1*05a0b428SJohn Marino /*
2*05a0b428SJohn Marino * ====================================================
3*05a0b428SJohn Marino * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4*05a0b428SJohn Marino *
5*05a0b428SJohn Marino * Developed at SunPro, a Sun Microsystems, Inc. business.
6*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this
7*05a0b428SJohn Marino * software is freely granted, provided that this notice
8*05a0b428SJohn Marino * is preserved.
9*05a0b428SJohn Marino * ====================================================
10*05a0b428SJohn Marino */
11*05a0b428SJohn Marino
12*05a0b428SJohn Marino /*
13*05a0b428SJohn Marino * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
14*05a0b428SJohn Marino *
15*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this software for any
16*05a0b428SJohn Marino * purpose with or without fee is hereby granted, provided that the above
17*05a0b428SJohn Marino * copyright notice and this permission notice appear in all copies.
18*05a0b428SJohn Marino *
19*05a0b428SJohn Marino * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
20*05a0b428SJohn Marino * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
21*05a0b428SJohn Marino * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
22*05a0b428SJohn Marino * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
23*05a0b428SJohn Marino * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
24*05a0b428SJohn Marino * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
25*05a0b428SJohn Marino * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
26*05a0b428SJohn Marino */
27*05a0b428SJohn Marino
28*05a0b428SJohn Marino /* lgammal(x)
29*05a0b428SJohn Marino * Reentrant version of the logarithm of the Gamma function
30*05a0b428SJohn Marino * with user provide pointer for the sign of Gamma(x).
31*05a0b428SJohn Marino *
32*05a0b428SJohn Marino * Method:
33*05a0b428SJohn Marino * 1. Argument Reduction for 0 < x <= 8
34*05a0b428SJohn Marino * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
35*05a0b428SJohn Marino * reduce x to a number in [1.5,2.5] by
36*05a0b428SJohn Marino * lgamma(1+s) = log(s) + lgamma(s)
37*05a0b428SJohn Marino * for example,
38*05a0b428SJohn Marino * lgamma(7.3) = log(6.3) + lgamma(6.3)
39*05a0b428SJohn Marino * = log(6.3*5.3) + lgamma(5.3)
40*05a0b428SJohn Marino * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
41*05a0b428SJohn Marino * 2. Polynomial approximation of lgamma around its
42*05a0b428SJohn Marino * minimun ymin=1.461632144968362245 to maintain monotonicity.
43*05a0b428SJohn Marino * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
44*05a0b428SJohn Marino * Let z = x-ymin;
45*05a0b428SJohn Marino * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
46*05a0b428SJohn Marino * 2. Rational approximation in the primary interval [2,3]
47*05a0b428SJohn Marino * We use the following approximation:
48*05a0b428SJohn Marino * s = x-2.0;
49*05a0b428SJohn Marino * lgamma(x) = 0.5*s + s*P(s)/Q(s)
50*05a0b428SJohn Marino * Our algorithms are based on the following observation
51*05a0b428SJohn Marino *
52*05a0b428SJohn Marino * zeta(2)-1 2 zeta(3)-1 3
53*05a0b428SJohn Marino * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
54*05a0b428SJohn Marino * 2 3
55*05a0b428SJohn Marino *
56*05a0b428SJohn Marino * where Euler = 0.5771... is the Euler constant, which is very
57*05a0b428SJohn Marino * close to 0.5.
