lib/libm/noieee_src/n_cbrt.c

   1 /*      $NetBSD: n_cbrt.c,v 1.5 2003/08/07 16:44:50 agc 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 double
  66 cbrt(double x)
  67 {
  68         double r,s,t=0.0,w;
  69         unsigned long *px = (unsigned long *) &x,
  70                       *pt = (unsigned long *) &t,
  71                       mexp,sign;
  72
  73 #ifdef national /* ordering of words in a floating points number */
  74         const int n0=1,n1=0;
  75 #else   /* national */
  76         const int n0=0,n1=1;
  77 #endif  /* national */
  78
  79         mexp=px[n0]&0x7ff00000;
  80         if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
  81         if(x==0.0) return(x);           /* cbrt(0) is itself */
  82
  83         sign=px[n0]&0x80000000; /* sign= sign(x) */
  84         px[n0] ^= sign;         /* x=|x| */
  85
  86
  87     /* rough cbrt to 5 bits */
  88         if(mexp==0)             /* subnormal number */
  89           {pt[n0]=0x43500000;   /* set t= 2**54 */
  90            t*=x; pt[n0]=pt[n0]/3+B2;
  91           }
  92         else
  93           pt[n0]=px[n0]/3+B1;
  94
  95
  96     /* new cbrt to 23 bits, may be implemented in single precision */
  97         r=t*t/x;
  98         s=C+r*t;
  99         t*=G+F/(s+E+D/s);
 100
 101     /* chopped to 20 bits and make it larger than cbrt(x) */
 102         pt[n1]=0; pt[n0]+=0x00000001;
 103
 104
 105     /* one step newton iteration to 53 bits with error less than 0.667 ulps */
 106         s=t*t;          /* t*t is exact */
 107         r=x/s;
 108         w=t+t;
 109         r=(r-t)/(w+r);  /* r-t is exact */
 110         t=t+t*r;
 111
 112
 113     /* retore the sign bit */
 114         pt[n0] |= sign;
 115         return(t);
 116 }
 117 #endif