1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
/* Double-precision vector (Advanced SIMD) erfc function
Copyright (C) 2024 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include "v_math.h"
#include "vecmath_config.h"
static const struct data
{
uint64x2_t offset, table_scale;
float64x2_t max, shift;
float64x2_t p20, p40, p41, p42;
float64x2_t p51, p52;
double qr5[2], qr6[2], qr7[2], qr8[2], qr9[2];
#if WANT_SIMD_EXCEPT
float64x2_t uflow_bound;
#endif
} data = {
/* Set an offset so the range of the index used for lookup is 3487, and it
can be clamped using a saturated add on an offset index.
Index offset is 0xffffffffffffffff - asuint64(shift) - 3487. */
.offset = V2 (0xbd3ffffffffff260),
.table_scale = V2 (0x37f0000000000000 << 1), /* asuint64 (2^-128) << 1. */
.max = V2 (0x1.b3ep+4), /* 3487/128. */
.shift = V2 (0x1p45),
.p20 = V2 (0x1.5555555555555p-2), /* 1/3, used to compute 2/3 and 1/6. */
.p40 = V2 (-0x1.999999999999ap-4), /* 1/10. */
.p41 = V2 (-0x1.999999999999ap-2), /* 2/5. */
.p42 = V2 (0x1.1111111111111p-3), /* 2/15. */
.p51 = V2 (-0x1.c71c71c71c71cp-3), /* 2/9. */
.p52 = V2 (0x1.6c16c16c16c17p-5), /* 2/45. */
/* Qi = (i+1) / i, Ri = -2 * i / ((i+1)*(i+2)), for i = 5, ..., 9. */
.qr5 = { 0x1.3333333333333p0, -0x1.e79e79e79e79ep-3 },
.qr6 = { 0x1.2aaaaaaaaaaabp0, -0x1.b6db6db6db6dbp-3 },
.qr7 = { 0x1.2492492492492p0, -0x1.8e38e38e38e39p-3 },
.qr8 = { 0x1.2p0, -0x1.6c16c16c16c17p-3 },
.qr9 = { 0x1.1c71c71c71c72p0, -0x1.4f2094f2094f2p-3 },
#if WANT_SIMD_EXCEPT
.uflow_bound = V2 (0x1.a8b12fc6e4892p+4),
#endif
};
#define TinyBound 0x4000000000000000 /* 0x1p-511 << 1. */
#define Off 0xfffffffffffff260 /* 0xffffffffffffffff - 3487. */
struct entry
{
float64x2_t erfc;
float64x2_t scale;
};
static inline struct entry
lookup (uint64x2_t i)
{
struct entry e;
float64x2_t e1
= vld1q_f64 (&__erfc_data.tab[vgetq_lane_u64 (i, 0) - Off].erfc);
float64x2_t e2
= vld1q_f64 (&__erfc_data.tab[vgetq_lane_u64 (i, 1) - Off].erfc);
e.erfc = vuzp1q_f64 (e1, e2);
e.scale = vuzp2q_f64 (e1, e2);
return e;
}
#if WANT_SIMD_EXCEPT
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t x, float64x2_t y, uint64x2_t cmp)
{
return v_call_f64 (erfc, x, y, cmp);
}
#endif
/* Optimized double-precision vector erfc(x).
Approximation based on series expansion near x rounded to
nearest multiple of 1/128.
Let d = x - r, and scale = 2 / sqrt(pi) * exp(-r^2). For x near r,
erfc(x) ~ erfc(r) - scale * d * poly(r, d), with
poly(r, d) = 1 - r d + (2/3 r^2 - 1/3) d^2 - r (1/3 r^2 - 1/2) d^3
+ (2/15 r^4 - 2/5 r^2 + 1/10) d^4
- r * (2/45 r^4 - 2/9 r^2 + 1/6) d^5
+ p6(r) d^6 + ... + p10(r) d^10
Polynomials p6(r) to p10(r) are computed using recurrence relation
2(i+1)p_i + 2r(i+2)p_{i+1} + (i+2)(i+3)p_{i+2} = 0,
with p0 = 1, and p1(r) = -r.
Values of erfc(r) and scale are read from lookup tables. Stored values
are scaled to avoid hitting the subnormal range.
Note that for x < 0, erfc(x) = 2.0 - erfc(-x).
