1/*
2 * Double-precision e^x - 1 function.
3 *
4 * Copyright (c) 2022-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8#include "poly_scalar_f64.h"
9#include "math_config.h"
10#include "pl_sig.h"
11#include "pl_test.h"
12
13#define InvLn2 0x1.71547652b82fep0
14#define Ln2hi 0x1.62e42fefa39efp-1
15#define Ln2lo 0x1.abc9e3b39803fp-56
16#define Shift 0x1.8p52
17/* 0x1p-51, below which expm1(x) is within 2 ULP of x.  */
18#define TinyBound 0x3cc0000000000000
19/* Above which expm1(x) overflows.  */
20#define BigBound 0x1.63108c75a1937p+9
21/* Below which expm1(x) rounds to 1.  */
22#define NegBound -0x1.740bf7c0d927dp+9
23#define AbsMask 0x7fffffffffffffff
24
25/* Approximation for exp(x) - 1 using polynomial on a reduced interval.
26   The maximum error observed error is 2.17 ULP:
27   expm1(0x1.63f90a866748dp-2) got 0x1.a9af56603878ap-2
28			      want 0x1.a9af566038788p-2.  */
29double
30expm1 (double x)
31{
32  uint64_t ix = asuint64 (x);
33  uint64_t ax = ix & AbsMask;
34
35  /* Tiny, +Infinity.  */
36  if (ax <= TinyBound || ix == 0x7ff0000000000000)
37    return x;
38
39  /* +/-NaN.  */
40  if (ax > 0x7ff0000000000000)
41    return __math_invalid (x);
42
43  /* Result is too large to be represented as a double.  */
44  if (x >= 0x1.63108c75a1937p+9)
45    return __math_oflow (0);
46
47  /* Result rounds to -1 in double precision.  */
48  if (x <= NegBound)
49    return -1;
50
51  /* Reduce argument to smaller range:
52     Let i = round(x / ln2)
53     and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
54     exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
55     where 2^i is exact because i is an integer.  */
56  double j = fma (InvLn2, x, Shift) - Shift;
57  int64_t i = j;
58  double f = fma (j, -Ln2hi, x);
59  f = fma (j, -Ln2lo, f);
60
61  /* Approximate expm1(f) using polynomial.
62     Taylor expansion for expm1(x) has the form:
63	 x + ax^2 + bx^3 + cx^4 ....
64     So we calculate the polynomial P(f) = a + bf + cf^2 + ...
65     and assemble the approximation expm1(f) ~= f + f^2 * P(f).  */
66  double f2 = f * f;
67  double f4 = f2 * f2;
68  double p = fma (f2, estrin_10_f64 (f, f2, f4, f4 * f4, __expm1_poly), f);
69
70  /* Assemble the result, using a slight rearrangement to achieve acceptable
71     accuracy.
72     expm1(x) ~= 2^i * (p + 1) - 1
73     Let t = 2^(i - 1).  */
74  double t = ldexp (0.5, i);
75  /* expm1(x) ~= 2 * (p * t + (t - 1/2)).  */
76  return 2 * fma (p, t, t - 0.5);
77}
78
79PL_SIG (S, D, 1, expm1, -9.9, 9.9)
80PL_TEST_ULP (expm1, 1.68)
81PL_TEST_SYM_INTERVAL (expm1, 0, 0x1p-51, 1000)
82PL_TEST_INTERVAL (expm1, 0x1p-51, 0x1.63108c75a1937p+9, 100000)
83PL_TEST_INTERVAL (expm1, -0x1p-51, -0x1.740bf7c0d927dp+9, 100000)
84PL_TEST_INTERVAL (expm1, 0x1.63108c75a1937p+9, inf, 100)
85PL_TEST_INTERVAL (expm1, -0x1.740bf7c0d927dp+9, -inf, 100)
86