1/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
2/*-
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 *
13 * The argument reduction and testing for exceptional cases was
14 * written by Steven G. Kargl with input from Bruce D. Evans
15 * and David A. Schultz.
16 */
17
18#include "libm.h"
19
20#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
21long double cbrtl(long double x)
22{
23 return cbrt(x);
24}
25#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
26static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
27
28long double cbrtl(long double x)
29{
30 union ldshape u = {x}, v;
31 union {float f; uint32_t i;} uft;
32 long double r, s, t, w;
33 double_t dr, dt, dx;
34 float_t ft;
35 int e = u.i.se & 0x7fff;
36 int sign = u.i.se & 0x8000;
37
38 /*
39 * If x = +-Inf, then cbrt(x) = +-Inf.
40 * If x = NaN, then cbrt(x) = NaN.
41 */
42 if (e == 0x7fff)
43 return x + x;
44 if (e == 0) {
45 /* Adjust subnormal numbers. */
46 u.f *= 0x1p120;
47 e = u.i.se & 0x7fff;
48 /* If x = +-0, then cbrt(x) = +-0. */
49 if (e == 0)
50 return x;
51 e -= 120;
52 }
53 e -= 0x3fff;
54 u.i.se = 0x3fff;
55 x = u.f;
56 switch (e % 3) {
57 case 1:
58 case -2:
59 x *= 2;
60 e--;
61 break;
62 case 2:
63 case -1:
64 x *= 4;
65 e -= 2;
66 break;
67 }
68 v.f = 1.0;
69 v.i.se = sign | (0x3fff + e/3);
70
71 /*
72 * The following is the guts of s_cbrtf, with the handling of
73 * special values removed and extra care for accuracy not taken,
74 * but with most of the extra accuracy not discarded.
75 */
76
77 /* ~5-bit estimate: */
78 uft.f = x;
79 uft.i = (uft.i & 0x7fffffff)/3 + B1;
80 ft = uft.f;
81
82 /* ~16-bit estimate: */
83 dx = x;
84 dt = ft;
85 dr = dt * dt * dt;
86 dt = dt * (dx + dx + dr) / (dx + dr + dr);
87
88 /* ~47-bit estimate: */
89 dr = dt * dt * dt;
90 dt = dt * (dx + dx + dr) / (dx + dr + dr);
91
92#if LDBL_MANT_DIG == 64
93 /*
94 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
95 * Round it away from zero to 32 bits (32 so that t*t is exact, and
96 * away from zero for technical reasons).
97 */
98 t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
99#elif LDBL_MANT_DIG == 113
100 /*
101 * Round dt away from zero to 47 bits. Since we don't trust the 47,
102 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
103 * might be avoidable in this case, since on most machines dt will
104 * have been evaluated in 53-bit precision and the technical reasons
105 * for rounding up might not apply to either case in cbrtl() since
106 * dt is much more accurate than needed.
107 */
108 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
109#endif
110
111 /*
112 * Final step Newton iteration to 64 or 113 bits with
113 * error < 0.667 ulps
114 */
115 s = t*t; /* t*t is exact */
116 r = x/s; /* error <= 0.5 ulps; |r| < |t| */
117 w = t+t; /* t+t is exact */
118 r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
119 t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
120
121 t *= v.f;
122 return t;
123}
124#endif