cbrt.c

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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
 *    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 (!cephes_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;
    x -= (x - (z / (x * x))) * 0.33333333333333333333;
 
    if (sign < 0)
        x = -x;
    return (x);
}