1 /*
2 * Double-precision e^x function.
3 *
4 * Copyright (c) 2018-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #ifndef PL_MATH_EXP_INLINE_H
9 #define PL_MATH_EXP_INLINE_H
10
11 #include <float.h>
12 #include <math.h>
13 #include <stdint.h>
14 #include "math_config.h"
15
16 #define N (1 << EXP_TABLE_BITS)
17 #define InvLn2N __exp_data.invln2N
18 #define NegLn2hiN __exp_data.negln2hiN
19 #define NegLn2loN __exp_data.negln2loN
20 #define Shift __exp_data.shift
21 #define T __exp_data.tab
22 #define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
23 #define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
24 #define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
25 #define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
26 #define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
27
28 /* Handle cases that may overflow or underflow when computing the result that
29 is scale*(1+TMP) without intermediate rounding. The bit representation of
30 scale is in SBITS, however it has a computed exponent that may have
31 overflown into the sign bit so that needs to be adjusted before using it as
32 a double. (int32_t)KI is the k used in the argument reduction and exponent
33 adjustment of scale, positive k here means the result may overflow and
34 negative k means the result may underflow. */
35 static inline double
exp_inline_special_case(double_t tmp,uint64_t sbits,uint64_t ki)36 exp_inline_special_case (double_t tmp, uint64_t sbits, uint64_t ki)
37 {
38 double_t scale, y;
39
40 if ((ki & 0x80000000) == 0)
41 {
42 /* k > 0, the exponent of scale might have overflowed by <= 460. */
43 sbits -= 1009ull << 52;
44 scale = asdouble (sbits);
45 y = 0x1p1009 * (scale + scale * tmp);
46 return check_oflow (eval_as_double (y));
47 }
48 /* k < 0, need special care in the subnormal range. */
49 sbits += 1022ull << 52;
50 scale = asdouble (sbits);
51 y = scale + scale * tmp;
52 if (y < 1.0)
53 {
54 /* Round y to the right precision before scaling it into the subnormal
55 range to avoid double rounding that can cause 0.5+E/2 ulp error where
56 E is the worst-case ulp error outside the subnormal range. So this
57 is only useful if the goal is better than 1 ulp worst-case error. */
58 double_t hi, lo;
59 lo = scale - y + scale * tmp;
60 hi = 1.0 + y;
61 lo = 1.0 - hi + y + lo;
62 y = eval_as_double (hi + lo) - 1.0;
63 /* Avoid -0.0 with downward rounding. */
64 if (WANT_ROUNDING && y == 0.0)
65 y = 0.0;
66 /* The underflow exception needs to be signaled explicitly. */
67 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
68 }
69 y = 0x1p-1022 * y;
70 return check_uflow (eval_as_double (y));
71 }
72
73 /* Top 12 bits of a double (sign and exponent bits). */
74 static inline uint32_t
top12(double x)75 top12 (double x)
76 {
77 return asuint64 (x) >> 52;
78 }
79
80 /* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
81 If hastail is 0 then xtail is assumed to be 0 too. */
82 static inline double
exp_inline(double x,double xtail)83 exp_inline (double x, double xtail)
84 {
85 uint32_t abstop;
86 uint64_t ki, idx, top, sbits;
87 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
88 double_t kd, z, r, r2, scale, tail, tmp;
89
90 abstop = top12 (x) & 0x7ff;
91 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
92 {
93 if (abstop - top12 (0x1p-54) >= 0x80000000)
94 /* Avoid spurious underflow for tiny x. */
95 /* Note: 0 is common input. */
96 return WANT_ROUNDING ? 1.0 + x : 1.0;
97 if (abstop >= top12 (1024.0))
98 {
99 if (asuint64 (x) == asuint64 (-INFINITY))
100 return 0.0;
101 if (abstop >= top12 (INFINITY))
102 return 1.0 + x;
103 if (asuint64 (x) >> 63)
104 return __math_uflow (0);
105 else
106 return __math_oflow (0);
107 }
108 /* Large x is special cased below. */
109 abstop = 0;
110 }
111
112 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
113 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
114 z = InvLn2N * x;
115 #if TOINT_INTRINSICS
116 kd = roundtoint (z);
117 ki = converttoint (z);
118 #elif EXP_USE_TOINT_NARROW
119 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
120 kd = eval_as_double (z + Shift);
121 ki = asuint64 (kd) >> 16;
122 kd = (double_t) (int32_t) ki;
123 #else
124 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
125 kd = eval_as_double (z + Shift);
126 ki = asuint64 (kd);
127 kd -= Shift;
128 #endif
129 r = x + kd * NegLn2hiN + kd * NegLn2loN;
130 /* The code assumes 2^-200 < |xtail| < 2^-8/N. */
131 if (!__builtin_constant_p (xtail) || xtail != 0.0)
132 r += xtail;
133 /* 2^(k/N) ~= scale * (1 + tail). */
134 idx = 2 * (ki % N);
135 top = ki << (52 - EXP_TABLE_BITS);
136 tail = asdouble (T[idx]);
137 /* This is only a valid scale when -1023*N < k < 1024*N. */
138 sbits = T[idx + 1] + top;
139 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
140 /* Evaluation is optimized assuming superscalar pipelined execution. */
141 r2 = r * r;
142 /* Without fma the worst case error is 0.25/N ulp larger. */
143 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
144 #if EXP_POLY_ORDER == 4
145 tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
146 #elif EXP_POLY_ORDER == 5
147 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
148 #elif EXP_POLY_ORDER == 6
149 tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
150 #endif
151 if (unlikely (abstop == 0))
152 return exp_inline_special_case (tmp, sbits, ki);
153 scale = asdouble (sbits);
154 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
155 is no spurious underflow here even without fma. */
156 return eval_as_double (scale + scale * tmp);
157 }
158
159 #endif
160