4.3BSD/usr/src/usr.lib/libm/IEEE/cbrt.c

/* 
 * Copyright (c) 1985 Regents of the University of California.
 * 
 * Use and reproduction of this software are granted  in  accordance  with
 * the terms and conditions specified in  the  Berkeley  Software  License
 * Agreement (in particular, this entails acknowledgement of the programs'
 * source, and inclusion of this notice) with the additional understanding
 * that  all  recipients  should regard themselves as participants  in  an
 * ongoing  research  project and hence should  feel  obligated  to report
 * their  experiences (good or bad) with these elementary function  codes,
 * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
 */

#ifndef lint
static char sccsid[] = "@(#)cbrt.c	1.1 (Berkeley) 5/23/85";
#endif not lint

/* 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. 
 */
#ifndef VAX

static unsigned long B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
	             B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
static 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 */
	int n0=1,n1=0;
#else
	int n0=0,n1=1;
#endif

	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-s is exact */
	t=t+t*r;


    /* retore the sign bit */
	pt[n0] |= sign;
	return(t);
}
#endif