1/*
2 * Double-precision 2^x function.
3 *
4 * Copyright (c) 2018-2019, 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
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#define C6 __exp_data.exp2_poly[5]
22
23/* Handle cases that may overflow or underflow when computing the result that
24   is scale*(1+TMP) without intermediate rounding.  The bit representation of
25   scale is in SBITS, however it has a computed exponent that may have
26   overflown into the sign bit so that needs to be adjusted before using it as
27   a double.  (int32_t)KI is the k used in the argument reduction and exponent
28   adjustment of scale, positive k here means the result may overflow and
29   negative k means the result may underflow.  */
30static inline double
31specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
32{
33  double_t scale, y;
34
35  if ((ki & 0x80000000) == 0)
36    {
37      /* k > 0, the exponent of scale might have overflowed by 1.  */
38      sbits -= 1ull << 52;
39      scale = asdouble (sbits);
40      y = 2 * (scale + scale * tmp);
41      return check_oflow (eval_as_double (y));
42    }
43  /* k < 0, need special care in the subnormal range.  */
44  sbits += 1022ull << 52;
45  scale = asdouble (sbits);
46  y = scale + scale * tmp;
47  if (y < 1.0)
48    {
49      /* Round y to the right precision before scaling it into the subnormal
50	 range to avoid double rounding that can cause 0.5+E/2 ulp error where
51	 E is the worst-case ulp error outside the subnormal range.  So this
52	 is only useful if the goal is better than 1 ulp worst-case error.  */
53      double_t hi, lo;
54      lo = scale - y + scale * tmp;
55      hi = 1.0 + y;
56      lo = 1.0 - hi + y + lo;
57      y = eval_as_double (hi + lo) - 1.0;
58      /* Avoid -0.0 with downward rounding.  */
59      if (WANT_ROUNDING && y == 0.0)
60	y = 0.0;
61      /* The underflow exception needs to be signaled explicitly.  */
62      force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
63    }
64  y = 0x1p-1022 * y;
65  return check_uflow (eval_as_double (y));
66}
67
68/* Top 12 bits of a double (sign and exponent bits).  */
69static inline uint32_t
70top12 (double x)
71{
72  return asuint64 (x) >> 52;
73}
74
75double
76exp2 (double x)
77{
78  uint32_t abstop;
79  uint64_t ki, idx, top, sbits;
80  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
81  double_t kd, r, r2, scale, tail, tmp;
82
83  abstop = top12 (x) & 0x7ff;
84  if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
85    {
86      if (abstop - top12 (0x1p-54) >= 0x80000000)
87	/* Avoid spurious underflow for tiny x.  */
88	/* Note: 0 is common input.  */
89	return WANT_ROUNDING ? 1.0 + x : 1.0;
90      if (abstop >= top12 (1024.0))
91	{
92	  if (asuint64 (x) == asuint64 (-INFINITY))
93	    return 0.0;
94	  if (abstop >= top12 (INFINITY))
95	    return 1.0 + x;
96	  if (!(asuint64 (x) >> 63))
97	    return __math_oflow (0);
98	  else if (asuint64 (x) >= asuint64 (-1075.0))
99	    return __math_uflow (0);
100	}
101      if (2 * asuint64 (x) > 2 * asuint64 (928.0))
102	/* Large x is special cased below.  */
103	abstop = 0;
104    }
105
106  /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */
107  /* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */
108  kd = eval_as_double (x + Shift);
109  ki = asuint64 (kd); /* k.  */
110  kd -= Shift; /* k/N for int k.  */
111  r = x - kd;
112  /* 2^(k/N) ~= scale * (1 + tail).  */
113  idx = 2 * (ki % N);
114  top = ki << (52 - EXP_TABLE_BITS);
115  tail = asdouble (T[idx]);
116  /* This is only a valid scale when -1023*N < k < 1024*N.  */
117  sbits = T[idx + 1] + top;
118  /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */
119  /* Evaluation is optimized assuming superscalar pipelined execution.  */
120  r2 = r * r;
121  /* Without fma the worst case error is 0.5/N ulp larger.  */
122  /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */
123#if EXP2_POLY_ORDER == 4
124  tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
125#elif EXP2_POLY_ORDER == 5
126  tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
127#elif EXP2_POLY_ORDER == 6
128  tmp = tail + r * C1 + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
129#endif
130  if (unlikely (abstop == 0))
131    return specialcase (tmp, sbits, ki);
132  scale = asdouble (sbits);
133  /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
134     is no spurious underflow here even without fma.  */
135  return eval_as_double (scale + scale * tmp);
136}
137#if USE_GLIBC_ABI
138strong_alias (exp2, __exp2_finite)
139hidden_alias (exp2, __ieee754_exp2)
140# if LDBL_MANT_DIG == 53
141long double exp2l (long double x) { return exp2 (x); }
142# endif
143#endif
144