common/Q/sqrtl.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.
 */

#pragma weak __sqrtl = sqrtl

#include "libm.h"
#include "longdouble.h"

extern int __swapTE(int);
extern int __swapEX(int);
extern enum fp_direction_type __swapRD(enum fp_direction_type);

/*
 * in struct longdouble, msw consists of
 *  unsigned short  sgn:1;
 *  unsigned short  exp:15;
 *  unsigned short  frac1:16;
 */

#ifdef __LITTLE_ENDIAN

/* array indices used to access words within a double */
#define HIWORD  1
#define LOWORD  0

/* structure used to access words within a quad */
union longdouble {
    struct {
        unsigned int    frac4;
        unsigned int    frac3;
        unsigned int    frac2;
        unsigned int    msw;
    } l;
    long double d;
};

/* default NaN returned for sqrt(neg) */
static const union longdouble
    qnan = { 0xffffffff, 0xffffffff, 0xffffffff, 0x7fffffff };

/* signalling NaN used to raise invalid */
static const union {
    unsigned u[2];
    double d;
} snan = { 0, 0x7ff00001 };

#else

/* array indices used to access words within a double */
#define HIWORD  0
#define LOWORD  1

/* structure used to access words within a quad */
union longdouble {
    struct {
        unsigned int    msw;
        unsigned int    frac2;
        unsigned int    frac3;
        unsigned int    frac4;
    } l;
    long double d;
};

/* default NaN returned for sqrt(neg) */
static const union longdouble
    qnan = { 0x7fffffff, 0xffffffff, 0xffffffff, 0xffffffff };

/* signalling NaN used to raise invalid */
static const union {
    unsigned u[2];
    double d;
} snan = { 0x7ff00001, 0 };

#endif /* __LITTLE_ENDIAN */


static const double
    zero = 0.0,
    half = 0.5,
    one = 1.0,
    huge = 1.0e300,
    tiny = 1.0e-300,
    two36 = 6.87194767360000000000e+10,
    two30 = 1.07374182400000000000e+09,
    two6 = 6.40000000000000000000e+01,
    two4 = 1.60000000000000000000e+01,
    twom18 = 3.81469726562500000000e-06,
    twom28 = 3.72529029846191406250e-09,
    twom42 = 2.27373675443232059479e-13,
    twom60 = 8.67361737988403547206e-19,
    twom62 = 2.16840434497100886801e-19,
    twom66 = 1.35525271560688054251e-20,
    twom90 = 8.07793566946316088742e-28,
    twom113 = 9.62964972193617926528e-35,
    twom124 = 4.70197740328915003187e-38;


/*
*   Extract the exponent and normalized significand (represented as
*   an array of five doubles) from a finite, nonzero quad.
*/
static int
__q_unpack(const union longdouble *x, double *s)
{
    union {
        double          d;
        unsigned int    l[2];
    } u;
    double          b;
    unsigned int    lx, w[3];
    int             ex;

    /* get the normalized significand and exponent */
    ex = (int) ((x->l.msw & 0x7fffffff) >> 16);
    lx = x->l.msw & 0xffff;
    if (ex)
    {
        lx |= 0x10000;
        w[0] = x->l.frac2;
        w[1] = x->l.frac3;
        w[2] = x->l.frac4;
    }
    else
    {
        if (lx | (x->l.frac2 & 0xfffe0000))
        {
            w[0] = x->l.frac2;
            w[1] = x->l.frac3;
            w[2] = x->l.frac4;
            ex = 1;
        }
        else if (x->l.frac2 | (x->l.frac3 & 0xfffe0000))
        {
            lx = x->l.frac2;
            w[0] = x->l.frac3;
            w[1] = x->l.frac4;
            w[2] = 0;
            ex = -31;
        }
        else if (x->l.frac3 | (x->l.frac4 & 0xfffe0000))
        {
            lx = x->l.frac3;
            w[0] = x->l.frac4;
            w[1] = w[2] = 0;
            ex = -63;
        }
        else
        {
            lx = x->l.frac4;
            w[0] = w[1] = w[2] = 0;
            ex = -95;
        }
        while ((lx & 0x10000) == 0)
        {
            lx = (lx << 1) | (w[0] >> 31);
            w[0] = (w[0] << 1) | (w[1] >> 31);
            w[1] = (w[1] << 1) | (w[2] >> 31);
            w[2] <<= 1;
            ex--;
        }
    }

