GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/ld80/e_lgammal.c Lines: 38 109 34.9 %
Date: 2017-11-13 Branches: 21 64 32.8 %

Line Branch Exec Source
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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/*
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 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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 *
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 * Permission to use, copy, modify, and distribute this software for any
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 * purpose with or without fee is hereby granted, provided that the above
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 * copyright notice and this permission notice appear in all copies.
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 *
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 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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 */
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/* lgammal(x)
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 * Reentrant version of the logarithm of the Gamma function
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 * with user provide pointer for the sign of Gamma(x).
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 *
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 * Method:
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 *   1. Argument Reduction for 0 < x <= 8
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 *	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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 *	reduce x to a number in [1.5,2.5] by
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 *		lgamma(1+s) = log(s) + lgamma(s)
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 *	for example,
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 *		lgamma(7.3) = log(6.3) + lgamma(6.3)
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 *			    = log(6.3*5.3) + lgamma(5.3)
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 *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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 *   2. Polynomial approximation of lgamma around its
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 *	minimun ymin=1.461632144968362245 to maintain monotonicity.
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 *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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 *		Let z = x-ymin;
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 *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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 *   2. Rational approximation in the primary interval [2,3]
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 *	We use the following approximation:
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 *		s = x-2.0;
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 *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
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 *	Our algorithms are based on the following observation
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 *
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 *                             zeta(2)-1    2    zeta(3)-1    3
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 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
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 *                                 2                 3
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 *
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 *	where Euler = 0.5771... is the Euler constant, which is very
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 *	close to 0.5.
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 *
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 *   3. For x>=8, we have
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 *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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 *	(better formula:
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 *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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 *	Let z = 1/x, then we approximation
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 *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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 *	by
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 *				    3       5             11
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 *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
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 *
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 *   4. For negative x, since (G is gamma function)
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 *		-x*G(-x)*G(x) = pi/sin(pi*x),
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 *	we have
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 *		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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 *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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 *	Hence, for x<0, signgam = sign(sin(pi*x)) and
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 *		lgamma(x) = log(|Gamma(x)|)
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 *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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 *	Note: one should avoid compute pi*(-x) directly in the
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 *	      computation of sin(pi*(-x)).
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 *
