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/* @(#)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.

 * ====================================================

 */

#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"

#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] */

R02  =  1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */

R03  = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */

R04  =  1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */

R05  = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */

S01  =  1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */

S02  =  1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */

S03  =  5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */

S04  =  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;

        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 */

                s = sin(x);

                c = cos(x);

                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)/sqrt(x);

                else {

                    u = pzero(x); v = qzero(x);

                    z = invsqrtpi*(u*cc-v*ss)/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;

        r =  z*(R02+z*(R03+z*(R04+z*R05)));

        s =  one+z*(S01+z*(S02+z*(S03+z*S04)));

        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

u00  = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */

u01  =  1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */

u02  = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */

u03  =  3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */

u04  = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */

u05  =  1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */

u06  = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */

v01  =  1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */

v02  =  7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */

v03  =  2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */

v04  =  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;

        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  */

        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 */

        /* 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(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)/sqrt(x);

                else {

                    u = pzero(x); v = qzero(x);

                    z = invsqrtpi*(u*ss+v*cc)/sqrt(x);

                }

                return z;

        }

        if(ix<=0x3e400000) {    /* x < 2**-27 */

            return(u00 + tpi*__ieee754_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*(__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

static double pS8[5] = {

#endif

  1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */

  3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */

  4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */

  1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */

  4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */

};

#ifdef __STDC__

static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */

#else

static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */

#endif

 -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */

 -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */

 -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */

 -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */

 -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */

 -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */

};

#ifdef __STDC__

static const double pS5[5] = {

#else

static double pS5[5] = {

#endif

  6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */

  1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */

  5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */

  9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */

  2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */

};

#ifdef __STDC__

static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

#else

static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

#endif

 -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */

 -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */

 -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */

 -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */

 -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */

 -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */

};

#ifdef __STDC__

static const double pS3[5] = {

#else

static double pS3[5] = {

#endif

  3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */

  3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */

  1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */

  1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */

  1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */

};

#ifdef __STDC__

static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */

#else

static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */

#endif

 -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */

 -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */

 -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */

 -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */

 -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */

 -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */

};

#ifdef __STDC__

static const double pS2[5] = {

#else

static double pS2[5] = {

#endif

  2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */

  1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */

  2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */

  1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */

  1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */

};

#ifdef __STDC__

        static double pzero(double x)

#else

        static double pzero(x)

        double x;

#endif

{

#ifdef __STDC__

        const double *p = 0,*q = 0;

#else

        double *p,*q;

#endif

        double z,r,s;

        int32_t ix;

        GET_HIGH_WORD(ix,x);

        ix &= 0x7fffffff;

        if(ix>=0x40200000)     {p = pR8; q= pS8;}

        else if(ix>=0x40122E8B){p = pR5; q= pS5;}

        else if(ix>=0x4006DB6D){p = pR3; q= pS3;}

        else if(ix>=0x40000000){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)

 */

#ifdef __STDC__

static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */

#else

static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */

#endif

  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */

  7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */

  1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */

  5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */

  8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */

  3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */

};

#ifdef __STDC__

static const double qS8[6] = {

#else

static double qS8[6] = {

#endif

  1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */

  8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */

  1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */

  8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */

  8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */

 -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */

};

#ifdef __STDC__

static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */

#else

static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */

#endif

  1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */

  7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */

  5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */

  1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */

  1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */

  1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */

};

#ifdef __STDC__

static const double qS5[6] = {

#else

static double qS5[6] = {

#endif

  8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */

  2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */

  1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */

  5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */

  3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */

 -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */

};

#ifdef __STDC__

static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

#else

static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */

#endif

  4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */

  7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */

  3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */

  4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */

  1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */

  1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */

};

#ifdef __STDC__

static const double qS3[6] = {

#else

static double qS3[6] = {

#endif

  4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */

  7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */

  3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */

  6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */

  2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */

 -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */

};

#ifdef __STDC__

static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */

#else

static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */

#endif

  1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */

  7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */

  1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */

  1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */

  3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */

  1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */

};

#ifdef __STDC__

static const double qS2[6] = {

#else

static double qS2[6] = {

#endif

  3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */

  2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */

  8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */

  8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */

  2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */

 -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */

};

#ifdef __STDC__

        static double qzero(double x)

#else

        static double qzero(x)

        double x;

#endif

{

#ifdef __STDC__

        const double *p=0,*q=0;

#else

        double *p,*q;

#endif

        double s,r,z;

        int32_t ix;

        GET_HIGH_WORD(ix,x);

        ix &= 0x7fffffff;

        if(ix>=0x40200000)     {p = qR8; q= qS8;}

        else if(ix>=0x40122E8B){p = qR5; q= qS5;}

        else if(ix>=0x4006DB6D){p = qR3; q= qS3;}

        else if(ix>=0x40000000){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;

}