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/*
* Double-precision vector sinh(x) function.
*
* Copyright (c) 2022-2023, 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[11];
float64x2_t inv_ln2, m_ln2_hi, m_ln2_lo, shift;
uint64x2_t halff;
int64x2_t onef;
#if WANT_SIMD_EXCEPT
uint64x2_t tiny_bound, thresh;
#else
uint64x2_t large_bound;
#endif
} data = {
/* Generated using Remez, deg=12 in [-log(2)/2, log(2)/2]. */
.poly = { V2 (0x1p-1), V2 (0x1.5555555555559p-3), V2 (0x1.555555555554bp-5),
V2 (0x1.111111110f663p-7), V2 (0x1.6c16c16c1b5f3p-10),
V2 (0x1.a01a01affa35dp-13), V2 (0x1.a01a018b4ecbbp-16),
V2 (0x1.71ddf82db5bb4p-19), V2 (0x1.27e517fc0d54bp-22),
V2 (0x1.af5eedae67435p-26), V2 (0x1.1f143d060a28ap-29), },
.inv_ln2 = V2 (0x1.71547652b82fep0),
.m_ln2_hi = V2 (-0x1.62e42fefa39efp-1),
.m_ln2_lo = V2 (-0x1.abc9e3b39803fp-56),
.shift = V2 (0x1.8p52),
.halff = V2 (0x3fe0000000000000),
.onef = V2 (0x3ff0000000000000),
#if WANT_SIMD_EXCEPT
/* 2^-26, below which sinh(x) rounds to x. */
.tiny_bound = V2 (0x3e50000000000000),
/* asuint(large_bound) - asuint(tiny_bound). */
.thresh = V2 (0x0230000000000000),
#else
/* 2^9. expm1 helper overflows for large input. */
.large_bound = V2 (0x4080000000000000),
#endif
};
static inline float64x2_t
expm1_inline (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
/* Reduce argument:
exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
where i = round(x / ln2)
and f = x - i * ln2 (f in [-ln2/2, ln2/2]). */
float64x2_t j = vsubq_f64 (vfmaq_f64 (d->shift, d->inv_ln2, x), d->shift);
int64x2_t i = vcvtq_s64_f64 (j);
float64x2_t f = vfmaq_f64 (x, j, d->m_ln2_hi);
f = vfmaq_f64 (f, j, d->m_ln2_lo);
/* Approximate expm1(f) using polynomial. */
float64x2_t f2 = vmulq_f64 (f, f);
float64x2_t f4 = vmulq_f64 (f2, f2);
float64x2_t f8 = vmulq_f64 (f4, f4);
float64x2_t p = vfmaq_f64 (f, f2, v_estrin_10_f64 (f, f2, f4, f8, d->poly));
/* t = 2^i. */
float64x2_t t = vreinterpretq_f64_u64 (
vreinterpretq_u64_s64 (vaddq_s64 (vshlq_n_s64 (i, 52), d->onef)));
/* expm1(x) ~= p * t + (t - 1). */
return vfmaq_f64 (vsubq_f64 (t, v_f64 (1.0)), p, t);
}
static float64x2_t NOINLINE VPCS_ATTR
special_case (float64x2_t x)
{
return v_call_f64 (sinh, x, x, v_u64 (-1));
}
/* Approximation for vector double-precision sinh(x) using expm1.
sinh(x) = (exp(x) - exp(-x)) / 2.
The greatest observed error is 2.57 ULP:
_ZGVnN2v_sinh (0x1.9fb1d49d1d58bp-2) got 0x1.ab34e59d678dcp-2
want 0x1.ab34e59d678d9p-2. */
float64x2_t VPCS_ATTR V_NAME_D1 (sinh) (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
float64x2_t ax = vabsq_f64 (x);
uint64x2_t sign
= veorq_u64 (vreinterpretq_u64_f64 (x), vreinterpretq_u64_f64 (ax));
float64x2_t halfsign = vreinterpretq_f64_u64 (vorrq_u64 (sign, d->halff));
#if WANT_SIMD_EXCEPT
uint64x2_t special = vcgeq_u64 (
vsubq_u64 (vreinterpretq_u64_f64 (ax), d->tiny_bound), d->thresh);
#else
uint64x2_t special = vcgeq_u64 (vreinterpretq_u64_f64 (ax), d->large_bound);
#endif
/* Fall back to scalar variant for all lanes if any of them are special. */
if (unlikely (v_any_u64 (special)))
return special_case (x);
/* Up to the point that expm1 overflows, we can use it to calculate sinh
using a slight rearrangement of the definition of sinh. This allows us to
retain acceptable accuracy for very small inputs. */
float64x2_t t = expm1_inline (ax);
t = vaddq_f64 (t, vdivq_f64 (t, vaddq_f64 (t, v_f64 (1.0))));
return vmulq_f64 (t, halfsign);
}
PL_SIG (V, D, 1, sinh, -10.0, 10.0)
PL_TEST_ULP (V_NAME_D1 (sinh), 2.08)
PL_TEST_EXPECT_FENV (V_NAME_D1 (sinh), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), 0, 0x1p-26, 1000)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), -0, -0x1p-26, 1000)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), 0x1p-26, 0x1p9, 500000)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), -0x1p-26, -0x1p9, 500000)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), 0x1p9, inf, 1000)
PL_TEST_INTERVAL (V_NAME_D1 (sinh), -0x1p9, -inf, 1000)
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