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 *   5. Special Cases
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 *		lgamma(2+s) ~ s*(1-Euler) for tiny s
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 *		lgamma(1)=lgamma(2)=0
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 *		lgamma(x) ~ -log(x) for tiny x
84
 *		lgamma(0) = lgamma(inf) = inf
85
 *		lgamma(-integer) = +-inf
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 *
87
 */
88
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#include <math.h>
90
91
#include "math_private.h"
92
93
static const long double
94
  half = 0.5L,
95
  one = 1.0L,
96
  pi = 3.14159265358979323846264L,
97
  two63 = 9.223372036854775808e18L,
98
99
  /* lgam(1+x) = 0.5 x + x a(x)/b(x)
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     -0.268402099609375 <= x <= 0
101
     peak relative error 6.6e-22 */
102
  a0 = -6.343246574721079391729402781192128239938E2L,
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  a1 =  1.856560238672465796768677717168371401378E3L,
104
  a2 =  2.404733102163746263689288466865843408429E3L,
105
  a3 =  8.804188795790383497379532868917517596322E2L,
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  a4 =  1.135361354097447729740103745999661157426E2L,
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  a5 =  3.766956539107615557608581581190400021285E0L,
108
109
  b0 =  8.214973713960928795704317259806842490498E3L,
110
  b1 =  1.026343508841367384879065363925870888012E4L,
111
  b2 =  4.553337477045763320522762343132210919277E3L,
112
  b3 =  8.506975785032585797446253359230031874803E2L,
113
  b4 =  6.042447899703295436820744186992189445813E1L,
114
  /* b5 =  1.000000000000000000000000000000000000000E0 */
115
116
117
  tc =  1.4616321449683623412626595423257213284682E0L,
118
  tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
119
/* tt = (tail of tf), i.e. tf + tt has extended precision. */
120
  tt = 3.3649914684731379602768989080467587736363E-18L,
121
  /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
122
-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
123
124
  /* lgam (x + tc) = tf + tt + x g(x)/h(x)
125
     - 0.230003726999612341262659542325721328468 <= x
126
     <= 0.2699962730003876587373404576742786715318
127
     peak relative error 2.1e-21 */
128
  g0 = 3.645529916721223331888305293534095553827E-18L,
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  g1 = 5.126654642791082497002594216163574795690E3L,
130
  g2 = 8.828603575854624811911631336122070070327E3L,
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  g3 = 5.464186426932117031234820886525701595203E3L,
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  g4 = 1.455427403530884193180776558102868592293E3L,
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  g5 = 1.541735456969245924860307497029155838446E2L,
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  g6 = 4.335498275274822298341872707453445815118E0L,
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136
  h0 = 1.059584930106085509696730443974495979641E4L,
137
  h1 =  2.147921653490043010629481226937850618860E4L,
138
  h2 = 1.643014770044524804175197151958100656728E4L,
139
  h3 =  5.869021995186925517228323497501767586078E3L,
140
  h4 =  9.764244777714344488787381271643502742293E2L,
141
  h5 =  6.442485441570592541741092969581997002349E1L,
142
  /* h6 = 1.000000000000000000000000000000000000000E0 */
143
144
145
  /* lgam (x+1) = -0.5 x + x u(x)/v(x)
146
     -0.100006103515625 <= x <= 0.231639862060546875
147
     peak relative error 1.3e-21 */
148
  u0 = -8.886217500092090678492242071879342025627E1L,
149
  u1 =  6.840109978129177639438792958320783599310E2L,
150
  u2 =  2.042626104514127267855588786511809932433E3L,
151
  u3 =  1.911723903442667422201651063009856064275E3L,
152
  u4 =  7.447065275665887457628865263491667767695E2L,
153
  u5 =  1.132256494121790736268471016493103952637E2L,
154
  u6 =  4.484398885516614191003094714505960972894E0L,
155
156
  v0 =  1.150830924194461522996462401210374632929E3L,
157
  v1 =  3.399692260848747447377972081399737098610E3L,
158
  v2 =  3.786631705644460255229513563657226008015E3L,
159
  v3 =  1.966450123004478374557778781564114347876E3L,
160
  v4 =  4.741359068914069299837355438370682773122E2L,
161
  v5 =  4.508989649747184050907206782117647852364E1L,
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  /* v6 =  1.000000000000000000000000000000000000000E0 */
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164
165
  /* lgam (x+2) = .5 x + x s(x)/r(x)
166
     0 <= x <= 1
167
     peak relative error 7.2e-22 */
168
  s0 =  1.454726263410661942989109455292824853344E6L,
169
  s1 = -3.901428390086348447890408306153378922752E6L,
170
  s2 = -6.573568698209374121847873064292963089438E6L,
171
  s3 = -3.319055881485044417245964508099095984643E6L,
172
  s4 = -7.094891568758439227560184618114707107977E5L,
173
  s5 = -6.263426646464505837422314539808112478303E4L,
174
  s6 = -1.684926520999477529949915657519454051529E3L,
175
176
  r0 = -1.883978160734303518163008696712983134698E7L,
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  r1 = -2.815206082812062064902202753264922306830E7L,
178
  r2 = -1.600245495251915899081846093343626358398E7L,
179
  r3 = -4.310526301881305003489257052083370058799E6L,
180
  r4 = -5.563807682263923279438235987186184968542E5L,
181
  r5 = -3.027734654434169996032905158145259713083E4L,
182
  r6 = -4.501995652861105629217250715790764371267E2L,
183
  /* r6 =  1.000000000000000000000000000000000000000E0 */
184
185
186
/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
187
   x >= 8
188
   Peak relative error 1.51e-21
189
   w0 = LS2PI - 0.5 */
190
  w0 =  4.189385332046727417803e-1L,
191
  w1 =  8.333333333333331447505E-2L,
192
