1 /* $NetBSD: n_cbrt.c,v 1.4 2002/06/15 00:10:17 matt 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 70 cbrt(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