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n_cbrt.c revision 1.6 
      1 /*	$NetBSD: n_cbrt.c,v 1.6 2013/11/24 14:49:00 martin 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. Neither the name of the University nor the names of its contributors
     15  *    may be used to endorse or promote products derived from this software
     16  *    without specific prior written permission.
     17  *
     18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
     19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
     22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     28  * SUCH DAMAGE.
     29  */
     30 
     31 #ifndef lint
     32 static char sccsid[] = "@(#)cbrt.c	8.1 (Berkeley) 6/4/93";
     33 #endif /* not lint */
     34 
     35 #include <sys/cdefs.h>
     36 
     37 /* kahan's cube root (53 bits IEEE double precision)
     38  * for IEEE machines only
     39  * coded in C by K.C. Ng, 4/30/85
     40  *
     41  * Accuracy:
     42  *	better than 0.667 ulps according to an error analysis. Maximum
     43  * error observed was 0.666 ulps in an 1,000,000 random arguments test.
     44  *
     45  * Warning: this code is semi machine dependent; the ordering of words in
     46  * a floating point number must be known in advance. I assume that the
     47  * long interger at the address of a floating point number will be the
     48  * leading 32 bits of that floating point number (i.e., sign, exponent,
     49  * and the 20 most significant bits).
     50  * On a National machine, it has different ordering; therefore, this code
     51  * must be compiled with flag -DNATIONAL.
     52  */
     53 #if !defined(__vax__)&&!defined(tahoe)
     54 
     55 static const unsigned long
     56 		     B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
     57 	             B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
     58 static const double
     59 	    C= 19./35.,
     60 	    D= -864./1225.,
     61 	    E= 99./70.,
     62 	    F= 45./28.,
     63 	    G= 5./14.;
     64 
     65 float
     66 cbrtf(float x)
     67 {
     68 	return (float)cbrt(x);
     69 }
     70 
     71 long double
     72 cbrtl(long double x)
     73 {
     74 	return cbrt((double)x);
     75 }
     76 
     77 double
     78 cbrt(double x)
     79 {
     80 	double r,s,t=0.0,w;
     81 	unsigned long *px = (unsigned long *) &x,
     82 	              *pt = (unsigned long *) &t,
     83 		      mexp,sign;
     84 
     85 #ifdef national /* ordering of words in a floating points number */
     86 	const int n0=1,n1=0;
     87 #else	/* national */
     88 	const int n0=0,n1=1;
     89 #endif	/* national */
     90 
     91 	mexp=px[n0]&0x7ff00000;
     92 	if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
     93 	if(x==0.0) return(x);		/* cbrt(0) is itself */
     94 
     95 	sign=px[n0]&0x80000000; /* sign= sign(x) */
     96 	px[n0] ^= sign;		/* x=|x| */
     97 
     98 
     99     /* rough cbrt to 5 bits */
    100 	if(mexp==0) 		/* subnormal number */
    101 	  {pt[n0]=0x43500000; 	/* set t= 2**54 */
    102 	   t*=x; pt[n0]=pt[n0]/3+B2;
    103 	  }
    104 	else
    105 	  pt[n0]=px[n0]/3+B1;
    106 
    107 
    108     /* new cbrt to 23 bits, may be implemented in single precision */
    109 	r=t*t/x;
    110 	s=C+r*t;
    111 	t*=G+F/(s+E+D/s);
    112 
    113     /* chopped to 20 bits and make it larger than cbrt(x) */
    114 	pt[n1]=0; pt[n0]+=0x00000001;
    115 
    116 
    117     /* one step newton iteration to 53 bits with error less than 0.667 ulps */
    118 	s=t*t;		/* t*t is exact */
    119 	r=x/s;
    120 	w=t+t;
    121 	r=(r-t)/(w+r);	/* r-t is exact */
    122 	t=t+t*r;
    123 
    124 
    125     /* retore the sign bit */
    126 	pt[n0] |= sign;
    127 	return(t);
    128 }
    129 #endif
    130