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/* Single-Precision vector (Advanced SIMD) inverse cospi function
Copyright (C) 2025 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
{
float32x4_t c0, c2, c4, inv_pi;
float c1, c3, c5, null;
} data = {
/* Coefficients of polynomial P such that asin(x)/pi~ x/pi + x^3 * poly(x^2)
on [ 0x1p-126 0x1p-2 ]. rel error: 0x1.ef9f94b1p-33. Generated using
iterative approach for minimisation of relative error in asinpif Sollya
file. */
.c0 = V4 (0x1.b2995ep-5f), .c1 = 0x1.8724ep-6f,
.c2 = V4 (0x1.d1301ep-7f), .c3 = 0x1.446d3cp-7f,
.c4 = V4 (0x1.654848p-8f), .c5 = 0x1.5fdaa8p-7f,
.inv_pi = V4 (0x1.45f306p-2f),
};
#define AbsMask 0x7fffffff
/* Single-precision implementation of vector acospi(x).
For |x| in [0, 0.5], use order 5 polynomial P to approximate asinpi
such that the final approximation of acospi is an odd polynomial:
acospi(x) ~ 1/2 - (x/pi + x^3 P(x^2)).
The largest observed error in this region is 1.23 ulps,
_ZGVnN4v_acospif (0x1.fee13ep-2) got 0x1.55beb4p-2 want 0x1.55beb2p-2.
For |x| in [0.5, 1.0], use same approximation with a change of variable
acospi(x) = y/pi + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
The largest observed error in this region is 2.53 ulps,
_ZGVnN4v_acospif (0x1.6ad644p-1) got 0x1.fe8f96p-3
want 0x1.fe8f9cp-3. */
float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (acospi) (float32x4_t x)
{
const struct data *d = ptr_barrier (&data);
uint32x4_t ix = vreinterpretq_u32_f32 (x);
uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
float32x4_t ax = vreinterpretq_f32_u32 (ia);
uint32x4_t a_le_half = vcaltq_f32 (x, v_f32 (0.5f));
/* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
z2 = x ^ 2 and z = |x| , if |x| < 0.5
z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x),
vfmsq_n_f32 (v_f32 (0.5f), ax, 0.5f));
float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
/* Use a single polynomial approximation P for both intervals. */
/* Order-5 Estrin evaluation scheme. */
float32x4_t z4 = vmulq_f32 (z2, z2);
float32x4_t z8 = vmulq_f32 (z4, z4);
float32x4_t c135 = vld1q_f32 (&d->c1);
float32x4_t p01 = vfmaq_laneq_f32 (d->c0, z2, c135, 0);
float32x4_t p23 = vfmaq_laneq_f32 (d->c2, z2, c135, 1);
float32x4_t p03 = vfmaq_f32 (p01, z4, p23);
float32x4_t p45 = vfmaq_laneq_f32 (d->c4, z2, c135, 2);
float32x4_t p = vfmaq_f32 (p03, z8, p45);
/* Add 1/pi as final coeff. */
p = vfmaq_f32 (d->inv_pi, z2, p);
/* Finalize polynomial: z * P(z^2). */
p = vmulq_f32 (z, p);
/* acospi(|x|)
= 1/2 - sign(x) * Q(|x|), for |x| < 0.5
= 2 Q(|x|) , for 0.5 < x < 1.0
= 1 - 2 Q(|x|) , for -1.0 < x < -0.5. */
float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x);
uint32x4_t is_neg = vcltzq_f32 (x);
float32x4_t off = vreinterpretq_f32_u32 (
vandq_u32 (vreinterpretq_u32_f32 (v_f32 (1.0f)), is_neg));
float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (1.0f), v_f32 (-2.0f));
float32x4_t add = vbslq_f32 (a_le_half, v_f32 (0.5f), off);
return vfmsq_f32 (add, mul, y);
}
libmvec_hidden_def (V_NAME_F1 (acospi))
HALF_WIDTH_ALIAS_F1 (acospi)
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