path: root/src/cephes.c

#include <stdio.h>
#include <math.h>
#include "include/cephes.h"

static const double	rel_error = 1E-12;

double MACHEP = 1.11022302462515654042E-16;		// 2**-53
double MAXLOG = 7.09782712893383996732224E2;	// log(MAXNUM)
double MAXNUM = 1.7976931348623158E308;			// 2**1024*(1-MACHEP)
double PI     = 3.14159265358979323846;			// pi, duh!

static double big = 4.503599627370496e15;
static double biginv =  2.22044604925031308085e-16;

int sgngam = 0;

double
cephes_igamc(double a, double x)
{
	double ans, ax, c, yc, r, t, y, z;
	double pk, pkm1, pkm2, qk, qkm1, qkm2;

	if ( (x <= 0) || ( a <= 0) )
		return( 1.0 );

	if ( (x < 1.0) || (x < a) )
		return( 1.e0 - cephes_igam(a,x) );

	ax = a * log(x) - x - cephes_lgam(a);

	if ( ax < -MAXLOG ) {
		printf("igamc: UNDERFLOW\n");
		return 0.0;
	}
	ax = exp(ax);

	/* continued fraction */
	y = 1.0 - a;
	z = x + y + 1.0;
	c = 0.0;
	pkm2 = 1.0;
	qkm2 = x;
	pkm1 = x + 1.0;
	qkm1 = z * x;
	ans = pkm1/qkm1;

	do {
		c += 1.0;
		y += 1.0;
		z += 2.0;
		yc = y * c;
		pk = pkm1 * z  -  pkm2 * yc;
		qk = qkm1 * z  -  qkm2 * yc;
		if ( qk != 0 ) {
			r = pk/qk;
			t = fabs( (ans - r)/r );
			ans = r;
		}
		else
			t = 1.0;
		pkm2 = pkm1;
		pkm1 = pk;
		qkm2 = qkm1;
		qkm1 = qk;
		if ( fabs(pk) > big ) {
			pkm2 *= biginv;
			pkm1 *= biginv;
			qkm2 *= biginv;
			qkm1 *= biginv;
		}
	} while ( t > MACHEP );

	return ans*ax;
}

double
cephes_igam(double a, double x)
{
	double ans, ax, c, r;

	if ( (x <= 0) || ( a <= 0) )
		return 0.0;

	if ( (x > 1.0) && (x > a ) )
		return 1.e0 - cephes_igamc(a,x);

	/* Compute  x**a * exp(-x) / gamma(a)  */
	ax = a * log(x) - x - cephes_lgam(a);
	if ( ax < -MAXLOG ) {
		printf("igam: UNDERFLOW\n");
		return 0.0;
	}
	ax = exp(ax);

	/* power series */
	r = a;
	c = 1.0;
	ans = 1.0;

	do {
		r += 1.0;
		c *= x/r;
		ans += c;
	} while ( c/ans > MACHEP );

	return ans * ax/a;
}

union A_Array {
    double d[5];
    unsigned short us[20];
};


union BC_Array {
    double d[6];
    unsigned short us[24];
};

/* A[]: Stirling's formula expansion of log gamma
 * B[], C[]: log gamma function between 2 and 3
 */

/**/
static union A_Array A = {
	.us = {
	0x6661,0x2733,0x9850,0x3f4a,
	0xe943,0xb580,0x7fbd,0xbf43,
	0x5ebb,0x20dc,0x019f,0x3f4a,
	0xa5a1,0x16b0,0xc16c,0xbf66,
	0x554b,0x5555,0x5555,0x3fb5
	}
};
static union BC_Array B = {
	.us = {
	0x6761,0x8ff3,0x8901,0xc095,
	0xb93e,0x355b,0xf234,0xc0e2,
	0x89e5,0xf890,0x3d73,0xc114,
	0xdb51,0xf994,0xbc82,0xc131,
	0xf20b,0x0219,0x4589,0xc13a,
	0x055e,0x5418,0x0c67,0xc12a
	}
};
static union BC_Array C  = {
	/*0x0000,0x0000,0x0000,0x3ff0,*/
	.us = {
	0x12b2,0x1cf3,0xfd0d,0xc075,
	0xd757,0x7b89,0xaa0d,0xc0d0,
	0x4c9b,0xb974,0xeb84,0xc10a,
	0x0043,0x7195,0x6286,0xc131,
	0xf34c,0x892f,0x5255,0xc143,
	0xe14a,0x6a11,0xce4b,0xc13e
	}
};

