1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = "@(#)cbrt.c 1.1 (Berkeley) 05/23/85"; 16 #endif not lint 17 18 /* kahan's cube root (53 bits IEEE double precision) 19 * for IEEE machines only 20 * coded in C by K.C. Ng, 4/30/85 21 * 22 * Accuracy: 23 * better than 0.667 ulps according to an error analysis. Maximum 24 * error observed was 0.666 ulps in an 1,000,000 random arguments test. 25 * 26 * Warning: this code is semi machine dependent; the ordering of words in 27 * a floating point number must be known in advance. I assume that the 28 * long interger at the address of a floating point number will be the 29 * leading 32 bits of that floating point number (i.e., sign, exponent, 30 * and the 20 most significant bits). 31 * On a National machine, it has different ordering; therefore, this code 32 * must be compiled with flag -DNATIONAL. 33 */ 34 #ifndef VAX 35 36 static unsigned long B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 37 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 38 static double 39 C= 19./35., 40 D= -864./1225., 41 E= 99./70., 42 F= 45./28., 43 G= 5./14.; 44 45 double cbrt(x) 46 double x; 47 { 48 double r,s,t=0.0,w; 49 unsigned long *px = (unsigned long *) &x, 50 *pt = (unsigned long *) &t, 51 mexp,sign; 52 53 #ifdef NATIONAL /* ordering of words in a floating points number */ 54 int n0=1,n1=0; 55 #else 56 int n0=0,n1=1; 57 #endif 58 59 mexp=px[n0]&0x7ff00000; 60 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */ 61 if(x==0.0) return(x); /* cbrt(0) is itself */ 62 63 sign=px[n0]&0x80000000; /* sign= sign(x) */ 64 px[n0] ^= sign; /* x=|x| */ 65 66 67 /* rough cbrt to 5 bits */ 68 if(mexp==0) /* subnormal number */ 69 {pt[n0]=0x43500000; /* set t= 2**54 */ 70 t*=x; pt[n0]=pt[n0]/3+B2; 71 } 72 else 73 pt[n0]=px[n0]/3+B1; 74 75 76 /* new cbrt to 23 bits, may be implemented in single precision */ 77 r=t*t/x; 78 s=C+r*t; 79 t*=G+F/(s+E+D/s); 80 81 /* chopped to 20 bits and make it larger than cbrt(x) */ 82 pt[n1]=0; pt[n0]+=0x00000001; 83 84 85 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 86 s=t*t; /* t*t is exact */ 87 r=x/s; 88 w=t+t; 89 r=(r-t)/(w+r); /* r-s is exact */ 90 t=t+t*r; 91 92 93 /* retore the sign bit */ 94 pt[n0] |= sign; 95 return(t); 96 } 97 #endif 98