1  /*							cbrtq.c
       2   *
       3   *	Cube root, long double precision
       4   *
       5   *
       6   *
       7   * SYNOPSIS:
       8   *
       9   * long double x, y, cbrtq();
      10   *
      11   * y = cbrtq( x );
      12   *
      13   *
      14   *
      15   * DESCRIPTION:
      16   *
      17   * Returns the cube root of the argument, which may be negative.
      18   *
      19   * Range reduction involves determining the power of 2 of
      20   * the argument.  A polynomial of degree 2 applied to the
      21   * mantissa, and multiplication by the cube root of 1, 2, or 4
      22   * approximates the root to within about 0.1%.  Then Newton's
      23   * iteration is used three times to converge to an accurate
      24   * result.
      25   *
      26   *
      27   *
      28   * ACCURACY:
      29   *
      30   *                      Relative error:
      31   * arithmetic   domain     # trials      peak         rms
      32   *    IEEE       -8,8       100000      1.3e-34     3.9e-35
      33   *    IEEE    exp(+-707)    100000      1.3e-34     4.3e-35
      34   *
      35   */
      36  
      37  /*
      38  Cephes Math Library Release 2.2: January, 1991
      39  Copyright 1984, 1991 by Stephen L. Moshier
      40  Adapted for glibc October, 2001.
      41  
      42      This library is free software; you can redistribute it and/or
      43      modify it under the terms of the GNU Lesser General Public
      44      License as published by the Free Software Foundation; either
      45      version 2.1 of the License, or (at your option) any later version.
      46  
      47      This library is distributed in the hope that it will be useful,
      48      but WITHOUT ANY WARRANTY; without even the implied warranty of
      49      MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
      50      Lesser General Public License for more details.
      51  
      52      You should have received a copy of the GNU Lesser General Public
      53      License along with this library; if not, see
      54      <http://www.gnu.org/licenses/>.  */
      55  
      56  #include "quadmath-imp.h"
      57  
      58  static const __float128 CBRT2 = 1.259921049894873164767210607278228350570251Q;
      59  static const __float128 CBRT4 = 1.587401051968199474751705639272308260391493Q;
      60  static const __float128 CBRT2I = 0.7937005259840997373758528196361541301957467Q;
      61  static const __float128 CBRT4I = 0.6299605249474365823836053036391141752851257Q;
      62  
      63  
      64  __float128
      65  cbrtq (__float128 x)
      66  {
      67    int e, rem, sign;
      68    __float128 z;
      69  
      70    if (!finiteq (x))
      71      return x + x;
      72  
      73    if (x == 0)
      74      return (x);
      75  
      76    if (x > 0)
      77      sign = 1;
      78    else
      79      {
      80        sign = -1;
      81        x = -x;
      82      }
      83  
      84    z = x;
      85   /* extract power of 2, leaving mantissa between 0.5 and 1  */
      86    x = frexpq (x, &e);
      87  
      88    /* Approximate cube root of number between .5 and 1,
      89       peak relative error = 1.2e-6  */
      90    x = ((((1.3584464340920900529734e-1Q * x
      91  	  - 6.3986917220457538402318e-1Q) * x
      92  	 + 1.2875551670318751538055e0Q) * x
      93  	- 1.4897083391357284957891e0Q) * x
      94         + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
      95  
      96    /* exponent divided by 3 */
      97    if (e >= 0)
      98      {
      99        rem = e;
     100        e /= 3;
     101        rem -= 3 * e;
     102        if (rem == 1)
     103  	x *= CBRT2;
     104        else if (rem == 2)
     105  	x *= CBRT4;
     106      }
     107    else
     108      {				/* argument less than 1 */
     109        e = -e;
     110        rem = e;
     111        e /= 3;
     112        rem -= 3 * e;
     113        if (rem == 1)
     114  	x *= CBRT2I;
     115        else if (rem == 2)
     116  	x *= CBRT4I;
     117        e = -e;
     118      }
     119  
     120    /* multiply by power of 2 */
     121    x = ldexpq (x, e);
     122  
     123    /* Newton iteration */
     124    x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
     125    x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
     126    x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
     127  
     128    if (sign < 0)
     129      x = -x;
     130    return (x);
     131  }