/*
 *     Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, Gamma();
 *
 * y = Gamma( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Gamma function of the argument.  The result is
 * correctly signed.
 *
 * Arguments |x| <= 34 are reduced by recurrence and the function
 * approximated by a rational function of degree 6/7 in the
 * interval (2,3).  Large arguments are handled by Stirling's
 * formula. Large negative arguments are made positive using
 * a reflection formula.  
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE    -170,-33      20000       2.3e-15     3.3e-16
 *    IEEE     -33,  33     20000       9.4e-16     2.2e-16
 *    IEEE      33, 171.6   20000       2.3e-15     3.2e-16
 *
 * Error for arguments outside the test range will be larger
 * owing to error amplification by the exponential function.
 *
 */

/*                                                     lgam()
 *
 *     Natural logarithm of Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, lgam();
 *
 * y = lgam( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base e (2.718...) logarithm of the absolute
 * value of the Gamma function of the argument.
 *
 * For arguments greater than 13, the logarithm of the Gamma
 * function is approximated by the logarithmic version of
 * Stirling's formula using a polynomial approximation of
 * degree 4. Arguments between -33 and +33 are reduced by
 * recurrence to the interval [2,3] of a rational approximation.
 * The cosecant reflection formula is employed for arguments
 * less than -33.
 *
 * Arguments greater than MAXLGM return NPY_INFINITY and an error
 * message.  MAXLGM = 2.556348e305 for IEEE arithmetic.
 *
 *
 *
 * ACCURACY:
 *
 *
 * arithmetic      domain        # trials     peak         rms
 *    IEEE    0, 3                 28000     5.4e-16     1.1e-16
 *    IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
 * The error criterion was relative when the function magnitude
 * was greater than one but absolute when it was less than one.
 *
 * The following test used the relative error criterion, though
 * at certain points the relative error could be much higher than
 * indicated.
 *    IEEE    -200, -4             10000     4.8e-16     1.3e-16
 *
 */

/*
 * Cephes Math Library Release 2.2:  July, 1992
 * Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */


#include "mconf.h"

static double P[] = {
    1.60119522476751861407E-4,
    1.19135147006586384913E-3,
    1.04213797561761569935E-2,
    4.76367800457137231464E-2,
    2.07448227648435975150E-1,
    4.94214826801497100753E-1,
    9.99999999999999996796E-1
};

static double Q[] = {
    -2.31581873324120129819E-5,
    5.39605580493303397842E-4,
    -4.45641913851797240494E-3,
    1.18139785222060435552E-2,
    3.58236398605498653373E-2,
    -2.34591795718243348568E-1,
    7.14304917030273074085E-2,
    1.00000000000000000320E0
};

#define MAXGAM 171.624376956302725
static double LOGPI = 1.14472988584940017414;

/* Stirling's formula for the Gamma function */
static double STIR[5] = {
    7.87311395793093628397E-4,
    -2.29549961613378126380E-4,
    -2.68132617805781232825E-3,
    3.47222221605458667310E-3,
    8.33333333333482257126E-2,
};

#define MAXSTIR 143.01608
static double SQTPI = 2.50662827463100050242E0;

extern double MAXLOG;
static double stirf(double);

/* Gamma function computed by Stirling's formula.
 * The polynomial STIR is valid for 33 <= x <= 172.
 */
static double stirf(double x)
{
    double y, w, v;

    if (x >= MAXGAM) {
	return (NPY_INFINITY);
    }
    w = 1.0 / x;
    w = 1.0 + w * polevl(w, STIR, 4);
    y = exp(x);
    if (x > MAXSTIR) {		/* Avoid overflow in pow() */
	v = pow(x, 0.5 * x - 0.25);
	y = v * (v / y);
    }
    else {
	y = pow(x, x - 0.5) / y;
    }
    y = SQTPI * y * w;
    return (y);
}


double Gamma(double x)
{
    double p, q, z;
    int i;
    int sgngam = 1;

    if (!cephes_isfinite(x)) {
	return x;
    }
    q = fabs(x);

