1 /*
2 * Double-precision 2^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 Shift __exp_data.exp2_shift
15 #define T __exp_data.tab
16 #define C1 __exp_data.exp2_poly[0]
17 #define C2 __exp_data.exp2_poly[1]
18 #define C3 __exp_data.exp2_poly[2]
19 #define C4 __exp_data.exp2_poly[3]
20 #define C5 __exp_data.exp2_poly[4]
21 22 /* Handle cases that may overflow or underflow when computing the result that
23 is scale*(1+TMP) without intermediate rounding. The bit representation of
24 scale is in SBITS, however it has a computed exponent that may have
25 overflown into the sign bit so that needs to be adjusted before using it as
26 a double. (int32_t)KI is the k used in the argument reduction and exponent
27 adjustment of scale, positive k here means the result may overflow and
28 negative k means the result may underflow. */
29 static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
30 {
31 double_t scale, y;
32 33 if ((ki & 0x80000000) == 0) {
34 /* k > 0, the exponent of scale might have overflowed by 1. */
35 sbits -= 1ull << 52;
36 scale = asdouble(sbits);
37 y = 2 * (scale + scale * tmp);
38 return eval_as_double(y);
39 }
40 /* k < 0, need special care in the subnormal range. */
41 sbits += 1022ull << 52;
42 scale = asdouble(sbits);
43 y = scale + scale * tmp;
44 if (y < 1.0) {
45 /* Round y to the right precision before scaling it into the subnormal
46 range to avoid double rounding that can cause 0.5+E/2 ulp error where
47 E is the worst-case ulp error outside the subnormal range. So this
48 is only useful if the goal is better than 1 ulp worst-case error. */
49 double_t hi, lo;
50 lo = scale - y + scale * tmp;
51 hi = 1.0 + y;
52 lo = 1.0 - hi + y + lo;
53 y = eval_as_double(hi + lo) - 1.0;
54 /* Avoid -0.0 with downward rounding. */
55 if (WANT_ROUNDING && y == 0.0)
56 y = 0.0;
57 /* The underflow exception needs to be signaled explicitly. */
58 fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
59 }
60 y = 0x1p-1022 * y;
61 return eval_as_double(y);
62 }
63 64 /* Top 12 bits of a double (sign and exponent bits). */
65 static inline uint32_t top12(double x)
66 {
67 return asuint64(x) >> 52;
68 }
69 70 double exp2(double x)
71 {
72 uint32_t abstop;
73 uint64_t ki, idx, top, sbits;
74 double_t kd, r, r2, scale, tail, tmp;
75 76 abstop = top12(x) & 0x7ff;
77 if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
78 if (abstop - top12(0x1p-54) >= 0x80000000)
79 /* Avoid spurious underflow for tiny x. */
80 /* Note: 0 is common input. */
81 return WANT_ROUNDING ? 1.0 + x : 1.0;
82 if (abstop >= top12(1024.0)) {
83 if (asuint64(x) == asuint64(-INFINITY))
84 return 0.0;
85 if (abstop >= top12(INFINITY))
86 return 1.0 + x;
87 if (!(asuint64(x) >> 63))
88 return __math_oflow(0);
89 else if (asuint64(x) >= asuint64(-1075.0))
90 return __math_uflow(0);
91 }
92 if (2 * asuint64(x) > 2 * asuint64(928.0))
93 /* Large x is special cased below. */
94 abstop = 0;
95 }
96 97 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
98 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
99 kd = eval_as_double(x + Shift);
100 ki = asuint64(kd); /* k. */
101 kd -= Shift; /* k/N for int k. */
102 r = x - kd;
103 /* 2^(k/N) ~= scale * (1 + tail). */
104 idx = 2 * (ki % N);
105 top = ki << (52 - EXP_TABLE_BITS);
106 tail = asdouble(T[idx]);
107 /* This is only a valid scale when -1023*N < k < 1024*N. */
108 sbits = T[idx + 1] + top;
109 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
110 /* Evaluation is optimized assuming superscalar pipelined execution. */
111 r2 = r * r;
112 /* Without fma the worst case error is 0.5/N ulp larger. */
113 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
114 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
115 if (predict_false(abstop == 0))
116 return specialcase(tmp, sbits, ki);
117 scale = asdouble(sbits);
118 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
119 is no spurious underflow here even without fma. */
120 return eval_as_double(scale + scale * tmp);
121 }
122