cbrt.c

/*							cbrt.c
 *
 *	Cube root
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, cbrt();
 *
 * y = cbrt( 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
 *    DEC        -10,10     200000      1.8e-17     6.2e-18
 *    IEEE       0,1e308     30000      1.5e-16     5.0e-17
 *
 */
/*							cbrt.c  */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1991, 2000 by Stephen L. Moshier
*/


#include "mconf.h"

const static double CBRT2  = 1.2599210498948731647672;
const static double CBRT4  = 1.5874010519681994747517;
const static double CBRT2I = 0.79370052598409973737585;
const static double CBRT4I = 0.62996052494743658238361;

#ifdef ANSIPROT
extern double frexp ( double, int * );
extern double ldexp ( double, int );
extern int isnan ( double );
extern int isfinite ( double );
#else
double frexp(), ldexp();
int isnan(), isfinite();
#endif

double cbrt(x)
double x;
{
int e, rem, sign;
double z;

#ifdef NANS
if( isnan(x) )
  return x;
#endif
#ifdef INFINITIES
if( !isfinite(x) )
  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 = frexp( x, &e );

/* Approximate cube root of number between .5 and 1,
 * peak relative error = 9.2e-6
 */
x = (((-1.3466110473359520655053e-1  * x
      + 5.4664601366395524503440e-1) * x
      - 9.5438224771509446525043e-1) * x
      + 1.1399983354717293273738e0 ) * x
      + 4.0238979564544752126924e-1;

/* 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;
	}


/* argument less than 1 */

else
	{
	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 = ldexp( x, e );

/* Newton iteration */
x -= ( x - (z/(x*x)) )*0.33333333333333333333;
#ifdef DEC
x -= ( x - (z/(x*x)) )/3.0;
#else
x -= ( x - (z/(x*x)) )*0.33333333333333333333;
#endif

if( sign < 0 )
	x = -x;
return(x);
}