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/* @(#)er_lgamma.c 5.1 93/09/24 */

/*

 * ====================================================

 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.

 *

 * Developed at SunPro, a Sun Microsystems, Inc. business.

 * Permission to use, copy, modify, and distribute this

 * software is freely granted, provided that this notice

 * is preserved.

 * ====================================================

 *

 */

/*

FUNCTION

        <<gamma>>, <<gammaf>>, <<lgamma>>, <<lgammaf>>, <<gamma_r>>,

        <<gammaf_r>>, <<lgamma_r>>, <<lgammaf_r>>---logarithmic gamma

        function

INDEX

gamma

INDEX

gammaf

INDEX

lgamma

INDEX

lgammaf

INDEX

gamma_r

INDEX

gammaf_r

INDEX

lgamma_r

INDEX

lgammaf_r

ANSI_SYNOPSIS

#include <math.h>

double gamma(double <[x]>);

float gammaf(float <[x]>);

double lgamma(double <[x]>);

float lgammaf(float <[x]>);

double gamma_r(double <[x]>, int *<[signgamp]>);

float gammaf_r(float <[x]>, int *<[signgamp]>);

double lgamma_r(double <[x]>, int *<[signgamp]>);

float lgammaf_r(float <[x]>, int *<[signgamp]>);

TRAD_SYNOPSIS

#include <math.h>

double gamma(<[x]>)

double <[x]>;

float gammaf(<[x]>)

float <[x]>;

double lgamma(<[x]>)

double <[x]>;

float lgammaf(<[x]>)

float <[x]>;

double gamma_r(<[x]>, <[signgamp]>)

double <[x]>;

int <[signgamp]>;

float gammaf_r(<[x]>, <[signgamp]>)

float <[x]>;

int <[signgamp]>;

double lgamma_r(<[x]>, <[signgamp]>)

double <[x]>;

int <[signgamp]>;

float lgammaf_r(<[x]>, <[signgamp]>)

float <[x]>;

int <[signgamp]>;

DESCRIPTION

<<gamma>> calculates

@tex

$\mit ln\bigl(\Gamma(x)\bigr)$,

@end tex

the natural logarithm of the gamma function of <[x]>.  The gamma function

(<<exp(gamma(<[x]>))>>) is a generalization of factorial, and retains

the property that

@ifnottex

<<exp(gamma(N))>> is equivalent to <<N*exp(gamma(N-1))>>.

@end ifnottex

@tex

$\mit \Gamma(N)\equiv N\times\Gamma(N-1)$.

@end tex

Accordingly, the results of the gamma function itself grow very

quickly.  <<gamma>> is defined as

@tex

$\mit ln\bigl(\Gamma(x)\bigr)$ rather than simply $\mit \Gamma(x)$

@end tex

@ifnottex

the natural log of the gamma function, rather than the gamma function

itself,

@end ifnottex

to extend the useful range of results representable.

The sign of the result is returned in the global variable <<signgam>>,

which is declared in math.h.

<<gammaf>> performs the same calculation as <<gamma>>, but uses and

returns <<float>> values.

<<lgamma>> and <<lgammaf>> are alternate names for <<gamma>> and

<<gammaf>>.  The use of <<lgamma>> instead of <<gamma>> is a reminder

that these functions compute the log of the gamma function, rather

than the gamma function itself.

The functions <<gamma_r>>, <<gammaf_r>>, <<lgamma_r>>, and

<<lgammaf_r>> are just like <<gamma>>, <<gammaf>>, <<lgamma>>, and

<<lgammaf>>, respectively, but take an additional argument.  This

additional argument is a pointer to an integer.  This additional

argument is used to return the sign of the result, and the global

variable <<signgam>> is not used.  These functions may be used for

reentrant calls (but they will still set the global variable <<errno>>

if an error occurs).

RETURNS

Normally, the computed result is returned.

When <[x]> is a nonpositive integer, <<gamma>> returns <<HUGE_VAL>>

and <<errno>> is set to <<EDOM>>.  If the result overflows, <<gamma>>

returns <<HUGE_VAL>> and <<errno>> is set to <<ERANGE>>.

