1 /*
2 * Double-precision 2^x function.
3 *
4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
5 * See https://llvm.org/LICENSE.txt for license information.
6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
7 */
8
9 #include <float.h>
10 #include <math.h>
11 #include <stdint.h>
12 #include "math_config.h"
13
14 #define N (1 << EXP_TABLE_BITS)
15 #define Shift __exp_data.exp2_shift
16 #define T __exp_data.tab
17 #define C1 __exp_data.exp2_poly[0]
18 #define C2 __exp_data.exp2_poly[1]
19 #define C3 __exp_data.exp2_poly[2]
20 #define C4 __exp_data.exp2_poly[3]
21 #define C5 __exp_data.exp2_poly[4]
22 #define C6 __exp_data.exp2_poly[5]
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 specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
33 {
34 double_t scale, y;
35
36 if ((ki & 0x80000000) == 0)
37 {
38 /* k > 0, the exponent of scale might have overflowed by 1. */
39 sbits -= 1ull << 52;
40 scale = asdouble (sbits);
41 y = 2 * (scale + scale * tmp);
42 return check_oflow (eval_as_double (y));
43 }
44 /* k < 0, need special care in the subnormal range. */
45 sbits += 1022ull << 52;
46 scale = asdouble (sbits);
47 y = scale + scale * tmp;
48 if (y < 1.0)
49 {
50 /* Round y to the right precision before scaling it into the subnormal
51 range to avoid double rounding that can cause 0.5+E/2 ulp error where
52 E is the worst-case ulp error outside the subnormal range. So this
53 is only useful if the goal is better than 1 ulp worst-case error. */
54 double_t hi, lo;
55 lo = scale - y + scale * tmp;
56 hi = 1.0 + y;
57 lo = 1.0 - hi + y + lo;
58 y = eval_as_double (hi + lo) - 1.0;
59 /* Avoid -0.0 with downward rounding. */
60 if (WANT_ROUNDING && y == 0.0)
61 y = 0.0;
62 /* The underflow exception needs to be signaled explicitly. */
63 force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
64 }
65 y = 0x1p-1022 * y;
66 return check_uflow (eval_as_double (y));
67 }
68
69 /* Top 12 bits of a double (sign and exponent bits). */
70 static inline uint32_t
top12(double x)71 top12 (double x)
72 {
73 return asuint64 (x) >> 52;
74 }
75
76 double
exp2(double x)77 exp2 (double x)
78 {
79 uint32_t abstop;
80 uint64_t ki, idx, top, sbits;
81 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
82 double_t kd, r, r2, scale, tail, tmp;
83
84 abstop = top12 (x) & 0x7ff;
85 if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
86 {
87 if (abstop - top12 (0x1p-54) >= 0x80000000)
88 /* Avoid spurious underflow for tiny x. */
89 /* Note: 0 is common input. */
90 return WANT_ROUNDING ? 1.0 + x : 1.0;
91 if (abstop >= top12 (1024.0))
92 {
93 if (asuint64 (x) == asuint64 (-INFINITY))
94 return 0.0;
95 if (abstop >= top12 (INFINITY))
96 return 1.0 + x;
97 if (!(asuint64 (x) >> 63))
98 return __math_oflow (0);
99 else if (asuint64 (x) >= asuint64 (-1075.0))
100 return __math_uflow (0);
101 }
102 if (2 * asuint64 (x) > 2 * asuint64 (928.0))
103 /* Large x is special cased below. */
104 abstop = 0;
105 }
106
107 /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
108 /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
109 kd = eval_as_double (x + Shift);
110 ki = asuint64 (kd); /* k. */
111 kd -= Shift; /* k/N for int k. */
112 r = x - kd;
113 /* 2^(k/N) ~= scale * (1 + tail). */
114 idx = 2 * (ki % N);
115 top = ki << (52 - EXP_TABLE_BITS);
116 tail = asdouble (T[idx]);
117 /* This is only a valid scale when -1023*N < k < 1024*N. */
118 sbits = T[idx + 1] + top;
119 /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
120 /* Evaluation is optimized assuming superscalar pipelined execution. */
121 r2 = r * r;
122 /* Without fma the worst case error is 0.5/N ulp larger. */
123 /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
124 #if EXP2_POLY_ORDER == 4
125 tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
126 #elif EXP2_POLY_ORDER == 5
127 tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
128 #elif EXP2_POLY_ORDER == 6
129 tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
130 #endif
131 if (unlikely (abstop == 0))
132 return specialcase (tmp, sbits, ki);
133 scale = asdouble (sbits);
134 /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
135 is no spurious underflow here even without fma. */
136 return eval_as_double (scale + scale * tmp);
137 }
138 #if USE_GLIBC_ABI
strong_alias(exp2,__exp2_finite)139 strong_alias (exp2, __exp2_finite)
140 hidden_alias (exp2, __ieee754_exp2)
141 # if LDBL_MANT_DIG == 53
142 long double exp2l (long double x) { return exp2 (x); }
143 # endif
144 #endif
145