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 uint32_t hx, lx;
35 uint16_t expsign, es;
36 int k;
37 volatile double vd1, vd2;
38
39 GET_LDOUBLE_EXP(expsign,x);
40 k = expsign & 0x7fff;
41
42 /*
43 * If x = +-Inf, then cbrt(x) = +-Inf.
44 * If x = NaN, then cbrt(x) = NaN.
45 */
46 if (k == BIAS + LDBL_MAX_EXP)
47 return (x + x);
48
49 if (k == 0) {
50 /* If x = +-0, then cbrt(x) = +-0. */
51 GET_LDOUBLE_WORDS(es,hx,lx,x);
52 if ((hx|lx) == 0) {
53 return (x);
54 }
55 /* Adjust subnormal numbers. */
56 x *= 0x1.0p514;
57 GET_LDOUBLE_EXP(k,x);
58 k &= 0x7fff;
59 k -= BIAS + 514;
60 } else
61 k -= BIAS;
62 SET_LDOUBLE_EXP(x,BIAS);
63 v = 1;
64
65 switch (k % 3) {
66 case 1:
67 case -2:
68 x = 2*x;
69 k--;
70 break;
71 case 2:
72 case -1:
73 x = 4*x;
74 k -= 2;
75 break;
76 }
77 SET_LDOUBLE_EXP(v, (expsign & 0x8000) | (BIAS + k / 3));
78
79 /*
80 * The following is the guts of s_cbrtf, with the handling of
81 * special values removed and extra care for accuracy not taken,
82 * but with most of the extra accuracy not discarded.
83 */
84
85 /* ~5-bit estimate: */
86 fx = x;
87 GET_FLOAT_WORD(hx, fx);
88 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
89
90 /* ~16-bit estimate: */
91 dx = x;
92 dt = ft;
93 dr = dt * dt * dt;
94 dt = dt * (dx + dx + dr) / (dx + dr + dr);
95
96 /* ~47-bit estimate: */
97 dr = dt * dt * dt;
98 dt = dt * (dx + dx + dr) / (dx + dr + dr);
99
100 /*
101 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
102 * Round it away from zero to 32 bits (32 so that t*t is exact, and
103 * away from zero for technical reasons).
104 */
105 vd2 = 0x1.0p32;
106 vd1 = 0x1.0p-31;
107 #define vd ((long double)vd2 + vd1)
108
109 t = dt + vd - 0x1.0p32;
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;
122 return (t);
123 }
124