    if (q > 33.0) {
	if (x < 0.0) {
	    p = floor(q);
	    if (p == q) {
	      gamnan:
		mtherr("Gamma", OVERFLOW);
		return (NPY_INFINITY);
	    }
	    i = p;
	    if ((i & 1) == 0)
		sgngam = -1;
	    z = q - p;
	    if (z > 0.5) {
		p += 1.0;
		z = q - p;
	    }
	    z = q * sin(NPY_PI * z);
	    if (z == 0.0) {
		return (sgngam * NPY_INFINITY);
	    }
	    z = fabs(z);
	    z = NPY_PI / (z * stirf(q));
	}
	else {
	    z = stirf(x);
	}
	return (sgngam * z);
    }

    z = 1.0;
    while (x >= 3.0) {
	x -= 1.0;
	z *= x;
    }

    while (x < 0.0) {
	if (x > -1.E-9)
	    goto small;
	z /= x;
	x += 1.0;
    }

    while (x < 2.0) {
	if (x < 1.e-9)
	    goto small;
	z /= x;
	x += 1.0;
    }

    if (x == 2.0)
	return (z);

    x -= 2.0;
    p = polevl(x, P, 6);
    q = polevl(x, Q, 7);
    return (z * p / q);

  small:
    if (x == 0.0) {
	goto gamnan;
    }
    else
	return (z / ((1.0 + 0.5772156649015329 * x) * x));
}



/* A[]: Stirling's formula expansion of log Gamma
 * B[], C[]: log Gamma function between 2 and 3
 */
static double A[] = {
    8.11614167470508450300E-4,
    -5.95061904284301438324E-4,
    7.93650340457716943945E-4,
    -2.77777777730099687205E-3,
    8.33333333333331927722E-2
};

static double B[] = {
    -1.37825152569120859100E3,
    -3.88016315134637840924E4,
    -3.31612992738871184744E5,
    -1.16237097492762307383E6,
    -1.72173700820839662146E6,
    -8.53555664245765465627E5
};

static double C[] = {
    /* 1.00000000000000000000E0, */
    -3.51815701436523470549E2,
    -1.70642106651881159223E4,
    -2.20528590553854454839E5,
    -1.13933444367982507207E6,
    -2.53252307177582951285E6,
    -2.01889141433532773231E6
};

/* log( sqrt( 2*pi ) ) */
static double LS2PI = 0.91893853320467274178;

#define MAXLGM 2.556348e305


/* Logarithm of Gamma function */
double lgam(double x)
{
    int sign;
    return lgam_sgn(x, &sign);
}

double lgam_sgn(double x, int *sign)
{
    double p, q, u, w, z;
    int i;

    *sign = 1;

    if (!cephes_isfinite(x))
	return x;

    if (x < -34.0) {
	q = -x;
	w = lgam_sgn(q, sign);
	p = floor(q);
	if (p == q) {
	  lgsing:
	    mtherr("lgam", SING);
	    return (NPY_INFINITY);
	}
	i = p;
	if ((i & 1) == 0)
	    *sign = -1;
	else
	    *sign = 1;
	z = q - p;
	if (z > 0.5) {
	    p += 1.0;
	    z = p - q;
	}
	z = q * sin(NPY_PI * z);
	if (z == 0.0)
	    goto lgsing;
	/*     z = log(NPY_PI) - log( z ) - w; */
	z = LOGPI - log(z) - w;
	return (z);
    }

    if (x < 13.0) {
	z = 1.0;
	p = 0.0;
	u = x;
	while (u >= 3.0) {
	    p -= 1.0;
	    u = x + p;
	    z *= u;
	}
	while (u < 2.0) {
	    if (u == 0.0)
		goto lgsing;
	    z /= u;
	    p += 1.0;
	    u = x + p;
	}
	if (z < 0.0) {
	    *sign = -1;
	    z = -z;
	}
	else
	    *sign = 1;
	if (u == 2.0)
	    return (log(z));
	p -= 2.0;
	x = x + p;
	p = x * polevl(x, B, 5) / p1evl(x, C, 6);
	return (log(z) + p);
    }

    if (x > MAXLGM) {
	return (*sign * NPY_INFINITY);
    }

    q = (x - 0.5) * log(x) - x + LS2PI;
    if (x > 1.0e8)
	return (q);

    p = 1.0 / (x * x);
    if (x >= 1000.0)
	q += ((7.9365079365079365079365e-4 * p
	       - 2.7777777777777777777778e-3) * p
	      + 0.0833333333333333333333) / x;
    else
	q += polevl(p, A, 4) / x;
    return (q);
}
