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 "libm.h"
11 #include "exp_data.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 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 {
33 double_t scale, y;
34 35 if ((ki & 0x80000000) == 0) {
36 /* k > 0, the exponent of scale might have overflowed by <= 460. */
37 sbits -= 1009ull << 52;
38 scale = asdouble(sbits);
39 y = 0x1p1009 * (scale + scale * tmp);
40 return eval_as_double(y);
41 }
42 /* k < 0, need special care in the subnormal range. */
43 sbits += 1022ull << 52;
44 scale = asdouble(sbits);
45 y = scale + scale * tmp;
46 if (y < 1.0) {
47 /* Round y to the right precision before scaling it into the subnormal
48 range to avoid double rounding that can cause 0.5+E/2 ulp error where
49 E is the worst-case ulp error outside the subnormal range. So this
50 is only useful if the goal is better than 1 ulp worst-case error. */
51 double_t hi, lo;
52 lo = scale - y + scale * tmp;
53 hi = 1.0 + y;
54 lo = 1.0 - hi + y + lo;
55 y = eval_as_double(hi + lo) - 1.0;
56 /* Avoid -0.0 with downward rounding. */
57 if (WANT_ROUNDING && y == 0.0)
58 y = 0.0;
59 /* The underflow exception needs to be signaled explicitly. */
60 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
61 }
62 y = 0x1p-1022 * y;
63 return eval_as_double(y);
64 }
65 66 /* Top 12 bits of a double (sign and exponent bits). */
67 static inline uint32_t top12(double x)
68 {
69 return asuint64(x) >> 52;
70 }
71 72 double exp(double x)
73 {
74 uint32_t abstop;
75 uint64_t ki, idx, top, sbits;
76 double_t kd, z, r, r2, scale, tail, tmp;
77 78 abstop = top12(x) & 0x7ff;
79 if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
80 if (abstop - top12(0x1p-54) >= 0x80000000)
81 /* Avoid spurious underflow for tiny x. */
82 /* Note: 0 is common input. */
83 return WANT_ROUNDING ? 1.0 + x : 1.0;
84 if (abstop >= top12(1024.0)) {
85 if (asuint64(x) == asuint64(-INFINITY))
86 return 0.0;
87 if (abstop >= top12(INFINITY))
88 return 1.0 + x;
89 if (asuint64(x) >> 63)
90 return __math_uflow(0);
91 else
92 return __math_oflow(0);
93 }
94 /* Large x is special cased below. */
95 abstop = 0;
96 }
97 98 /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
99 /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
100 z = InvLn2N * x;
101 #if TOINT_INTRINSICS
102 kd = roundtoint(z);
103 ki = converttoint(z);
104 #elif EXP_USE_TOINT_NARROW
105 /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
106 kd = eval_as_double(z + Shift);
107 ki = asuint64(kd) >> 16;
108 kd = (double_t)(int32_t)ki;
109 #else
110 /* z - kd is in [-1, 1] in non-nearest rounding modes. */
111 kd = eval_as_double(z + Shift);
112 ki = asuint64(kd);
113 kd -= Shift;
114 #endif
115 r = x + kd * NegLn2hiN + kd * NegLn2loN;
116 /* 2^(k/N) ~= scale * (1 + tail). */
117 idx = 2 * (ki % N);
118 top = ki << (52 - EXP_TABLE_BITS);
119 tail = asdouble(T[idx]);
120 /* This is only a valid scale when -1023*N < k < 1024*N. */
121 sbits = T[idx + 1] + top;
122 /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
123 /* Evaluation is optimized assuming superscalar pipelined execution. */
124 r2 = r * r;
125 /* Without fma the worst case error is 0.25/N ulp larger. */
126 /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
127 tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
128 if (predict_false(abstop == 0))
129 return specialcase(tmp, sbits, ki);
130 scale = asdouble(sbits);
131 /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
132 is no spurious underflow here even without fma. */
133 return eval_as_double(scale + scale * tmp);
134 }
135