common/C/j1.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: j1(x),y1(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 __j1 = j1
#pragma weak __y1 = y1

#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-20,
one = 1.0,
invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
tpi = 0.636619772367581343075535053490057448;

static GENERIC pone(GENERIC), qone(GENERIC);
static const GENERIC r0[4] = {
    -6.250000000000002203053200981413218949548e-0002,
    1.600998455640072901321605101981501263762e-0003,
    -1.963888815948313758552511884390162864930e-0005,
    8.263917341093549759781339713418201620998e-0008,
};
static const GENERIC s0[7] = {
    1.0e0,
    1.605069137643004242395356851797873766927e-0002,
    1.149454623251299996428500249509098499383e-0004,
    3.849701673735260970379681807910852327825e-0007,
};
static const GENERIC r1[12] = {
    4.999999999999999995517408894340485471724e-0001,
    -6.003825028120475684835384519945468075423e-0002,
    2.301719899263321828388344461995355419832e-0003,
    -4.208494869238892934859525221654040304068e-0005,
    4.377745135188837783031540029700282443388e-0007,
    -2.854106755678624335145364226735677754179e-0009,
    1.234002865443952024332943901323798413689e-0011,
    -3.645498437039791058951273508838177134310e-0014,
    7.404320596071797459925377103787837414422e-0017,
    -1.009457448277522275262808398517024439084e-0019,
    8.520158355824819796968771418801019930585e-0023,
    -3.458159926081163274483854614601091361424e-0026,
};
static const GENERIC s1[5] = {
    1.0e0,
    4.923499437590484879081138588998986303306e-0003,
    1.054389489212184156499666953501976688452e-0005,
    1.180768373106166527048240364872043816050e-0008,
    5.942665743476099355323245707680648588540e-0012,
};

GENERIC
j1(GENERIC x) {
    GENERIC z, d, s, c, ss, cc, r;
    int i, sgn;

    if (!finite(x))
        return (one/x);
    sgn = signbit(x);
    x = fabs(x);
    if (x > 8.00) {
        s = sin(x);
        c = cos(x);
    /*
     * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(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 (x > 8.9e307) {  /* x+x may overflow */
            ss = -s-c;
            cc =  s-c;
        } else if (signbit(s) != signbit(c)) {
            cc = s - c;
            ss = cos(x+x)/cc;
        } else {
            ss = -s-c;
            cc = cos(x+x)/ss;
        }
    /*
     * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
     * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
     */
        if (x > 1.0e40)
            d = (invsqrtpi*cc)/sqrt(x);
        else
            d =  invsqrtpi*(pone(x)*cc-qone(x)*ss)/sqrt(x);

        if (x > X_TLOSS) {
            if (sgn != 0) { d = -d; x = -x; }
            return (_SVID_libm_err(x, d, 36));
        } else
            if (sgn == 0)
                return (d);
            else
                return (-d);
    }
    if (x <= small) {
        if (x <= tiny)
            d = 0.5*x;
        else
            d =  x*(0.5-x*x*0.125);
        if (sgn == 0)
            return (d);
        else
            return (-d);
    }
    z = x*x;
    if (x < 1.28) {
        r = r0[3];
        s = s0[3];
        for (i = 2; i >= 0; i--) {
        r = r*z + r0[i];
        s = s*z + s0[i];
        }
        d = x*0.5+x*(z*(r/s));
    } else {
        r = r1[11];
        for (i = 10; i >= 0; i--) r = r*z + r1[i];
        s = s1[0]+z*(s1[1]+z*(s1[2]+z*(s1[3]+z*s1[4])));
        d = x*(r/s);
    }
    if (sgn == 0)
        return (d);
    else
        return (-d);
}

static const GENERIC u0[4] = {
    -1.960570906462389461018983259589655961560e-0001,
    4.931824118350661953459180060007970291139e-0002,
    -1.626975871565393656845930125424683008677e-0003,
    1.359657517926394132692884168082224258360e-0005,
};
static const GENERIC v0[5] = {
    1.0e0,
    2.565807214838390835108224713630901653793e-0002,
    3.374175208978404268650522752520906231508e-0004,
    2.840368571306070719539936935220728843177e-0006,
    1.396387402048998277638900944415752207592e-0008,
};
static const GENERIC u1[12] = {
    -1.960570906462389473336339614647555351626e-0001,
    5.336268030335074494231369159933012844735e-0002,
    -2.684137504382748094149184541866332033280e-0003,
    5.737671618979185736981543498580051903060e-0005,
    -6.642696350686335339171171785557663224892e-0007,
    4.692417922568160354012347591960362101664e-0009,
    -2.161728635907789319335231338621412258355e-0011,
    6.727353419738316107197644431844194668702e-0014,
    -1.427502986803861372125234355906790573422e-0016,
    2.020392498726806769468143219616642940371e-0019,
    -1.761371948595104156753045457888272716340e-0022,
    7.352828391941157905175042420249225115816e-0026,
};
static const GENERIC v1[5] = {
    1.0e0,
    5.029187436727947764916247076102283399442e-0003,
    1.102693095808242775074856548927801750627e-0005,
    1.268035774543174837829534603830227216291e-0008,
    6.579416271766610825192542295821308730206e-0012,
};


