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[] = 16 "@(#)cbrt.c 1.1 (Berkeley) 5/23/85; 1.2 (ucb.elefunt) 09/12/85"; 17 #endif not lint 18 19 /* kahan's cube root (53 bits IEEE double precision) 20 * for IEEE machines only 21 * coded in C by K.C. Ng, 4/30/85 22 * 23 * Accuracy: 24 * better than 0.667 ulps according to an error analysis. Maximum 25 * error observed was 0.666 ulps in an 1,000,000 random arguments test. 26 * 27 * Warning: this code is semi machine dependent; the ordering of words in 28 * a floating point number must be known in advance. I assume that the 29 * long interger at the address of a floating point number will be the 30 * leading 32 bits of that floating point number (i.e., sign, exponent, 31 * and the 20 most significant bits). 32 * On a National machine, it has different ordering; therefore, this code 33 * must be compiled with flag -DNATIONAL. 34 */ 35 #ifndef VAX 36 37 static unsigned long B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 38 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 39 static double 40 C= 19./35., 41 D= -864./1225., 42 E= 99./70., 43 F= 45./28., 44 G= 5./14.; 45 46 double cbrt(x) 47 double x; 48 { 49 double r,s,t=0.0,w; 50 unsigned long *px = (unsigned long *) &x, 51 *pt = (unsigned long *) &t, 52 mexp,sign; 53 54 #ifdef NATIONAL /* ordering of words in a floating points number */ 55 int n0=1,n1=0; 56 #else 57 int n0=0,n1=1; 58 #endif 59 60 mexp=px[n0]&0x7ff00000; 61 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */ 62 if(x==0.0) return(x); /* cbrt(0) is itself */ 63 64 sign=px[n0]&0x80000000; /* sign= sign(x) */ 65 px[n0] ^= sign; /* x=|x| */ 66 67 68 /* rough cbrt to 5 bits */ 69 if(mexp==0) /* subnormal number */ 70 {pt[n0]=0x43500000; /* set t= 2**54 */ 71 t*=x; pt[n0]=pt[n0]/3+B2; 72 } 73 else 74 pt[n0]=px[n0]/3+B1; 75 76 77 /* new cbrt to 23 bits, may be implemented in single precision */ 78 r=t*t/x; 79 s=C+r*t; 80 t*=G+F/(s+E+D/s); 81 82 /* chopped to 20 bits and make it larger than cbrt(x) */ 83 pt[n1]=0; pt[n0]+=0x00000001; 84 85 86 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 87 s=t*t; /* t*t is exact */ 88 r=x/s; 89 w=t+t; 90 r=(r-t)/(w+r); /* r-s is exact */ 91 t=t+t*r; 92 93 94 /* retore the sign bit */ 95 pt[n0] |= sign; 96 return(t); 97 } 98 #endif 99