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/* cbrtq.c
*
* Cube root, __float128 precision
*
*
*
* SYNOPSIS:
*
* __float128 x, y, cbrtq();
*
* y = cbrtq( 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 -8,8 100000 1.3e-34 3.9e-35
* IEEE exp(+-707) 100000 1.3e-34 4.3e-35
*
*/
/*
Cephes Math Library Release 2.2: January, 1991
Copyright 1984, 1991 by Stephen L. Moshier
Adapted for glibc October, 2001.
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, see
<http://www.gnu.org/licenses/>. */
#include "quadmath-imp.h"
static const long double CBRT2 = 1.259921049894873164767210607278228350570251Q;
static const long double CBRT4 = 1.587401051968199474751705639272308260391493Q;
static const long double CBRT2I = 0.7937005259840997373758528196361541301957467Q;
static const long double CBRT4I = 0.6299605249474365823836053036391141752851257Q;
__float128
cbrtq ( __float128 x)
{
int e, rem, sign;
__float128 z;
if (!finiteq (x))
return x + 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 = frexpq (x, &e);
/* Approximate cube root of number between .5 and 1,
peak relative error = 1.2e-6 */
x = ((((1.3584464340920900529734e-1Q * x
- 6.3986917220457538402318e-1Q) * x
+ 1.2875551670318751538055e0Q) * x
- 1.4897083391357284957891e0Q) * x
+ 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
/* 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 = ldexpq (x, e);
/* Newton iteration */
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
if (sign < 0)
x = -x;
return (x);
}
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