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_private.h> |
33 | |
34 | #include "t_exp2.h" |
35 | |
36 | static const double TWO1023 = 8.988465674311579539e+307; |
37 | static const double TWOM1000 = 9.3326361850321887899e-302; |
38 | |
39 | double |
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 | } |
133 | strong_alias (__ieee754_exp2, __exp2_finite) |
134 | |