Maximum measured error: 1.71 ULP
V_NAME_D1 (erfc)(0x1.46cfe976733p+4) got 0x1.e15fcbea3e7afp-608
want 0x1.e15fcbea3e7adp-608. */
VPCS_ATTR
float64x2_t V_NAME_D1 (erfc) (float64x2_t x)
{
const struct data *dat = ptr_barrier (&data);
#if WANT_SIMD_EXCEPT
/* |x| < 2^-511. Avoid fabs by left-shifting by 1. */
uint64x2_t ix = vreinterpretq_u64_f64 (x);
uint64x2_t cmp = vcltq_u64 (vaddq_u64 (ix, ix), v_u64 (TinyBound));
/* x >= ~26.54 (into subnormal case and uflow case). Comparison is done in
integer domain to avoid raising exceptions in presence of nans. */
uint64x2_t uflow = vcgeq_s64 (vreinterpretq_s64_f64 (x),
vreinterpretq_s64_f64 (dat->uflow_bound));
cmp = vorrq_u64 (cmp, uflow);
float64x2_t xm = x;
/* If any lanes are special, mask them with 0 and retain a copy of x to allow
special case handler to fix special lanes later. This is only necessary if
fenv exceptions are to be triggered correctly. */
if (__glibc_unlikely (v_any_u64 (cmp)))
x = v_zerofy_f64 (x, cmp);
#endif
float64x2_t a = vabsq_f64 (x);
a = vminq_f64 (a, dat->max);
/* Lookup erfc(r) and scale(r) in tables, e.g. set erfc(r) to 0 and scale to
2/sqrt(pi), when x reduced to r = 0. */
float64x2_t shift = dat->shift;
float64x2_t z = vaddq_f64 (a, shift);
/* Clamp index to a range of 3487. A naive approach would use a subtract and
min. Instead we offset the table address and the index, then use a
saturating add. */
uint64x2_t i = vqaddq_u64 (vreinterpretq_u64_f64 (z), dat->offset);
struct entry e = lookup (i);
/* erfc(x) ~ erfc(r) - scale * d * poly(r, d). */
float64x2_t r = vsubq_f64 (z, shift);
float64x2_t d = vsubq_f64 (a, r);
float64x2_t d2 = vmulq_f64 (d, d);
float64x2_t r2 = vmulq_f64 (r, r);
float64x2_t p1 = r;
float64x2_t p2 = vfmsq_f64 (dat->p20, r2, vaddq_f64 (dat->p20, dat->p20));
float64x2_t p3 = vmulq_f64 (r, vfmaq_f64 (v_f64 (-0.5), r2, dat->p20));
float64x2_t p4 = vfmaq_f64 (dat->p41, r2, dat->p42);
p4 = vfmsq_f64 (dat->p40, r2, p4);
float64x2_t p5 = vfmaq_f64 (dat->p51, r2, dat->p52);
p5 = vmulq_f64 (r, vfmaq_f64 (vmulq_f64 (v_f64 (0.5), dat->p20), r2, p5));
/* Compute p_i using recurrence relation:
p_{i+2} = (p_i + r * Q_{i+1} * p_{i+1}) * R_{i+1}. */
float64x2_t qr5 = vld1q_f64 (dat->qr5), qr6 = vld1q_f64 (dat->qr6),
qr7 = vld1q_f64 (dat->qr7), qr8 = vld1q_f64 (dat->qr8),
qr9 = vld1q_f64 (dat->qr9);
float64x2_t p6 = vfmaq_f64 (p4, p5, vmulq_laneq_f64 (r, qr5, 0));
p6 = vmulq_laneq_f64 (p6, qr5, 1);
float64x2_t p7 = vfmaq_f64 (p5, p6, vmulq_laneq_f64 (r, qr6, 0));
p7 = vmulq_laneq_f64 (p7, qr6, 1);
float64x2_t p8 = vfmaq_f64 (p6, p7, vmulq_laneq_f64 (r, qr7, 0));
p8 = vmulq_laneq_f64 (p8, qr7, 1);
float64x2_t p9 = vfmaq_f64 (p7, p8, vmulq_laneq_f64 (r, qr8, 0));
p9 = vmulq_laneq_f64 (p9, qr8, 1);
float64x2_t p10 = vfmaq_f64 (p8, p9, vmulq_laneq_f64 (r, qr9, 0));
p10 = vmulq_laneq_f64 (p10, qr9, 1);
/* Compute polynomial in d using pairwise Horner scheme. */
float64x2_t p90 = vfmaq_f64 (p9, d, p10);
float64x2_t p78 = vfmaq_f64 (p7, d, p8);
float64x2_t p56 = vfmaq_f64 (p5, d, p6);
float64x2_t p34 = vfmaq_f64 (p3, d, p4);
float64x2_t p12 = vfmaq_f64 (p1, d, p2);
float64x2_t y = vfmaq_f64 (p78, d2, p90);
y = vfmaq_f64 (p56, d2, y);
y = vfmaq_f64 (p34, d2, y);
y = vfmaq_f64 (p12, d2, y);
y = vfmsq_f64 (e.erfc, e.scale, vfmsq_f64 (d, d2, y));
/* Offset equals 2.0 if sign, else 0.0. */
uint64x2_t sign = vshrq_n_u64 (vreinterpretq_u64_f64 (x), 63);
float64x2_t off = vreinterpretq_f64_u64 (vshlq_n_u64 (sign, 62));
/* Copy sign and scale back in a single fma. Since the bit patterns do not
overlap, then logical or and addition are equivalent here. */
float64x2_t fac = vreinterpretq_f64_u64 (
vsraq_n_u64 (vshlq_n_u64 (sign, 63), dat->table_scale, 1));
#if WANT_SIMD_EXCEPT
if (__glibc_unlikely (v_any_u64 (cmp)))
return special_case (xm, vfmaq_f64 (off, fac, y), cmp);
#endif
return vfmaq_f64 (off, fac, y);
}
|