1 /*
2 * Double-precision 2^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 #include <float.h>
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.h"
12 #include "test_defs.h"
13 #include "test_sig.h"
14
15 #define N (1 << EXP_TABLE_BITS)
16 #define Shift __exp_data.exp2_shift
17 #define T __exp_data.tab
18 #define C1 __exp_data.exp2_poly[0]
19 #define C2 __exp_data.exp2_poly[1]
20 #define C3 __exp_data.exp2_poly[2]
21 #define C4 __exp_data.exp2_poly[3]
22 #define C5 __exp_data.exp2_poly[4]
23 #define C6 __exp_data.exp2_poly[5]
24
25 /* Handle cases that may overflow or underflow when computing the result that
26 is scale*(1+TMP) without intermediate rounding. The bit representation of
27 scale is in SBITS, however it has a computed exponent that may have
28 overflown into the sign bit so that needs to be adjusted before using it as
29 a double. (int32_t)KI is the k used in the argument reduction and exponent
30 adjustment of scale, positive k here means the result may overflow and
31 negative k means the result may underflow. */
32 static inline double
specialcase(double_t tmp,uint64_t sbits,uint64_t ki)33 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
34 {
35 double_t scale, y;
36
37 if ((ki & 0x80000000) == 0)
38 {
39 /* k > 0, the exponent of scale might have overflowed by 1. */
40 sbits -= 1ull << 52;
41 scale = asdouble (sbits);
42 y = 2 * (scale + scale * tmp);
43 return check_oflow (eval_as_double (y));
44 }
45 /* k < 0, need special care in the subnormal range. */
46 sbits += 1022ull << 52;
47 scale = asdouble (sbits);
48 y = scale + scale * tmp;
49 if (y < 1.0)
50 {
51 /* Round y to the right precision before scaling it into the subnormal
52 range to avoid double rounding that can cause 0.5+E/2 ulp error where
53 E is the worst-case ulp error outside the subnormal range. So this
54 is only useful if the goal is better than 1 ulp worst-case error. */
55 double_t hi, lo;
56 lo = scale - y + scale * tmp;
57 hi = 1.0 + y;
58 lo = 1.0 - hi + y + lo;
59 y = eval_as_double (hi + lo) - 1.0;
60 /* Avoid -0.0 with downward rounding. */
61 if (WANT_ROUNDING && y == 0.0)
62 y = 0.0;
63 /* The underflow exception needs to be signaled explicitly. */
64 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
65 }
66 y = 0x1p-1022 * y;
67 return check_uflow (eval_as_double (y));
68 }
69
70 /* Top 12 bits of a double (sign and exponent bits). */
71 static inline uint32_t
top12(double x)72 top12 (double x)
73 {
74 return asuint64 (x) >> 52;
75 }
76
77 double
exp2(double x)78 exp2 (double x)
79 {
80 uint32_t abstop;
81 uint64_t ki, idx, top, sbits;
82 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
83 double_t kd, r, r2, scale, tail, tmp;
84
85 abstop = top12 (x) & 0x7ff;
86 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
87 {
88 if (abstop - top12 (0x1p-54) >= 0x80000000)
89 /* Avoid spurious underflow for tiny x. */
90 /* Note: 0 is common input. */
91 return WANT_ROUNDING ? 1.0 + x : 1.0;
92 if (abstop >= top12 (1024.0))
93 {
94 if (asuint64 (x) == asuint64 (-INFINITY))
95 return 0.0;
96 if (abstop >= top12 (INFINITY))
97 return 1.0 + x;
98 if (!(asuint64 (x) >> 63))
99 return __math_oflow (0);
100 else if (asuint64 (x) >= asuint64 (-1075.0))
101 return __math_uflow (0);
102 }
103 if (2 * asuint64 (x) > 2 * asuint64 (928.0))
104 /* Large x is special cased below. */
105 abstop = 0;
106 }
107
108 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
109 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
110 kd = eval_as_double (x + Shift);
111 ki = asuint64 (kd); /* k. */
112 kd -= Shift; /* k/N for int k. */
113 r = x - kd;
114 /* 2^(k/N) ~= scale * (1 + tail). */
115 idx = 2 * (ki % N);
116 top = ki << (52 - EXP_TABLE_BITS);
117 tail = asdouble (T[idx]);
118 /* This is only a valid scale when -1023*N < k < 1024*N. */
119 sbits = T[idx + 1] + top;
120 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
121 /* Evaluation is optimized assuming superscalar pipelined execution. */
122 r2 = r * r;
123 /* Without fma the worst case error is 0.5/N ulp larger. */
124 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
125 #if EXP2_POLY_ORDER == 4
126 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
127 #elif EXP2_POLY_ORDER == 5
128 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
129 #elif EXP2_POLY_ORDER == 6
130 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
131 #endif
132 if (unlikely (abstop == 0))
133 return specialcase (tmp, sbits, ki);
134 scale = asdouble (sbits);
135 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
136 is no spurious underflow here even without fma. */
137 return eval_as_double (scale + scale * tmp);
138 }
139 #if USE_GLIBC_ABI
strong_alias(exp2,__exp2_finite)140 strong_alias (exp2, __exp2_finite)
141 hidden_alias (exp2, __ieee754_exp2)
142 # if LDBL_MANT_DIG == 53
143 long double exp2l (long double x) { return exp2 (x); }
144 # endif
145 #endif
146
147 TEST_SIG (S, D, 1, exp2, -9.9, 9.9)
148 TEST_ULP (exp2, 0.01)
149 TEST_ULP_NONNEAREST (exp2, 0.5)
150 TEST_INTERVAL (exp2, 0, 0xffff000000000000, 10000)
151 TEST_SYM_INTERVAL (exp2, 0x1p-6, 0x1p6, 40000)
152 TEST_SYM_INTERVAL (exp2, 633.3, 733.3, 10000)
153