common/C/j0.c

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License (the "License").
 * You may not use this file except in compliance with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
 * information: Portions Copyright [yyyy] [name of copyright owner]
 *
 * CDDL HEADER END
 */

/*
 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
 */
/*
 * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 */

/*
 * Floating point Bessel's function of the first and second kinds
 * of order zero: j0(x),y0(x);
 *
 * Special cases:
 *  y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
 *  y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
 */

#pragma weak __j0 = j0
#pragma weak __y0 = y0

#include "libm.h"
#include "libm_protos.h"
#include <math.h>
#include <values.h>

#define GENERIC double
static const GENERIC
zero    = 0.0,
small   = 1.0e-5,
tiny    = 1.0e-18,
one = 1.0,
eight   = 8.0,
invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
tpi = 0.636619772367581343075535053490057448;

static GENERIC pzero(GENERIC), qzero(GENERIC);
static const GENERIC r0[4] = {  /* [1.e-5, 1.28] */
    -2.500000000000003622131880894830476755537e-0001,
    1.095597547334830263234433855932375353303e-0002,
    -1.819734750463320921799187258987098087697e-0004,
    9.977001946806131657544212501069893930846e-0007,
};
static const GENERIC s0[4] = {  /* [1.e-5, 1.28] */
    1.0,
    1.867609810662950169966782360588199673741e-0002,
    1.590389206181565490878430827706972074208e-0004,
    6.520867386742583632375520147714499522721e-0007,
};
static const GENERIC r1[9] = {  /* [1.28,8] */
    9.999999999999999942156495584397047660949e-0001,
    -2.389887722731319130476839836908143731281e-0001,
    1.293359476138939027791270393439493640570e-0002,
    -2.770985642343140122168852400228563364082e-0004,
    2.905241575772067678086738389169625218912e-0006,
    -1.636846356264052597969042009265043251279e-0008,
    5.072306160724884775085431059052611737827e-0011,
    -8.187060730684066824228914775146536139112e-0014,
    5.422219326959949863954297860723723423842e-0017,
};
static const GENERIC s1[9] = {  /* [1.28,8] */
    1.0,
    1.101122772686807702762104741932076228349e-0002,
    6.140169310641649223411427764669143978228e-0005,
    2.292035877515152097976946119293215705250e-0007,
    6.356910426504644334558832036362219583789e-0010,
    1.366626326900219555045096999553948891401e-0012,
    2.280399586866739522891837985560481180088e-0015,
    2.801559820648939665270492520004836611187e-0018,
    2.073101088320349159764410261466350732968e-0021,
};

GENERIC
j0(GENERIC x) {
    GENERIC z, s, c, ss, cc, r, u, v, ox;
    int i;

    if (isnan(x))
        return (x*x);   /* + -> * for Cheetah */
    ox = x;
    x = fabs(x);
    if (x > 8.0) {
        if (!finite(x))
            return (zero);
        s = sin(x);
        c = cos(x);
    /*
     * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
     * where x0 = x-pi/4
     *  Better formula:
     *      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.
     */
        if (x > 8.9e307) {  /* x+x may overflow */
            ss = s-c;
            cc = s+c;
        } else if (signbit(s) != signbit(c)) {
            ss = s - c;
            cc = -cos(x+x)/ss;
        } else {
            cc = s + c;
            ss = -cos(x+x)/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 (x > 1.0e40) z = (invsqrtpi*cc)/sqrt(x);
        else {
            u = pzero(x); v = qzero(x);
            z = invsqrtpi*(u*cc-v*ss)/sqrt(x);
        }
    /* force to pass SVR4 even the result is wrong (sign) */
        if (x > X_TLOSS)
            return (_SVID_libm_err(ox, z, 34));
        else
            return (z);
    }
    if (x <= small) {
        if (x <= tiny)
            return (one-x);
        else
            return (one-x*x*0.25);
    }
    z = x*x;
    if (x <= 1.28) {
        r =  r0[0]+z*(r0[1]+z*(r0[2]+z*r0[3]));
        s =  s0[0]+z*(s0[1]+z*(s0[2]+z*s0[3]));
        return (one + z*(r/s));
    } else {
        for (r = r1[8], s = s1[8], i = 7; i >= 0; i--) {
        r = r*z + r1[i];
        s = s*z + s1[i];
        }
        return (r/s);
    }
}

static const GENERIC u0[13] = {
    -7.380429510868722526754723020704317641941e-0002,
    1.772607102684869924301459663049874294814e-0001,
    -1.524370666542713828604078090970799356306e-0002,
    4.650819100693891757143771557629924591915e-0004,
    -7.125768872339528975036316108718239946022e-0006,
    6.411017001656104598327565004771515257146e-0008,
    -3.694275157433032553021246812379258781665e-0010,
    1.434364544206266624252820889648445263842e-0012,
    -3.852064731859936455895036286874139896861e-0015,
    7.182052899726138381739945881914874579696e-0018,
    -9.060556574619677567323741194079797987200e-0021,
    7.124435467408860515265552217131230511455e-0024,
    -2.709726774636397615328813121715432044771e-0027,
};
static const GENERIC v0[5] = {
    1.0,
    4.678678931512549002587702477349214886475e-0003,
    9.486828955529948534822800829497565178985e-0006,
    1.001495929158861646659010844136682454906e-0008,
    4.725338116256021660204443235685358593611e-0012,
};

