1 /*-
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
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 * The argument reduction and testing for exceptional cases was
13 * written by Steven G. Kargl with input from Bruce D. Evans
14 * and David A. Schultz.
15 */
16
17 #include <sys/cdefs.h>
18 __RCSID("$NetBSD: s_cbrtl.c,v 1.1 2013/11/19 19:24:34 joerg Exp $");
19 #if 0
20 __FBSDID("$FreeBSD: head/lib/msun/src/s_cbrtl.c 238924 2012-07-30 21:58:28Z kargl $");
21 #endif
22
23 #include "namespace.h"
24 #include <machine/ieee.h>
25 #include <float.h>
26
27 #include "math.h"
28 #include "math_private.h"
29
30 #ifdef __HAVE_LONG_DOUBLE
31 __weak_alias(cbrtl, _cbrtl)
32
33 static const unsigned
34 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
35
36 long double
cbrtl(long double x)37 cbrtl(long double x)
38 {
39 union ieee_ext_u ux, vx;
40 long double r, s, t, w;
41 double dr, dt, dx;
42 float ft, fx;
43 uint32_t hx;
44 int k;
45
46 ux.extu_ld = x;
47
48
49 /*
50 * If x = +-Inf, then cbrt(x) = +-Inf.
51 * If x = NaN, then cbrt(x) = NaN.
52 */
53 if (ux.extu_exp == EXT_EXP_INFNAN)
54 return (x + x);
55 if ((ux.extu_frach | ux.extu_fracl | ux.extu_exp) == 0)
56 return (x);
57
58 vx.extu_ld = 1;
59 vx.extu_ext.ext_sign = ux.extu_ext.ext_sign;
60 ux.extu_ext.ext_sign = 0;
61 if (ux.extu_exp == 0) {
62 /* Adjust subnormal numbers. */
63 ux.extu_ld *= 0x1.0p514;
64 k = ux.extu_exp - EXT_EXP_BIAS - 514;
65 } else {
66 k = ux.extu_exp - EXT_EXP_BIAS;
67 }
68
69 ux.extu_exp = EXT_EXP_BIAS;
70 x = ux.extu_ld;
71 switch (k % 3) {
72 case 1:
73 case -2:
74 x = 2*x;
75 k--;
76 break;
77 case 2:
78 case -1:
79 x = 4*x;
80 k -= 2;
81 break;
82 }
83 vx.extu_exp = EXT_EXP_BIAS + k / 3;
84
85 /*
86 * The following is the guts of s_cbrtf, with the handling of
87 * special values removed and extra care for accuracy not taken,
88 * but with most of the extra accuracy not discarded.
89 */
90
91 /* ~5-bit estimate: */
92 fx = x;
93 GET_FLOAT_WORD(hx, fx);
94 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
95
96 /* ~16-bit estimate: */
97 dx = x;
98 dt = ft;
99 dr = dt * dt * dt;
100 dt = dt * (dx + dx + dr) / (dx + dr + dr);
101
102 /* ~47-bit estimate: */
103 dr = dt * dt * dt;
104 dt = dt * (dx + dx + dr) / (dx + dr + dr);
105
106 #if LDBL_MANT_DIG == 64
107 /*
108 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
109 * Round it away from zero to 32 bits (32 so that t*t is exact, and
110 * away from zero for technical reasons).
111 */
112 volatile double vd2 = 0x1.0p32;
113 volatile double vd1 = 0x1.0p-31;
114 #define vd ((long double)vd2 + vd1)
115
116 t = dt + vd - 0x1.0p32;
117 #elif LDBL_MANT_DIG == 113
118 /*
119 * Round dt away from zero to 47 bits. Since we don't trust the 47,
120 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
121 * might be avoidable in this case, since on most machines dt will
122 * have been evaluated in 53-bit precision and the technical reasons
123 * for rounding up might not apply to either case in cbrtl() since
124 * dt is much more accurate than needed.
125 */
126 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
127 #else
128 #error "Unsupported long double format"
129 #endif
130
131 /*
132 * Final step Newton iteration to 64 or 113 bits with
133 * error < 0.667 ulps
134 */
135 s=t*t; /* t*t is exact */
136 r=x/s; /* error <= 0.5 ulps; |r| < |t| */
137 w=t+t; /* t+t is exact */
138 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
139 t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
140
141 t *= vx.extu_ld;
142 return t;
143 }
144
145 #endif /* __HAVE_LONG_DOUBLE */
146