/*							zetacf.c
 *
 *	Riemann zeta function
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, y, zetacf();
 *
 * y = zetacf( x );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 *
 *                inf.
 *                 -    -x
 *   zetac(x)  =   >   k   ,   x > 1,
 *                 -
 *                k=2
 *
 * is related to the Riemann zeta function by
 *
 *	Riemann zeta(x) = zetac(x) + 1.
 *
 * Extension of the function definition for x < 1 is implemented.
 * Zero is returned for x > log2(MAXNUM).
 *
 * An overflow error may occur for large negative x, due to the
 * gamma function in the reflection formula.
 *
 * ACCURACY:
 *
 * Tabulated values have full machine accuracy.
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      1,50        30000       5.5e-7      7.5e-8
 *
 *
 */

/*
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 <math.h>


/* Riemann zeta(x) - 1
 * for integer arguments between 0 and 30.
 */
static float azetacf[] = {
-1.50000000000000000000E0,
 1.70141183460469231730E38, /* infinity. */
 6.44934066848226436472E-1,
 2.02056903159594285400E-1,
 8.23232337111381915160E-2,
 3.69277551433699263314E-2,
 1.73430619844491397145E-2,
 8.34927738192282683980E-3,
 4.07735619794433937869E-3,
 2.00839282608221441785E-3,
 9.94575127818085337146E-4,
 4.94188604119464558702E-4,
 2.46086553308048298638E-4,
 1.22713347578489146752E-4,
 6.12481350587048292585E-5,
 3.05882363070204935517E-5,
 1.52822594086518717326E-5,
 7.63719763789976227360E-6,
 3.81729326499983985646E-6,
 1.90821271655393892566E-6,
 9.53962033872796113152E-7,
 4.76932986787806463117E-7,
 2.38450502727732990004E-7,
 1.19219925965311073068E-7,
 5.96081890512594796124E-8,
 2.98035035146522801861E-8,
 1.49015548283650412347E-8,
 7.45071178983542949198E-9,
 3.72533402478845705482E-9,
 1.86265972351304900640E-9,
 9.31327432419668182872E-10
};


/* 2**x (1 - 1/x) (zeta(x) - 1) = P(1/x)/Q(1/x), 1 <= x <= 10 */
static float P[9] = {
  5.85746514569725319540E11,
  2.57534127756102572888E11,
  4.87781159567948256438E10,
  5.15399538023885770696E9,
  3.41646073514754094281E8,
  1.60837006880656492731E7,
  5.92785467342109522998E5,
  1.51129169964938823117E4,
  2.01822444485997955865E2,
};
static float Q[8] = {
/*  1.00000000000000000000E0,*/
  3.90497676373371157516E11,
  5.22858235368272161797E10,
  5.64451517271280543351E9,
  3.39006746015350418834E8,
  1.79410371500126453702E7,
  5.66666825131384797029E5,
  1.60382976810944131506E4,
  1.96436237223387314144E2,
};

/* log(zeta(x) - 1 - 2**-x), 10 <= x <= 50 */
static float A[11] = {
 8.70728567484590192539E6,
 1.76506865670346462757E8,
 2.60889506707483264896E10,
 5.29806374009894791647E11,
 2.26888156119238241487E13,
 3.31884402932705083599E14,
 5.13778997975868230192E15,
-1.98123688133907171455E15,
-9.92763810039983572356E16,
 7.82905376180870586444E16,
 9.26786275768927717187E16,
};
static float B[10] = {
/* 1.00000000000000000000E0,*/
-7.92625410563741062861E6,
-1.60529969932920229676E8,
-2.37669260975543221788E10,
-4.80319584350455169857E11,
-2.07820961754173320170E13,
-2.96075404507272223680E14,
-4.86299103694609136686E15,
 5.34589509675789930199E15,
 5.71464111092297631292E16,
-1.79915597658676556828E16,
};

/* (1-x) (zeta(x) - 1), 0 <= x <= 1 */

static float R[6] = {
-3.28717474506562731748E-1,
 1.55162528742623950834E1,
-2.48762831680821954401E2,
 1.01050368053237678329E3,
 1.26726061410235149405E4,
-1.11578094770515181334E5,
};
static float S[5] = {
/* 1.00000000000000000000E0,*/
 1.95107674914060531512E1,
 3.17710311750646984099E2,
 3.03835500874445748734E3,
 2.03665876435770579345E4,
 7.43853965136767874343E4,
};


#define MAXL2 127

/*
 * Riemann zeta function, minus one
 */

extern float MACHEPF, PIO2F, MAXNUMF, PIF;

#ifdef ANSIC
extern float sinf ( float xx );
extern float floorf ( float x );
extern float gammaf ( float xx );
extern float powf ( float x, float y );
extern float expf ( float xx );
extern float polevlf ( float xx, float *coef, int N );
extern float p1evlf ( float xx, float *coef, int N );
#else
float sinf(), floorf(), gammaf(), powf(), expf();
float polevlf(), p1evlf();
#endif

float zetacf(float xx)
{
int i;
float x, a, b, s, w;

x = xx;
if( x < 0.0 )
	{
	if( x < -30.8148 )
		{
		mtherr( "zetacf", OVERFLOW );
		return(0.0);
		}
	s = 1.0 - x;
	w = zetacf( s );
	b = sinf(PIO2F*x) * powf(2.0*PIF, x) * gammaf(s) * (1.0 + w) / PIF;
	return(b - 1.0);
	}

if( x >= MAXL2 )
	return(0.0);	/* because first term is 2**-x */

/* Tabulated values for integer argument */
w = floorf(x);
if( w == x )
	{
	i = x;
	if( i < 31 )
		{
		return( azetacf[i] );
		}
	}


if( x < 1.0 )
	{
	w = 1.0 - x;
	a = polevlf( x, R, 5 ) / ( w * p1evlf( x, S, 5 ));
	return( a );
	}

if( x == 1.0 )
	{
	mtherr( "zetacf", SING );
	return( MAXNUMF );
	}

if( x <= 10.0 )
	{
	b = powf( 2.0, x ) * (x - 1.0);
	w = 1.0/x;
	s = (x * polevlf( w, P, 8 )) / (b * p1evlf( w, Q, 8 ));
	return( s );
	}

if( x <= 50.0 )
	{
	b = powf( 2.0, -x );
	w = polevlf( x, A, 10 ) / p1evlf( x, B, 10 );
	w = expf(w) + b;
	return(w);
	}


/* Basic sum of inverse powers */


s = 0.0;
a = 1.0;
do
	{
	a += 2.0;
	b = powf( a, -x );
	s += b;
	}
while( b/s > MACHEPF );

b = powf( 2.0, -x );
s = (s + b)/(1.0-b);
return(s);
}