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
16 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.10 2005/12/13 20:17:23 bde Exp $";
17 #endif
18
19 #include "math.h"
20 #include "math_private.h"
21
22 /* cbrt(x)
23 * Return cube root of x
24 */
25 static 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
29 static const double
30 C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
31 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
32 E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
33 F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
34 G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
35
36 double
cbrt(double x)37 cbrt(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 /* new cbrt to 23 bits; may be implemented in single precision */
76 r=t*t/x;
77 s=C+r*t;
78 t*=G+F/(s+E+D/s);
79
80 /* chop t to 20 bits and make it larger in magnitude than cbrt(x) */
81 GET_HIGH_WORD(high,t);
82 INSERT_WORDS(t,high+0x00000001,0);
83
84 /* one step Newton iteration to 53 bits with error less than 0.667 ulps */
85 s=t*t; /* t*t is exact */
86 r=x/s;
87 w=t+t;
88 r=(r-t)/(w+r); /* r-t is exact */
89 t=t+t*r;
90
91 return(t);
92 }
93