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 <float.h>
18 #include <ieeefp.h>
19 #include <math.h>
20
21 #include "math_private.h"
22
23 #define BIAS (LDBL_MAX_EXP - 1)
24
25 static const unsigned
26 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
27
28 long double
cbrtl(long double x)29 cbrtl(long double x)
30 {
31 long double v, r, s, t, w;
32 double dr, dt, dx;
33 float ft, fx;
34 uint64_t hx, lx;
35 uint16_t expsign;
36 int k;
37
38 GET_LDOUBLE_MSW64(hx,x);
39 k = (hx>>48)&0x7fff;
40
41 /*
42 * If x = +-Inf, then cbrt(x) = +-Inf.
43 * If x = NaN, then cbrt(x) = NaN.
44 */
45 if (k == BIAS + LDBL_MAX_EXP)
46 return (x + x);
47
48 if (k == 0) {
49 /* If x = +-0, then cbrt(x) = +-0. */
50 GET_LDOUBLE_WORDS64(hx,lx,x);
51 if (((hx&0x7fffffffffffffffLL)|lx) == 0) {
52 return (x);
53 }
54 /* Adjust subnormal numbers. */
55 x *= 0x1.0p514;
56 GET_LDOUBLE_MSW64(hx,x);
57 k = (hx>>48)&0x7fff;
58 k -= BIAS + 514;
59 } else
60 k -= BIAS;
61 GET_LDOUBLE_MSW64(hx,x);
62 hx = (hx&0x8000ffffffffffffLL)|((uint64_t)BIAS<<48);
63 SET_LDOUBLE_MSW64(x,hx);
64 v = 1;
65
66 switch (k % 3) {
67 case 1:
68 case -2:
69 x = 2*x;
70 k--;
71 break;
72 case 2:
73 case -1:
74 x = 4*x;
75 k -= 2;
76 break;
77 }
78 GET_LDOUBLE_MSW64(hx,x);
79 expsign = ((hx>>48) & 0x8000) | (BIAS + k / 3);
80 hx = (hx&0x8000ffffffffffffLL)|((uint64_t)expsign<<48);
81 SET_LDOUBLE_MSW64(x,hx);
82
83 /*
84 * The following is the guts of s_cbrtf, with the handling of
85 * special values removed and extra care for accuracy not taken,
86 * but with most of the extra accuracy not discarded.
87 */
88
89 /* ~5-bit estimate: */
90 fx = x;
91 GET_FLOAT_WORD(hx, fx);
92 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
93
94 /* ~16-bit estimate: */
95 dx = x;
96 dt = ft;
97 dr = dt * dt * dt;
98 dt = dt * (dx + dx + dr) / (dx + dr + dr);
99
100 /* ~47-bit estimate: */
101 dr = dt * dt * dt;
102 dt = dt * (dx + dx + dr) / (dx + dr + dr);
103
104 /*
105 * Round dt away from zero to 47 bits. Since we don't trust the 47,
106 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
107 * might be avoidable in this case, since on most machines dt will
108 * have been evaluated in 53-bit precision and the technical reasons
109 * for rounding up might not apply to either case in cbrtl() since
110 * dt is much more accurate than needed.
111 */
112 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
113
114 /*
115 * Final step Newton iteration to 64 or 113 bits with
116 * error < 0.667 ulps
117 */
118 s=t*t; /* t*t is exact */
119 r=x/s; /* error <= 0.5 ulps; |r| < |t| */
120 w=t+t; /* t+t is exact */
121 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
122 t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
123
124 t *= v;
125 return (t);
126 }
127