xref: /csrg-svn/lib/libm/common_source/j0.c (revision 64988)

/*-
 * Copyright (c) 1992, 1993
 *	The Regents of the University of California.  All rights reserved.
 *
 * %sccs.include.redist.c%
 */

#ifndef lint
static char sccsid[] = "@(#)j0.c	8.2 (Berkeley) 11/30/93";
#endif /* not lint */

/*
 * 16 December 1992
 * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
 */

/*
 * ====================================================
 * Copyright (C) 1992 by Sun Microsystems, Inc.
 *
 * 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.
 * ====================================================
 *
 * ******************* WARNING ********************
 * This is an alpha version of SunPro's FDLIBM (Freely
 * Distributable Math Library) for IEEE double precision
 * arithmetic. FDLIBM is a basic math library written
 * in C that runs on machines that conform to IEEE
 * Standard 754/854. This alpha version is distributed
 * for testing purpose. Those who use this software
 * should report any bugs to
 *
 *		fdlibm-comments@sunpro.eng.sun.com
 *
 * -- K.C. Ng, Oct 12, 1992
 * ************************************************
 */

/* double j0(double x), y0(double x)
 * Bessel function of the first and second kinds of order zero.
 * Method -- j0(x):
 *	1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
 *	2. Reduce x to |x| since j0(x)=j0(-x),  and
 *	   for x in (0,2)
 *		j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
 *	   (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
 *	   for x in (2,inf)
 * 		j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
 * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
 *	   as follow:
 *		cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 *			= 1/sqrt(2) * (cos(x) + sin(x))
 *		sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
 *			= 1/sqrt(2) * (sin(x) - cos(x))
 * 	   (To avoid cancellation, use
 *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 * 	    to compute the worse one.)
 *
 *	3 Special cases
 *		j0(nan)= nan
 *		j0(0) = 1
 *		j0(inf) = 0
 *
 * Method -- y0(x):
 *	1. For x<2.
 *	   Since
 *		y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
 *	   therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
 *	   We use the following function to approximate y0,
 *		y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
 *	   where
 *		U(z) = u0 + u1*z + ... + u6*z^6
 *		V(z) = 1  + v1*z + ... + v4*z^4
 *	   with absolute approximation error bounded by 2**-72.
 *	   Note: For tiny x, U/V = u0 and j0(x)~1, hence
 *		y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
 *	2. For x>=2.
 * 		y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
 * 	   where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
 *	   by the method mentioned above.
 *	3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
 */

#include <math.h>
#include <float.h>
#if defined(vax) || defined(tahoe)
#define _IEEE	0
#else
#define _IEEE	1
#define infnan(x) (0.0)
#endif

static double pzero __P((double)), qzero __P((double));

static double
huge 	= 1e300,
zero    = 0.0,
one	= 1.0,
invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
tpi	= 0.636619772367581343075535053490057448,
 		/* R0/S0 on [0, 2.00] */
r02 =   1.562499999999999408594634421055018003102e-0002,
r03 =  -1.899792942388547334476601771991800712355e-0004,
r04 =   1.829540495327006565964161150603950916854e-0006,
r05 =  -4.618326885321032060803075217804816988758e-0009,
s01 =   1.561910294648900170180789369288114642057e-0002,
s02 =   1.169267846633374484918570613449245536323e-0004,
s03 =   5.135465502073181376284426245689510134134e-0007,
s04 =   1.166140033337900097836930825478674320464e-0009;

double
j0(x)
	double x;
{
	double z, s,c,ss,cc,r,u,v;

	if (!finite(x))
		if (_IEEE) return one/(x*x);
		else return (0);
	x = fabs(x);
	if (x >= 2.0) {	/* |x| >= 2.0 */
		s = sin(x);
		c = cos(x);
		ss = s-c;
		cc = s+c;
		if (x < .5 * DBL_MAX) {  /* make sure x+x not overflow */
		    z = -cos(x+x);
		    if ((s*c)<zero) cc = z/ss;
		    else 	    ss = z/cc;
		}
	/*
	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
	 */
		if (_IEEE && x> 6.80564733841876927e+38) /* 2^129 */
			z = (invsqrtpi*cc)/sqrt(x);
		else {
		    u = pzero(x); v = qzero(x);
		    z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
		}
		return z;
	}
	if (x < 1.220703125e-004) {		   /* |x| < 2**-13 */
	    if (huge+x > one) {			   /* raise inexact if x != 0 */
	        if (x < 7.450580596923828125e-009) /* |x|<2**-27 */
			return one;
	        else return (one - 0.25*x*x);
	    }
	}
	z = x*x;
	r =  z*(r02+z*(r03+z*(r04+z*r05)));
	s =  one+z*(s01+z*(s02+z*(s03+z*s04)));
	if (x < one) {			/* |x| < 1.00 */
	    return (one + z*(-0.25+(r/s)));
	} else {
	    u = 0.5*x;
	    return ((one+u)*(one-u)+z*(r/s));
	}
}

