1/* Double-precision 2^x function.
2 Copyright (C) 2018-2022 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
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.
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
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19#include <math.h>
20#include <stdint.h>
21#include <math-barriers.h>
22#include <math-narrow-eval.h>
23#include <math-svid-compat.h>
24#include <libm-alias-finite.h>
25#include <libm-alias-double.h>
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. */
44static inline double
45specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
46{
47 double_t scale, y;
48
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}
81
82/* Top 12 bits of a double (sign and exponent bits). */
83static inline uint32_t
84top12 (double x)
85{
86 return asuint64 (x) >> 52;
87}
88
89double
90__exp2 (double x)
91{
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)))
100 {
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))
106 {
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);
115 }
116 if (2 * asuint64 (x) > 2 * asuint64 (928.0))
117 /* Large x is special cased below. */
118 abstop = 0;
119 }
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;
145}
146#ifndef __exp2
147strong_alias (__exp2, __ieee754_exp2)
148libm_alias_finite (__ieee754_exp2, __exp2)
149# if LIBM_SVID_COMPAT
150versioned_symbol (libm, __exp2, exp2, GLIBC_2_29);
151libm_alias_double_other (__exp2, exp2)
152# else
153libm_alias_double (__exp2, exp2)
154# endif
155#endif
156