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e_lgamma_r.c
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00001 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
00002  *
00003  * ***** BEGIN LICENSE BLOCK *****
00004  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
00005  *
00006  * The contents of this file are subject to the Mozilla Public License Version
00007  * 1.1 (the "License"); you may not use this file except in compliance with
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00010  *
00011  * Software distributed under the License is distributed on an "AS IS" basis,
00012  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
00013  * for the specific language governing rights and limitations under the
00014  * License.
00015  *
00016  * The Original Code is Mozilla Communicator client code, released
00017  * March 31, 1998.
00018  *
00019  * The Initial Developer of the Original Code is
00020  * Sun Microsystems, Inc.
00021  * Portions created by the Initial Developer are Copyright (C) 1998
00022  * the Initial Developer. All Rights Reserved.
00023  *
00024  * Contributor(s):
00025  *
00026  * Alternatively, the contents of this file may be used under the terms of
00027  * either of the GNU General Public License Version 2 or later (the "GPL"),
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00038  * ***** END LICENSE BLOCK ***** */
00039 
00040 /* @(#)e_lgamma_r.c 1.3 95/01/18 */
00041 /*
00042  * ====================================================
00043  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00044  *
00045  * Developed at SunSoft, a Sun Microsystems, Inc. business.
00046  * Permission to use, copy, modify, and distribute this
00047  * software is freely granted, provided that this notice 
00048  * is preserved.
00049  * ====================================================
00050  *
00051  */
00052 
00053 /* __ieee754_lgamma_r(x, signgamp)
00054  * Reentrant version of the logarithm of the Gamma function 
00055  * with user provide pointer for the sign of Gamma(x). 
00056  *
00057  * Method:
00058  *   1. Argument Reduction for 0 < x <= 8
00059  *     Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 
00060  *     reduce x to a number in [1.5,2.5] by
00061  *            lgamma(1+s) = log(s) + lgamma(s)
00062  *     for example,
00063  *            lgamma(7.3) = log(6.3) + lgamma(6.3)
00064  *                       = log(6.3*5.3) + lgamma(5.3)
00065  *                       = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
00066  *   2. Polynomial approximation of lgamma around its
00067  *     minimun ymin=1.461632144968362245 to maintain monotonicity.
00068  *     On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
00069  *            Let z = x-ymin;
00070  *            lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
00071  *     where
00072  *            poly(z) is a 14 degree polynomial.
00073  *   2. Rational approximation in the primary interval [2,3]
00074  *     We use the following approximation:
00075  *            s = x-2.0;
00076  *            lgamma(x) = 0.5*s + s*P(s)/Q(s)
00077  *     with accuracy
00078  *            |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
00079  *     Our algorithms are based on the following observation
00080  *
00081  *                             zeta(2)-1    2    zeta(3)-1    3
00082  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
00083  *                                 2                 3
00084  *
00085  *     where Euler = 0.5771... is the Euler constant, which is very
00086  *     close to 0.5.
00087  *
00088  *   3. For x>=8, we have
00089  *     lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
00090  *     (better formula:
00091  *        lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
00092  *     Let z = 1/x, then we approximation
00093  *            f(z) = lgamma(x) - (x-0.5)(log(x)-1)
00094  *     by
00095  *                              3       5             11
00096  *            w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
00097  *     where 
00098  *            |w - f(z)| < 2**-58.74
00099  *            
00100  *   4. For negative x, since (G is gamma function)
00101  *            -x*G(-x)*G(x) = pi/sin(pi*x),
00102  *     we have
00103  *            G(x) = pi/(sin(pi*x)*(-x)*G(-x))
00104  *     since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
00105  *     Hence, for x<0, signgam = sign(sin(pi*x)) and 
00106  *            lgamma(x) = log(|Gamma(x)|)
00107  *                     = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
00108  *     Note: one should avoid compute pi*(-x) directly in the 
00109  *           computation of sin(pi*(-x)).