58*05a0b428SJohn Marino *
59*05a0b428SJohn Marino * 3. For x>=8, we have
60*05a0b428SJohn Marino * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
61*05a0b428SJohn Marino * (better formula:
62*05a0b428SJohn Marino * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
63*05a0b428SJohn Marino * Let z = 1/x, then we approximation
64*05a0b428SJohn Marino * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
65*05a0b428SJohn Marino * by
66*05a0b428SJohn Marino * 3 5 11
67*05a0b428SJohn Marino * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
68*05a0b428SJohn Marino *
69*05a0b428SJohn Marino * 4. For negative x, since (G is gamma function)
70*05a0b428SJohn Marino * -x*G(-x)*G(x) = pi/sin(pi*x),
71*05a0b428SJohn Marino * we have
72*05a0b428SJohn Marino * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
73*05a0b428SJohn Marino * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
74*05a0b428SJohn Marino * Hence, for x<0, signgam = sign(sin(pi*x)) and
75*05a0b428SJohn Marino * lgamma(x) = log(|Gamma(x)|)
76*05a0b428SJohn Marino * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
77*05a0b428SJohn Marino * Note: one should avoid compute pi*(-x) directly in the
78*05a0b428SJohn Marino * computation of sin(pi*(-x)).
79*05a0b428SJohn Marino *
80*05a0b428SJohn Marino * 5. Special Cases
81*05a0b428SJohn Marino * lgamma(2+s) ~ s*(1-Euler) for tiny s
82*05a0b428SJohn Marino * lgamma(1)=lgamma(2)=0
83*05a0b428SJohn Marino * lgamma(x) ~ -log(x) for tiny x
84*05a0b428SJohn Marino * lgamma(0) = lgamma(inf) = inf
85*05a0b428SJohn Marino * lgamma(-integer) = +-inf
86*05a0b428SJohn Marino *
87*05a0b428SJohn Marino */
88*05a0b428SJohn Marino
89*05a0b428SJohn Marino #include <math.h>
90*05a0b428SJohn Marino
91*05a0b428SJohn Marino #include "math_private.h"
92*05a0b428SJohn Marino
93*05a0b428SJohn Marino static const long double
94*05a0b428SJohn Marino half = 0.5L,
95*05a0b428SJohn Marino one = 1.0L,
96*05a0b428SJohn Marino pi = 3.14159265358979323846264L,
97*05a0b428SJohn Marino two63 = 9.223372036854775808e18L,
98*05a0b428SJohn Marino
99*05a0b428SJohn Marino /* lgam(1+x) = 0.5 x + x a(x)/b(x)
100*05a0b428SJohn Marino -0.268402099609375 <= x <= 0
101*05a0b428SJohn Marino peak relative error 6.6e-22 */
102*05a0b428SJohn Marino a0 = -6.343246574721079391729402781192128239938E2L,
103*05a0b428SJohn Marino a1 = 1.856560238672465796768677717168371401378E3L,
104*05a0b428SJohn Marino a2 = 2.404733102163746263689288466865843408429E3L,
105*05a0b428SJohn Marino a3 = 8.804188795790383497379532868917517596322E2L,
106*05a0b428SJohn Marino a4 = 1.135361354097447729740103745999661157426E2L,
107*05a0b428SJohn Marino a5 = 3.766956539107615557608581581190400021285E0L,
108*05a0b428SJohn Marino
109*05a0b428SJohn Marino b0 = 8.214973713960928795704317259806842490498E3L,
110*05a0b428SJohn Marino b1 = 1.026343508841367384879065363925870888012E4L,
111*05a0b428SJohn Marino b2 = 4.553337477045763320522762343132210919277E3L,
112*05a0b428SJohn Marino b3 = 8.506975785032585797446253359230031874803E2L,
