1 /* 2 * Copyright (c) 1985, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)cbrt.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 #include <sys/cdefs.h> 13 14 /* kahan's cube root (53 bits IEEE double precision) 15 * for IEEE machines only 16 * coded in C by K.C. Ng, 4/30/85 17 * 18 * Accuracy: 19 * better than 0.667 ulps according to an error analysis. Maximum 20 * error observed was 0.666 ulps in an 1,000,000 random arguments test. 21 * 22 * Warning: this code is semi machine dependent; the ordering of words in 23 * a floating point number must be known in advance. I assume that the 24 * long interger at the address of a floating point number will be the 25 * leading 32 bits of that floating point number (i.e., sign, exponent, 26 * and the 20 most significant bits). 27 * On a National machine, it has different ordering; therefore, this code 28 * must be compiled with flag -DNATIONAL. 29 */ 30 #if !defined(vax)&&!defined(tahoe) 31 32 static const unsigned long 33 B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */ 34 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */ 35 static const double 36 C= 19./35., 37 D= -864./1225., 38 E= 99./70., 39 F= 45./28., 40 G= 5./14.; 41 42 double cbrt(x) 43 double x; 44 { 45 double r,s,t=0.0,w; 46 unsigned long *px = (unsigned long *) &x, 47 *pt = (unsigned long *) &t, 48 mexp,sign; 49 50 #ifdef national /* ordering of words in a floating points number */ 51 const int n0=1,n1=0; 52 #else /* national */ 53 const int n0=0,n1=1; 54 #endif /* national */ 55 56 mexp=px[n0]&0x7ff00000; 57 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */ 58 if(x==0.0) return(x); /* cbrt(0) is itself */ 59 60 sign=px[n0]&0x80000000; /* sign= sign(x) */ 61 px[n0] ^= sign; /* x=|x| */ 62 63 64 /* rough cbrt to 5 bits */ 65 if(mexp==0) /* subnormal number */ 66 {pt[n0]=0x43500000; /* set t= 2**54 */ 67 t*=x; pt[n0]=pt[n0]/3+B2; 68 } 69 else 70 pt[n0]=px[n0]/3+B1; 71 72 73 /* new cbrt to 23 bits, may be implemented in single precision */ 74 r=t*t/x; 75 s=C+r*t; 76 t*=G+F/(s+E+D/s); 77 78 /* chopped to 20 bits and make it larger than cbrt(x) */ 79 pt[n1]=0; pt[n0]+=0x00000001; 80 81 82 /* one step newton iteration to 53 bits with error less than 0.667 ulps */ 83 s=t*t; /* t*t is exact */ 84 r=x/s; 85 w=t+t; 86 r=(r-t)/(w+r); /* r-t is exact */ 87 t=t+t*r; 88 89 90 /* retore the sign bit */ 91 pt[n0] |= sign; 92 return(t); 93 } 94 #endif 95