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/* Double-precision vector (Advanced SIMD) erf 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"
static const struct data
{
float64x2_t third;
float64x2_t tenth, two_over_five, two_over_fifteen;
float64x2_t two_over_nine, two_over_fortyfive;
float64x2_t max, shift;
#if WANT_SIMD_EXCEPT
float64x2_t tiny_bound, huge_bound, scale_minus_one;
#endif
} data = {
.third = V2 (0x1.5555555555556p-2), /* used to compute 2/3 and 1/6 too. */
.two_over_fifteen = V2 (0x1.1111111111111p-3),
.tenth = V2 (-0x1.999999999999ap-4),
.two_over_five = V2 (-0x1.999999999999ap-2),
.two_over_nine = V2 (-0x1.c71c71c71c71cp-3),
.two_over_fortyfive = V2 (0x1.6c16c16c16c17p-5),
.max = V2 (5.9921875), /* 6 - 1/128. */
.shift = V2 (0x1p45),
#if WANT_SIMD_EXCEPT
.huge_bound = V2 (0x1p205),
.tiny_bound = V2 (0x1p-226),
.scale_minus_one = V2 (0x1.06eba8214db69p-3), /* 2/sqrt(pi) - 1.0. */
#endif
};
#define AbsMask 0x7fffffffffffffff
struct entry
{
float64x2_t erf;
float64x2_t scale;
};
static inline struct entry
lookup (uint64x2_t i)
{
struct entry e;
float64x2_t e1 = vld1q_f64 (&__erf_data.tab[vgetq_lane_u64 (i, 0)].erf),
e2 = vld1q_f64 (&__erf_data.tab[vgetq_lane_u64 (i, 1)].erf);
e.erf = vuzp1q_f64 (e1, e2);
e.scale = vuzp2q_f64 (e1, e2);
return e;
}
/* Double-precision implementation of vector erf(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,
erf(x) ~ erf(r) + scale * d * [
+ 1
- r d
+ 1/3 (2 r^2 - 1) d^2
- 1/6 (r (2 r^2 - 3)) d^3
+ 1/30 (4 r^4 - 12 r^2 + 3) d^4
- 1/90 (4 r^4 - 20 r^2 + 15) d^5
]
Maximum measure error: 2.29 ULP
V_NAME_D1 (erf)(-0x1.00003c924e5d1p-8) got -0x1.20dd59132ebadp-8
want -0x1.20dd59132ebafp-8. */
float64x2_t VPCS_ATTR V_NAME_D1 (erf) (float64x2_t x)
{
const struct data *dat = ptr_barrier (&data);
float64x2_t a = vabsq_f64 (x);
/* Reciprocal conditions that do not catch NaNs so they can be used in BSLs
to return expected results. */
uint64x2_t a_le_max = vcleq_f64 (a, dat->max);
uint64x2_t a_gt_max = vcgtq_f64 (a, dat->max);
#if WANT_SIMD_EXCEPT
/* |x| huge or tiny. */
uint64x2_t cmp1 = vcgtq_f64 (a, dat->huge_bound);
uint64x2_t cmp2 = vcltq_f64 (a, dat->tiny_bound);
uint64x2_t cmp = vorrq_u64 (cmp1, cmp2);
/* If any lanes are special, mask them with 1 for small x or 8 for large
values and retain a copy of a 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)))
{
a = vbslq_f64 (cmp1, v_f64 (8.0), a);
a = vbslq_f64 (cmp2, v_f64 (1.0), a);
}
#endif
/* Set r to multiple of 1/128 nearest to |x|. */
float64x2_t shift = dat->shift;
float64x2_t z = vaddq_f64 (a, shift);
/* Lookup erf(r) and scale(r) in table, without shortcut for small values,
but with saturated indices for large values and NaNs in order to avoid
segfault. */
uint64x2_t i
= vsubq_u64 (vreinterpretq_u64_f64 (z), vreinterpretq_u64_f64 (shift));
i = vbslq_u64 (a_le_max, i, v_u64 (768));
struct entry e = lookup (i);
float64x2_t r = vsubq_f64 (z, shift);
/* erf(x) ~ erf(r) + scale * d * poly (r, d). */
float64x2_t d = vsubq_f64 (a, r);
float64x2_t d2 = vmulq_f64 (d, d);
float64x2_t r2 = vmulq_f64 (r, r);
/* poly (d, r) = 1 + p1(r) * d + p2(r) * d^2 + ... + p5(r) * d^5. */
float64x2_t p1 = r;
float64x2_t p2
= vfmsq_f64 (dat->third, r2, vaddq_f64 (dat->third, dat->third));
float64x2_t p3 = vmulq_f64 (r, vfmaq_f64 (v_f64 (-0.5), r2, dat->third));
float64x2_t p4 = vfmaq_f64 (dat->two_over_five, r2, dat->two_over_fifteen);
p4 = vfmsq_f64 (dat->tenth, r2, p4);
float64x2_t p5 = vfmaq_f64 (dat->two_over_nine, r2, dat->two_over_fortyfive);
p5 = vmulq_f64 (r, vfmaq_f64 (vmulq_f64 (v_f64 (0.5), dat->third), r2, p5));
float64x2_t p34 = vfmaq_f64 (p3, d, p4);
float64x2_t p12 = vfmaq_f64 (p1, d, p2);
float64x2_t y = vfmaq_f64 (p34, d2, p5);
y = vfmaq_f64 (p12, d2, y);
y = vfmaq_f64 (e.erf, e.scale, vfmsq_f64 (d, d2, y));
/* Solves the |x| = inf and NaN cases. */
y = vbslq_f64 (a_gt_max, v_f64 (1.0), y);
/* Copy sign. */
y = vbslq_f64 (v_u64 (AbsMask), y, x);
#if WANT_SIMD_EXCEPT
if (__glibc_unlikely (v_any_u64 (cmp2)))
{
/* Neutralise huge values of x before fixing small values. */
x = vbslq_f64 (cmp1, v_f64 (1.0), x);
/* Fix tiny values that trigger spurious underflow. */
return vbslq_f64 (cmp2, vfmaq_f64 (x, dat->scale_minus_one, x), y);
}
#endif
return y;
}
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