1/* Double-precision floating point 2^x.
2 Copyright (C) 1997-2018 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-barriers.h>
33#include <math_private.h>
34#include <math-underflow.h>
35
36#include "t_exp2.h"
37
38static const double TWO1023 = 8.988465674311579539e+307;
39static const double TWOM1000 = 9.3326361850321887899e-302;
40
41double
42__ieee754_exp2 (double x)
43{
44 static const double himark = (double) DBL_MAX_EXP;
45 static const double lomark = (double) (DBL_MIN_EXP - DBL_MANT_DIG - 1);
46
47 /* Check for usual case. */
48 if (__glibc_likely (isless (x, himark)))
49 {
50 /* Exceptional cases: */
51 if (__glibc_unlikely (!isgreaterequal (x, lomark)))
52 {
53 if (isinf (x))
54 /* e^-inf == 0, with no error. */
55 return 0;
56 else
57 /* Underflow */
58 return TWOM1000 * TWOM1000;
59 }
60
61 static const double THREEp42 = 13194139533312.0;
62 int tval, unsafe;
63 double rx, x22, result;
64 union ieee754_double ex2_u, scale_u;
65
66 if (fabs (x) < DBL_EPSILON / 4.0)
67 return 1.0 + x;
68
69 {
70 SET_RESTORE_ROUND_NOEX (FE_TONEAREST);
71
72 /* 1. Argument reduction.
73 Choose integers ex, -256 <= t < 256, and some real
74 -1/1024 <= x1 <= 1024 so that
75 x = ex + t/512 + x1.
76
77 First, calculate rx = ex + t/512. */
78 rx = x + THREEp42;
79 rx -= THREEp42;
80 x -= rx; /* Compute x=x1. */
81 /* Compute tval = (ex*512 + t)+256.
82 Now, t = (tval mod 512)-256 and ex=tval/512 [that's mod, NOT %;
83 and /-round-to-nearest not the usual c integer /]. */
84 tval = (int) (rx * 512.0 + 256.0);
85
86 /* 2. Adjust for accurate table entry.
87 Find e so that
88 x = ex + t/512 + e + x2
89 where -1e6 < e < 1e6, and
90 (double)(2^(t/512+e))
91 is accurate to one part in 2^-64. */
92
93 /* 'tval & 511' is the same as 'tval%512' except that it's always
94 positive.
95 Compute x = x2. */
96 x -= exp2_deltatable[tval & 511];
97
98 /* 3. Compute ex2 = 2^(t/512+e+ex). */
99 ex2_u.d = exp2_accuratetable[tval & 511];
100 tval >>= 9;
101 /* x2 is an integer multiple of 2^-54; avoid intermediate
102 underflow from the calculation of x22 * x. */
103 unsafe = abs (tval) >= -DBL_MIN_EXP - 56;
104 ex2_u.ieee.exponent += tval >> unsafe;
105 scale_u.d = 1.0;
106 scale_u.ieee.exponent += tval - (tval >> unsafe);
107
108 /* 4. Approximate 2^x2 - 1, using a fourth-degree polynomial,
109 with maximum error in [-2^-10-2^-30,2^-10+2^-30]
110 less than 10^-19. */
111
112 x22 = (((.0096181293647031180
113 * x + .055504110254308625)
114 * x + .240226506959100583)
115 * x + .69314718055994495) * ex2_u.d;
116 math_opt_barrier (x22);
117 }
118
119 /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex). */
120 result = x22 * x + ex2_u.d;
121
122 if (!unsafe)
123 return result;
124 else
125 {
126 result *= scale_u.d;
127 math_check_force_underflow_nonneg (result);
128 return result;
129 }
130 }
131 else
132 /* Return x, if x is a NaN or Inf; or overflow, otherwise. */
133 return TWO1023 * x;
134}
135strong_alias (__ieee754_exp2, __exp2_finite)
136