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/* 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 2.2: January, 1991
Copyright 1984, 1991 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include "mconf.h"
static double CBRT2 = 1.2599210498948731647672;
static double CBRT4 = 1.5874010519681994747517;
static double CBRT2I = 0.79370052598409973737585;
static double CBRT4I = 0.62996052494743658238361;
double cbrt(double x)
{
int e, rem, sign;
double z;
if( !npy_isfinite(x) )
return x;
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);
}
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