e_j0.cpp

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/* See the import.pl script for potential modifications */
/* @(#)e_j0.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * 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.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
   for performance improvement on pipelined processors.
*/

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_j0.c,v 1.8 1995/05/10 20:45:23 jtc Exp $";
#endif

/* __ieee754_j0(x), __ieee754_y0(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) = u00 + u01*z + ... + u06*z^6
 *		V(z) = 1  + v01*z + ... + v04*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 "math_private.h"

namespace streflop_libm {
#ifdef __STDC__
static Double pzero(Double), qzero(Double);
#else
static Double pzero(), qzero();
#endif

#ifdef __STDC__
static const Double
#else
static Double
#endif
huge 	= 1e300,
one	= 1.0,
invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
 		/* R0/S0 on [0, 2.00] */
R[]  =  {0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
 -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
 -4.61832688532103189199e-09}, /* 0xBE33D5E7, 0x73D63FCE */
S[]  =  {0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  1.16614003333790000205e-09}; /* 0x3E1408BC, 0xF4745D8F */

#ifdef __STDC__
static const Double zero = 0.0;
#else
static Double zero = 0.0;
#endif

#ifdef __STDC__
	Double __ieee754_j0(Double x)
#else
	Double __ieee754_j0(x)
	Double x;
#endif
{
	Double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
	int32_t hx,ix;

	GET_HIGH_WORD(hx,x);
	ix = hx&0x7fffffff;
	if(ix>=0x7ff00000) return one/(x*x);
	x = fabs(x);
	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
		__sincos (x, &s, &c);
		ss = s-c;
		cc = s+c;
		if(ix<0x7fe00000) {  /* 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(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(x);
		else {
		    u = pzero(x); v = qzero(x);
		    z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(x);
		}
		return z;
	}
	if(ix<0x3f200000) {	/* |x| < 2**-13 */
	    if(huge+x>one) {	/* raise inexact if x != 0 */
	        if(ix<0x3e400000) return one;	/* |x|<2**-27 */
	        else 	      return one - 0.25*x*x;
	    }
	}
	z = x*x;
#ifdef DO_NOT_USE_THIS
	r =  z*(R02+z*(R03+z*(R04+z*R05)));
	s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
#else
	r1 = z*R[2]; z2=z*z;
	r2 = R[3]+z*R[4]; z4=z2*z2;
	r  = r1 + z2*r2 + z4*R[5];
	s1 = one+z*S[1];
	s2 = S[2]+z*S[3];
	s = s1 + z2*s2 + z4*S[4];
#endif
	if(ix < 0x3FF00000) {	/* |x| < 1.00 */
	    return one + z*(-0.25+(r/s));
	} else {
	    u = 0.5*x;
	    return((one+u)*(one-u)+z*(r/s));
	}
}

#ifdef __STDC__
static const Double
#else
static Double
#endif
U[]  = {-7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 -3.98205194132103398453e-11}, /* 0xBDC5E43D, 0x693FB3C8 */
V[]  =  {1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
  4.41110311332675467403e-10}; /* 0x3DFE5018, 0x3BD6D9EF */

#ifdef __STDC__
	Double __ieee754_y0(Double x)
#else
	Double __ieee754_y0(x)
	Double x;
#endif
{
	Double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
	int32_t hx,ix,lx;

	EXTRACT_WORDS(hx,lx,x);
        ix = 0x7fffffff&hx;
    /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf.  */
	if(ix>=0x7ff00000) return  one/(x+x*x);
        if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */
        if(hx<0) return zero/(zero*x);
        if(ix >= 0x40000000) {  /* |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.
         */
		__sincos (x, &s, &c);
                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(ix<0x7fe00000) {  /* make sure x+x not overflow */
                    z = -__cos(x+x);
                    if ((s*c)<zero) cc = z/ss;
                    else            ss = z/cc;
                }
                if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
                else {
                    u = pzero(x); v = qzero(x);
                    z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
                }
                return z;
	}
	if(ix<=0x3e400000) {	/* x < 2**-27 */
	    return(U[0] + tpi*__ieee754_log(x));
	}
	z = x*x;
#ifdef DO_NOT_USE_THIS
	u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
	v = one+z*(v01+z*(v02+z*(v03+z*v04)));
#else
	u1 = U[0]+z*U[1]; z2=z*z;
	u2 = U[2]+z*U[3]; z4=z2*z2;
	u3 = U[4]+z*U[5]; z6=z4*z2;
	u = u1 + z2*u2 + z4*u3 + z6*U[6];
	v1 = one+z*V[0];
	v2 = V[1]+z*V[2];
	v = v1 + z2*v2 + z4*V[3];
#endif
	return(u/v + tpi*(__ieee754_j0(x)*__ieee754_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)
 */
#ifdef __STDC__
static const Double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#else
static Double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#endif
  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
};
#ifdef __STDC__
static const Double pS8[5] = {
#else