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
cbrt(x)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