You can modify this error treatment using <<matherr>>.

PORTABILITY

Neither <<gamma>> nor <<gammaf>> is ANSI C.  */

/* lgamma_r(x, signgamp)

 * Reentrant version of the logarithm of the Gamma function

 * with user provide pointer for the sign of Gamma(x).

 *

 * Method:

 *   1. Argument Reduction for 0 < x <= 8

 *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may

 *      reduce x to a number in [1.5,2.5] by

 *              lgamma(1+s) = log(s) + lgamma(s)

 *      for example,

 *              lgamma(7.3) = log(6.3) + lgamma(6.3)

 *                          = log(6.3*5.3) + lgamma(5.3)

 *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)

 *   2. Polynomial approximation of lgamma around its

 *      minimun ymin=1.461632144968362245 to maintain monotonicity.

 *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use

 *              Let z = x-ymin;

 *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)

 *      where

 *              poly(z) is a 14 degree polynomial.

 *   2. Rational approximation in the primary interval [2,3]

 *      We use the following approximation:

 *              s = x-2.0;

 *              lgamma(x) = 0.5*s + s*P(s)/Q(s)

 *      with accuracy

 *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71

 *      Our algorithms are based on the following observation

 *

 *                             zeta(2)-1    2    zeta(3)-1    3

 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...

 *                                 2                 3

 *

 *      where Euler = 0.5771... is the Euler constant, which is very

 *      close to 0.5.

 *

 *   3. For x>=8, we have

 *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....

 *      (better formula:

 *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)

 *      Let z = 1/x, then we approximation

 *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)

 *      by

 *                                  3       5             11

 *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z

 *      where

 *              |w - f(z)| < 2**-58.74

 *

 *   4. For negative x, since (G is gamma function)

 *              -x*G(-x)*G(x) = pi/sin(pi*x),

 *      we have

 *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))

 *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0

 *      Hence, for x<0, signgam = sign(sin(pi*x)) and

 *              lgamma(x) = log(|Gamma(x)|)

 *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);

 *      Note: one should avoid compute pi*(-x) directly in the

 *            computation of sin(pi*(-x)).

 *

 *   5. Special Cases

 *              lgamma(2+s) ~ s*(1-Euler) for tiny s

 *              lgamma(1)=lgamma(2)=0

 *              lgamma(x) ~ -log(x) for tiny x

 *              lgamma(0) = lgamma(inf) = inf

 *              lgamma(-integer) = +-inf

 *

 */

#include "fdlibm.h"

#ifdef __STDC__

static const double

#else

static double

#endif

two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */

half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */

one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */

pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */

a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */

a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */

a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */

a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */

a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */

a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */

a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */

a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */

a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */

a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */

a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */

a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */

tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */

tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */

/* tt = -(tail of tf) */

tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */

t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */

t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */

t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */

t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */

t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */

t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */

t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */

t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */

t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */

t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */

t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */

t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */

t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */

t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */

t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */

u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */

u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */

u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */

u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */

u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */

u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */

v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */

v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */

v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */

v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */

v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */

s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */

s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */

s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */

s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */

s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */

s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */

s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */

r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */

r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */

r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */

r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */

r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */

r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */

w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */

w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */

w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */

w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */

w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */

w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */

w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */

#ifdef __STDC__

static const double zero=  0.00000000000000000000e+00;

#else

static double zero=  0.00000000000000000000e+00;

#endif

#ifdef __STDC__

        static double sin_pi(double x)

#else

        static double sin_pi(x)

        double x;

#endif

{

        double y,z;

