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/*
* Double-precision vector asin(x) function.
*
* Copyright (c) 2023-2024, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "v_math.h"
#include "poly_advsimd_f64.h"
#include "pl_sig.h"
#include "pl_test.h"
static const struct data
{
float64x2_t poly[12];
float64x2_t pi_over_2;
uint64x2_t abs_mask;
} data = {
/* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
.poly = { V2 (0x1.555555555554ep-3), V2 (0x1.3333333337233p-4),
V2 (0x1.6db6db67f6d9fp-5), V2 (0x1.f1c71fbd29fbbp-6),
V2 (0x1.6e8b264d467d6p-6), V2 (0x1.1c5997c357e9dp-6),
V2 (0x1.c86a22cd9389dp-7), V2 (0x1.856073c22ebbep-7),
V2 (0x1.fd1151acb6bedp-8), V2 (0x1.087182f799c1dp-6),
V2 (-0x1.6602748120927p-7), V2 (0x1.cfa0dd1f9478p-6), },
.pi_over_2 = V2 (0x1.921fb54442d18p+0),
.abs_mask = V2 (0x7fffffffffffffff),
};
#define AllMask v_u64 (0xffffffffffffffff)
#define One 0x3ff0000000000000
#define Small 0x3e50000000000000 /* 2^-12. */
#if WANT_SIMD_EXCEPT
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
{
return v_call_f64 (asin, x, y, special);
}
#endif
/* Double-precision implementation of vector asin(x).
For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct
rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the
following approximation.
For |x| in [Small, 0.5], use an order 11 polynomial P such that the final
approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
The largest observed error in this region is 1.01 ulps,
_ZGVnN2v_asin (0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2
want 0x1.ed78525a927eep-2.
For |x| in [0.5, 1.0], use same approximation with a change of variable
asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
The largest observed error in this region is 2.69 ulps,
_ZGVnN2v_asin (0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1
want 0x1.110d7e85fdd53p-1. */
float64x2_t VPCS_ATTR V_NAME_D1 (asin) (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
float64x2_t ax = vabsq_f64 (x);
#if WANT_SIMD_EXCEPT
/* Special values need to be computed with scalar fallbacks so
that appropriate exceptions are raised. */
uint64x2_t special
= vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)),
v_u64 (One - Small));
if (unlikely (v_any_u64 (special)))
return special_case (x, x, AllMask);
#endif
uint64x2_t a_lt_half = vcltq_f64 (ax, v_f64 (0.5));
/* Evaluate polynomial Q(x) = y + y * z * P(z) with
z = x ^ 2 and y = |x| , if |x| < 0.5
z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
float64x2_t z2 = vbslq_f64 (a_lt_half, vmulq_f64 (x, x),
vfmsq_n_f64 (v_f64 (0.5), ax, 0.5));
float64x2_t z = vbslq_f64 (a_lt_half, ax, vsqrtq_f64 (z2));
/* Use a single polynomial approximation P for both intervals. */
float64x2_t z4 = vmulq_f64 (z2, z2);
float64x2_t z8 = vmulq_f64 (z4, z4);
float64x2_t z16 = vmulq_f64 (z8, z8);
float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly);
/* Finalize polynomial: z + z * z2 * P(z2). */
p = vfmaq_f64 (z, vmulq_f64 (z, z2), p);
/* asin(|x|) = Q(|x|) , for |x| < 0.5
= pi/2 - 2 Q(|x|), for |x| >= 0.5. */
float64x2_t y = vbslq_f64 (a_lt_half, p, vfmsq_n_f64 (d->pi_over_2, p, 2.0));
/* Copy sign. */
return vbslq_f64 (d->abs_mask, y, x);
}
PL_SIG (V, D, 1, asin, -1.0, 1.0)
PL_TEST_ULP (V_NAME_D1 (asin), 2.19)
PL_TEST_EXPECT_FENV (V_NAME_D1 (asin), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME_D1 (asin), 0, Small, 5000)
PL_TEST_INTERVAL (V_NAME_D1 (asin), Small, 0.5, 50000)
PL_TEST_INTERVAL (V_NAME_D1 (asin), 0.5, 1.0, 50000)
PL_TEST_INTERVAL (V_NAME_D1 (asin), 1.0, 0x1p11, 50000)
PL_TEST_INTERVAL (V_NAME_D1 (asin), 0x1p11, inf, 20000)
PL_TEST_INTERVAL (V_NAME_D1 (asin), -0, -inf, 20000)
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