1/* Double-precision SVE expm1
2
3 Copyright (C) 2023-2024 Free Software Foundation, Inc.
4 This file is part of the GNU C Library.
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20#include "sv_math.h"
21#include "poly_sve_f64.h"
22
23#define SpecialBound 0x1.62b7d369a5aa9p+9
24#define ExponentBias 0x3ff0000000000000
25
26static const struct data
27{
28 double poly[11];
29 double shift, inv_ln2, special_bound;
30 /* To be loaded in one quad-word. */
31 double ln2_hi, ln2_lo;
32} data = {
33 /* Generated using fpminimax. */
34 .poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5,
35 0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13,
36 0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22,
37 0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, },
38
39 .special_bound = SpecialBound,
40 .inv_ln2 = 0x1.71547652b82fep0,
41 .ln2_hi = 0x1.62e42fefa39efp-1,
42 .ln2_lo = 0x1.abc9e3b39803fp-56,
43 .shift = 0x1.8p52,
44};
45
46static svfloat64_t NOINLINE
47special_case (svfloat64_t x, svfloat64_t y, svbool_t pg)
48{
49 return sv_call_f64 (f: expm1, x, y, cmp: pg);
50}
51
52/* Double-precision vector exp(x) - 1 function.
53 The maximum error observed error is 2.18 ULP:
54 _ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
55 want 0x1.a8b9ea8d66e2p-2. */
56svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg)
57{
58 const struct data *d = ptr_barrier (&data);
59
60 /* Large, Nan/Inf. */
61 svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound));
62
63 /* Reduce argument to smaller range:
64 Let i = round(x / ln2)
65 and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
66 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
67 where 2^i is exact because i is an integer. */
68 svfloat64_t shift = sv_f64 (x: d->shift);
69 svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift);
70 svint64_t i = svcvt_s64_x (pg, n);
71 svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
72 svfloat64_t f = svmls_lane (x, n, ln2, 0);
73 f = svmls_lane (f, n, ln2, 1);
74
75 /* Approximate expm1(f) using polynomial.
76 Taylor expansion for expm1(x) has the form:
77 x + ax^2 + bx^3 + cx^4 ....
78 So we calculate the polynomial P(f) = a + bf + cf^2 + ...
79 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
80 svfloat64_t f2 = svmul_x (pg, f, f);
81 svfloat64_t f4 = svmul_x (pg, f2, f2);
82 svfloat64_t f8 = svmul_x (pg, f4, f4);
83 svfloat64_t p
84 = svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, x: f, x2: f2, x4: f4, x8: f8, poly: d->poly));
85
86 /* Assemble the result.
87 expm1(x) ~= 2^i * (p + 1) - 1
88 Let t = 2^i. */
89 svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias);
90 svfloat64_t t = svreinterpret_f64 (u);
91
92 /* expm1(x) ~= p * t + (t - 1). */
93 svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t);
94
95 if (__glibc_unlikely (svptest_any (pg, special)))
96 return special_case (x, y, pg: special);
97
98 return y;
99}
100

source code of glibc/sysdeps/aarch64/fpu/expm1_sve.c