113*05a0b428SJohn Marino b4 = 6.042447899703295436820744186992189445813E1L,
114*05a0b428SJohn Marino /* b5 = 1.000000000000000000000000000000000000000E0 */
115*05a0b428SJohn Marino
116*05a0b428SJohn Marino
117*05a0b428SJohn Marino tc = 1.4616321449683623412626595423257213284682E0L,
118*05a0b428SJohn Marino tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
119*05a0b428SJohn Marino /* tt = (tail of tf), i.e. tf + tt has extended precision. */
120*05a0b428SJohn Marino tt = 3.3649914684731379602768989080467587736363E-18L,
121*05a0b428SJohn Marino /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
122*05a0b428SJohn Marino -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
123*05a0b428SJohn Marino
124*05a0b428SJohn Marino /* lgam (x + tc) = tf + tt + x g(x)/h(x)
125*05a0b428SJohn Marino - 0.230003726999612341262659542325721328468 <= x
126*05a0b428SJohn Marino <= 0.2699962730003876587373404576742786715318
127*05a0b428SJohn Marino peak relative error 2.1e-21 */
128*05a0b428SJohn Marino g0 = 3.645529916721223331888305293534095553827E-18L,
129*05a0b428SJohn Marino g1 = 5.126654642791082497002594216163574795690E3L,
130*05a0b428SJohn Marino g2 = 8.828603575854624811911631336122070070327E3L,
131*05a0b428SJohn Marino g3 = 5.464186426932117031234820886525701595203E3L,
132*05a0b428SJohn Marino g4 = 1.455427403530884193180776558102868592293E3L,
133*05a0b428SJohn Marino g5 = 1.541735456969245924860307497029155838446E2L,
134*05a0b428SJohn Marino g6 = 4.335498275274822298341872707453445815118E0L,
135*05a0b428SJohn Marino
136*05a0b428SJohn Marino h0 = 1.059584930106085509696730443974495979641E4L,
137*05a0b428SJohn Marino h1 = 2.147921653490043010629481226937850618860E4L,
138*05a0b428SJohn Marino h2 = 1.643014770044524804175197151958100656728E4L,
139*05a0b428SJohn Marino h3 = 5.869021995186925517228323497501767586078E3L,
140*05a0b428SJohn Marino h4 = 9.764244777714344488787381271643502742293E2L,
141*05a0b428SJohn Marino h5 = 6.442485441570592541741092969581997002349E1L,
142*05a0b428SJohn Marino /* h6 = 1.000000000000000000000000000000000000000E0 */
143*05a0b428SJohn Marino
144*05a0b428SJohn Marino
145*05a0b428SJohn Marino /* lgam (x+1) = -0.5 x + x u(x)/v(x)
146*05a0b428SJohn Marino -0.100006103515625 <= x <= 0.231639862060546875
147*05a0b428SJohn Marino peak relative error 1.3e-21 */
148*05a0b428SJohn Marino u0 = -8.886217500092090678492242071879342025627E1L,
149*05a0b428SJohn Marino u1 = 6.840109978129177639438792958320783599310E2L,
150*05a0b428SJohn Marino u2 = 2.042626104514127267855588786511809932433E3L,
151*05a0b428SJohn Marino u3 = 1.911723903442667422201651063009856064275E3L,
152*05a0b428SJohn Marino u4 = 7.447065275665887457628865263491667767695E2L,
153*05a0b428SJohn Marino u5 = 1.132256494121790736268471016493103952637E2L,
154*05a0b428SJohn Marino u6 = 4.484398885516614191003094714505960972894E0L,
155*05a0b428SJohn Marino
156*05a0b428SJohn Marino v0 = 1.150830924194461522996462401210374632929E3L,
157*05a0b428SJohn Marino v1 = 3.399692260848747447377972081399737098610E3L,