    /* extract the significand into five doubles */
    u.l[HIWORD] = 0x42300000;
    u.l[LOWORD] = 0;
    b = u.d;
    u.l[LOWORD] = lx;
    s[0] = u.d - b;

    u.l[HIWORD] = 0x40300000;
    u.l[LOWORD] = 0;
    b = u.d;
    u.l[LOWORD] = w[0] & 0xffffff00;
    s[1] = u.d - b;

    u.l[HIWORD] = 0x3e300000;
    u.l[LOWORD] = 0;
    b = u.d;
    u.l[HIWORD] |= w[0] & 0xff;
    u.l[LOWORD] = w[1] & 0xffff0000;
    s[2] = u.d - b;

    u.l[HIWORD] = 0x3c300000;
    u.l[LOWORD] = 0;
    b = u.d;
    u.l[HIWORD] |= w[1] & 0xffff;
    u.l[LOWORD] = w[2] & 0xff000000;
    s[3] = u.d - b;

    u.l[HIWORD] = 0x3c300000;
    u.l[LOWORD] = 0;
    b = u.d;
    u.l[LOWORD] = w[2] & 0xffffff;
    s[4] = u.d - b;

    return ex - 0x3fff;
}


/*
*   Pack an exponent and array of three doubles representing a finite,
*   nonzero number into a quad.  Assume the sign is already there and
*   the rounding mode has been fudged accordingly.
*/
static void
__q_pack(const double *z, int exp, enum fp_direction_type rm,
    union longdouble *x, int *inexact)
{
    union {
        double          d;
        unsigned int    l[2];
    } u;
    double          s[3], t, t2;
    unsigned int    msw, frac2, frac3, frac4;

    /* bias exponent and strip off integer bit */
    exp += 0x3fff;
    s[0] = z[0] - one;
    s[1] = z[1];
    s[2] = z[2];

    /*
     * chop the significand to obtain the fraction;
     * use round-to-minus-infinity to ensure chopping
     */
    (void) __swapRD(fp_negative);

    /* extract the first eighty bits of fraction */
    t = s[1] + s[2];
    u.d = two36 + (s[0] + t);
    msw = u.l[LOWORD];
    s[0] -= (u.d - two36);

    u.d = two4 + (s[0] + t);
    frac2 = u.l[LOWORD];
    s[0] -= (u.d - two4);

    u.d = twom28 + (s[0] + t);
    frac3 = u.l[LOWORD];
    s[0] -= (u.d - twom28);

    /* condense the remaining fraction; errors here won't matter */
    t = s[0] + s[1];
    s[1] = ((s[0] - t) + s[1]) + s[2];
    s[0] = t;

    /* get the last word of fraction */
    u.d = twom60 + (s[0] + s[1]);
    frac4 = u.l[LOWORD];
    s[0] -= (u.d - twom60);

    /*
     * keep track of what's left for rounding; note that
     * t2 will be non-negative due to rounding mode
     */
    t = s[0] + s[1];
    t2 = (s[0] - t) + s[1];

    if (t != zero)
    {
        *inexact = 1;

        /* decide whether to round the fraction up */
        if (rm == fp_positive || (rm == fp_nearest && (t > twom113 ||
            (t == twom113 && (t2 != zero || frac4 & 1)))))
        {
            /* round up and renormalize if necessary */
            if (++frac4 == 0)
                if (++frac3 == 0)
                    if (++frac2 == 0)
                        if (++msw == 0x10000)
                        {
                            msw = 0;
                            exp++;
                        }
        }
    }

    /* assemble the result */
    x->l.msw |= msw | (exp << 16);
    x->l.frac2 = frac2;
    x->l.frac3 = frac3;
    x->l.frac4 = frac4;
}


/*
*   Compute the square root of x and place the TP result in s.
*/
static void
__q_tp_sqrt(const double *x, double *s)
{
    double  c, rr, r[3], tt[3], t[5];

    /* approximate the divisor for the Newton iteration */
    c = sqrt((x[0] + x[1]) + x[2]);
    rr = half / c;

