dp_sqrt.c

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/* IEEE754 floating point arithmetic
 * double precision square root
 */
/*
 * MIPS floating point support
 * Copyright (C) 1994-2000 Algorithmics Ltd.
 *
 * ########################################################################
 *
 *  This program is free software; you can distribute it and/or modify it
 *  under the terms of the GNU General Public License (Version 2) as
 *  published by the Free Software Foundation.
 *
 *  This program is distributed in the hope it will be useful, but WITHOUT
 *  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 *  FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
 *  for more details.
 *
 *  You should have received a copy of the GNU General Public License along
 *  with this program; if not, write to the Free Software Foundation, Inc.,
 *  59 Temple Place - Suite 330, Boston MA 02111-1307, USA.
 *
 * ########################################################################
 */


#include "ieee754dp.h"

static const unsigned table[] = {
    0, 1204, 3062, 5746, 9193, 13348, 18162, 23592,
    29598, 36145, 43202, 50740, 58733, 67158, 75992,
    85215, 83599, 71378, 60428, 50647, 41945, 34246,
    27478, 21581, 16499, 12183, 8588, 5674, 3403,
    1742, 661, 130
};

ieee754dp ieee754dp_sqrt(ieee754dp x)
{
    struct _ieee754_csr oldcsr;
    ieee754dp y, z, t;
    unsigned scalx, yh;
    COMPXDP;

    EXPLODEXDP;
    CLEARCX;
    FLUSHXDP;

    /* x == INF or NAN? */
    switch (xc) {
    case IEEE754_CLASS_QNAN:
        /* sqrt(Nan) = Nan */
        return ieee754dp_nanxcpt(x, "sqrt");
    case IEEE754_CLASS_SNAN:
        SETCX(IEEE754_INVALID_OPERATION);
        return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
    case IEEE754_CLASS_ZERO:
        /* sqrt(0) = 0 */
        return x;
    case IEEE754_CLASS_INF:
        if (xs) {
            /* sqrt(-Inf) = Nan */
            SETCX(IEEE754_INVALID_OPERATION);
            return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
        }
        /* sqrt(+Inf) = Inf */
        return x;
    case IEEE754_CLASS_DNORM:
        DPDNORMX;
        /* fall through */
    case IEEE754_CLASS_NORM:
        if (xs) {
            /* sqrt(-x) = Nan */
            SETCX(IEEE754_INVALID_OPERATION);
            return ieee754dp_nanxcpt(ieee754dp_indef(), "sqrt");
        }
        break;
    }

    /* save old csr; switch off INX enable & flag; set RN rounding */
    oldcsr = ieee754_csr;
    ieee754_csr.mx &= ~IEEE754_INEXACT;
    ieee754_csr.sx &= ~IEEE754_INEXACT;
    ieee754_csr.rm = IEEE754_RN;

    /* adjust exponent to prevent overflow */
    scalx = 0;
    if (xe > 512) {     /* x > 2**-512? */
        xe -= 512;  /* x = x / 2**512 */
        scalx += 256;
    } else if (xe < -512) { /* x < 2**-512? */
        xe += 512;  /* x = x * 2**512 */
        scalx -= 256;
    }

    y = x = builddp(0, xe + DP_EBIAS, xm & ~DP_HIDDEN_BIT);

    /* magic initial approximation to almost 8 sig. bits */
    yh = y.bits >> 32;
    yh = (yh >> 1) + 0x1ff80000;
    yh = yh - table[(yh >> 15) & 31];
    y.bits = ((u64) yh << 32) | (y.bits & 0xffffffff);

    /* Heron's rule once with correction to improve to ~18 sig. bits */
    /* t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0; */
    t = ieee754dp_div(x, y);
    y = ieee754dp_add(y, t);
    y.bits -= 0x0010000600000000LL;
    y.bits &= 0xffffffff00000000LL;

    /* triple to almost 56 sig. bits: y ~= sqrt(x) to within 1 ulp */
    /* t=y*y; z=t;  pt[n0]+=0x00100000; t+=z; z=(x-z)*y; */
    z = t = ieee754dp_mul(y, y);
    t.parts.bexp += 0x001;
    t = ieee754dp_add(t, z);
    z = ieee754dp_mul(ieee754dp_sub(x, z), y);

    /* t=z/(t+x) ;  pt[n0]+=0x00100000; y+=t; */
    t = ieee754dp_div(z, ieee754dp_add(t, x));
    t.parts.bexp += 0x001;
    y = ieee754dp_add(y, t);

    /* twiddle last bit to force y correctly rounded */

    /* set RZ, clear INEX flag */
    ieee754_csr.rm = IEEE754_RZ;
    ieee754_csr.sx &= ~IEEE754_INEXACT;

    /* t=x/y; ...chopped quotient, possibly inexact */
    t = ieee754dp_div(x, y);

    if (ieee754_csr.sx & IEEE754_INEXACT || t.bits != y.bits) {

        if (!(ieee754_csr.sx & IEEE754_INEXACT))
            /* t = t-ulp */
            t.bits -= 1;

        /* add inexact to result status */
        oldcsr.cx |= IEEE754_INEXACT;
        oldcsr.sx |= IEEE754_INEXACT;

        switch (oldcsr.rm) {
        case IEEE754_RP:
            y.bits += 1;
            /* drop through */
        case IEEE754_RN:
            t.bits += 1;
            break;
        }

        /* y=y+t; ...chopped sum */
        y = ieee754dp_add(y, t);

        /* adjust scalx for correctly rounded sqrt(x) */
        scalx -= 1;
    }

    /* py[n0]=py[n0]+scalx; ...scale back y */
    y.parts.bexp += scalx;

    /* restore rounding mode, possibly set inexact */
    ieee754_csr = oldcsr;

    return y;
}