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 this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)cbrt.c 5.3 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* kahan's cube root (53 bits IEEE double precision) 23 * for IEEE machines only 24 * coded in C by K.C. Ng, 4/30/85 25 * 26 * Accuracy: 27 * better than 0.667 ulps according to an error analysis. Maximum 28 * error observed was 0.666 ulps in an 1,000,000 random arguments test. 29 * 30 * Warning: this code is semi machine dependent; the ordering of words in 31 * a floating point number must be known in advance. I assume that the 32 * long interger at the address of a floating point number will be the 33 * leading 32 bits of that floating point number (i.e., sign, exponent, 34 * and the 20 most significant bits). 35 * On a National machine, it has different ordering; therefore, this code 36 * must be compiled with flag -DNATIONAL. 37 */ 38 #if !defined(vax)&&!defined(tahoe) 39 40 static unsigned long B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 41 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 42 static double 43 C= 19./35., 44 D= -864./1225., 45 E= 99./70., 46 F= 45./28., 47 G= 5./14.; 48 49 double cbrt(x) 50 double x; 51 { 52 double r,s,t=0.0,w; 53 unsigned long *px = (unsigned long *) &x, 54 *pt = (unsigned long *) &t, 55 mexp,sign; 56 57 #ifdef national /* ordering of words in a floating points number */ 58 int n0=1,n1=0; 59 #else /* national */ 60 int n0=0,n1=1; 61 #endif /* national */ 62 63 mexp=px[n0]&0x7ff00000; 64 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */ 65 if(x==0.0) return(x); /* cbrt(0) is itself */ 66 67 sign=px[n0]&0x80000000; /* sign= sign(x) */ 68 px[n0] ^= sign; /* x=|x| */ 69 70 71 /* rough cbrt to 5 bits */ 72 if(mexp==0) /* subnormal number */ 73 {pt[n0]=0x43500000; /* set t= 2**54 */ 74 t*=x; pt[n0]=pt[n0]/3+B2; 75 } 76 else 77 pt[n0]=px[n0]/3+B1; 78 79 80 /* new cbrt to 23 bits, may be implemented in single precision */ 81 r=t*t/x; 82 s=C+r*t; 83 t*=G+F/(s+E+D/s); 84 85 /* chopped to 20 bits and make it larger than cbrt(x) */ 86 pt[n1]=0; pt[n0]+=0x00000001; 87 88 89 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 90 s=t*t; /* t*t is exact */ 91 r=x/s; 92 w=t+t; 93 r=(r-t)/(w+r); /* r-t is exact */ 94 t=t+t*r; 95 96 97 /* retore the sign bit */ 98 pt[n0] |= sign; 99 return(t); 100 } 101 #endif 102