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/* Single-precision AdvSIMD inverse tan
Copyright (C) 2023 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 "poly_advsimd_f32.h"
static const struct data
{
float32x4_t poly[8];
float32x4_t pi_over_2;
} data = {
/* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
[2**-128, 1.0].
Generated using fpminimax between FLT_MIN and 1. */
.poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
.pi_over_2 = V4 (0x1.921fb6p+0f),
};
#define SignMask v_u32 (0x80000000)
#define P(i) d->poly[i]
#define TinyBound 0x30800000 /* asuint(0x1p-30). */
#define BigBound 0x4e800000 /* asuint(0x1p30). */
#if WANT_SIMD_EXCEPT
static float32x4_t VPCS_ATTR NOINLINE
special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
{
return v_call_f32 (atanf, x, y, special);
}
#endif
/* Fast implementation of vector atanf based on
atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps:
_ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */
float32x4_t VPCS_ATTR V_NAME_F1 (atan) (float32x4_t x)
{
const struct data *d = ptr_barrier (&data);
/* Small cases, infs and nans are supported by our approximation technique,
but do not set fenv flags correctly. Only trigger special case if we need
fenv. */
uint32x4_t ix = vreinterpretq_u32_f32 (x);
uint32x4_t sign = vandq_u32 (ix, SignMask);
#if WANT_SIMD_EXCEPT
uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000));
uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)),
v_u32 (BigBound - TinyBound));
/* If any lane is special, fall back to the scalar routine for all lanes. */
if (__glibc_unlikely (v_any_u32 (special)))
return special_case (x, x, v_u32 (-1));
#endif
/* Argument reduction:
y := arctan(x) for x < 1
y := pi/2 + arctan(-1/x) for x > 1
Hence, use z=-1/a if x>=1, otherwise z=a. */
uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0));
/* Avoid dependency in abs(x) in division (and comparison). */
float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x);
float32x4_t shift = vreinterpretq_f32_u32 (
vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2)));
/* Use absolute value only when needed (odd powers of z). */
float32x4_t az = vbslq_f32 (
SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z);
/* Calculate the polynomial approximation.
Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
a standard implementation using z8 creates spurious underflow
in the very last fma (when z^8 is small enough).
Therefore, we split the last fma into a mul and an fma.
Horner and single-level Estrin have higher errors that exceed
threshold. */
float32x4_t z2 = vmulq_f32 (z, z);
float32x4_t z4 = vmulq_f32 (z2, z2);
float32x4_t y = vfmaq_f32 (
v_pairwise_poly_3_f32 (z2, z4, d->poly), z4,
vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4)));
/* y = shift + z * P(z^2). */
y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift);
/* y = atan(x) if x>0, -atan(-x) otherwise. */
y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign));
return y;
}
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