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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"

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 (isnan(x))
      return x;

    if (!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;
    } 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 = ldexp(x, e);

    /* Newton iteration */
    x -= (x - (z/(x*x))) * 0.33333333333333333333;
    x -= (x - (z/(x*x))) * 0.33333333333333333333;

    return (sign < 0)? -x : x;
}

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