/*							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        0,8        50000       1.8e-17     5.6e-18
 *    IEEE       0,1e308    30000       1.5e-16     5.0e-17
 *
 */
/*							cbrt.c  */

/*
Cephes Math Library Release 2.1:  December, 1988
Copyright 1984, 1987, 1988 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/


#include "mconf.h"

#ifdef UNK
static double CBRT2 = 1.25992104989487316477;
static double CBRT4 = 1.58740105196819947475;
#endif

#ifdef DEC
static short CBT2[] = {040241,042427,0146153,0112127};
static short CBT4[] = {040313,027765,024753,070744};
#define CBRT2 *(double *)CBT2
#define CBRT4 *(double *)CBT4
#endif

#ifdef IBMPC
static short CBT2[] = {0x728b,0xf98d,0x28a2,0x3ff4};
static short CBT4[] = {0x6e3d,0xa53d,0x65fe,0x3ff9};
#define CBRT2 *(double *)CBT2
#define CBRT4 *(double *)CBT4
#endif

#ifdef MIEEE
static short CBT2[] = {
0x3ff4,0x28a2,0xf98d,0x728b
};
static short CBT4[] = {
0x3ff9,0x65fe,0xa53d,0x6e3d,
};
#define CBRT2 *(double *)CBT2
#define CBRT4 *(double *)CBT4
#endif

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

if( x == 0 )
	return( 0.0 );
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 = 6.36e-4
 */
x = (-1.9150215751434832257e-1  * x
    + 6.9757045195246484393e-1) * x
    + 4.9329566506409572486e-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 /= CBRT2;
	else if( rem == 2 )
		x /= CBRT4;
	e = -e;
	}

/* multiply by power of 2 */
x = ldexp( x, e );

/* Newton iteration */
x -= ( x - (z/(x*x)) )/3.0;
x -= ( x - (z/(x*x)) )/3.0;
x -= ( x - (z/(x*x)) )/3.0;

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