1/* Double-precision AdvSIMD 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 "v_math.h"
21#include "poly_advsimd_f64.h"
22
23static const struct data
24{
25 float64x2_t poly[11];
26 float64x2_t invln2;
27 double ln2[2];
28 float64x2_t shift;
29 int64x2_t exponent_bias;
30#if WANT_SIMD_EXCEPT
31 uint64x2_t thresh, tiny_bound;
32#else
33 float64x2_t oflow_bound;
34#endif
35} data = {
36 /* Generated using fpminimax, with degree=12 in [log(2)/2, log(2)/2]. */
37 .poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
38 V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
39 V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
40 V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
41 V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29) },
42 .invln2 = V2 (0x1.71547652b82fep0),
43 .ln2 = { 0x1.62e42fefa39efp-1, 0x1.abc9e3b39803fp-56 },
44 .shift = V2 (0x1.8p52),
45 .exponent_bias = V2 (0x3ff0000000000000),
46#if WANT_SIMD_EXCEPT
47 /* asuint64(oflow_bound) - asuint64(0x1p-51), shifted left by 1 for abs
48 compare. */
49 .thresh = V2 (0x78c56fa6d34b552),
50 /* asuint64(0x1p-51) << 1. */
51 .tiny_bound = V2 (0x3cc0000000000000 << 1),
52#else
53 /* Value above which expm1(x) should overflow. Absolute value of the
54 underflow bound is greater than this, so it catches both cases - there is
55 a small window where fallbacks are triggered unnecessarily. */
56 .oflow_bound = V2 (0x1.62b7d369a5aa9p+9),
57#endif
58};
59
60static float64x2_t VPCS_ATTR NOINLINE
61special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
62{
63 return v_call_f64 (expm1, x, y, special);
64}
65
66/* Double-precision vector exp(x) - 1 function.
67 The maximum error observed error is 2.18 ULP:
68 _ZGVnN2v_expm1 (0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
69 want 0x1.a8b9ea8d66e2p-2. */
70float64x2_t VPCS_ATTR V_NAME_D1 (expm1) (float64x2_t x)
71{
72 const struct data *d = ptr_barrier (&data);
73
74 uint64x2_t ix = vreinterpretq_u64_f64 (x);
75
76#if WANT_SIMD_EXCEPT
77 /* If fp exceptions are to be triggered correctly, fall back to scalar for
78 |x| < 2^-51, |x| > oflow_bound, Inf & NaN. Add ix to itself for
79 shift-left by 1, and compare with thresh which was left-shifted offline -
80 this is effectively an absolute compare. */
81 uint64x2_t special
82 = vcgeq_u64 (vsubq_u64 (vaddq_u64 (ix, ix), d->tiny_bound), d->thresh);
83 if (__glibc_unlikely (v_any_u64 (special)))
84 x = v_zerofy_f64 (x, special);
85#else
86 /* Large input, NaNs and Infs. */
87 uint64x2_t special = vcageq_f64 (x, d->oflow_bound);
88#endif
89
90 /* Reduce argument to smaller range:
91 Let i = round(x / ln2)
92 and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
93 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
94 where 2^i is exact because i is an integer. */
95 float64x2_t n = vsubq_f64 (vfmaq_f64 (d->shift, d->invln2, x), d->shift);
96 int64x2_t i = vcvtq_s64_f64 (n);
97 float64x2_t ln2 = vld1q_f64 (&d->ln2[0]);
98 float64x2_t f = vfmsq_laneq_f64 (x, n, ln2, 0);
99 f = vfmsq_laneq_f64 (f, n, ln2, 1);
100
101 /* Approximate expm1(f) using polynomial.
102 Taylor expansion for expm1(x) has the form:
103 x + ax^2 + bx^3 + cx^4 ....
104 So we calculate the polynomial P(f) = a + bf + cf^2 + ...
105 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
106 float64x2_t f2 = vmulq_f64 (f, f);
107 float64x2_t f4 = vmulq_f64 (f2, f2);
108 float64x2_t f8 = vmulq_f64 (f4, f4);
109 float64x2_t p = vfmaq_f64 (f, f2, v_estrin_10_f64 (f, f2, f4, f8, d->poly));
110
111 /* Assemble the result.
112 expm1(x) ~= 2^i * (p + 1) - 1
113 Let t = 2^i. */
114 int64x2_t u = vaddq_s64 (vshlq_n_s64 (i, 52), d->exponent_bias);
115 float64x2_t t = vreinterpretq_f64_s64 (u);
116
117 if (__glibc_unlikely (v_any_u64 (special)))
118 return special_case (vreinterpretq_f64_u64 (ix),
119 vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t),
120 special);
121
122 /* expm1(x) ~= p * t + (t - 1). */
123 return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
124}
125

source code of glibc/sysdeps/aarch64/fpu/expm1_advsimd.c