.0625f;
      float x_adj = x - xm;
      size_t deg = poly_deg[j];
      size_t end = poly_end[j];
      float g = poly_coeff[end];
      for (size_t j = 1; j <= deg; j++)
	g = g * x_adj + poly_coeff[end - j];
      return __log1pf (g * xdiff / (x - xn));
    }

  /* The result we want is log (sinpi (X0) / sinpi (X))
     + log (gamma (1 - X0) / gamma (1 - X)).  */
  float x_idiff = fabsf (xn - x), x0_idiff = fabsf (xn - x0_hi - x0_lo);
  float log_sinpi_ratio;
  if (x0_idiff < x_idiff * 0.5f)
    /* Use log not log1p to avoid inaccuracy from log1p of arguments
       close to -1.  */
    log_sinpi_ratio = __ieee754_logf (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.  */
      float x0diff2 = ((i & 1) == 0 ? xdiff : -xdiff) * 0.5f;
      float sx0d2 = lg_sinpi (x0diff2);
      float cx0d2 = lg_cospi (x0diff2);
      log_sinpi_ratio = __log1pf (2 * sx0d2
				  * (-sx0d2 + cx0d2 * lg_cotpi (x_idiff)));
    }

  float log_gamma_ratio;
  float y0 = math_narrow_eval (1 - x0_hi);
  float y0_eps = -x0_hi + (1 - y0) - x0_lo;
  float y = math_narrow_eval (1 - x);
  float 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.  */
  float log_gamma_high
    = (xdiff * __log1pf ((y0 - e_hi - e_lo + y0_eps) / e_hi)
       + (y - 0.5f + y_eps) * __log1pf (xdiff / y));
  /* Compute the sum of (B_2k / 2k(2k-1))(Y0^-(2k-1) - Y^-(2k-1)).  */
  float y0r = 1 / y0, yr = 1 / y;
  float y0r2 = y0r * y0r, yr2 = yr * yr;
  float rdiff = -xdiff / (y * y0);
  float bterm[NCOEFF];
  float dlast = rdiff, elast = rdiff * yr * (yr + y0r);
  bterm[0] = dlast * lgamma_coeff[0];
  for (size_t j = 1; j < NCOEFF; j++)
    {
      float dnext = dlast * y0r2 + elast;
      float enext = elast * yr2;
      bterm[j] = dnext * lgamma_coeff[j];
      dlast = dnext;
      elast = enext;
    }
  float 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;
}