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/* sqrtl.c
*
* Square root, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, sqrtl();
*
* y = sqrtl( x );
*
*
*
* DESCRIPTION:
*
* Returns the square root of x.
*
* Range reduction involves isolating the power of two of the
* argument and using a polynomial approximation to obtain
* a rough value for the square root. Then Heron's iteration
* is used three times to converge to an accurate value.
*
* Note, some arithmetic coprocessors such as the 8087 and
* 68881 produce correctly rounded square roots, which this
* routine will not.
*
* ACCURACY:
*
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,10 30000 8.1e-20 3.1e-20
*
*
* ERROR MESSAGES:
*
* message condition value returned
* sqrt domain x < 0 0.0
*
*/
/*
Cephes Math Library Release 2.2: December, 1990
Copyright 1984, 1990 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
#define SQRT2 1.4142135623730950488017E0L
#ifdef ANSIPROT
extern long double frexpl ( long double, int * );
extern long double ldexpl ( long double, int );
#else
long double frexpl(), ldexpl();
#endif
long double sqrtl(x)
long double x;
{
int e;
long double z, w;
#ifndef UNK
short *q;
#endif
if( x <= 0.0 )
{
if( x < 0.0 )
mtherr( "sqrtl", DOMAIN );
return( 0.0 );
}
w = x;
/* separate exponent and significand */
#ifdef UNK
z = frexpl( x, &e );
#endif
/* Note, frexp and ldexp are used in order to
* handle denormal numbers properly.
*/
#ifdef IBMPC
z = frexpl( x, &e );
q = (short *)&x; /* point to the exponent word */
q += 4;
/*
e = ((*q >> 4) & 0x0fff) - 0x3fe;
*q &= 0x000f;
*q |= 0x3fe0;
z = x;
*/
#endif
#ifdef MIEEE
z = frexpl( x, &e );
q = (short *)&x;
/*
e = ((*q >> 4) & 0x0fff) - 0x3fe;
*q &= 0x000f;
*q |= 0x3fe0;
z = x;
*/
#endif
/* approximate square root of number between 0.5 and 1
* relative error of linear approximation = 7.47e-3
*/
/*
x = 0.4173075996388649989089L + 0.59016206709064458299663L * z;
*/
/* quadratic approximation, relative error 6.45e-4 */
x = ( -0.20440583154734771959904L * z
+ 0.89019407351052789754347L) * z
+ 0.31356706742295303132394L;
/* adjust for odd powers of 2 */
if( (e & 1) != 0 )
x *= SQRT2;
/* re-insert exponent */
#ifdef UNK
x = ldexpl( x, (e >> 1) );
#endif
#ifdef IBMPC
x = ldexpl( x, (e >> 1) );
/*
*q += ((e >>1) & 0x7ff) << 4;
*q &= 077777;
*/
#endif
#ifdef MIEEE
x = ldexpl( x, (e >> 1) );
/*
*q += ((e >>1) & 0x7ff) << 4;
*q &= 077777;
*/
#endif
/* Newton iterations: */
#ifdef UNK
x += w/x;
x = ldexpl( x, -1 ); /* divide by 2 */
x += w/x;
x = ldexpl( x, -1 );
x += w/x;
x = ldexpl( x, -1 );
#endif
/* Note, assume the square root cannot be denormal,
* so it is safe to use integer exponent operations here.
*/
#ifdef IBMPC
x += w/x;
*q -= 1;
x += w/x;
*q -= 1;
x += w/x;
*q -= 1;
#endif
#ifdef MIEEE
x += w/x;
*q -= 1;
x += w/x;
*q -= 1;
x += w/x;
*q -= 1;
#endif
return(x);
}
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