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e70c1768 | 1 | /* Double-precision 2^x function. |
04277e02 | 2 | Copyright (C) 2018-2019 Free Software Foundation, Inc. |
63640cb7 | 3 | This file is part of the GNU C Library. |
63640cb7 UD |
4 | |
5 | The GNU C Library is free software; you can redistribute it and/or | |
41bdb6e2 AJ |
6 | modify it under the terms of the GNU Lesser General Public |
7 | License as published by the Free Software Foundation; either | |
8 | version 2.1 of the License, or (at your option) any later version. | |
63640cb7 UD |
9 | |
10 | The GNU C Library is distributed in the hope that it will be useful, | |
11 | but WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
41bdb6e2 | 13 | Lesser General Public License for more details. |
63640cb7 | 14 | |
41bdb6e2 | 15 | You should have received a copy of the GNU Lesser General Public |
59ba27a6 | 16 | License along with the GNU C Library; if not, see |
5a82c748 | 17 | <https://www.gnu.org/licenses/>. */ |
63640cb7 | 18 | |
63640cb7 | 19 | #include <math.h> |
e70c1768 | 20 | #include <stdint.h> |
b4d5b8b0 | 21 | #include <math-barriers.h> |
e70c1768 | 22 | #include <math-narrow-eval.h> |
c20a1056 SN |
23 | #include <math-svid-compat.h> |
24 | #include <shlib-compat.h> | |
25 | #include <libm-alias-double.h> | |
e70c1768 SN |
26 | #include "math_config.h" |
27 | ||
28 | #define N (1 << EXP_TABLE_BITS) | |
29 | #define Shift __exp_data.exp2_shift | |
30 | #define T __exp_data.tab | |
31 | #define C1 __exp_data.exp2_poly[0] | |
32 | #define C2 __exp_data.exp2_poly[1] | |
33 | #define C3 __exp_data.exp2_poly[2] | |
34 | #define C4 __exp_data.exp2_poly[3] | |
35 | #define C5 __exp_data.exp2_poly[4] | |
36 | ||
37 | /* Handle cases that may overflow or underflow when computing the result that | |
38 | is scale*(1+TMP) without intermediate rounding. The bit representation of | |
39 | scale is in SBITS, however it has a computed exponent that may have | |
40 | overflown into the sign bit so that needs to be adjusted before using it as | |
41 | a double. (int32_t)KI is the k used in the argument reduction and exponent | |
42 | adjustment of scale, positive k here means the result may overflow and | |
43 | negative k means the result may underflow. */ | |
44 | static inline double | |
45 | specialcase (double_t tmp, uint64_t sbits, uint64_t ki) | |
46 | { | |
47 | double_t scale, y; | |
63640cb7 | 48 | |
e70c1768 SN |
49 | if ((ki & 0x80000000) == 0) |
50 | { | |
51 | /* k > 0, the exponent of scale might have overflowed by 1. */ | |
52 | sbits -= 1ull << 52; | |
53 | scale = asdouble (sbits); | |
54 | y = 2 * (scale + scale * tmp); | |
55 | return check_oflow (y); | |
56 | } | |
57 | /* k < 0, need special care in the subnormal range. */ | |
58 | sbits += 1022ull << 52; | |
59 | scale = asdouble (sbits); | |
60 | y = scale + scale * tmp; | |
61 | if (y < 1.0) | |
62 | { | |
63 | /* Round y to the right precision before scaling it into the subnormal | |
64 | range to avoid double rounding that can cause 0.5+E/2 ulp error where | |
65 | E is the worst-case ulp error outside the subnormal range. So this | |
66 | is only useful if the goal is better than 1 ulp worst-case error. */ | |
67 | double_t hi, lo; | |
68 | lo = scale - y + scale * tmp; | |
69 | hi = 1.0 + y; | |
70 | lo = 1.0 - hi + y + lo; | |
71 | y = math_narrow_eval (hi + lo) - 1.0; | |
