/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)cbrt.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
#include
/* kahan's cube root (53 bits IEEE double precision)
* for IEEE machines only
* coded in C by K.C. Ng, 4/30/85
*
* Accuracy:
* better than 0.667 ulps according to an error analysis. Maximum
* error observed was 0.666 ulps in an 1,000,000 random arguments test.
*
* Warning: this code is semi machine dependent; the ordering of words in
* a floating point number must be known in advance. I assume that the
* long interger at the address of a floating point number will be the
* leading 32 bits of that floating point number (i.e., sign, exponent,
* and the 20 most significant bits).
* On a National machine, it has different ordering; therefore, this code
* must be compiled with flag -DNATIONAL.
*/
#if !defined(vax)&&!defined(tahoe)
static const unsigned long
B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
static const double
C= 19./35.,
D= -864./1225.,
E= 99./70.,
F= 45./28.,
G= 5./14.;
double cbrt(x)
double x;
{
double r,s,t=0.0,w;
unsigned long *px = (unsigned long *) &x,
*pt = (unsigned long *) &t,
mexp,sign;
#ifdef national /* ordering of words in a floating points number */
const int n0=1,n1=0;
#else /* national */
const int n0=0,n1=1;
#endif /* national */
mexp=px[n0]&0x7ff00000;
if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
if(x==0.0) return(x); /* cbrt(0) is itself */
sign=px[n0]&0x80000000; /* sign= sign(x) */
px[n0] ^= sign; /* x=|x| */
/* rough cbrt to 5 bits */
if(mexp==0) /* subnormal number */
{pt[n0]=0x43500000; /* set t= 2**54 */
t*=x; pt[n0]=pt[n0]/3+B2;
}
else
pt[n0]=px[n0]/3+B1;
/* new cbrt to 23 bits, may be implemented in single precision */
r=t*t/x;
s=C+r*t;
t*=G+F/(s+E+D/s);
/* chopped to 20 bits and make it larger than cbrt(x) */
pt[n1]=0; pt[n0]+=0x00000001;
/* one step newton iteration to 53 bits with error less than 0.667 ulps */
s=t*t; /* t*t is exact */
r=x/s;
w=t+t;
r=(r-t)/(w+r); /* r-t is exact */
t=t+t*r;
/* retore the sign bit */
pt[n0] |= sign;
return(t);
}
#endif