e_j1.c

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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
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 * The Original Code is Mozilla Communicator client code, released
 * March 31, 1998.
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/* @(#)e_j1.c 1.3 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* __ieee754_j1(x), __ieee754_y1(x)
 * Bessel function of the first and second kinds of order zero.
 * Method -- j1(x):
 *	1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
 *	2. Reduce x to |x| since j1(x)=-j1(-x),  and
 *	   for x in (0,2)
 *		j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
 *	   (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
 *	   for x in (2,inf)
 * 		j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   as follow:
 *		cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 *			=  1/sqrt(2) * (sin(x) - cos(x))
 *		sin(x1) =  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.)
 *	   
 *	3 Special cases
 *		j1(nan)= nan
 *		j1(0) = 0
 *		j1(inf) = 0
 *		
 * Method -- y1(x):
 *	1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 
 *	2. For x<2.
 *	   Since 
 *		y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
 *	   therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
 *	   We use the following function to approximate y1,
 *		y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
 *	   where for x in [0,2] (abs err less than 2**-65.89)
 *		U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
 *		V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
 *	   Note: For tiny x, 1/x dominate y1 and hence
 *		y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
 *	3. For x>=2.
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   by method mentioned above.
 */

#include "fdlibm.h"

#ifdef __STDC__
static double pone(double), qone(double);
#else
static double pone(), qone();
#endif

#ifdef __STDC__
static const double 
#else
static double 
#endif
really_big    = 1e300,
one	= 1.0,
invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
	/* R0/S0 on [0,2] */
r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */

static double zero    = 0.0;

#ifdef __STDC__
	double __ieee754_j1(double x) 
#else
	double __ieee754_j1(x) 
	double x;
#endif
{
        fd_twoints un;
	double z, s,c,ss,cc,r,u,v,y;
	int hx,ix;

        un.d = x;
	hx = __HI(un);
	ix = hx&0x7fffffff;
	if(ix>=0x7ff00000) return one/x;
	y = fd_fabs(x);
	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
		s = fd_sin(y);
		c = fd_cos(y);
		ss = -s-c;
		cc = s-c;
		if(ix<0x7fe00000) {  /* make sure y+y not overflow */
		    z = fd_cos(y+y);
		    if ((s*c)>zero) cc = z/ss;
		    else 	    ss = z/cc;
		}
	/*
	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
	 */
		if(ix>0x48000000) z = (invsqrtpi*cc)/fd_sqrt(y);
		else {
		    u = pone(y); v = qone(y);
		    z = invsqrtpi*(u*cc-v*ss)/fd_sqrt(y);
		}
		if(hx<0) return -z;
		else  	 return  z;
	}
	if(ix<0x3e400000) {	/* |x|<2**-27 */
	    if(really_big+x>one) return 0.5*x;/* inexact if x!=0 necessary */
	}
	z = x*x;
	r =  z*(r00+z*(r01+z*(r02+z*r03)));
	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
	r *= x;
	return(x*0.5+r/s);
}

#ifdef __STDC__
static const double U0[5] = {
#else
static double U0[5] = {
#endif
 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
};
#ifdef __STDC__
static const double V0[5] = {
#else
static double V0[5] = {
#endif
  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
};

#ifdef __STDC__
	double __ieee754_y1(double x) 
#else
	double __ieee754_y1(x) 
	double x;
#endif
{
        fd_twoints un;
	double z, s,c,ss,cc,u,v;
	int hx,ix,lx;

        un.d = x;
        hx = __HI(un);
        ix = 0x7fffffff&hx;
        lx = __LO(un);
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
	if(ix>=0x7ff00000) return  one/(x+x*x); 
        if((ix|lx)==0) return -one/zero;
        if(hx<0) return zero/zero;
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
                s = fd_sin(x);
                c = fd_cos(x);
                ss = -s-c;
                cc = s-c;
                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
                    z = fd_cos(x+x);
                    if ((s*c)>zero) cc = z/ss;
                    else            ss = z/cc;
                }
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
         * where x0 = x-3pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
                if(ix>0x48000000) z = (invsqrtpi*ss)/fd_sqrt(x);
                else {
                    u = pone(x); v = qone(x);
                    z = invsqrtpi*(u*ss+v*cc)/fd_sqrt(x);
                }
                return z;
        } 
        if(ix<=0x3c900000) {    /* x < 2**-54 */
            return(-tpi/x);
        } 
        z = x*x;
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
        return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
}

/* For x >= 8, the asymptotic expansions of pone is
 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
 * We approximate pone by
 * 	pone(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
 *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
 */