158*05a0b428SJohn Marino v2 = 3.786631705644460255229513563657226008015E3L,
159*05a0b428SJohn Marino v3 = 1.966450123004478374557778781564114347876E3L,
160*05a0b428SJohn Marino v4 = 4.741359068914069299837355438370682773122E2L,
161*05a0b428SJohn Marino v5 = 4.508989649747184050907206782117647852364E1L,
162*05a0b428SJohn Marino /* v6 = 1.000000000000000000000000000000000000000E0 */
163*05a0b428SJohn Marino
164*05a0b428SJohn Marino
165*05a0b428SJohn Marino /* lgam (x+2) = .5 x + x s(x)/r(x)
166*05a0b428SJohn Marino 0 <= x <= 1
167*05a0b428SJohn Marino peak relative error 7.2e-22 */
168*05a0b428SJohn Marino s0 = 1.454726263410661942989109455292824853344E6L,
169*05a0b428SJohn Marino s1 = -3.901428390086348447890408306153378922752E6L,
170*05a0b428SJohn Marino s2 = -6.573568698209374121847873064292963089438E6L,
171*05a0b428SJohn Marino s3 = -3.319055881485044417245964508099095984643E6L,
172*05a0b428SJohn Marino s4 = -7.094891568758439227560184618114707107977E5L,
173*05a0b428SJohn Marino s5 = -6.263426646464505837422314539808112478303E4L,
174*05a0b428SJohn Marino s6 = -1.684926520999477529949915657519454051529E3L,
175*05a0b428SJohn Marino
176*05a0b428SJohn Marino r0 = -1.883978160734303518163008696712983134698E7L,
177*05a0b428SJohn Marino r1 = -2.815206082812062064902202753264922306830E7L,
178*05a0b428SJohn Marino r2 = -1.600245495251915899081846093343626358398E7L,
179*05a0b428SJohn Marino r3 = -4.310526301881305003489257052083370058799E6L,
180*05a0b428SJohn Marino r4 = -5.563807682263923279438235987186184968542E5L,
181*05a0b428SJohn Marino r5 = -3.027734654434169996032905158145259713083E4L,
182*05a0b428SJohn Marino r6 = -4.501995652861105629217250715790764371267E2L,
183*05a0b428SJohn Marino /* r6 = 1.000000000000000000000000000000000000000E0 */
184*05a0b428SJohn Marino
185*05a0b428SJohn Marino
186*05a0b428SJohn Marino /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
187*05a0b428SJohn Marino x >= 8
188*05a0b428SJohn Marino Peak relative error 1.51e-21
189*05a0b428SJohn Marino w0 = LS2PI - 0.5 */
190*05a0b428SJohn Marino w0 = 4.189385332046727417803e-1L,
191*05a0b428SJohn Marino w1 = 8.333333333333331447505E-2L,
192*05a0b428SJohn Marino w2 = -2.777777777750349603440E-3L,
193*05a0b428SJohn Marino w3 = 7.936507795855070755671E-4L,
194*05a0b428SJohn Marino w4 = -5.952345851765688514613E-4L,
195*05a0b428SJohn Marino w5 = 8.412723297322498080632E-4L,
196*05a0b428SJohn Marino w6 = -1.880801938119376907179E-3L,
197*05a0b428SJohn Marino w7 = 4.885026142432270781165E-3L;
198*05a0b428SJohn Marino
199*05a0b428SJohn Marino static const long double zero = 0.0L;
200*05a0b428SJohn Marino
201*05a0b428SJohn Marino static long double
sin_pi(long double x)202*05a0b428SJohn Marino sin_pi(long double x)
203*05a0b428SJohn Marino {
204*05a0b428SJohn Marino long double y, z;
205*05a0b428SJohn Marino int n, ix;
206*05a0b428SJohn Marino u_int32_t se, i0, i1;
207*05a0b428SJohn Marino
208*05a0b428SJohn Marino GET_LDOUBLE_WORDS (se, i0, i1, x);