GENERIC
y1(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)
            /* return -one/zero;  */
            return (_SVID_libm_err(x, x, 10));
        else
            /* return zero/zero; */
            return (_SVID_libm_err(x, x, 11));
    }
    if (x > 8.0) {
        if (!finite(x))
            return (zero);
        s = sin(x);
        c = cos(x);
    /*
     * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(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 (x > 8.9e307) {  /* x+x may overflow */
            ss = -s-c;
            cc =  s-c;
        } else if (signbit(s) != signbit(c)) {
            cc = s - c;
            ss = cos(x+x)/cc;
        } else {
            ss = -s-c;
            cc = cos(x+x)/ss;
        }
    /*
     * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss)
     * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc)
     */
        if (x > 1.0e91)
            d =  (invsqrtpi*ss)/sqrt(x);
        else
            d = invsqrtpi*(pone(x)*ss+qone(x)*cc)/sqrt(x);

        if (x > X_TLOSS)
            return (_SVID_libm_err(x, d, 37));
        else
            return (d);
    }
        if (x <= tiny) {
            return (-tpi/x);
        }
    z = x*x;
    if (x < 1.28) {
        u = u0[3]; v = v0[3]+z*v0[4];
        for (i = 2; i >= 0; i--) {
        u = u*z + u0[i];
        v = v*z + v0[i];
        }
    } else {
        for (u = u1[11], i = 10; i >= 0; i--) u = u*z+u1[i];
        v = v1[0]+z*(v1[1]+z*(v1[2]+z*(v1[3]+z*v1[4])));
    }
    return (x*(u/v) + tpi*(j1(x)*log(x)-one/x));
}

static const GENERIC pr0[6] = {
    -.4435757816794127857114720794e7,
    -.9942246505077641195658377899e7,
    -.6603373248364939109255245434e7,
    -.1523529351181137383255105722e7,
    -.1098240554345934672737413139e6,
    -.1611616644324610116477412898e4,
};
static const GENERIC ps0[6] = {
    -.4435757816794127856828016962e7,
    -.9934124389934585658967556309e7,
    -.6585339479723087072826915069e7,
    -.1511809506634160881644546358e7,
    -.1072638599110382011903063867e6,
    -.1455009440190496182453565068e4,
};
static const GENERIC huge    = 1.0e10;

static GENERIC
pone(GENERIC x) {
    GENERIC s, r, t, z;
    int i;
        /* assume x > 8 */
    if (x > huge)
        return (one);

    t = 8.0/x; z = t*t;
    r = pr0[5]; s = ps0[5]+z;
    for (i = 4; i >= 0; i--) {
        r = z*r + pr0[i];
        s = z*s + ps0[i];
    }
    return (r/s);
}


static const GENERIC qr0[6] = {
    0.3322091340985722351859704442e5,
    0.8514516067533570196555001171e5,
    0.6617883658127083517939992166e5,
    0.1849426287322386679652009819e5,
    0.1706375429020768002061283546e4,
    0.3526513384663603218592175580e2,
};
static const GENERIC qs0[6] = {
    0.7087128194102874357377502472e6,
    0.1819458042243997298924553839e7,
    0.1419460669603720892855755253e7,
    0.4002944358226697511708610813e6,
    0.3789022974577220264142952256e5,
    0.8638367769604990967475517183e3,
};

static GENERIC
qone(GENERIC x) {
    GENERIC s, r, t, z;
    int i;
    if (x > huge)
        return (0.375/x);

    t = 8.0/x; z = t*t;
        /* assume x > 8 */
    r = qr0[5]; s = qs0[5]+z;
    for (i = 4; i >= 0; i--) {
        r = z*r + qr0[i];
        s = z*s + qs0[i];
    }
    return (t*(r/s));
}