1/*
2 * Double-precision e^x function.
3 *
4 * Copyright (c) 2018-2023, 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 InvLn2N __exp_data.invln2N
15#define NegLn2hiN __exp_data.negln2hiN
16#define NegLn2loN __exp_data.negln2loN
17#define Shift __exp_data.shift
18#define T __exp_data.tab
19#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
20#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
21#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
22#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
23#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
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.  */
32static inline double
33specialcase (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 <= 460.  */
40      sbits -= 1009ull << 52;
41      scale = asdouble (sbits);
42      y = 0x1p1009 * (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).  */
71static inline uint32_t
72top12 (double x)
73{
74  return asuint64 (x) >> 52;
75}
76
77/* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
78   If hastail is 0 then xtail is assumed to be 0 too.  */
79static inline double
80exp_inline (double x, double xtail, int hastail)
81{
82  uint32_t abstop;
83  uint64_t ki, idx, top, sbits;
84  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
85  double_t kd, z, r, r2, scale, tail, tmp;
86
87  abstop = top12 (x) & 0x7ff;
88  if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
89    {
90      if (abstop - top12 (0x1p-54) >= 0x80000000)
91	/* Avoid spurious underflow for tiny x.  */
92	/* Note: 0 is common input.  */
93	return WANT_ROUNDING ? 1.0 + x : 1.0;
94      if (abstop >= top12 (1024.0))
95	{
96	  if (asuint64 (x) == asuint64 (-INFINITY))
97	    return 0.0;
98	  if (abstop >= top12 (INFINITY))
99	    return 1.0 + x;
100	  if (asuint64 (x) >> 63)
101	    return __math_uflow (0);
102	  else
103	    return __math_oflow (0);
104	}
105      /* Large x is special cased below.  */
106      abstop = 0;
107    }
108
109  /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
110  /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
111  z = InvLn2N * x;
112#if TOINT_INTRINSICS
113  kd = roundtoint (z);
114  ki = converttoint (z);
115#elif EXP_USE_TOINT_NARROW
116  /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
117  kd = eval_as_double (z + Shift);
118  ki = asuint64 (kd) >> 16;
119  kd = (double_t) (int32_t) ki;
120#else
121  /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
122  kd = eval_as_double (z + Shift);
123  ki = asuint64 (kd);
124  kd -= Shift;
125#endif
126  r = x + kd * NegLn2hiN + kd * NegLn2loN;
127  /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
128  if (hastail)
129    r += xtail;
130  /* 2^(k/N) ~= scale * (1 + tail).  */
131  idx = 2 * (ki % N);
132  top = ki << (52 - EXP_TABLE_BITS);
133  tail = asdouble (T[idx]);
134  /* This is only a valid scale when -1023*N < k < 1024*N.  */
135  sbits = T[idx + 1] + top;
136  /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
137  /* Evaluation is optimized assuming superscalar pipelined execution.  */
138  r2 = r * r;
139  /* Without fma the worst case error is 0.25/N ulp larger.  */
140  /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
141#if EXP_POLY_ORDER == 4
142  tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
143#elif EXP_POLY_ORDER == 5
144  tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
145#elif EXP_POLY_ORDER == 6
146  tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
147#endif
148  if (unlikely (abstop == 0))
149    return specialcase (tmp, sbits, ki);
150  scale = asdouble (sbits);
151  /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
152     is no spurious underflow here even without fma.  */
153  return eval_as_double (scale + scale * tmp);
154}
155
156/* May be useful for implementing pow where more than double
157   precision input is needed.  */
158double
159__exp_dd (double x, double xtail)
160{
161  return exp_inline (x, xtail, 1);
162}
163
164