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1 207 jeremybenn
 
2
/* @(#)er_lgamma.c 5.1 93/09/24 */
3
/*
4
 * ====================================================
5
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6
 *
7
 * Developed at SunPro, a Sun Microsystems, Inc. business.
8
 * Permission to use, copy, modify, and distribute this
9
 * software is freely granted, provided that this notice
10
 * is preserved.
11
 * ====================================================
12
 *
13
 */
14
 
15
/* __ieee754_lgamma_r(x, signgamp)
16
 * Reentrant version of the logarithm of the Gamma function
17
 * with user provide pointer for the sign of Gamma(x).
18
 *
19
 * Method:
20
 *   1. Argument Reduction for 0 < x <= 8
21
 *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
22
 *      reduce x to a number in [1.5,2.5] by
23
 *              lgamma(1+s) = log(s) + lgamma(s)
24
 *      for example,
25
 *              lgamma(7.3) = log(6.3) + lgamma(6.3)
26
 *                          = log(6.3*5.3) + lgamma(5.3)
27
 *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
28
 *   2. Polynomial approximation of lgamma around its
29
 *      minimun ymin=1.461632144968362245 to maintain monotonicity.
30
 *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
31
 *              Let z = x-ymin;
32
 *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
33
 *      where
34
 *              poly(z) is a 14 degree polynomial.
35
 *   2. Rational approximation in the primary interval [2,3]
36
 *      We use the following approximation:
37
 *              s = x-2.0;
38
 *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
39
 *      with accuracy
40
 *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
41
 *      Our algorithms are based on the following observation
42
 *
43
 *                             zeta(2)-1    2    zeta(3)-1    3
44
 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
45
 *                                 2                 3
46
 *
47
 *      where Euler = 0.5771... is the Euler constant, which is very
48
 *      close to 0.5.
49
 *
50
 *   3. For x>=8, we have
51
 *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
52
 *      (better formula:
53
 *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
54
 *      Let z = 1/x, then we approximation
55
 *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
56
 *      by
57
 *                                  3       5             11
58
 *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
59
 *      where
60
 *              |w - f(z)| < 2**-58.74
61
 *
62
 *   4. For negative x, since (G is gamma function)
63
 *              -x*G(-x)*G(x) = pi/sin(pi*x),
64
 *      we have
65
 *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
66
 *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
67
 *      Hence, for x<0, signgam = sign(sin(pi*x)) and
68
 *              lgamma(x) = log(|Gamma(x)|)
69
 *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
70
 *      Note: one should avoid compute pi*(-x) directly in the
71
 *            computation of sin(pi*(-x)).
72
 *
73
 *   5. Special Cases
74
 *              lgamma(2+s) ~ s*(1-Euler) for tiny s
75
 *              lgamma(1)=lgamma(2)=0
76
 *              lgamma(x) ~ -log(x) for tiny x
77
 *              lgamma(0) = lgamma(inf) = inf
78
 *              lgamma(-integer) = +-inf
79
 *
80
 */
81
 
82
#include "fdlibm.h"
83
 
84
#ifdef __STDC__
85
static const double
86
#else
87
static double
88
#endif
89
two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
90
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
91
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
92
pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
93
a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
94
a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
95
a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
96
a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
97
a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
98
a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
99
a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
100
a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
101
a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
102
a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
103
a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
104
a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
105
tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
106
tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
107
/* tt = -(tail of tf) */
108
tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
109
t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
110
t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
111
t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
112
t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
113
t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
114
t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
115
t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
116
t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
117
t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
118
t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
119
t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
120
t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
121
t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
122
t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
123
t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
124
u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
125
u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
126
u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
127
u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
128
u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
129
u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
130
v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
131
v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
132
v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
133
v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
134
v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
135
s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
136
s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
137
s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
138
s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
139
s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
140
s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
141
s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
142
r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
143
r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
144
r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
145
r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
146
r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
147
r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
148
w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
149
w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
150
w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
151
w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
152
w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
153
w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
154
w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
155
 
156
#ifdef __STDC__
157
static const double zero=  0.00000000000000000000e+00;
158
#else
159
static double zero=  0.00000000000000000000e+00;
160
#endif
161
 
162
#ifdef __STDC__
163
        static double sin_pi(double x)
164
#else
165
        static double sin_pi(x)
166
        double x;
167
#endif
168
{
169
        double y,z;
170
        __int32_t n,ix;
171
 
172
        GET_HIGH_WORD(ix,x);
173
        ix &= 0x7fffffff;
174
 
175
        if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
176
        y = -x;         /* x is assume negative */
177
 