    /* compute the first five "digits" of the square root */
    t[0] = (c + two30) - two30;
    tt[0] = t[0] + t[0];
    r[0] = ((x[0] - t[0] * t[0]) + x[1]) + x[2];

    t[1] = (rr * (r[0] + x[3]) + two6) - two6;
    tt[1] = t[1] + t[1];
    r[0] -= tt[0] * t[1];
    r[1] = x[3] - t[1] * t[1];
    c = (r[1] + twom18) - twom18;
    r[0] += c;
    r[1] = (r[1] - c) + x[4];

    t[2] = (rr * (r[0] + r[1]) + twom18) - twom18;
    tt[2] = t[2] + t[2];
    r[0] -= tt[0] * t[2];
    r[1] -= tt[1] * t[2];
    c = (r[1] + twom42) - twom42;
    r[0] += c;
    r[1] = (r[1] - c) - t[2] * t[2];

    t[3] = (rr * (r[0] + r[1]) + twom42) - twom42;
    r[0] = ((r[0] - tt[0] * t[3]) + r[1]) - tt[1] * t[3];
    r[1] = -tt[2] * t[3];
    c = (r[1] + twom90) - twom90;
    r[0] += c;
    r[1] = (r[1] - c) - t[3] * t[3];

    t[4] = (rr * (r[0] + r[1]) + twom66) - twom66;

    /* here we just need to get the sign of the remainder */
    c = (((((r[0] - tt[0] * t[4]) - tt[1] * t[4]) + r[1])
        - tt[2] * t[4]) - (t[3] + t[3]) * t[4]) - t[4] * t[4];

    /* reduce to three doubles */
    t[0] += t[1];
    t[1] = t[2] + t[3];
    t[2] = t[4];

    /* if the third term might lie on a rounding boundary, perturb it */
    if (c != zero && t[2] == (twom62 + t[2]) - twom62)
    {
        if (c < zero)
            t[2] -= twom124;
        else
            t[2] += twom124;
    }

    /* condense the square root */
    c = t[1] + t[2];
    t[2] += (t[1] - c);
    t[1] = c;
    c = t[0] + t[1];
    s[1] = t[1] + (t[0] - c);
    s[0] = c;
    if (s[1] == zero)
    {
        c = s[0] + t[2];
        s[1] = t[2] + (s[0] - c);
        s[0] = c;
        s[2] = zero;
    }
    else
    {
        c = s[1] + t[2];
        s[2] = t[2] + (s[1] - c);
        s[1] = c;
    }
}


long double
sqrtl(long double ldx)
{
    union   longdouble      x;
    volatile double         t;
    double                  xx[5], zz[3];
    enum fp_direction_type  rm;
    int             ex, inexact, exc, traps;

    /* clear cexc */
    t = zero;
    t -= zero;

    /* check for zero operand */
    x.d = ldx;
    if (!((x.l.msw & 0x7fffffff) | x.l.frac2 | x.l.frac3 | x.l.frac4))
        return ldx;

    /* handle nan and inf cases */
    if ((x.l.msw & 0x7fffffff) >= 0x7fff0000)
    {
        if ((x.l.msw & 0xffff) | x.l.frac2 | x.l.frac3 | x.l.frac4)
        {
            if (!(x.l.msw & 0x8000))
            {
                /* snan, signal invalid */
                t += snan.d;
            }
            x.l.msw |= 0x8000;
            return x.d;
        }
        if (x.l.msw & 0x80000000)
        {
            /* sqrt(-inf), signal invalid */
            t = -one;
            t = sqrt(t);
            return qnan.d;
        }
        /* sqrt(inf), return inf */
        return x.d;
    }

    /* handle negative numbers */
    if (x.l.msw & 0x80000000)
    {
        t = -one;
        t = sqrt(t);
        return qnan.d;
    }

    /* now x is finite, positive */

    traps = __swapTE(0);
    exc = __swapEX(0);
    rm = __swapRD(fp_nearest);

    ex = __q_unpack(&x, xx);
    if (ex & 1)
    {
        /* make exponent even */
        xx[0] += xx[0];
        xx[1] += xx[1];
        xx[2] += xx[2];
        xx[3] += xx[3];
        xx[4] += xx[4];
        ex--;
    }
    __q_tp_sqrt(xx, zz);

    /* put everything together */
    x.l.msw = 0;
    inexact = 0;
    __q_pack(zz, ex >> 1, rm, &x, &inexact);

    (void) __swapRD(rm);
    (void) __swapEX(exc);
    (void) __swapTE(traps);
    if (inexact)
    {
        t = huge;
        t += tiny;
    }
    return x.d;
}