00110  *            
00111  *   5. Special Cases
00112  *            lgamma(2+s) ~ s*(1-Euler) for tiny s
00113  *            lgamma(1)=lgamma(2)=0
00114  *            lgamma(x) ~ -log(x) for tiny x
00115  *            lgamma(0) = lgamma(inf) = inf
00116  *            lgamma(-integer) = +-inf
00117  *     
00118  */
00119 
00120 #include "fdlibm.h"
00121 
00122 #ifdef __STDC__
00123 static const double 
00124 #else
00125 static double 
00126 #endif
00127 two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
00128 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
00129 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00130 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
00131 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
00132 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
00133 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
00134 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
00135 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
00136 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
00137 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
00138 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
00139 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
00140 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
00141 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
00142 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
00143 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
00144 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
00145 /* tt = -(tail of tf) */
00146 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
00147 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
00148 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
00149 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
00150 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
00151 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
00152 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
00153 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
00154 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
00155 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
00156 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
00157 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
00158 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
00159 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
00160 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
00161 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
00162 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
00163 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
00164 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
00165 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
00166 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
00167 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
00168 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
00169 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
00170 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
00171 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
00172 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
00173 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
00174 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
00175 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
00176 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
00177 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
00178 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
00179 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
00180 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
00181 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
00182 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
00183 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
00184 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
00185 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
00186 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
00187 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
00188 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
00189 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
00190 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
00191 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
00192 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
00193 
00194 static double zero=  0.00000000000000000000e+00;
00195 
00196 #ifdef __STDC__
00197        static double sin_pi(double x)
00198 #else
00199        static double sin_pi(x)
00200        double x;
00201 #endif
00202 {
00203         fd_twoints u;
00204        double y,z;
00205        int n,ix;
00206 
00207         u.d = x;
00208        ix = 0x7fffffff&__HI(u);
00209 
00210        if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
00211        y = -x;              /* x is assume negative */
00212 
00213     /*
00214      * argument reduction, make sure inexact flag not raised if input
00215      * is an integer
00216      */
00217        z = fd_floor(y);
00218        if(z!=y) {                         /* inexact anyway */
00219            y  *= 0.5;
00220            y   = 2.0*(y - fd_floor(y));          /* y = |x| mod 2.0 */
00221            n   = (int) (y*4.0);
00222        } else {
00223             if(ix>=0x43400000) {
00224                 y = zero; n = 0;                 /* y must be even */
00225             } else {
00226                 if(ix<0x43300000) z = y+two52;   /* exact */
00227                 u.d = z;
00228                 n   = __LO(u)&1;        /* lower word of z */
00229                 y  = n;
00230                 n<<= 2;
00231             }
00232         }
00233        switch (n) {
00234            case 0:   y =  __kernel_sin(pi*y,zero,0); break;
00235            case 1:   
00236            case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
00237            case 3:  
00238            case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
00239            case 5:
00240            case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
00241            default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
00242            }
00243        return -y;
00244 }
00245 
00246 
00247 #ifdef __STDC__
00248        double __ieee754_lgamma_r(double x, int *signgamp)
00249 #else
00250        double __ieee754_lgamma_r(x,signgamp)
00251        double x; int *signgamp;
00252 #endif
00253 {
00254         fd_twoints u;
00255        double t,y,z,nadj,p,p1,p2,p3,q,r,w;
00256        int i,hx,lx,ix;
00257 
00258         u.d = x;
00259        hx = __HI(u);
00260        lx = __LO(u);
00261 
00262     /* purge off +-inf, NaN, +-0, and negative arguments */
00263        *signgamp = 1;
00264        ix = hx&0x7fffffff;
00265        if(ix>=0x7ff00000) return x*x;
00266        if((ix|lx)==0) return one/zero;
00267        if(ix<0x3b900000) {  /* |x|<2**-70, return -log(|x|) */
00268            if(hx<0) {
00269                *signgamp = -1;
00270                return -__ieee754_log(-x);
00271            } else return -__ieee754_log(x);
00272        }
00273        if(hx<0) {
00274            if(ix>=0x43300000)      /* |x|>=2**52, must be -integer */
00275               return one/zero;
00276            t = sin_pi(x);
00277            if(t==zero) return one/zero; /* -integer */
00278            nadj = __ieee754_log(pi/fd_fabs(t*x));
00279            if(t<zero) *signgamp = -1;
00280            x = -x;
00281        }
00282 
00283     /* purge off 1 and 2 */
00284        if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
00285     /* for x < 2.0 */
00286        else if(ix<0x40000000) {
00287            if(ix<=0x3feccccc) {    /* lgamma(x) = lgamma(x+1)-log(x) */
00288               r = -__ieee754_log(x);
00289               if(ix>=0x3FE76944) {y = one-x; i= 0;}
00290               else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
00291               else {y = x; i=2;}
00292            } else {
00293               r = zero;
00294                if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
00295                else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
00296               else {y=x-one;i=2;}
00297            }
00298            switch(i) {
00299              case 0:
00300               z = y*y;
00301               p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
00302               p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
00303               p  = y*p1+p2;
00304               r  += (p-0.5*y); break;
00305              case 1:
00306               z = y*y;
00307               w = z*y;
00308               p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));      /* parallel comp */
00309               p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
00310               p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
00311               p  = z*p1-(tt-w*(p2+y*p3));
00312               r += (tf + p); break;
00313              case 2: 
00314               p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
00315               p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
00316               r += (-0.5*y + p1/p2);
00317            }
00318        }
00319        else if(ix<0x40200000) {                  /* x < 8.0 */
00320            i = (int)x;
00321            t = zero;
00322            y = x-(double)i;
00323            p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
00324            q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
00325            r = half*y+p/q;
00326            z = one;  /* lgamma(1+s) = log(s) + lgamma(s) */
00327            switch(i) {
00328            case 7: z *= (y+6.0);   /* FALLTHRU */
00329            case 6: z *= (y+5.0);   /* FALLTHRU */
00330            case 5: z *= (y+4.0);   /* FALLTHRU */
00331            case 4: z *= (y+3.0);   /* FALLTHRU */
00332            case 3: z *= (y+2.0);   /* FALLTHRU */
00333                   r += __ieee754_log(z); break;
00334            }
00335     /* 8.0 <= x < 2**58 */
00336        } else if (ix < 0x43900000) {
00337            t = __ieee754_log(x);
00338            z = one/x;
00339            y = z*z;
00340            w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
00341            r = (x-half)*(t-one)+w;
00342        } else 
00343     /* 2**58 <= x <= inf */
00344            r =  x*(__ieee754_log(x)-one);
00345        if(hx<0) r = nadj - r;
00346        return r;
00347 }