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n_cbrt.c revision 1.3 
      1 /*	$NetBSD: n_cbrt.c,v 1.3 1999/07/02 15:37:36 simonb Exp $	*/
      2 /*
      3  * Copyright (c) 1985, 1993
      4  *	The Regents of the University of California.  All rights reserved.
      5  *
      6  * Redistribution and use in source and binary forms, with or without
      7  * modification, are permitted provided that the following conditions
      8  * are met:
      9  * 1. Redistributions of source code must retain the above copyright
     10  *    notice, this list of conditions and the following disclaimer.
     11  * 2. Redistributions in binary form must reproduce the above copyright
     12  *    notice, this list of conditions and the following disclaimer in the
     13  *    documentation and/or other materials provided with the distribution.
     14  * 3. All advertising materials mentioning features or use of this software
     15  *    must display the following acknowledgement:
     16  *	This product includes software developed by the University of
     17  *	California, Berkeley and its contributors.
     18  * 4. Neither the name of the University nor the names of its contributors
     19  *    may be used to endorse or promote products derived from this software
     20  *    without specific prior written permission.
     21  *
     22  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     23  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     24  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     25  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     26  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     27  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     28  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     29  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     30  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     31  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     32  * SUCH DAMAGE.
     33  */
     34 
     35 #ifndef lint
     36 static char sccsid[] = "@(#)cbrt.c	8.1 (Berkeley) 6/4/93";
     37 #endif /* not lint */
     38 
     39 #include <sys/cdefs.h>
     40 
     41 /* kahan's cube root (53 bits IEEE double precision)
     42  * for IEEE machines only
     43  * coded in C by K.C. Ng, 4/30/85
     44  *
     45  * Accuracy:
     46  *	better than 0.667 ulps according to an error analysis. Maximum
     47  * error observed was 0.666 ulps in an 1,000,000 random arguments test.
     48  *
     49  * Warning: this code is semi machine dependent; the ordering of words in
     50  * a floating point number must be known in advance. I assume that the
     51  * long interger at the address of a floating point number will be the
     52  * leading 32 bits of that floating point number (i.e., sign, exponent,
     53  * and the 20 most significant bits).
     54  * On a National machine, it has different ordering; therefore, this code
     55  * must be compiled with flag -DNATIONAL.
     56  */
     57 #if !defined(__vax__)&&!defined(tahoe)
     58 
     59 static const unsigned long
     60 		     B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
     61 	             B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
     62 static const double
     63 	    C= 19./35.,
     64 	    D= -864./1225.,
     65 	    E= 99./70.,
     66 	    F= 45./28.,
     67 	    G= 5./14.;
     68 
     69 double cbrt(x)
     70 double x;
     71 {
     72 	double r,s,t=0.0,w;
     73 	unsigned long *px = (unsigned long *) &x,
     74 	              *pt = (unsigned long *) &t,
     75 		      mexp,sign;
     76 
     77 #ifdef national /* ordering of words in a floating points number */
     78 	const int n0=1,n1=0;
     79 #else	/* national */
     80 	const int n0=0,n1=1;
     81 #endif	/* national */
     82 
     83 	mexp=px[n0]&0x7ff00000;
     84 	if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
     85 	if(x==0.0) return(x);		/* cbrt(0) is itself */
     86 
     87 	sign=px[n0]&0x80000000; /* sign= sign(x) */
     88 	px[n0] ^= sign;		/* x=|x| */
     89 
     90 
     91     /* rough cbrt to 5 bits */
     92 	if(mexp==0) 		/* subnormal number */
     93 	  {pt[n0]=0x43500000; 	/* set t= 2**54 */
     94 	   t*=x; pt[n0]=pt[n0]/3+B2;
     95 	  }
     96 	else
     97 	  pt[n0]=px[n0]/3+B1;
     98 
     99 
    100     /* new cbrt to 23 bits, may be implemented in single precision */
    101 	r=t*t/x;
    102 	s=C+r*t;
    103 	t*=G+F/(s+E+D/s);
    104 
    105     /* chopped to 20 bits and make it larger than cbrt(x) */
    106 	pt[n1]=0; pt[n0]+=0x00000001;
    107 
    108 
    109     /* one step newton iteration to 53 bits with error less than 0.667 ulps */
    110 	s=t*t;		/* t*t is exact */
    111 	r=x/s;
    112 	w=t+t;
    113 	r=(r-t)/(w+r);	/* r-t is exact */
    114 	t=t+t*r;
    115 
    116 
    117     /* retore the sign bit */
    118 	pt[n0] |= sign;
    119 	return(t);
    120 }
    121 #endif
    122