1 /*
2 * Double-precision e^x function.
3 *
4 * Copyright (c) 2018-2019, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include <float.h>
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.h"
12
13 #define N (1 << EXP_TABLE_BITS)
14 #define InvLn2N __exp_data.invln2N
15 #define NegLn2hiN __exp_data.negln2hiN
16 #define NegLn2loN __exp_data.negln2loN
17 #define Shift __exp_data.shift
18 #define T __exp_data.tab
19 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
20 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
21 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
22 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
23 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
24
25 /* Handle cases that may overflow or underflow when computing the result that
26 is scale*(1+TMP) without intermediate rounding. The bit representation of
27 scale is in SBITS, however it has a computed exponent that may have
28 overflown into the sign bit so that needs to be adjusted before using it as
29 a double. (int32_t)KI is the k used in the argument reduction and exponent
30 adjustment of scale, positive k here means the result may overflow and
31 negative k means the result may underflow. */
32 static inline double
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)33 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
34 {
35 double_t scale, y;
36
37 if ((ki & 0x80000000) == 0)
38 {
39 /* k > 0, the exponent of scale might have overflowed by <= 460. */
40 sbits -= 1009ull << 52;
41 scale = asdouble (sbits);
42 y = 0x1p1009 * (scale + scale * tmp);
43 return check_oflow (eval_as_double (y));
44 }
45 /* k < 0, need special care in the subnormal range. */
46 sbits += 1022ull << 52;
47 scale = asdouble (sbits);
48 y = scale + scale * tmp;
49 if (y < 1.0)
50 {
51 /* Round y to the right precision before scaling it into the subnormal
52 range to avoid double rounding that can cause 0.5+E/2 ulp error where
53 E is the worst-case ulp error outside the subnormal range. So this
54 is only useful if the goal is better than 1 ulp worst-case error. */
55 double_t hi, lo;
56 lo = scale - y + scale * tmp;
57 hi = 1.0 + y;
58 lo = 1.0 - hi + y + lo;
59 y = eval_as_double (hi + lo) - 1.0;
60 /* Avoid -0.0 with downward rounding. */
61 if (WANT_ROUNDING && y == 0.0)
62 y = 0.0;
63 /* The underflow exception needs to be signaled explicitly. */
64 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
65 }
66 y = 0x1p-1022 * y;
67 return check_uflow (eval_as_double (y));
68 }
69
70 /* Top 12 bits of a double (sign and exponent bits). */
71 static inline uint32_t
top12(double x)72 top12 (double x)
73 {
74 return asuint64 (x) >> 52;
75 }
76
77 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
78 If hastail is 0 then xtail is assumed to be 0 too. */
79 static inline double
exp_inline(double x,double xtail,int hastail)80 exp_inline (double x, double xtail, int hastail)
81 {
82 uint32_t abstop;
83 uint64_t ki, idx, top, sbits;
84 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
85 double_t kd, z, r, r2, scale, tail, tmp;
86
87 abstop = top12 (x) & 0x7ff;
88 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
89 {
90 if (abstop - top12 (0x1p-54) >= 0x80000000)
91 /* Avoid spurious underflow for tiny x. */
92 /* Note: 0 is common input. */
93 return WANT_ROUNDING ? 1.0 + x : 1.0;
94 if (abstop >= top12 (1024.0))
95 {
96 if (asuint64 (x) == asuint64 (-INFINITY))
97 return 0.0;
98 if (abstop >= top12 (INFINITY))
99 return 1.0 + x;
100 if (asuint64 (x) >> 63)
101 return __math_uflow (0);
102 else
103 return __math_oflow (0);
104 }
105 /* Large x is special cased below. */
106 abstop = 0;
107 }
108
109 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
110 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
111 z = InvLn2N * x;
112 #if TOINT_INTRINSICS
113 kd = roundtoint (z);
114 ki = converttoint (z);
115 #elif EXP_USE_TOINT_NARROW
116 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
117 kd = eval_as_double (z + Shift);
118 ki = asuint64 (kd) >> 16;
119 kd = (double_t) (int32_t) ki;
120 #else
121 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
122 kd = eval_as_double (z + Shift);
123 ki = asuint64 (kd);
124 kd -= Shift;
125 #endif
126 r = x + kd * NegLn2hiN + kd * NegLn2loN;
127 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */
128 if (hastail)
129 r += xtail;
130 /* 2^(k/N) ~= scale * (1 + tail). */
131 idx = 2 * (ki % N);
132 top = ki << (52 - EXP_TABLE_BITS);
133 tail = asdouble (T[idx]);
134 /* This is only a valid scale when -1023*N < k < 1024*N. */
135 sbits = T[idx + 1] + top;
136 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
137 /* Evaluation is optimized assuming superscalar pipelined execution. */
138 r2 = r * r;
139 /* Without fma the worst case error is 0.25/N ulp larger. */
140 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
141 #if EXP_POLY_ORDER == 4
142 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
143 #elif EXP_POLY_ORDER == 5
144 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
145 #elif EXP_POLY_ORDER == 6
146 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
147 #endif
148 if (unlikely (abstop == 0))
149 return specialcase (tmp, sbits, ki);
150 scale = asdouble (sbits);
151 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
152 is no spurious underflow here even without fma. */
153 return eval_as_double (scale + scale * tmp);
154 }
155
156 double
exp(double x)157 exp (double x)
158 {
159 return exp_inline (x, 0, 0);
160 }
161
162 /* May be useful for implementing pow where more than double
163 precision input is needed. */
164 double
__exp_dd(double x,double xtail)165 __exp_dd (double x, double xtail)
166 {
167 return exp_inline (x, xtail, 1);
168 }
169 #if USE_GLIBC_ABI
strong_alias(exp,__exp_finite)170 strong_alias (exp, __exp_finite)
171 hidden_alias (exp, __ieee754_exp)
172 hidden_alias (__exp_dd, __exp1)
173 # if LDBL_MANT_DIG == 53
174 long double expl (long double x) { return exp (x); }
175 # endif
176 #endif
177