static double
u00 =  -7.380429510868722527422411862872999615628e-0002,
u01 =   1.766664525091811069896442906220827182707e-0001,
u02 =  -1.381856719455968955440002438182885835344e-0002,
u03 =   3.474534320936836562092566861515617053954e-0004,
u04 =  -3.814070537243641752631729276103284491172e-0006,
u05 =   1.955901370350229170025509706510038090009e-0008,
u06 =  -3.982051941321034108350630097330144576337e-0011,
v01 =   1.273048348341237002944554656529224780561e-0002,
v02 =   7.600686273503532807462101309675806839635e-0005,
v03 =   2.591508518404578033173189144579208685163e-0007,
v04 =   4.411103113326754838596529339004302243157e-0010;

double
y0(x)
	double x;
{
	double z, s, c, ss, cc, u, v;
    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
	if (!finite(x))
		if (_IEEE)
			return (one/(x+x*x));
		else
			return (0);
        if (x == 0)
		if (_IEEE)	return (-one/zero);
		else		return(infnan(-ERANGE));
        if (x<0)
		if (_IEEE)	return (zero/zero);
		else		return (infnan(EDOM));
        if (x >= 2.00) {	/* |x| >= 2.0 */
        /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
         * where x0 = x-pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
         *                      =  1/sqrt(2) * (sin(x) + cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
                s = sin(x);
                c = cos(x);
                ss = s-c;
                cc = s+c;
	/*
	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
	 */
                if (x < .5 * DBL_MAX) {  /* make sure x+x not overflow */
                    z = -cos(x+x);
                    if ((s*c)<zero) cc = z/ss;
                    else            ss = z/cc;
                }
                if (_IEEE && x > 6.80564733841876927e+38) /* > 2^129 */
			z = (invsqrtpi*ss)/sqrt(x);
                else {
                    u = pzero(x); v = qzero(x);
                    z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
                }
                return z;
	}
	if (x <= 7.450580596923828125e-009) {		/* x < 2**-27 */
	    return (u00 + tpi*log(x));
	}
	z = x*x;
	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
	return (u/v + tpi*(j0(x)*log(x)));
}

/* The asymptotic expansions of pzero is
 *	1 - 9/128 s^2 + 11025/98304 s^4 - ...,	where s = 1/x.
 * For x >= 2, We approximate pzero by
 * 	pzero(x) = 1 + (R/S)
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
 * and
 *	| pzero(x)-1-R/S | <= 2  ** ( -60.26)
 */
static double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
   0.0,
  -7.031249999999003994151563066182798210142e-0002,
  -8.081670412753498508883963849859423939871e+0000,
  -2.570631056797048755890526455854482662510e+0002,
  -2.485216410094288379417154382189125598962e+0003,
  -5.253043804907295692946647153614119665649e+0003,
};
static double ps8[5] = {
   1.165343646196681758075176077627332052048e+0002,
   3.833744753641218451213253490882686307027e+0003,
   4.059785726484725470626341023967186966531e+0004,
   1.167529725643759169416844015694440325519e+0005,
   4.762772841467309430100106254805711722972e+0004,
};

static double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  -1.141254646918944974922813501362824060117e-0011,
  -7.031249408735992804117367183001996028304e-0002,
  -4.159610644705877925119684455252125760478e+0000,
  -6.767476522651671942610538094335912346253e+0001,
  -3.312312996491729755731871867397057689078e+0002,
  -3.464333883656048910814187305901796723256e+0002,
};
static double ps5[5] = {
   6.075393826923003305967637195319271932944e+0001,
   1.051252305957045869801410979087427910437e+0003,
   5.978970943338558182743915287887408780344e+0003,
   9.625445143577745335793221135208591603029e+0003,
   2.406058159229391070820491174867406875471e+0003,
};

static double pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  -2.547046017719519317420607587742992297519e-0009,
  -7.031196163814817199050629727406231152464e-0002,
  -2.409032215495295917537157371488126555072e+0000,
  -2.196597747348830936268718293366935843223e+0001,
  -5.807917047017375458527187341817239891940e+0001,
  -3.144794705948885090518775074177485744176e+0001,
};
static double ps3[5] = {
   3.585603380552097167919946472266854507059e+0001,
   3.615139830503038919981567245265266294189e+0002,
   1.193607837921115243628631691509851364715e+0003,
   1.127996798569074250675414186814529958010e+0003,
   1.735809308133357510239737333055228118910e+0002,
};