209*05a0b428SJohn Marino ix = se & 0x7fff;
210*05a0b428SJohn Marino ix = (ix << 16) | (i0 >> 16);
211*05a0b428SJohn Marino if (ix < 0x3ffd8000) /* 0.25 */
212*05a0b428SJohn Marino return sinl (pi * x);
213*05a0b428SJohn Marino y = -x; /* x is assume negative */
214*05a0b428SJohn Marino
215*05a0b428SJohn Marino /*
216*05a0b428SJohn Marino * argument reduction, make sure inexact flag not raised if input
217*05a0b428SJohn Marino * is an integer
218*05a0b428SJohn Marino */
219*05a0b428SJohn Marino z = floorl (y);
220*05a0b428SJohn Marino if (z != y)
221*05a0b428SJohn Marino { /* inexact anyway */
222*05a0b428SJohn Marino y *= 0.5;
223*05a0b428SJohn Marino y = 2.0*(y - floorl(y)); /* y = |x| mod 2.0 */
224*05a0b428SJohn Marino n = (int) (y*4.0);
225*05a0b428SJohn Marino }
226*05a0b428SJohn Marino else
227*05a0b428SJohn Marino {
228*05a0b428SJohn Marino if (ix >= 0x403f8000) /* 2^64 */
229*05a0b428SJohn Marino {
230*05a0b428SJohn Marino y = zero; n = 0; /* y must be even */
231*05a0b428SJohn Marino }
232*05a0b428SJohn Marino else
233*05a0b428SJohn Marino {
234*05a0b428SJohn Marino if (ix < 0x403e8000) /* 2^63 */
235*05a0b428SJohn Marino z = y + two63; /* exact */
236*05a0b428SJohn Marino GET_LDOUBLE_WORDS (se, i0, i1, z);
237*05a0b428SJohn Marino n = i1 & 1;
238*05a0b428SJohn Marino y = n;
239*05a0b428SJohn Marino n <<= 2;
240*05a0b428SJohn Marino }
241*05a0b428SJohn Marino }
242*05a0b428SJohn Marino
243*05a0b428SJohn Marino switch (n)
244*05a0b428SJohn Marino {
245*05a0b428SJohn Marino case 0:
246*05a0b428SJohn Marino y = sinl (pi * y);
247*05a0b428SJohn Marino break;
248*05a0b428SJohn Marino case 1:
249*05a0b428SJohn Marino case 2:
250*05a0b428SJohn Marino y = cosl (pi * (half - y));
251*05a0b428SJohn Marino break;
252*05a0b428SJohn Marino case 3:
253*05a0b428SJohn Marino case 4:
254*05a0b428SJohn Marino y = sinl (pi * (one - y));
255*05a0b428SJohn Marino break;
256*05a0b428SJohn Marino case 5:
257*05a0b428SJohn Marino case 6:
258*05a0b428SJohn Marino y = -cosl (pi * (y - 1.5));
259*05a0b428SJohn Marino break;
260*05a0b428SJohn Marino default:
261*05a0b428SJohn Marino y = sinl (pi * (y - 2.0));
262*05a0b428SJohn Marino break;
263*05a0b428SJohn Marino }
264*05a0b428SJohn Marino return -y;
265*05a0b428SJohn Marino }
266*05a0b428SJohn Marino
267*05a0b428SJohn Marino
268*05a0b428SJohn Marino long double
lgammal(long double x)269*05a0b428SJohn Marino lgammal(long double x)
270*05a0b428SJohn Marino {
271*05a0b428SJohn Marino long double t, y, z, nadj, p, p1, p2, q, r, w;
272*05a0b428SJohn Marino int i, ix;
273*05a0b428SJohn Marino u_int32_t se, i0, i1;
274*05a0b428SJohn Marino
275*05a0b428SJohn Marino signgam = 1;
276*05a0b428SJohn Marino GET_LDOUBLE_WORDS (se, i0, i1, x);
277*05a0b428SJohn Marino ix = se & 0x7fff;
278*05a0b428SJohn Marino
279*05a0b428SJohn Marino if ((ix | i0 | i1) == 0)
280*05a0b428SJohn Marino {
281*05a0b428SJohn Marino if (se & 0x8000)
282*05a0b428SJohn Marino signgam = -1;
283*05a0b428SJohn Marino return one / fabsl (x);
284*05a0b428SJohn Marino }
