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/* cbrtl.c
*
* Cube root, long double precision
*
*
*
* SYNOPSIS:
*
* long double x, y, cbrtl();
*
* y = cbrtl( x );
*
*
*
* DESCRIPTION:
*
* Returns the cube root of the argument, which may be negative.
*
* Range reduction involves determining the power of 2 of
* the argument. A polynomial of degree 2 applied to the
* mantissa, and multiplication by the cube root of 1, 2, or 4
* approximates the root to within about 0.1%. Then Newton's
* iteration is used three times to converge to an accurate
* result.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE .125,8 80000 7.0e-20 2.2e-20
* IEEE exp(+-707) 100000 7.0e-20 2.4e-20
*
*/
/*
Cephes Math Library Release 2.2: January, 1991
Copyright 1984, 1991 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
static long double CBRT2 = 1.2599210498948731647672L;
static long double CBRT4 = 1.5874010519681994747517L;
static long double CBRT2I = 0.79370052598409973737585L;
static long double CBRT4I = 0.62996052494743658238361L;
#ifdef ANSIPROT
extern long double frexpl ( long double, int * );
extern long double ldexpl ( long double, int );
extern int isnanl ( long double );
#else
long double frexpl(), ldexpl();
extern int isnanl();
#endif
#ifdef INFINITIES
extern long double INFINITYL;
#endif
long double cbrtl(x)
long double x;
{
int e, rem, sign;
long double z;
#ifdef NANS
if(isnanl(x))
return(x);
#endif
#ifdef INFINITIES
if( x == INFINITYL)
return(x);
if( x == -INFINITYL)
return(x);
#endif
if( x == 0 )
return( x );
if( x > 0 )
sign = 1;
else
{
sign = -1;
x = -x;
}
z = x;
/* extract power of 2, leaving
* mantissa between 0.5 and 1
*/
x = frexpl( x, &e );
/* Approximate cube root of number between .5 and 1,
* peak relative error = 1.2e-6
*/
x = (((( 1.3584464340920900529734e-1L * x
- 6.3986917220457538402318e-1L) * x
+ 1.2875551670318751538055e0L) * x
- 1.4897083391357284957891e0L) * x
+ 1.3304961236013647092521e0L) * x
+ 3.7568280825958912391243e-1L;
/* exponent divided by 3 */
if( e >= 0 )
{
rem = e;
e /= 3;
rem -= 3*e;
if( rem == 1 )
x *= CBRT2;
else if( rem == 2 )
x *= CBRT4;
}
else
{ /* argument less than 1 */
e = -e;
rem = e;
e /= 3;
rem -= 3*e;
if( rem == 1 )
x *= CBRT2I;
else if( rem == 2 )
x *= CBRT4I;
e = -e;
}
/* multiply by power of 2 */
x = ldexpl( x, e );
/* Newton iteration */
x -= ( x - (z/(x*x)) )*0.3333333333333333333333L;
x -= ( x - (z/(x*x)) )*0.3333333333333333333333L;
if( sign < 0 )
x = -x;
return(x);
}
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