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. */
29 double
expm1(double x)30 expm1 (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
79 PL_SIG (S, D, 1, expm1, -9.9, 9.9)
80 PL_TEST_ULP (expm1, 1.68)
81 PL_TEST_SYM_INTERVAL (expm1, 0, 0x1p-51, 1000)
82 PL_TEST_INTERVAL (expm1, 0x1p-51, 0x1.63108c75a1937p+9, 100000)
83 PL_TEST_INTERVAL (expm1, -0x1p-51, -0x1.740bf7c0d927dp+9, 100000)
84 PL_TEST_INTERVAL (expm1, 0x1.63108c75a1937p+9, inf, 100)
85 PL_TEST_INTERVAL (expm1, -0x1.740bf7c0d927dp+9, -inf, 100)
86