133 lines
3.2 KiB
C
133 lines
3.2 KiB
C
/* cbrtq.c
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*
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* Cube root, __float128 precision
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*
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*
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*
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* SYNOPSIS:
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*
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* __float128 x, y, cbrtq();
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*
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* y = cbrtq( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the cube root of the argument, which may be negative.
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*
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* Range reduction involves determining the power of 2 of
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* the argument. A polynomial of degree 2 applied to the
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* mantissa, and multiplication by the cube root of 1, 2, or 4
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* approximates the root to within about 0.1%. Then Newton's
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* iteration is used three times to converge to an accurate
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* result.
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*
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE -8,8 100000 1.3e-34 3.9e-35
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* IEEE exp(+-707) 100000 1.3e-34 4.3e-35
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*
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*/
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/*
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Cephes Math Library Release 2.2: January, 1991
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Copyright 1984, 1991 by Stephen L. Moshier
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Adapted for glibc October, 2001.
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<http://www.gnu.org/licenses/>. */
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#include "quadmath-imp.h"
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static const long double CBRT2 = 1.259921049894873164767210607278228350570251Q;
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static const long double CBRT4 = 1.587401051968199474751705639272308260391493Q;
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static const long double CBRT2I = 0.7937005259840997373758528196361541301957467Q;
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static const long double CBRT4I = 0.6299605249474365823836053036391141752851257Q;
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__float128
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cbrtq ( __float128 x)
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{
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int e, rem, sign;
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__float128 z;
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if (!finiteq (x))
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return x + x;
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if (x == 0)
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return (x);
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if (x > 0)
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sign = 1;
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else
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{
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sign = -1;
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x = -x;
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}
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z = x;
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/* extract power of 2, leaving mantissa between 0.5 and 1 */
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x = frexpq (x, &e);
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/* Approximate cube root of number between .5 and 1,
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peak relative error = 1.2e-6 */
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x = ((((1.3584464340920900529734e-1L * x
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- 6.3986917220457538402318e-1L) * x
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+ 1.2875551670318751538055e0L) * x
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- 1.4897083391357284957891e0L) * x
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+ 1.3304961236013647092521e0L) * x + 3.7568280825958912391243e-1L;
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/* exponent divided by 3 */
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if (e >= 0)
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{
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rem = e;
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e /= 3;
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rem -= 3 * e;
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if (rem == 1)
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x *= CBRT2;
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else if (rem == 2)
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x *= CBRT4;
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}
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else
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{ /* argument less than 1 */
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e = -e;
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rem = e;
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e /= 3;
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rem -= 3 * e;
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if (rem == 1)
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x *= CBRT2I;
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else if (rem == 2)
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x *= CBRT4I;
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e = -e;
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}
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/* multiply by power of 2 */
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x = ldexpq (x, e);
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/* Newton iteration */
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x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
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x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
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x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
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if (sign < 0)
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x = -x;
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return (x);
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}
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