285*05a0b428SJohn Marino
286*05a0b428SJohn Marino ix = (ix << 16) | (i0 >> 16);
287*05a0b428SJohn Marino
288*05a0b428SJohn Marino /* purge off +-inf, NaN, +-0, and negative arguments */
289*05a0b428SJohn Marino if (ix >= 0x7fff0000)
290*05a0b428SJohn Marino return x * x;
291*05a0b428SJohn Marino
292*05a0b428SJohn Marino if (ix < 0x3fc08000) /* 2^-63 */
293*05a0b428SJohn Marino { /* |x|<2**-63, return -log(|x|) */
294*05a0b428SJohn Marino if (se & 0x8000)
295*05a0b428SJohn Marino {
296*05a0b428SJohn Marino signgam = -1;
297*05a0b428SJohn Marino return -logl (-x);
298*05a0b428SJohn Marino }
299*05a0b428SJohn Marino else
300*05a0b428SJohn Marino return -logl (x);
301*05a0b428SJohn Marino }
302*05a0b428SJohn Marino if (se & 0x8000)
303*05a0b428SJohn Marino {
304*05a0b428SJohn Marino t = sin_pi (x);
305*05a0b428SJohn Marino if (t == zero)
306*05a0b428SJohn Marino return one / fabsl (t); /* -integer */
307*05a0b428SJohn Marino nadj = logl (pi / fabsl (t * x));
308*05a0b428SJohn Marino if (t < zero)
309*05a0b428SJohn Marino signgam = -1;
310*05a0b428SJohn Marino x = -x;
311*05a0b428SJohn Marino }
312*05a0b428SJohn Marino
313*05a0b428SJohn Marino /* purge off 1 and 2 */
314*05a0b428SJohn Marino if ((((ix - 0x3fff8000) | i0 | i1) == 0)
315*05a0b428SJohn Marino || (((ix - 0x40008000) | i0 | i1) == 0))
316*05a0b428SJohn Marino r = 0;
317*05a0b428SJohn Marino else if (ix < 0x40008000) /* 2.0 */
318*05a0b428SJohn Marino {
319*05a0b428SJohn Marino /* x < 2.0 */
320*05a0b428SJohn Marino if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
321*05a0b428SJohn Marino {
322*05a0b428SJohn Marino /* lgamma(x) = lgamma(x+1) - log(x) */
323*05a0b428SJohn Marino r = -logl (x);
324*05a0b428SJohn Marino if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
325*05a0b428SJohn Marino {
326*05a0b428SJohn Marino y = x - one;
327*05a0b428SJohn Marino i = 0;
328*05a0b428SJohn Marino }
329*05a0b428SJohn Marino else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
330*05a0b428SJohn Marino {
331*05a0b428SJohn Marino y = x - (tc - one);
332*05a0b428SJohn Marino i = 1;
333*05a0b428SJohn Marino }
334*05a0b428SJohn Marino else
335*05a0b428SJohn Marino {
336*05a0b428SJohn Marino /* x < 0.23 */
337*05a0b428SJohn Marino y = x;
338*05a0b428SJohn Marino i = 2;
339*05a0b428SJohn Marino }
340*05a0b428SJohn Marino }
341*05a0b428SJohn Marino else
342*05a0b428SJohn Marino {
343*05a0b428SJohn Marino r = zero;
344*05a0b428SJohn Marino if (ix >= 0x3fffdda6) /* 1.73162841796875 */
345*05a0b428SJohn Marino {
346*05a0b428SJohn Marino /* [1.7316,2] */
347*05a0b428SJohn Marino y = x - 2.0;
348*05a0b428SJohn Marino i = 0;
349*05a0b428SJohn Marino }
350*05a0b428SJohn Marino else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
351*05a0b428SJohn Marino {
352*05a0b428SJohn Marino /* [1.23,1.73] */
353*05a0b428SJohn Marino y = x - tc;
354*05a0b428SJohn Marino i = 1;
355*05a0b428SJohn Marino }
356*05a0b428SJohn Marino else
357*05a0b428SJohn Marino {
358*05a0b428SJohn Marino /* [0.9, 1.23] */
