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      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 }
    132