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/*
* Single-precision vector acos(x) function.
*
* Copyright (c) 2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "v_math.h"
#include "poly_advsimd_f32.h"
#include "pl_sig.h"
#include "pl_test.h"
static const struct data
{
float32x4_t poly[5];
float32x4_t pi_over_2f, pif;
} data = {
/* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on
[ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */
.poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5),
V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) },
.pi_over_2f = V4 (0x1.921fb6p+0f),
.pif = V4 (0x1.921fb6p+1f),
};
#define AbsMask 0x7fffffff
#define Half 0x3f000000
#define One 0x3f800000
#define Small 0x32800000 /* 2^-26. */
#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 (acosf, x, y, special);
}
#endif
/* Single-precision implementation of vector acos(x).
For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
rounding.
If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
approximation.
For |x| in [Small, 0.5], use order 4 polynomial P such that the final
approximation of asin is an odd polynomial:
acos(x) ~ pi/2 - (x + x^3 P(x^2)).
The largest observed error in this region is 1.26 ulps,
_ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0.
For |x| in [0.5, 1.0], use same approximation with a change of variable
acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
The largest observed error in this region is 1.32 ulps,
_ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
want 0x1.feb32ep-1. */
float32x4_t VPCS_ATTR V_NAME_F1 (acos) (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));
#if WANT_SIMD_EXCEPT
/* A single comparison for One, Small and QNaN. */
uint32x4_t special
= vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
if (unlikely (v_any_u32 (special)))
return special_case (x, x, v_u32 (0xffffffff));
#endif
float32x4_t ax = vreinterpretq_f32_u32 (ia);
uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half));
/* 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 (0.5), ax, 0.5));
float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
/* Use a single polynomial approximation P for both intervals. */
float32x4_t p = v_horner_4_f32 (z2, d->poly);
/* Finalize polynomial: z + z * z2 * P(z2). */
p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
/* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
= 2 Q(|x|) , for 0.5 < x < 1.0
= pi - 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 (d->pif), is_neg));
float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0));
float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off);
return vfmaq_f32 (add, mul, y);
}
PL_SIG (V, F, 1, acos, -1.0, 1.0)
PL_TEST_ULP (V_NAME_F1 (acos), 0.82)
PL_TEST_EXPECT_FENV (V_NAME_F1 (acos), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME_F1 (acos), 0, 0x1p-26, 5000)
PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p-26, 0.5, 50000)
PL_TEST_INTERVAL (V_NAME_F1 (acos), 0.5, 1.0, 50000)
PL_TEST_INTERVAL (V_NAME_F1 (acos), 1.0, 0x1p11, 50000)
PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p11, inf, 20000)
PL_TEST_INTERVAL (V_NAME_F1 (acos), -0, -inf, 20000)
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