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33 * @(#)cbrt.c 8.1 (Berkeley) 6/4/93
36 #include <sys/cdefs.h>
38 /* kahan's cube root (53 bits IEEE double precision)
39 * for IEEE machines only
40 * coded in C by K.C. Ng, 4/30/85
43 * better than 0.667 ulps according to an error analysis. Maximum
44 * error observed was 0.666 ulps in an 1,000,000 random arguments test.
46 * Warning: this code is semi machine dependent; the ordering of words in
47 * a floating point number must be known in advance. I assume that the
48 * long interger at the address of a floating point number will be the
49 * leading 32 bits of that floating point number (i.e., sign, exponent,
50 * and the 20 most significant bits).
51 * On a National machine, it has different ordering; therefore, this code
52 * must be compiled with flag -DNATIONAL.
54 #if !defined(vax)&&!defined(tahoe)
56 static const unsigned long
57 B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
58 B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
70 unsigned long *px = (unsigned long *) &x,
71 *pt = (unsigned long *) &t,
74 #ifdef national /* ordering of words in a floating points number */
80 mexp=px[n0]&0x7ff00000;
81 if(mexp==0x7ff00000) return(x); /* cbrt(NaN,INF) is itself */
82 if(x==0.0) return(x); /* cbrt(0) is itself */
84 sign=px[n0]&0x80000000; /* sign= sign(x) */
85 px[n0] ^= sign; /* x=|x| */
88 /* rough cbrt to 5 bits */
89 if(mexp==0) /* subnormal number */
90 {pt[n0]=0x43500000; /* set t= 2**54 */
91 t*=x; pt[n0]=pt[n0]/3+B2;
97 /* new cbrt to 23 bits, may be implemented in single precision */
102 /* chopped to 20 bits and make it larger than cbrt(x) */
103 pt[n1]=0; pt[n0]+=0x00000001;
106 /* one step newton iteration to 53 bits with error less than 0.667 ulps */
107 s=t*t; /* t*t is exact */
110 r=(r-t)/(w+r); /* r-t is exact */
114 /* retore the sign bit */