72 | /* Avoid -0.0 with downward rounding. */ | |
73 | if (WANT_ROUNDING && y == 0.0) | |
74 | y = 0.0; | |
75 | /* The underflow exception needs to be signaled explicitly. */ | |
76 | math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022); | |
77 | } | |
78 | y = 0x1p-1022 * y; | |
79 | return check_uflow (y); | |
80 | } | |
63640cb7 | 81 | |
e70c1768 SN |
82 | /* Top 12 bits of a double (sign and exponent bits). */ |
83 | static inline uint32_t | |
84 | top12 (double x) | |
85 | { | |
86 | return asuint64 (x) >> 52; | |
87 | } | |
63640cb7 UD |
88 | |
89 | double | |
c20a1056 | 90 | __exp2 (double x) |
63640cb7 | 91 | { |
e70c1768 SN |
92 | uint32_t abstop; |
93 | uint64_t ki, idx, top, sbits; | |
94 | /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ | |
95 | double_t kd, r, r2, scale, tail, tmp; | |
96 | ||
97 | abstop = top12 (x) & 0x7ff; | |
98 | if (__glibc_unlikely (abstop - top12 (0x1p-54) | |
99 | >= top12 (512.0) - top12 (0x1p-54))) | |
63640cb7 | 100 | { |
e70c1768 SN |
101 | if (abstop - top12 (0x1p-54) >= 0x80000000) |
102 | /* Avoid spurious underflow for tiny x. */ | |
103 | /* Note: 0 is common input. */ | |
104 | return WANT_ROUNDING ? 1.0 + x : 1.0; | |
105 | if (abstop >= top12 (1024.0)) | |
903af5af | 106 | { |
e70c1768 SN |
107 | if (asuint64 (x) == asuint64 (-INFINITY)) |
108 | return 0.0; | |
109 | if (abstop >= top12 (INFINITY)) | |
110 | return 1.0 + x; | |
111 | if (!(asuint64 (x) >> 63)) | |
112 | return __math_oflow (0); | |
113 | else if (asuint64 (x) >= asuint64 (-1075.0)) | |
114 | return __math_uflow (0); | |
903af5af | 115 | } |
e70c1768 SN |
116 | if (2 * asuint64 (x) > 2 * asuint64 (928.0)) |
117 | /* Large x is special cased below. */ | |
118 | abstop = 0; | |
63640cb7 | 119 | } |
e70c1768 SN |
120 | |
121 | /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */ | |
122 | /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */ | |
123 | kd = math_narrow_eval (x + Shift); | |
124 | ki = asuint64 (kd); /* k. */ | |
125 | kd -= Shift; /* k/N for int k. */ | |
126 | r = x - kd; | |
127 | /* 2^(k/N) ~= scale * (1 + tail). */ | |
128 | idx = 2 * (ki % N); | |
129 | top = ki << (52 - EXP_TABLE_BITS); | |
130 | tail = asdouble (T[idx]); | |
131 | /* This is only a valid scale when -1023*N < k < 1024*N. */ | |
132 | sbits = T[idx + 1] + top; | |
133 | /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */ | |
134 | /* Evaluation is optimized assuming superscalar pipelined execution. */ | |
135 | r2 = r * r; | |
136 | /* Without fma the worst case error is 0.5/N ulp larger. */ | |
137 | /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */ | |
138 | tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); | |
139 | if (__glibc_unlikely (abstop == 0)) | |
140 | return specialcase (tmp, sbits, ki); | |
141 | scale = asdouble (sbits); | |
142 | /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there | |
143 | is no spurious underflow here even without fma. */ | |
144 | return scale + scale * tmp; | |
63640cb7 | 145 | } |
c20a1056 SN |
146 | #ifndef __exp2 |
147 | strong_alias (__exp2, __ieee754_exp2) | |
148 | strong_alias (__exp2, __exp2_finite) | |
149 | # if LIBM_SVID_COMPAT | |
150 | versioned_symbol (libm, __exp2, exp2, GLIBC_2_29); | |
151 | libm_alias_double_other (__exp2, exp2) | |
152 | # else | |
153 | libm_alias_double (__exp2, exp2) | |
154 | # endif | |
e70c1768 | 155 | #endif |