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