1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)cbrt.c 5.6 (Berkeley) 06/01/90"; 25 #endif /* not lint */ 26 27 #include <sys/stdc.h> 28 29 /* kahan's cube root (53 bits IEEE double precision) 30 * for IEEE machines only 31 * coded in C by K.C. Ng, 4/30/85 32 * 33 * Accuracy: 34 * better than 0.667 ulps according to an error analysis. Maximum 35 * error observed was 0.666 ulps in an 1,000,000 random arguments test. 36 * 37 * Warning: this code is semi machine dependent; the ordering of words in 38 * a floating point number must be known in advance. I assume that the 39 * long interger at the address of a floating point number will be the 40 * leading 32 bits of that floating point number (i.e., sign, exponent, 41 * and the 20 most significant bits). 42 * On a National machine, it has different ordering; therefore, this code 43 * must be compiled with flag -DNATIONAL. 44 */ 45 #if !defined(vax)&&!defined(tahoe) 46 47 static const unsigned long 48 B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 49 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 50 static const double 51 C= 19./35., 52 D= -864./1225., 53 E= 99./70., 54 F= 45./28., 55 G= 5./14.; 56 57 double cbrt(x) 58 double x; 59 { 60 double r,s,t=0.0,w; 61 unsigned long *px = (unsigned long *) &x, 62 *pt = (unsigned long *) &t, 63 mexp,sign; 64 65 #ifdef national /* ordering of words in a floating points number */ 66 const int n0=1,n1=0; 67 #else /* national */ 68 const int n0=0,n1=1; 69 #endif /* national */ 70 71 mexp=px[n0]&0x7ff00000; 72 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */ 73 if(x==0.0) return(x); /* cbrt(0) is itself */ 74 75 sign=px[n0]&0x80000000; /* sign= sign(x) */ 76 px[n0] ^= sign; /* x=|x| */ 77 78 79 /* rough cbrt to 5 bits */ 80 if(mexp==0) /* subnormal number */ 81 {pt[n0]=0x43500000; /* set t= 2**54 */ 82 t*=x; pt[n0]=pt[n0]/3+B2; 83 } 84 else 85 pt[n0]=px[n0]/3+B1; 86 87 88 /* new cbrt to 23 bits, may be implemented in single precision */ 89 r=t*t/x; 90 s=C+r*t; 91 t*=G+F/(s+E+D/s); 92 93 /* chopped to 20 bits and make it larger than cbrt(x) */ 94 pt[n1]=0; pt[n0]+=0x00000001; 95 96 97 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 98 s=t*t; /* t*t is exact */ 99 r=x/s; 100 w=t+t; 101 r=(r-t)/(w+r); /* r-t is exact */ 102 t=t+t*r; 103 104 105 /* retore the sign bit */ 106 pt[n0] |= sign; 107 return(t); 108 } 109 #endif 110