s_cbrt.c revision 153382 
1/* @(#)s_cbrt.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13#ifndef lint
14static char rcsid[] = "$FreeBSD: head/lib/msun/src/s_cbrt.c 153382 2005-12-13 18:22:00Z bde $";
15#endif
16
17#include "math.h"
18#include "math_private.h"
19
20/* cbrt(x)
21 * Return cube root of x
22 */
23static const u_int32_t
24	B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
25	B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
26
27static const double
28C =  5.42857142857142815906e-01, /* 19/35     = 0x3FE15F15, 0xF15F15F1 */
29D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
30E =  1.41428571428571436819e+00, /* 99/70     = 0x3FF6A0EA, 0x0EA0EA0F */
31F =  1.60714285714285720630e+00, /* 45/28     = 0x3FF9B6DB, 0x6DB6DB6E */
32G =  3.57142857142857150787e-01; /* 5/14      = 0x3FD6DB6D, 0xB6DB6DB7 */
33
34double
35cbrt(double x)
36{
37	int32_t	hx;
38	double r,s,t=0.0,w;
39	u_int32_t sign;
40	u_int32_t high,low;
41
42	GET_HIGH_WORD(hx,x);
43	sign=hx&0x80000000; 		/* sign= sign(x) */
44	hx  ^=sign;
45	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
46	GET_LOW_WORD(low,x);
47	if((hx|low)==0)
48	    return(x);		/* cbrt(0) is itself */
49
50	SET_HIGH_WORD(x,hx);	/* x <- |x| */
51    /*
52     * Rough cbrt to 5 bits:
53     *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
54     * where e is integral and >= 0, m is real and in [0, 1), and "/" and
55     * "%" are integer division and modulus with rounding towards minus
56     * infinity.  The RHS is always >= the LHS and has a maximum relative
57     * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
58     * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
59     * floating point representation, for finite positive normal values,
60     * ordinary integer divison of the value in bits magically gives
61     * almost exactly the RHS of the above provided we first subtract the
62     * exponent bias (1023 for doubles) and later add it back.  We do the
63     * subtraction virtually to keep e >= 0 so that ordinary integer
64     * division rounds towards minus infinity; this is also efficient.
65     */
66	if(hx<0x00100000) { 		/* subnormal number */
67	    SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
68	    t*=x;
69	    GET_HIGH_WORD(high,t);
70	    SET_HIGH_WORD(t,high/3+B2);
71	} else
72	    SET_HIGH_WORD(t,hx/3+B1);
73
74    /* new cbrt to 23 bits; may be implemented in single precision */
75	r=t*t/x;
76	s=C+r*t;
77	t*=G+F/(s+E+D/s);
78
79    /* chop t to 20 bits and make it larger than cbrt(x) */
80	GET_HIGH_WORD(high,t);
81	INSERT_WORDS(t,high+0x00000001,0);
82
83    /* one step Newton iteration to 53 bits with error less than 0.667 ulps */
84	s=t*t;		/* t*t is exact */
85	r=x/s;
86	w=t+t;
87	r=(r-t)/(w+r);	/* r-t is exact */
88	t=t+t*r;
89
90    /* restore the sign bit */
91	GET_HIGH_WORD(high,t);
92	SET_HIGH_WORD(t,high|sign);
93	return(t);
94}
95