static double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  -8.875343330325263874525704514800809730145e-0008,
  -7.030309954836247756556445443331044338352e-0002,
  -1.450738467809529910662233622603401167409e+0000,
  -7.635696138235277739186371273434739292491e+0000,
  -1.119316688603567398846655082201614524650e+0001,
  -3.233645793513353260006821113608134669030e+0000,
};
static double ps2[5] = {
   2.222029975320888079364901247548798910952e+0001,
   1.362067942182152109590340823043813120940e+0002,
   2.704702786580835044524562897256790293238e+0002,
   1.538753942083203315263554770476850028583e+0002,
   1.465761769482561965099880599279699314477e+0001,
};

static double pzero(x)
	double x;
{
	double *p,*q,z,r,s;
	if (x >= 8.00)			   {p = pr8; q= ps8;}
	else if (x >= 4.54545211791992188) {p = pr5; q= ps5;}
	else if (x >= 2.85714149475097656) {p = pr3; q= ps3;}
	else if (x >= 2.00)		   {p = pr2; q= ps2;}
	z = one/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
	return one+ r/s;
}


/* For x >= 8, the asymptotic expansions of qzero is
 *	-1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 * We approximate pzero by
 * 	qzero(x) = s*(-1.25 + (R/S))
 * where  R = qr0 + qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 * 	  S = 1 + qs0*s^2 + ... + qs5*s^12
 * and
 *	| qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 */
static double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
   0.0,
   7.324218749999350414479738504551775297096e-0002,
   1.176820646822526933903301695932765232456e+0001,
   5.576733802564018422407734683549251364365e+0002,
   8.859197207564685717547076568608235802317e+0003,
   3.701462677768878501173055581933725704809e+0004,
};
static double qs8[6] = {
   1.637760268956898345680262381842235272369e+0002,
   8.098344946564498460163123708054674227492e+0003,
   1.425382914191204905277585267143216379136e+0005,
   8.033092571195144136565231198526081387047e+0005,
   8.405015798190605130722042369969184811488e+0005,
  -3.438992935378666373204500729736454421006e+0005,
};

static double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
   1.840859635945155400568380711372759921179e-0011,
   7.324217666126847411304688081129741939255e-0002,
   5.835635089620569401157245917610984757296e+0000,
   1.351115772864498375785526599119895942361e+0002,
   1.027243765961641042977177679021711341529e+0003,
   1.989977858646053872589042328678602481924e+0003,
};
static double qs5[6] = {
   8.277661022365377058749454444343415524509e+0001,
   2.077814164213929827140178285401017305309e+0003,
   1.884728877857180787101956800212453218179e+0004,
   5.675111228949473657576693406600265778689e+0004,
   3.597675384251145011342454247417399490174e+0004,
  -5.354342756019447546671440667961399442388e+0003,
};

static double qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
   4.377410140897386263955149197672576223054e-0009,
   7.324111800429115152536250525131924283018e-0002,
   3.344231375161707158666412987337679317358e+0000,
   4.262184407454126175974453269277100206290e+0001,
   1.708080913405656078640701512007621675724e+0002,
   1.667339486966511691019925923456050558293e+0002,
};
static double qs3[6] = {
   4.875887297245871932865584382810260676713e+0001,
   7.096892210566060535416958362640184894280e+0002,
   3.704148226201113687434290319905207398682e+0003,
   6.460425167525689088321109036469797462086e+0003,
   2.516333689203689683999196167394889715078e+0003,
  -1.492474518361563818275130131510339371048e+0002,
};

static double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
   1.504444448869832780257436041633206366087e-0007,
   7.322342659630792930894554535717104926902e-0002,
   1.998191740938159956838594407540292600331e+0000,
   1.449560293478857407645853071687125850962e+0001,
   3.166623175047815297062638132537957315395e+0001,
   1.625270757109292688799540258329430963726e+0001,
};
static double qs2[6] = {
   3.036558483552191922522729838478169383969e+0001,
   2.693481186080498724211751445725708524507e+0002,
   8.447837575953201460013136756723746023736e+0002,
   8.829358451124885811233995083187666981299e+0002,
   2.126663885117988324180482985363624996652e+0002,
  -5.310954938826669402431816125780738924463e+0000,
};

static double qzero(x)
	double x;
{
	double *p,*q, s,r,z;
	if (x >= 8.00)			   {p = qr8; q= qs8;}
	else if (x >= 4.54545211791992188) {p = qr5; q= qs5;}
	else if (x >= 2.85714149475097656) {p = qr3; q= qs3;}
	else if (x >= 2.00)		   {p = qr2; q= qs2;}
	z = one/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
	return (-.125 + r/s)/x;
}