1/* Double-precision floating point 2^x.
2 Copyright (C) 1997-2016 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4 Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <http://www.gnu.org/licenses/>. */
19
20/* The basic design here is from
21 Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical
22 Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft.,
23 17 (1), March 1991, pp. 26-45.
24 It has been slightly modified to compute 2^x instead of e^x.
25 */
26#include <stdlib.h>
27#include <float.h>
28#include <ieee754.h>
29#include <math.h>
30#include <fenv.h>
31#include <inttypes.h>
32#include <math_private.h>
33
34#include "t_exp2.h"
35
36static const double TWO1023 = 8.988465674311579539e+307;
37static const double TWOM1000 = 9.3326361850321887899e-302;
38
39double
40__ieee754_exp2 (double x)
41{
42 static const double himark = (double) DBL_MAX_EXP;
43 static const double lomark = (double) (DBL_MIN_EXP - DBL_MANT_DIG - 1);
44
45 /* Check for usual case. */
46 if (__glibc_likely (isless (x, himark)))
47 {
48 /* Exceptional cases: */
49 if (__glibc_unlikely (!isgreaterequal (x, lomark)))
50 {
51 if (isinf (x))
52 /* e^-inf == 0, with no error. */
53 return 0;
54 else
55 /* Underflow */
56 return TWOM1000 * TWOM1000;
57 }
58
59 static const double THREEp42 = 13194139533312.0;
60 int tval, unsafe;
61 double rx, x22, result;
62 union ieee754_double ex2_u, scale_u;
63
64 if (fabs (x) < DBL_EPSILON / 4.0)
65 return 1.0 + x;
66
67 {
68 SET_RESTORE_ROUND_NOEX (FE_TONEAREST);
69
70 /* 1. Argument reduction.
71 Choose integers ex, -256 <= t < 256, and some real
72 -1/1024 <= x1 <= 1024 so that
73 x = ex + t/512 + x1.
74
75 First, calculate rx = ex + t/512. */
76 rx = x + THREEp42;
77 rx -= THREEp42;
78 x -= rx; /* Compute x=x1. */
79 /* Compute tval = (ex*512 + t)+256.
80 Now, t = (tval mod 512)-256 and ex=tval/512 [that's mod, NOT %;
81 and /-round-to-nearest not the usual c integer /]. */
82 tval = (int) (rx * 512.0 + 256.0);
83
84 /* 2. Adjust for accurate table entry.
85 Find e so that
86 x = ex + t/512 + e + x2
87 where -1e6 < e < 1e6, and
88 (double)(2^(t/512+e))
89 is accurate to one part in 2^-64. */
90
91 /* 'tval & 511' is the same as 'tval%512' except that it's always
92 positive.
93 Compute x = x2. */
94 x -= exp2_deltatable[tval & 511];
95
96 /* 3. Compute ex2 = 2^(t/512+e+ex). */
97 ex2_u.d = exp2_accuratetable[tval & 511];
98 tval >>= 9;
99 /* x2 is an integer multiple of 2^-54; avoid intermediate
100 underflow from the calculation of x22 * x. */
101 unsafe = abs (tval) >= -DBL_MIN_EXP - 56;
102 ex2_u.ieee.exponent += tval >> unsafe;
103 scale_u.d = 1.0;
104 scale_u.ieee.exponent += tval - (tval >> unsafe);
105
106 /* 4. Approximate 2^x2 - 1, using a fourth-degree polynomial,
107 with maximum error in [-2^-10-2^-30,2^-10+2^-30]
108 less than 10^-19. */
109
110 x22 = (((.0096181293647031180
111 * x + .055504110254308625)
112 * x + .240226506959100583)
113 * x + .69314718055994495) * ex2_u.d;
114 math_opt_barrier (x22);
115 }
116
117 /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex). */
118 result = x22 * x + ex2_u.d;
119
120 if (!unsafe)
121 return result;
122 else
123 {
124 result *= scale_u.d;
125 math_check_force_underflow_nonneg (result);
126 return result;
127 }
128 }
129 else
130 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
131 return TWO1023 * x;
132}
133strong_alias (__ieee754_exp2, __exp2_finite)
134