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 118