1/* Double-precision e^x function.
2 Copyright (C) 2018-2021 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 InvLn2N __exp_data.invln2N
30#define NegLn2hiN __exp_data.negln2hiN
31#define NegLn2loN __exp_data.negln2loN
32#define Shift __exp_data.shift
33#define T __exp_data.tab
34#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
35#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
36#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
37#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
38
39/* Handle cases that may overflow or underflow when computing the result that
40 is scale*(1+TMP) without intermediate rounding. The bit representation of
41 scale is in SBITS, however it has a computed exponent that may have
42 overflown into the sign bit so that needs to be adjusted before using it as
43 a double. (int32_t)KI is the k used in the argument reduction and exponent
44 adjustment of scale, positive k here means the result may overflow and
45 negative k means the result may underflow. */
46static inline double
47specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
48{
49 double_t scale, y;
50
51 if ((ki & 0x80000000) == 0)
52 {
53 /* k > 0, the exponent of scale might have overflowed by <= 460. */
54 sbits -= 1009ull << 52;
55 scale = asdouble (sbits);
56 y = 0x1p1009 * (scale + scale * tmp);
57 return check_oflow (y);
58 }
59 /* k < 0, need special care in the subnormal range. */
60 sbits += 1022ull << 52;
61 scale = asdouble (sbits);
62 y = scale + scale * tmp;
63 if (y < 1.0)
64 {
65 /* Round y to the right precision before scaling it into the subnormal
66 range to avoid double rounding that can cause 0.5+E/2 ulp error where
67 E is the worst-case ulp error outside the subnormal range. So this
68 is only useful if the goal is better than 1 ulp worst-case error. */
69 double_t hi, lo;
70 lo = scale - y + scale * tmp;
71 hi = 1.0 + y;
72 lo = 1.0 - hi + y + lo;
73 y = math_narrow_eval (hi + lo) - 1.0;
74 /* Avoid -0.0 with downward rounding. */
75 if (WANT_ROUNDING && y == 0.0)
76 y = 0.0;
77 /* The underflow exception needs to be signaled explicitly. */
78 math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022);
79 }
80 y = 0x1p-1022 * y;
81 return check_uflow (y);
82}
83
84/* Top 12 bits of a double (sign and exponent bits). */
85static inline uint32_t
86top12 (double x)
87{
88 return asuint64 (x) >> 52;
89}
90
91#ifndef SECTION
92# define SECTION
93#endif
94
95double
96SECTION
97__exp (double x)
98{
99 uint32_t abstop;
100 uint64_t ki, idx, top, sbits;
101 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
102 double_t kd, z, r, r2, scale, tail, tmp;
103
104 abstop = top12 (x) & 0x7ff;
105 if (__glibc_unlikely (abstop - top12 (0x1p-54)
106 >= top12 (512.0) - top12 (0x1p-54)))
107 {
108 if (abstop - top12 (0x1p-54) >= 0x80000000)
109 /* Avoid spurious underflow for tiny x. */
110 /* Note: 0 is common input. */
111 return WANT_ROUNDING ? 1.0 + x : 1.0;
112 if (abstop >= top12 (1024.0))
113 {
114 if (asuint64 (x) == asuint64 (-INFINITY))
115 return 0.0;
116 if (abstop >= top12 (INFINITY))
117 return 1.0 + x;
118 if (asuint64 (x) >> 63)
119 return __math_uflow (0);
120 else
121 return __math_oflow (0);
122 }
123 /* Large x is special cased below. */
124 abstop = 0;
125 }
126
127 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
128 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
129 z = InvLn2N * x;
130#if TOINT_INTRINSICS
131 kd = roundtoint (z);
132 ki = converttoint (z);
133#else
134 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
135 kd = math_narrow_eval (z + Shift);
136 ki = asuint64 (kd);
137 kd -= Shift;
138#endif
139 r = x + kd * NegLn2hiN + kd * NegLn2loN;
140 /* 2^(k/N) ~= scale * (1 + tail). */
141 idx = 2 * (ki % N);
142 top = ki << (52 - EXP_TABLE_BITS);
143 tail = asdouble (T[idx]);
144 /* This is only a valid scale when -1023*N < k < 1024*N. */
145 sbits = T[idx + 1] + top;
146 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
147 /* Evaluation is optimized assuming superscalar pipelined execution. */
148 r2 = r * r;
149 /* Without fma the worst case error is 0.25/N ulp larger. */
150 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
151 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
152 if (__glibc_unlikely (abstop == 0))
153 return specialcase (tmp, sbits, ki);
154 scale = asdouble (sbits);
155 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-739, so there
156 is no spurious underflow here even without fma. */
157 return scale + scale * tmp;
158}
159#ifndef __exp
160hidden_def (__exp)
161strong_alias (__exp, __ieee754_exp)
162libm_alias_finite (__ieee754_exp, __exp)
163# if LIBM_SVID_COMPAT
164versioned_symbol (libm, __exp, exp, GLIBC_2_29);
165libm_alias_double_other (__exp, exp)
166# else
167libm_alias_double (__exp, exp)
168# endif
169#endif
170