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 //__FBSDID("$FreeBSD: src/lib/msun/src/s_cbrtl.c,v 1.1 2011/03/12 19:37:35 kargl Exp $");
18
19
20 #define BIAS (LDBL_MAX_EXP - 1)
21
22 static const unsigned
23 B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
24
25 long double
cbrtl(long double x)26 cbrtl(long double x)
27 {
28 union IEEEl2bits u, v;
29 long double r, s, t, w;
30 double dr, dt, dx;
31 float ft, fx;
32 u_int32_t hx;
33 u_int16_t expsign;
34 int k;
35
36 u.e = x;
37 expsign = u.xbits.expsign;
38 k = expsign & 0x7fff;
39
40 /*
41 * If x = +-Inf, then cbrt(x) = +-Inf.
42 * If x = NaN, then cbrt(x) = NaN.
43 */
44 if (k == BIAS + LDBL_MAX_EXP)
45 return (x + x);
46
47 if (k == 0) {
48 /* If x = +-0, then cbrt(x) = +-0. */
49 if ((u.bits.manh | u.bits.manl) == 0) {
50 return (x);
51 }
52 /* Adjust subnormal numbers. */
53 u.e *= 0x1.0p514l;
54 k = u.bits.exp;
55 k -= BIAS + 514;
56 } else
57 k -= BIAS;
58 u.xbits.expsign = BIAS;
59 v.e = 1;
60
61 x = u.e;
62 switch (k % 3) {
63 case 1:
64 case -2:
65 x = 2*x;
66 k--;
67 break;
68 case 2:
69 case -1:
70 x = 4*x;
71 k -= 2;
72 break;
73 }
74 v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3);
75
76 /*
77 * The following is the guts of s_cbrtf, with the handling of
78 * special values removed and extra care for accuracy not taken,
79 * but with most of the extra accuracy not discarded.
80 */
81
82 /* ~5-bit estimate: */
83 fx = x;
84 GET_FLOAT_WORD(hx, fx);
85 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
86
87 /* ~16-bit estimate: */
88 dx = x;
89 dt = (double)ft;
90 dr = dt * dt * dt;
91 dt = dt * (dx + dx + dr) / (dx + dr + dr);
92
93 /* ~47-bit estimate: */
94 dr = dt * dt * dt;
95 dt = dt * (dx + dx + dr) / (dx + dr + dr);
96
97 #if LDBL_MANT_DIG == 64
98 /*
99 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
100 * Round it away from zero to 32 bits (32 so that t*t is exact, and
101 * away from zero for technical reasons).
102 */
103 volatile double vd2 = 0x1.0p32;
104 volatile double vd1 = 0x1.0p-31;
105 #define vd ((long double)vd2 + (long double)vd1)
106
107 t = (long double)dt + vd - 0x1.0p32l;
108 #elif LDBL_MANT_DIG == 113
109 /*
110 * Round dt away from zero to 47 bits. Since we don't trust the 47,
111 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
112 * might be avoidable in this case, since on most machines dt will
113 * have been evaluated in 53-bit precision and the technical reasons
114 * for rounding up might not apply to either case in cbrtl() since
115 * dt is much more accurate than needed.
116 */
117 t = (long double)dt + 0x2.0p-46L + 0x1.0p60L - 0x1.0p60L;
118 #else
119 #error "Unsupported long double format"
120 #endif
121
122 /*
123 * Final step Newton iteration to 64 or 113 bits with
124 * error < 0.667 ulps
125 */
126 s=t*t; /* t*t is exact */
127 r=x/s; /* error <= 0.5 ulps; |r| < |t| */
128 w=t+t; /* t+t is exact */
129 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
130 t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
131
132 t *= v.e;
133 return (t);
134 }
135