        __int32_t n,ix;

        GET_HIGH_WORD(ix,x);

        ix &= 0x7fffffff;

        if(ix<0x3fd00000) return sin(pi*x);

        y = -x;         /* x is assume negative */

    /*

     * argument reduction, make sure inexact flag not raised if input

     * is an integer

     */

        z = floor(y);

        if(z!=y) {                              /* inexact anyway */

            y  *= 0.5;

            y   = 2.0*(y - floor(y));           /* y = |x| mod 2.0 */

            n   = (__int32_t) (y*4.0);

        } else {

            if(ix>=0x43400000) {

                y = zero; n = 0;                 /* y must be even */

            } else {

                if(ix<0x43300000) z = y+two52;  /* exact */

                GET_LOW_WORD(n,z);

                n &= 1;

                y  = n;

                n<<= 2;

            }

        }

        switch (n) {

            case 0:   y =  sin(pi*y); break;

            case 1:

            case 2:   y =  cos(pi*(0.5-y)); break;

            case 3:

            case 4:   y =  sin(pi*(one-y)); break;

            case 5:

            case 6:   y = -cos(pi*(y-1.5)); break;

            default:  y =  sin(pi*(y-2.0)); break;

            }

        return -y;

}

#ifdef __STDC__

        double lgamma_r(double x, int *signgamp)

#else

        double lgamma_r(x,signgamp)

        double x; int *signgamp;

#endif

{

        double t,y,z,nadj,p,p1,p2,p3,q,r,w;

        __int32_t i,hx,lx,ix;

        nadj = 0;

        EXTRACT_WORDS(hx,lx,x);

    /* purge off +-inf, NaN, +-0, and negative arguments */

        *signgamp = 1;

        ix = hx&0x7fffffff;

        if(ix>=0x7ff00000) return x*x;

        if((ix|lx)==0) return one/zero;

        if(ix<0x3b900000) {     /* |x|<2**-70, return -log(|x|) */

            if(hx<0) {

                *signgamp = -1;

                return -log(-x);

            } else return -log(x);

        }

        if(hx<0) {

            if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */

                return one/zero;

            t = sin_pi(x);

            if(t==zero) return one/zero; /* -integer */

            nadj = log(pi/fabs(t*x));

            if(t<zero) *signgamp = -1;

            x = -x;

        }

    /* purge off 1 and 2 */

        if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;

    /* for x < 2.0 */

        else if(ix<0x40000000) {

            if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */

                r = -log(x);

                if(ix>=0x3FE76944) {y = one-x; i= 0;}

                else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}

                else {y = x; i=2;}

            } else {

                r = zero;

                if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */

                else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */

                else {y=x-one;i=2;}

            }

            switch(i) {

              case 0:

                z = y*y;

                p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));

                p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));

                p  = y*p1+p2;

                r  += (p-0.5*y); break;

              case 1:

                z = y*y;

                w = z*y;

                p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */

                p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));

                p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));

                p  = z*p1-(tt-w*(p2+y*p3));

                r += (tf + p); break;

              case 2:

                p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));

                p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));

                r += (-0.5*y + p1/p2);

            }

        }

        else if(ix<0x40200000) {                        /* x < 8.0 */

            i = (__int32_t)x;

            t = zero;

            y = x-(double)i;

            p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));

            q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));

            r = half*y+p/q;

            z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */

            switch(i) {

            case 7: z *= (y+6.0);       /* FALLTHRU */

            case 6: z *= (y+5.0);       /* FALLTHRU */

            case 5: z *= (y+4.0);       /* FALLTHRU */

            case 4: z *= (y+3.0);       /* FALLTHRU */

            case 3: z *= (y+2.0);       /* FALLTHRU */

                    r += log(z); break;

            }

    /* 8.0 <= x < 2**58 */

        } else if (ix < 0x43900000) {

            t = log(x);

            z = one/x;

            y = z*z;

            w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));

            r = (x-half)*(t-one)+w;

        } else

    /* 2**58 <= x <= inf */

            r =  x*(log(x)-one);

        if(hx<0) r = nadj - r;

        return r;

}

double

lgamma(double x)

{

  return lgamma_r(x, &(_REENT_SIGNGAM(_REENT)));

}