Branch data Line data Source code
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 : : #include "cdefs-compat.h"
16 : : //__FBSDID("$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.17 2011/03/12 16:50:39 kargl Exp $");
17 : :
18 : : #include <openlibm_math.h>
19 : :
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 : : /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
30 : : static const double
31 : : P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
32 : : P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */
33 : : P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
34 : : P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
35 : : P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
36 : :
37 : : OLM_DLLEXPORT double
38 : 11 : cbrt(double x)
39 : : {
40 : : int32_t hx;
41 : : union {
42 : : double value;
43 : : u_int64_t bits;
44 : : } u;
45 : 11 : double r,s,t=0.0,w;
46 : : u_int32_t sign;
47 : : u_int32_t high,low;
48 : :
49 : 11 : EXTRACT_WORDS(hx,low,x);
50 : 11 : sign=hx&0x80000000; /* sign= sign(x) */
51 : 11 : hx ^=sign;
52 [ + + ]: 11 : if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
53 : :
54 : : /*
55 : : * Rough cbrt to 5 bits:
56 : : * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
57 : : * where e is integral and >= 0, m is real and in [0, 1), and "/" and
58 : : * "%" are integer division and modulus with rounding towards minus
59 : : * infinity. The RHS is always >= the LHS and has a maximum relative
60 : : * error of about 1 in 16. Adding a bias of -0.03306235651 to the
61 : : * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
62 : : * floating point representation, for finite positive normal values,
63 : : * ordinary integer divison of the value in bits magically gives
64 : : * almost exactly the RHS of the above provided we first subtract the
65 : : * exponent bias (1023 for doubles) and later add it back. We do the
66 : : * subtraction virtually to keep e >= 0 so that ordinary integer
67 : : * division rounds towards minus infinity; this is also efficient.
68 : : */
69 [ + + ]: 8 : if(hx<0x00100000) { /* zero or subnormal? */
70 [ + - ]: 2 : if((hx|low)==0)
71 : 2 : return(x); /* cbrt(0) is itself */
72 : 0 : SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
73 : 0 : t*=x;
74 : 0 : GET_HIGH_WORD(high,t);
75 : 0 : INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
76 : : } else
77 : 6 : INSERT_WORDS(t,sign|(hx/3+B1),0);
78 : :
79 : : /*
80 : : * New cbrt to 23 bits:
81 : : * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
82 : : * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
83 : : * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
84 : : * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
85 : : * gives us bounds for r = t**3/x.
86 : : *
87 : : * Try to optimize for parallel evaluation as in k_tanf.c.
88 : : */
89 : 6 : r=(t*t)*(t/x);
90 : 6 : t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
91 : :
92 : : /*
93 : : * Round t away from zero to 23 bits (sloppily except for ensuring that
94 : : * the result is larger in magnitude than cbrt(x) but not much more than
95 : : * 2 23-bit ulps larger). With rounding towards zero, the error bound
96 : : * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
97 : : * in the rounded t, the infinite-precision error in the Newton
98 : : * approximation barely affects third digit in the final error
99 : : * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
100 : : * before the final error is larger than 0.667 ulps.
101 : : */
102 : 6 : u.value=t;
103 : 6 : u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
104 : 6 : t=u.value;
105 : :
106 : : /* one step Newton iteration to 53 bits with error < 0.667 ulps */
107 : 6 : s=t*t; /* t*t is exact */
108 : 6 : r=x/s; /* error <= 0.5 ulps; |r| < |t| */
109 : 6 : w=t+t; /* t+t is exact */
110 : 6 : r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
111 : 6 : t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
112 : :
113 : 6 : return(t);
114 : : }
115 : :
116 : : #if (LDBL_MANT_DIG == 53)
117 : : openlibm_weak_reference(cbrt, cbrtl);
118 : : #endif
|
