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.,
* 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 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
};
union ieee754dp ieee754dp_sqrt(union ieee754dp x)
{
struct _ieee754_csr oldcsr;
union ieee754dp y, z, t;
unsigned scalx, yh;
COMPXDP;
EXPLODEXDP;
ieee754_clearcx();
FLUSHXDP;
/* x == INF or NAN? */
switch (xc) {
case IEEE754_CLASS_SNAN:
return ieee754dp_nanxcpt(x);
case IEEE754_CLASS_QNAN:
/* sqrt(Nan) = Nan */
return x;
case IEEE754_CLASS_ZERO:
/* sqrt(0) = 0 */
return x;
case IEEE754_CLASS_INF:
if (xs) {
/* sqrt(-Inf) = Nan */
ieee754_setcx(IEEE754_INVALID_OPERATION);
return ieee754dp_indef();
}
/* sqrt(+Inf) = Inf */
return x;
case IEEE754_CLASS_DNORM:
DPDNORMX;
/* fall through */
case IEEE754_CLASS_NORM:
if (xs) {
/* sqrt(-x) = Nan */
ieee754_setcx(IEEE754_INVALID_OPERATION);
return ieee754dp_indef();
}
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 = FPU_CSR_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.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.bexp += 0x001;
y = ieee754dp_add(y, t);
/* twiddle last bit to force y correctly rounded */
/* set RZ, clear INEX flag */
ieee754_csr.rm = FPU_CSR_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 FPU_CSR_RU:
y.bits += 1;
/* drop through */
case FPU_CSR_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.bexp += scalx;
/* restore rounding mode, possibly set inexact */
ieee754_csr = oldcsr;
return y;
}