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 | |
38 | static const double TWO1023 = 8.988465674311579539e+307; |
39 | static const double TWOM1000 = 9.3326361850321887899e-302; |
40 | |
41 | double |
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 | } |
135 | strong_alias (__ieee754_exp2, __exp2_finite) |
136 | |