1 /*
2 * Double-precision vector exp(x) - 1 function.
3 *
4 * Copyright (c) 2022-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "v_math.h"
9 #include "poly_advsimd_f64.h"
10 #include "pl_sig.h"
11 #include "pl_test.h"
12
13 static const struct data
14 {
15 float64x2_t poly[11];
16 float64x2_t invln2;
17 double ln2[2];
18 float64x2_t shift;
19 int64x2_t exponent_bias;
20 #if WANT_SIMD_EXCEPT
21 uint64x2_t thresh, tiny_bound;
22 #else
23 float64x2_t oflow_bound;
24 #endif
25 } data = {
26 /* Generated using fpminimax, with degree=12 in [log(2)/2, log(2)/2]. */
27 .poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
28 V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
29 V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
30 V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
31 V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29) },
32 .invln2 = V2 (0x1.71547652b82fep0),
33 .ln2 = { 0x1.62e42fefa39efp-1, 0x1.abc9e3b39803fp-56 },
34 .shift = V2 (0x1.8p52),
35 .exponent_bias = V2 (0x3ff0000000000000),
36 #if WANT_SIMD_EXCEPT
37 /* asuint64(oflow_bound) - asuint64(0x1p-51), shifted left by 1 for abs
38 compare. */
39 .thresh = V2 (0x78c56fa6d34b552),
40 /* asuint64(0x1p-51) << 1. */
41 .tiny_bound = V2 (0x3cc0000000000000 << 1),
42 #else
43 /* Value above which expm1(x) should overflow. Absolute value of the
44 underflow bound is greater than this, so it catches both cases - there is
45 a small window where fallbacks are triggered unnecessarily. */
46 .oflow_bound = V2 (0x1.62b7d369a5aa9p+9),
47 #endif
48 };
49
50 static float64x2_t VPCS_ATTR NOINLINE
special_case(float64x2_t x,float64x2_t y,uint64x2_t special)51 special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
52 {
53 return v_call_f64 (expm1, x, y, special);
54 }
55
56 /* Double-precision vector exp(x) - 1 function.
57 The maximum error observed error is 2.18 ULP:
58 _ZGVnN2v_expm1 (0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
59 want 0x1.a8b9ea8d66e2p-2. */
V_NAME_D1(expm1)60 float64x2_t VPCS_ATTR V_NAME_D1 (expm1) (float64x2_t x)
61 {
62 const struct data *d = ptr_barrier (&data);
63
64 uint64x2_t ix = vreinterpretq_u64_f64 (x);
65
66 #if WANT_SIMD_EXCEPT
67 /* If fp exceptions are to be triggered correctly, fall back to scalar for
68 |x| < 2^-51, |x| > oflow_bound, Inf & NaN. Add ix to itself for
69 shift-left by 1, and compare with thresh which was left-shifted offline -
70 this is effectively an absolute compare. */
71 uint64x2_t special
72 = vcgeq_u64 (vsubq_u64 (vaddq_u64 (ix, ix), d->tiny_bound), d->thresh);
73 if (unlikely (v_any_u64 (special)))
74 x = v_zerofy_f64 (x, special);
75 #else
76 /* Large input, NaNs and Infs. */
77 uint64x2_t special = vcageq_f64 (x, d->oflow_bound);
78 #endif
79
80 /* Reduce argument to smaller range:
81 Let i = round(x / ln2)
82 and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
83 exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
84 where 2^i is exact because i is an integer. */
85 float64x2_t n = vsubq_f64 (vfmaq_f64 (d->shift, d->invln2, x), d->shift);
86 int64x2_t i = vcvtq_s64_f64 (n);
87 float64x2_t ln2 = vld1q_f64 (&d->ln2[0]);
88 float64x2_t f = vfmsq_laneq_f64 (x, n, ln2, 0);
89 f = vfmsq_laneq_f64 (f, n, ln2, 1);
90
91 /* Approximate expm1(f) using polynomial.
92 Taylor expansion for expm1(x) has the form:
93 x + ax^2 + bx^3 + cx^4 ....
94 So we calculate the polynomial P(f) = a + bf + cf^2 + ...
95 and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
96 float64x2_t f2 = vmulq_f64 (f, f);
97 float64x2_t f4 = vmulq_f64 (f2, f2);
98 float64x2_t f8 = vmulq_f64 (f4, f4);
99 float64x2_t p = vfmaq_f64 (f, f2, v_estrin_10_f64 (f, f2, f4, f8, d->poly));
100
101 /* Assemble the result.
102 expm1(x) ~= 2^i * (p + 1) - 1
103 Let t = 2^i. */
104 int64x2_t u = vaddq_s64 (vshlq_n_s64 (i, 52), d->exponent_bias);
105 float64x2_t t = vreinterpretq_f64_s64 (u);
106
107 if (unlikely (v_any_u64 (special)))
108 return special_case (vreinterpretq_f64_u64 (ix),
109 vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t),
110 special);
111
112 /* expm1(x) ~= p * t + (t - 1). */
113 return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
114 }
115
116 PL_SIG (V, D, 1, expm1, -9.9, 9.9)
117 PL_TEST_ULP (V_NAME_D1 (expm1), 1.68)
118 PL_TEST_EXPECT_FENV (V_NAME_D1 (expm1), WANT_SIMD_EXCEPT)
119 PL_TEST_SYM_INTERVAL (V_NAME_D1 (expm1), 0, 0x1p-51, 1000)
120 PL_TEST_SYM_INTERVAL (V_NAME_D1 (expm1), 0x1p-51, 0x1.62b7d369a5aa9p+9, 100000)
121 PL_TEST_SYM_INTERVAL (V_NAME_D1 (expm1), 0x1.62b7d369a5aa9p+9, inf, 100)
122