s_cbrt.c revision 153447 
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 * Optimized by Bruce D. Evans.
13 */
14
15#ifndef lint
16static char rcsid[] = "$FreeBSD: head/lib/msun/src/s_cbrt.c 153447 2005-12-15 16:23:22Z bde $";
17#endif
18
19#include "math.h"
20#include "math_private.h"
21
22/* cbrt(x)
23 * Return cube root of x
24 */
25static const u_int32_t
26	B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
27	B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
28
29static const double
30C =  5.42857142857142815906e-01, /* 19/35     = 0x3FE15F15, 0xF15F15F1 */
31D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
32E =  1.41428571428571436819e+00, /* 99/70     = 0x3FF6A0EA, 0x0EA0EA0F */
33F =  1.60714285714285720630e+00, /* 45/28     = 0x3FF9B6DB, 0x6DB6DB6E */
34G =  3.57142857142857150787e-01; /* 5/14      = 0x3FD6DB6D, 0xB6DB6DB7 */
35
36double
37cbrt(double x)
38{
39	int32_t	hx;
40	double r,s,t=0.0,w;
41	u_int32_t sign;
42	u_int32_t high,low;
43
44	GET_HIGH_WORD(hx,x);
45	sign=hx&0x80000000; 		/* sign= sign(x) */
46	hx  ^=sign;
47	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
48	GET_LOW_WORD(low,x);
49	if((hx|low)==0)
50	    return(x);		/* cbrt(0) is itself */
51
52    /*
53     * Rough cbrt to 5 bits:
54     *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
55     * where e is integral and >= 0, m is real and in [0, 1), and "/" and
56     * "%" are integer division and modulus with rounding towards minus
57     * infinity.  The RHS is always >= the LHS and has a maximum relative
58     * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
59     * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
60     * floating point representation, for finite positive normal values,
61     * ordinary integer divison of the value in bits magically gives
62     * almost exactly the RHS of the above provided we first subtract the
63     * exponent bias (1023 for doubles) and later add it back.  We do the
64     * subtraction virtually to keep e >= 0 so that ordinary integer
65     * division rounds towards minus infinity; this is also efficient.
66     */
67	if(hx<0x00100000) { 		/* subnormal number */
68	    SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
69	    t*=x;
70	    GET_HIGH_WORD(high,t);
71	    SET_HIGH_WORD(t,sign|((high&0x7fffffff)/3+B2));
72	} else
73	    SET_HIGH_WORD(t,sign|(hx/3+B1));
74
75    /*
76     * New cbrt to 26 bits; may be implemented in single precision:
77     *    cbrt(x) = t*cbrt(x/t**3) ~= t*R(x/t**3)
78     * where R(r) = (14*r**2 + 35*r + 5)/(5*r**2 + 35*r + 14) is the
79     * (2,2) Pade approximation to cbrt(r) at r = 1.  We replace
80     * r = x/t**3 by 1/r = t**3/x since the latter can be evaluated
81     * more efficiently, and rearrange the expression for R(r) to use
82     * 4 additions and 2 divisions instead of the 4 additions, 4
83     * multiplications and 1 division that would be required using
84     * Horner's rule on the numerator and denominator.  t being good
85     * to 32 bits means that |t/cbrt(x)-1| < 1/32, so |x/t**3-1| < 0.1
86     * and for R(r) we can use any approximation to cbrt(r) that is good
87     * to 20 bits on [0.9, 1.1].  The (2,2) Pade approximation is not an
88     * especially good choice.
89     */
90	r=t*t/x;
91	s=C+r*t;
92	t*=G+F/(s+E+D/s);
93
94    /* chop t to 20 bits and make it larger in magnitude than cbrt(x) */
95	GET_HIGH_WORD(high,t);
96	INSERT_WORDS(t,high+0x00000001,0);
97
98    /* one step Newton iteration to 53 bits with error less than 0.667 ulps */
99	s=t*t;		/* t*t is exact */
100	r=x/s;
101	w=t+t;
102	r=(r-t)/(w+r);	/* r-t is exact */
103	t=t+t*r;
104
105	return(t);
106}
107