359*05a0b428SJohn Marino y = x - one;
360*05a0b428SJohn Marino i = 2;
361*05a0b428SJohn Marino }
362*05a0b428SJohn Marino }
363*05a0b428SJohn Marino switch (i)
364*05a0b428SJohn Marino {
365*05a0b428SJohn Marino case 0:
366*05a0b428SJohn Marino p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
367*05a0b428SJohn Marino p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
368*05a0b428SJohn Marino r += half * y + y * p1/p2;
369*05a0b428SJohn Marino break;
370*05a0b428SJohn Marino case 1:
371*05a0b428SJohn Marino p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
372*05a0b428SJohn Marino p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
373*05a0b428SJohn Marino p = tt + y * p1/p2;
374*05a0b428SJohn Marino r += (tf + p);
375*05a0b428SJohn Marino break;
376*05a0b428SJohn Marino case 2:
377*05a0b428SJohn Marino p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
378*05a0b428SJohn Marino p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
379*05a0b428SJohn Marino r += (-half * y + p1 / p2);
380*05a0b428SJohn Marino }
381*05a0b428SJohn Marino }
382*05a0b428SJohn Marino else if (ix < 0x40028000) /* 8.0 */
383*05a0b428SJohn Marino {
384*05a0b428SJohn Marino /* x < 8.0 */
385*05a0b428SJohn Marino i = (int) x;
386*05a0b428SJohn Marino t = zero;
387*05a0b428SJohn Marino y = x - (double) i;
388*05a0b428SJohn Marino p = y *
389*05a0b428SJohn Marino (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
390*05a0b428SJohn Marino q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
391*05a0b428SJohn Marino r = half * y + p / q;
392*05a0b428SJohn Marino z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
393*05a0b428SJohn Marino switch (i)
394*05a0b428SJohn Marino {
395*05a0b428SJohn Marino case 7:
396*05a0b428SJohn Marino z *= (y + 6.0); /* FALLTHRU */
397*05a0b428SJohn Marino case 6:
398*05a0b428SJohn Marino z *= (y + 5.0); /* FALLTHRU */
399*05a0b428SJohn Marino case 5:
400*05a0b428SJohn Marino z *= (y + 4.0); /* FALLTHRU */
401*05a0b428SJohn Marino case 4:
402*05a0b428SJohn Marino z *= (y + 3.0); /* FALLTHRU */
403*05a0b428SJohn Marino case 3:
404*05a0b428SJohn Marino z *= (y + 2.0); /* FALLTHRU */
405*05a0b428SJohn Marino r += logl (z);
406*05a0b428SJohn Marino break;
407*05a0b428SJohn Marino }
408*05a0b428SJohn Marino }
409*05a0b428SJohn Marino else if (ix < 0x40418000) /* 2^66 */
410*05a0b428SJohn Marino {
411*05a0b428SJohn Marino /* 8.0 <= x < 2**66 */
412*05a0b428SJohn Marino t = logl (x);
413*05a0b428SJohn Marino z = one / x;
414*05a0b428SJohn Marino y = z * z;
415*05a0b428SJohn Marino w = w0 + z * (w1
416*05a0b428SJohn Marino + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
417*05a0b428SJohn Marino r = (x - half) * (t - one) + w;
418*05a0b428SJohn Marino }
419*05a0b428SJohn Marino else
420*05a0b428SJohn Marino /* 2**66 <= x <= inf */
421*05a0b428SJohn Marino r = x * (logl (x) - one);
422*05a0b428SJohn Marino if (se & 0x8000)
423*05a0b428SJohn Marino r = nadj - r;
424*05a0b428SJohn Marino return r;
425*05a0b428SJohn Marino }
426