1 /*
2 * Double-precision e^x function.
3 *
4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
5 * See https://llvm.org/LICENSE.txt for license information.
6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
7 */
8
9 #include <float.h>
10 #include <math.h>
11 #include <stdint.h>
12 #include "math_config.h"
13
14 #define N (1 << EXP_TABLE_BITS)
15 #define InvLn2N __exp_data.invln2N
16 #define NegLn2hiN __exp_data.negln2hiN
17 #define NegLn2loN __exp_data.negln2loN
18 #define Shift __exp_data.shift
19 #define T __exp_data.tab
20 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
21 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
22 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
23 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
24 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
25
26 /* Handle cases that may overflow or underflow when computing the result that
27 is scale*(1+TMP) without intermediate rounding. The bit representation of
28 scale is in SBITS, however it has a computed exponent that may have
29 overflown into the sign bit so that needs to be adjusted before using it as
30 a double. (int32_t)KI is the k used in the argument reduction and exponent
31 adjustment of scale, positive k here means the result may overflow and
32 negative k means the result may underflow. */
33 static inline double
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)34 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
35 {
36 double_t scale, y;
37
38 if ((ki & 0x80000000) == 0)
39 {
40 /* k > 0, the exponent of scale might have overflowed by <= 460. */
41 sbits -= 1009ull << 52;
42 scale = asdouble (sbits);
43 y = 0x1p1009 * (scale + scale * tmp);
44 return check_oflow (eval_as_double (y));
45 }
46 /* k < 0, need special care in the subnormal range. */
47 sbits += 1022ull << 52;
48 scale = asdouble (sbits);
49 y = scale + scale * tmp;
50 if (y < 1.0)
51 {
52 /* Round y to the right precision before scaling it into the subnormal
53 range to avoid double rounding that can cause 0.5+E/2 ulp error where
54 E is the worst-case ulp error outside the subnormal range. So this
55 is only useful if the goal is better than 1 ulp worst-case error. */
56 double_t hi, lo;
57 lo = scale - y + scale * tmp;
58 hi = 1.0 + y;
59 lo = 1.0 - hi + y + lo;
60 y = eval_as_double (hi + lo) - 1.0;
61 /* Avoid -0.0 with downward rounding. */
62 if (WANT_ROUNDING && y == 0.0)
63 y = 0.0;
64 /* The underflow exception needs to be signaled explicitly. */
65 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
66 }
67 y = 0x1p-1022 * y;
68 return check_uflow (eval_as_double (y));
69 }
70
71 /* Top 12 bits of a double (sign and exponent bits). */
72 static inline uint32_t
top12(double x)73 top12 (double x)
74 {
75 return asuint64 (x) >> 52;
76 }
77
78 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
79 If hastail is 0 then xtail is assumed to be 0 too. */
80 static inline double
exp_inline(double x,double xtail,int hastail)81 exp_inline (double x, double xtail, int hastail)
82 {
83 uint32_t abstop;
84 uint64_t ki, idx, top, sbits;
85 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
86 double_t kd, z, r, r2, scale, tail, tmp;
87
88 abstop = top12 (x) & 0x7ff;
89 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
90 {
91 if (abstop - top12 (0x1p-54) >= 0x80000000)
92 /* Avoid spurious underflow for tiny x. */
93 /* Note: 0 is common input. */
94 return WANT_ROUNDING ? 1.0 + x : 1.0;
95 if (abstop >= top12 (1024.0))
96 {
97 if (asuint64 (x) == asuint64 (-INFINITY))
98 return 0.0;
99 if (abstop >= top12 (INFINITY))
100 return 1.0 + x;
101 if (asuint64 (x) >> 63)
102 return __math_uflow (0);
103 else
104 return __math_oflow (0);
105 }
106 /* Large x is special cased below. */
107 abstop = 0;
108 }
109
110 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
111 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
112 z = InvLn2N * x;
113 #if TOINT_INTRINSICS
114 kd = roundtoint (z);
115 ki = converttoint (z);
116 #elif EXP_USE_TOINT_NARROW
117 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
118 kd = eval_as_double (z + Shift);
119 ki = asuint64 (kd) >> 16;
120 kd = (double_t) (int32_t) ki;
121 #else
122 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
123 kd = eval_as_double (z + Shift);
124 ki = asuint64 (kd);
125 kd -= Shift;
126 #endif
127 r = x + kd * NegLn2hiN + kd * NegLn2loN;
128 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */
129 if (hastail)
130 r += xtail;
131 /* 2^(k/N) ~= scale * (1 + tail). */
132 idx = 2 * (ki % N);
133 top = ki << (52 - EXP_TABLE_BITS);
134 tail = asdouble (T[idx]);
135 /* This is only a valid scale when -1023*N < k < 1024*N. */
136 sbits = T[idx + 1] + top;
137 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
138 /* Evaluation is optimized assuming superscalar pipelined execution. */
139 r2 = r * r;
140 /* Without fma the worst case error is 0.25/N ulp larger. */
141 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
142 #if EXP_POLY_ORDER == 4
143 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
144 #elif EXP_POLY_ORDER == 5
145 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
146 #elif EXP_POLY_ORDER == 6
147 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
148 #endif
149 if (unlikely (abstop == 0))
150 return specialcase (tmp, sbits, ki);
151 scale = asdouble (sbits);
152 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
153 is no spurious underflow here even without fma. */
154 return eval_as_double (scale + scale * tmp);
155 }
156
157 double
exp(double x)158 exp (double x)
159 {
160 return exp_inline (x, 0, 0);
161 }
162
163 /* May be useful for implementing pow where more than double
164 precision input is needed. */
165 double
__exp_dd(double x,double xtail)166 __exp_dd (double x, double xtail)
167 {
168 return exp_inline (x, xtail, 1);
169 }
170 #if USE_GLIBC_ABI
strong_alias(exp,__exp_finite)171 strong_alias (exp, __exp_finite)
172 hidden_alias (exp, __ieee754_exp)
173 hidden_alias (__exp_dd, __exp1)
174 # if LDBL_MANT_DIG == 53
175 long double expl (long double x) { return exp (x); }
176 # endif
177 #endif
178