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 "math_config.h"
11
12 #define N (1 << EXP_TABLE_BITS)
13 #define Shift __exp_data.exp2_shift
14 #define T __exp_data.tab
15 #define C1 __exp_data.exp2_poly[0]
16 #define C2 __exp_data.exp2_poly[1]
17 #define C3 __exp_data.exp2_poly[2]
18 #define C4 __exp_data.exp2_poly[3]
19 #define C5 __exp_data.exp2_poly[4]
20 #define C6 __exp_data.exp2_poly[5]
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 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
31 {
32 double_t scale, y;
33
34 if ((ki & 0x80000000) == 0)
35 {
36 /* k > 0, the exponent of scale might have overflowed by 1. */
37 sbits -= 1ull << 52;
38 scale = asdouble (sbits);
39 y = 2 * (scale + scale * tmp);
40 return check_oflow (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 {
48 /* Round y to the right precision before scaling it into the subnormal
49 range to avoid double rounding that can cause 0.5+E/2 ulp error where
50 E is the worst-case ulp error outside the subnormal range. So this
51 is only useful if the goal is better than 1 ulp worst-case error. */
52 double_t hi, lo;
53 lo = scale - y + scale * tmp;
54 hi = 1.0 + y;
55 lo = 1.0 - hi + y + lo;
56 y = eval_as_double (hi + lo) - 1.0;
57 /* Avoid -0.0 with downward rounding. */
58 if (WANT_ROUNDING && y == 0.0)
59 y = 0.0;
60 /* The underflow exception needs to be signaled explicitly. */
61 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
62 }
63 y = 0x1p-1022 * y;
64 return check_uflow (eval_as_double (y));
65 }
66
67 /* Top 12 bits of a double (sign and exponent bits). */
68 static inline uint32_t
top12(double x)69 top12 (double x)
70 {
71 return asuint64 (x) >> 52;
72 }
73
74 double
exp2(double x)75 exp2 (double x)
76 {
77 uint32_t abstop;
78 uint64_t ki, idx, top, sbits;
79 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
80 double_t kd, r, r2, scale, tail, tmp;
81
82 abstop = top12 (x) & 0x7ff;
83 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
84 {
85 if (abstop - top12 (0x1p-54) >= 0x80000000)
86 /* Avoid spurious underflow for tiny x. */
87 /* Note: 0 is common input. */
88 return WANT_ROUNDING ? 1.0 + x : 1.0;
89 if (abstop >= top12 (1024.0))
90 {
91 if (asuint64 (x) == asuint64 (-INFINITY))
92 return 0.0;
93 if (abstop >= top12 (INFINITY))
94 return 1.0 + x;
95 if (!(asuint64 (x) >> 63))
96 return __math_oflow (0);
97 else if (asuint64 (x) >= asuint64 (-1075.0))
98 return __math_uflow (0);
99 }
100 if (2 * asuint64 (x) > 2 * asuint64 (928.0))
101 /* Large x is special cased below. */
102 abstop = 0;
103 }
104
105 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
106 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
107 kd = eval_as_double (x + Shift);
108 ki = asuint64 (kd); /* k. */
109 kd -= Shift; /* k/N for int k. */
110 r = x - kd;
111 /* 2^(k/N) ~= scale * (1 + tail). */
112 idx = 2 * (ki % N);
113 top = ki << (52 - EXP_TABLE_BITS);
114 tail = asdouble (T[idx]);
115 /* This is only a valid scale when -1023*N < k < 1024*N. */
116 sbits = T[idx + 1] + top;
117 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
118 /* Evaluation is optimized assuming superscalar pipelined execution. */
119 r2 = r * r;
120 /* Without fma the worst case error is 0.5/N ulp larger. */
121 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
122 #if EXP2_POLY_ORDER == 4
123 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
124 #elif EXP2_POLY_ORDER == 5
125 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
126 #elif EXP2_POLY_ORDER == 6
127 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
128 #endif
129 if (unlikely (abstop == 0))
130 return specialcase (tmp, sbits, ki);
131 scale = asdouble (sbits);
132 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
133 is no spurious underflow here even without fma. */
134 return eval_as_double (scale + scale * tmp);
135 }
136 #if USE_GLIBC_ABI
137 strong_alias (exp2, __exp2_finite)
138 hidden_alias (exp2, __ieee754_exp2)
139 #endif
140