GENERIC
y0(GENERIC x) {
    GENERIC z, /* d, */ s, c, ss, cc, u, v;
    int i;

    if (isnan(x))
        return (x*x);   /* + -> * for Cheetah */
    if (x <= zero) {
        if (x == zero)
            /* d= -one/(x-x); */
            return (_SVID_libm_err(x, x, 8));
        else
            /* d = zero/(x-x); */
            return (_SVID_libm_err(x, x, 9));
    }
    if (x > 8.0) {
        if (!finite(x))
            return (zero);
        s = sin(x);
        c = cos(x);
    /*
     * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
     * where x0 = x-pi/4
     *  Better formula:
     *      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.
     */
        if (x > 8.9e307) {  /* x+x may overflow */
            ss = s-c;
            cc = s+c;
        } else if (signbit(s) != signbit(c)) {
            ss = s - c;
            cc = -cos(x+x)/ss;
        } else {
            cc = s + c;
            ss = -cos(x+x)/cc;
        }
    /*
     * j0(x) = 1/sqrt(pi*x) * (P(0,x)*cc - Q(0,x)*ss)
     * y0(x) = 1/sqrt(pi*x) * (P(0,x)*ss + Q(0,x)*cc)
     */
        if (x > 1.0e40)
            z = (invsqrtpi*ss)/sqrt(x);
        else
            z =  invsqrtpi*(pzero(x)*ss+qzero(x)*cc)/sqrt(x);
        if (x > X_TLOSS)
            return (_SVID_libm_err(x, z, 35));
        else
            return (z);

    }
    if (x <= tiny) {
        return (u0[0] + tpi*log(x));
    }
    z = x*x;
    for (u = u0[12], i = 11; i >= 0; i--) u = u*z + u0[i];
    v = v0[0]+z*(v0[1]+z*(v0[2]+z*(v0[3]+z*v0[4])));
    return (u/v + tpi*(j0(x)*log(x)));
}

static const GENERIC pr[7] = {  /* [8 -- inf]  pzero 6550 */
    .4861344183386052721391238447e5,
    .1377662549407112278133438945e6,
    .1222466364088289731869114004e6,
    .4107070084315176135583353374e5,
    .5026073801860637125889039915e4,
    .1783193659125479654541542419e3,
    .88010344055383421691677564e0,
};
static const GENERIC ps[7] = {  /* [8 -- inf] pzero 6550 */
    .4861344183386052721414037058e5,
    .1378196632630384670477582699e6,
    .1223967185341006542748936787e6,
    .4120150243795353639995862617e5,
    .5068271181053546392490184353e4,
    .1829817905472769960535671664e3,
    1.0,
};
static const GENERIC huge    = 1.0e10;

static GENERIC
pzero(GENERIC x) {
    GENERIC s, r, t, z;
    int i;
    if (x > huge)
        return (one);
    t = eight/x; z = t*t;
    r = pr[5]+z*pr[6];
    s = ps[5]+z;
    for (i = 4; i >= 0; i--) {
        r = r*z + pr[i];
        s = s*z + ps[i];
    }
    return (r/s);
}

static const GENERIC qr[7] = {  /* [8 -- inf]  qzero 6950 */
    -.1731210995701068539185611951e3,
    -.5522559165936166961235240613e3,
    -.5604935606637346590614529613e3,
    -.2200430300226009379477365011e3,
    -.323869355375648849771296746e2,
    -.14294979207907956223499258e1,
    -.834690374102384988158918e-2,
};
static const GENERIC qs[7] = {  /* [8 -- inf] qzero 6950 */
    .1107975037248683865326709645e5,
    .3544581680627082674651471873e5,
    .3619118937918394132179019059e5,
    .1439895563565398007471485822e5,
    .2190277023344363955930226234e4,
    .106695157020407986137501682e3,
    1.0,
};

static GENERIC
qzero(GENERIC x) {
    GENERIC s, r, t, z;
    int i;
    if (x > huge)
        return (-0.125/x);
    t = eight/x; z = t*t;
    r = qr[5]+z*qr[6];
    s = qs[5]+z;
    for (i = 4; i >= 0; i--) {
        r = r*z + qr[i];
        s = s*z + qs[i];
    }
    return (t*(r/s));
}