  w2 = -2.777777777750349603440E-3L,
193
  w3 =  7.936507795855070755671E-4L,
194
  w4 = -5.952345851765688514613E-4L,
195
  w5 =  8.412723297322498080632E-4L,
196
  w6 = -1.880801938119376907179E-3L,
197
  w7 =  4.885026142432270781165E-3L;
198
199
static const long double zero = 0.0L;
200
201
static long double
202
sin_pi(long double x)
203
{
204
  long double y, z;
205
  int n, ix;
206
  u_int32_t se, i0, i1;
207
208
  GET_LDOUBLE_WORDS (se, i0, i1, x);
209
  ix = se & 0x7fff;
210
  ix = (ix << 16) | (i0 >> 16);
211
  if (ix < 0x3ffd8000) /* 0.25 */
212
    return sinl (pi * x);
213
  y = -x;			/* x is assume negative */
214
215
  /*
216
   * argument reduction, make sure inexact flag not raised if input
217
   * is an integer
218
   */
219
  z = floorl (y);
220
  if (z != y)
221
    {				/* inexact anyway */
222
      y  *= 0.5;
223
      y = 2.0*(y - floorl(y));		/* y = |x| mod 2.0 */
224
      n = (int) (y*4.0);
225
    }
226
  else
227
    {
228
      if (ix >= 0x403f8000)  /* 2^64 */
229
	{
230
	  y = zero; n = 0;		/* y must be even */
231
	}
232
      else
233
	{
234
	if (ix < 0x403e8000)  /* 2^63 */
235
	  z = y + two63;	/* exact */
236
	GET_LDOUBLE_WORDS (se, i0, i1, z);
237
	n = i1 & 1;
238
	y  = n;
239
	n <<= 2;
240
      }
241
    }
242
243
  switch (n)
244
    {
245
    case 0:
246
      y = sinl (pi * y);
247
      break;
248
    case 1:
249
    case 2:
250
      y = cosl (pi * (half - y));
251
      break;
252
    case 3:
253
    case 4:
254
      y = sinl (pi * (one - y));
255
      break;
256
    case 5:
257
    case 6:
258
      y = -cosl (pi * (y - 1.5));
259
      break;
260
    default:
261
      y = sinl (pi * (y - 2.0));
262
      break;
263
    }
264
  return -y;
265
}
266
267
268
long double
269
lgammal(long double x)
270
{
271
  long double t, y, z, nadj, p, p1, p2, q, r, w;
272
  int i, ix;
273
  u_int32_t se, i0, i1;
274
275
36
  signgam = 1;
276
18
  GET_LDOUBLE_WORDS (se, i0, i1, x);
277
18
  ix = se & 0x7fff;
278
279
18
  if ((ix | i0 | i1) == 0)
280
    {
281
6
      if (se & 0x8000)
282
3
	signgam = -1;
283
6
      return one / fabsl (x);
284
    }
285
286
12
  ix = (ix << 16) | (i0 >> 16);
287
288
  /* purge off +-inf, NaN, +-0, and negative arguments */
289
12
  if (ix >= 0x7fff0000)
290
6
    return x * x;
291
292
6
  if (ix < 0x3fc08000) /* 2^-63 */
293
    {				/* |x|<2**-63, return -log(|x|) */
294
      if (se & 0x8000)
295
	{
296
	  signgam = -1;
297
	  return -logl (-x);
298
	}
299
      else
300
	return -logl (x);
301
    }
302
6
  if (se & 0x8000)
303
    {
304
      t = sin_pi (x);
305
      if (t == zero)
306
	return one / fabsl (t);	/* -integer */
307
      nadj = logl (pi / fabsl (t * x));
308
      if (t < zero)
309
	signgam = -1;
310
      x = -x;
311
    }
312
313
  /* purge off 1 and 2 */
314
12
  if ((((ix - 0x3fff8000) | i0 | i1) == 0)
315
12
      || (((ix - 0x40008000) | i0 | i1) == 0))
316
    r = 0;
317
6
  else if (ix < 0x40008000) /* 2.0 */
318
    {
319
      /* x < 2.0 */
320
3
      if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
321
	{
322
	  /* lgamma(x) = lgamma(x+1) - log(x) */
323
	  r = -logl (x);
324
	  if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
325
	    {
326
	      y = x - one;
327
	      i = 0;
328
	    }
329
	  else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
330
	    {
331
	      y = x - (tc - one);
332
	      i = 1;
333
	    }
334
	  else
335
	    {
336
	      /* x < 0.23 */
337
	      y = x;
338
	      i = 2;
339
	    }
340
	}
341
      else
342
	{
343
	  r = zero;
344
3
	  if (ix >= 0x3fffdda6) /* 1.73162841796875 */
345
	    {
346
	      /* [1.7316,2] */
347
	      y = x - 2.0;
348
	      i = 0;
349
	    }
350
3
	  else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
351
	    {
352
	      /* [1.23,1.73] */
353
	      y = x - tc;
354
	      i = 1;
355
	    }
356
	  else
357
	    {
358
	      /* [0.9, 1.23] */
359
3
	      y = x - one;
360
	      i = 2;
361
	    }
362
	}
363

6
      switch (i)
364
	{
365
	case 0:
366
	  p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
367
	  p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
368
	  r += half * y + y * p1/p2;
369
	  break;
370
	case 1:
371
    p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
372
    p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
373
    p = tt + y * p1/p2;
374
	  r += (tf + p);
375
	  break;
376
	case 2:
377
3
 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
378
3
      p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
379
3
	  r += (-half * y + p1 / p2);
380
3
	}
381
    }
382
3
  else if (ix < 0x40028000) /* 8.0 */
383
    {
384
      /* x < 8.0 */
385
6
      i = (int) x;
386
      t = zero;
387
6
      y = x - (double) i;
388
6
  p = y *
389
6
     (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
390
6
  q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
391
6
      r = half * y + p / q;
392
      z = one;			/* lgamma(1+s) = log(s) + lgamma(s) */
393

6
      switch (i)
394
	{
395
	case 7:
396
	  z *= (y + 6.0);	/* FALLTHRU */
397
	case 6:
398
	  z *= (y + 5.0);	/* FALLTHRU */
399
	case 5:
400
	  z *= (y + 4.0);	/* FALLTHRU */
401
	case 4:
402
	  z *= (y + 3.0);	/* FALLTHRU */
403
	case 3:
404
3
	  z *= (y + 2.0);	/* FALLTHRU */
405
3
	  r += logl (z);
406
3
	  break;
407
	}
408
    }
409
  else if (ix < 0x40418000) /* 2^66 */
410
    {
411
      /* 8.0 <= x < 2**66 */
412
      t = logl (x);
413
      z = one / x;
414
      y = z * z;
415
      w = w0 + z * (w1
416
	  + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
417
      r = (x - half) * (t - one) + w;
418
    }
419
  else
420
    /* 2**66 <= x <= inf */
421
    r = x * (logl (x) - one);
422
6
  if (se & 0x8000)
423
    r = nadj - r;
424
6
  return r;
425
18
}
426
DEF_STD(lgammal);