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dp_sqrt.c

/* IEEE754 floating point arithmetic
 * double precision square root
 */
/*
 * MIPS floating point support
 * Copyright (C) 1994-2000 Algorithmics Ltd.
 * http://www.algor.co.uk
 *
 * ########################################################################
 *
 *  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;
}

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