#define MAXLGM 2.556348e305


/* Logarithm of gamma function */
double
cephes_lgam(double x)
{
	double	p, q, u, w, z;
	int		i;

	sgngam = 1;

	if ( x < -34.0 ) {
		q = -x;
		w = cephes_lgam(q); /* note this modifies sgngam! */
		p = floor(q);
		if ( p == q ) {
lgsing:
			goto loverf;
		}
		i = (int)p;
		if ( (i & 1) == 0 )
			sgngam = -1;
		else
			sgngam = 1;
		z = q - p;
		if ( z > 0.5 ) {
			p += 1.0;
			z = p - q;
		}
		z = q * sin( PI * z );
		if ( z == 0.0 )
			goto lgsing;
		/*      z = log(PI) - log( z ) - w;*/
		z = log(PI) - 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 ) {
			sgngam = -1;
			z = -z;
		}
		else
			sgngam = 1;
		if ( u == 2.0 )
			return( log(z) );
		p -= 2.0;
		x = x + p;
		p = x * cephes_polevl( x, (double *)B.d, 5 ) / cephes_p1evl( x, (double *)C.d, 6);

		return log(z) + p;
	}

	if ( x > MAXLGM ) {
loverf:
		printf("lgam: OVERFLOW\n");

		return sgngam * MAXNUM;
	}

	q = ( x - 0.5 ) * log(x) - x + log( sqrt( 2*PI ) );
	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 += cephes_polevl( p, (double *)A.d, 4 ) / x;

	return q;
}

double
cephes_polevl(double x, double *coef, int N)
{
	double	ans;
	int		i;
	double	*p;

	p = coef;
	ans = *p++;
	i = N;

	do
		ans = ans * x  +  *p++;
	while ( --i );

	return ans;
}

double
cephes_p1evl(double x, double *coef, int N)
{
	double	ans;
	double	*p;
	int		i;

	p = coef;
	ans = x + *p++;
	i = N-1;

	do
		ans = ans * x  + *p++;
	while ( --i );

	return ans;
}

double
cephes_erf(double x)
{
	static const double two_sqrtpi = 1.128379167095512574;
	double	sum = x, term = x, xsqr = x * x;
	int		j = 1;

	if ( fabs(x) > 2.2 )
		return 1.0 - cephes_erfc(x);

	do {
		term *= xsqr/j;
		sum -= term/(2*j+1);
		j++;
		term *= xsqr/j;
		sum += term/(2*j+1);
		j++;
	} while ( fabs(term)/sum > rel_error );

	return two_sqrtpi*sum;
}

double
cephes_erfc(double x)
{
	static const double one_sqrtpi = 0.564189583547756287;
	double	a = 1, b = x, c = x, d = x*x + 0.5;
	double	q1, q2 = b/d, n = 1.0, t;

	if ( fabs(x) < 2.2 )
		return 1.0 - cephes_erf(x);
	if ( x < 0 )
		return 2.0 - cephes_erfc(-x);

	do {
		t = a*n + b*x;
		a = b;
		b = t;
		t = c*n + d*x;
		c = d;
		d = t;
		n += 0.5;
		q1 = q2;
		q2 = b/d;
	} while ( fabs(q1-q2)/q2 > rel_error );

	return one_sqrtpi*exp(-x*x)*q2;
}


double
cephes_normal(double x)
{
	double arg, result, sqrt2=1.414213562373095048801688724209698078569672;

	if (x > 0) {
		arg = x/sqrt2;
		result = 0.5 * ( 1 + erf(arg) );
	}
	else {
		arg = -x/sqrt2;
		result = 0.5 * (erfc(arg) );
	}

	return( result);
}