178
    /*
179
     * argument reduction, make sure inexact flag not raised if input
180
     * is an integer
181
     */
182
        z = floor(y);
183
        if(z!=y) {                              /* inexact anyway */
184
            y  *= 0.5;
185
            y   = 2.0*(y - floor(y));           /* y = |x| mod 2.0 */
186
            n   = (__int32_t) (y*4.0);
187
        } else {
188
            if(ix>=0x43400000) {
189
                y = zero; n = 0;                 /* y must be even */
190
            } else {
191
                if(ix<0x43300000) z = y+two52;  /* exact */
192
                GET_LOW_WORD(n,z);
193
                n &= 1;
194
                y  = n;
195
                n<<= 2;
196
            }
197
        }
198
        switch (n) {
199
            case 0:   y =  __kernel_sin(pi*y,zero,0); break;
200
            case 1:
201
            case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
202
            case 3:
203
            case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
204
            case 5:
205
            case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
206
            default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
207
            }
208
        return -y;
209
}
210
 
211
 
212
#ifdef __STDC__
213
        double __ieee754_lgamma_r(double x, int *signgamp)
214
#else
215
        double __ieee754_lgamma_r(x,signgamp)
216
        double x; int *signgamp;
217
#endif
218
{
219
        double t,y,z,nadj,p,p1,p2,p3,q,r,w;
220
        __int32_t i,hx,lx,ix;
221
 
222
        EXTRACT_WORDS(hx,lx,x);
223
 
224
    /* purge off +-inf, NaN, +-0, and negative arguments */
225
        *signgamp = 1;
226
        ix = hx&0x7fffffff;
227
        if(ix>=0x7ff00000) return x*x;
228
        if((ix|lx)==0) return one/zero;
229
        if(ix<0x3b900000) {     /* |x|<2**-70, return -log(|x|) */
230
            if(hx<0) {
231
                *signgamp = -1;
232
                return -__ieee754_log(-x);
233
            } else return -__ieee754_log(x);
234
        }
235
        if(hx<0) {
236
            if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
237
                return one/zero;
238
            t = sin_pi(x);
239
            if(t==zero) return one/zero; /* -integer */
240
            nadj = __ieee754_log(pi/fabs(t*x));
241
            if(t<zero) *signgamp = -1;
242
            x = -x;
243
        }
244
 
245
    /* purge off 1 and 2 */
246
        if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
247
    /* for x < 2.0 */
248
        else if(ix<0x40000000) {
249
            if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */
250
                r = -__ieee754_log(x);
251
                if(ix>=0x3FE76944) {y = one-x; i= 0;}
252
                else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
253
                else {y = x; i=2;}
254
            } else {
255
                r = zero;
256
                if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
257
                else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
258
                else {y=x-one;i=2;}
259
            }
260
            switch(i) {
261
              case 0:
262
                z = y*y;
263
                p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
264
                p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
265
                p  = y*p1+p2;
266
                r  += (p-0.5*y); break;
267
              case 1:
268
                z = y*y;
269
                w = z*y;
270
                p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
271
                p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
272
                p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
273
                p  = z*p1-(tt-w*(p2+y*p3));
274
                r += (tf + p); break;
275
              case 2:
276
                p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
277
                p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
278
                r += (-0.5*y + p1/p2);
279
            }
280
        }
281
        else if(ix<0x40200000) {                        /* x < 8.0 */
282
            i = (__int32_t)x;
283
            t = zero;
284
            y = x-(double)i;
285
            p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
286
            q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
287
            r = half*y+p/q;
288
            z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
289
            switch(i) {
290
            case 7: z *= (y+6.0);       /* FALLTHRU */
291
            case 6: z *= (y+5.0);       /* FALLTHRU */
292
            case 5: z *= (y+4.0);       /* FALLTHRU */
293
            case 4: z *= (y+3.0);       /* FALLTHRU */
294
            case 3: z *= (y+2.0);       /* FALLTHRU */
295
                    r += __ieee754_log(z); break;
296
            }
297
    /* 8.0 <= x < 2**58 */
298
        } else if (ix < 0x43900000) {
299
            t = __ieee754_log(x);
300
            z = one/x;
301
            y = z*z;
302
            w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
303
            r = (x-half)*(t-one)+w;
304
        } else
305
    /* 2**58 <= x <= inf */
306
            r =  x*(__ieee754_log(x)-one);
307
        if(hx<0) r = nadj - r;
308
        return r;
309
}

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