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/*
* Single-precision vector tan(x) function.
*
* Copyright (c) 2021-2023, 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[6];
float32x4_t neg_half_pi_1, neg_half_pi_2, neg_half_pi_3, two_over_pi, shift;
#if !WANT_SIMD_EXCEPT
float32x4_t range_val;
#endif
} data = {
/* Coefficients generated using FPMinimax. */
.poly = { V4 (0x1.55555p-2f), V4 (0x1.11166p-3f), V4 (0x1.b88a78p-5f),
V4 (0x1.7b5756p-6f), V4 (0x1.4ef4cep-8f), V4 (0x1.0e1e74p-7f) },
.neg_half_pi_1 = V4 (-0x1.921fb6p+0f),
.neg_half_pi_2 = V4 (0x1.777a5cp-25f),
.neg_half_pi_3 = V4 (0x1.ee59dap-50f),
.two_over_pi = V4 (0x1.45f306p-1f),
.shift = V4 (0x1.8p+23f),
#if !WANT_SIMD_EXCEPT
.range_val = V4 (0x1p15f),
#endif
};
#define RangeVal v_u32 (0x47000000) /* asuint32(0x1p15f). */
#define TinyBound v_u32 (0x30000000) /* asuint32 (0x1p-31f). */
#define Thresh v_u32 (0x16000000) /* asuint32(RangeVal) - TinyBound. */
/* Special cases (fall back to scalar calls). */
static float32x4_t VPCS_ATTR NOINLINE
special_case (float32x4_t x, float32x4_t y, uint32x4_t cmp)
{
return v_call_f32 (tanf, x, y, cmp);
}
/* Use a full Estrin scheme to evaluate polynomial. */
static inline float32x4_t
eval_poly (float32x4_t z, const struct data *d)
{
float32x4_t z2 = vmulq_f32 (z, z);
#if WANT_SIMD_EXCEPT
/* Tiny z (<= 0x1p-31) will underflow when calculating z^4. If fp exceptions
are to be triggered correctly, sidestep this by fixing such lanes to 0. */
uint32x4_t will_uflow
= vcleq_u32 (vreinterpretq_u32_f32 (vabsq_f32 (z)), TinyBound);
if (unlikely (v_any_u32 (will_uflow)))
z2 = vbslq_f32 (will_uflow, v_f32 (0), z2);
#endif
float32x4_t z4 = vmulq_f32 (z2, z2);
return v_estrin_5_f32 (z, z2, z4, d->poly);
}
/* Fast implementation of AdvSIMD tanf.
Maximum error is 3.45 ULP:
__v_tanf(-0x1.e5f0cap+13) got 0x1.ff9856p-1
want 0x1.ff9850p-1. */
float32x4_t VPCS_ATTR V_NAME_F1 (tan) (float32x4_t x)
{
const struct data *d = ptr_barrier (&data);
float32x4_t special_arg = x;
/* iax >= RangeVal means x, if not inf or NaN, is too large to perform fast
regression. */
#if WANT_SIMD_EXCEPT
uint32x4_t iax = vreinterpretq_u32_f32 (vabsq_f32 (x));
/* If fp exceptions are to be triggered correctly, also special-case tiny
input, as this will load to overflow later. Fix any special lanes to 1 to
prevent any exceptions being triggered. */
uint32x4_t special = vcgeq_u32 (vsubq_u32 (iax, TinyBound), Thresh);
if (unlikely (v_any_u32 (special)))
x = vbslq_f32 (special, v_f32 (1.0f), x);
#else
/* Otherwise, special-case large and special values. */
uint32x4_t special = vcageq_f32 (x, d->range_val);
#endif
/* n = rint(x/(pi/2)). */
float32x4_t q = vfmaq_f32 (d->shift, d->two_over_pi, x);
float32x4_t n = vsubq_f32 (q, d->shift);
/* Determine if x lives in an interval, where |tan(x)| grows to infinity. */
uint32x4_t pred_alt = vtstq_u32 (vreinterpretq_u32_f32 (q), v_u32 (1));
/* r = x - n * (pi/2) (range reduction into -pi./4 .. pi/4). */
float32x4_t r;
r = vfmaq_f32 (x, d->neg_half_pi_1, n);
r = vfmaq_f32 (r, d->neg_half_pi_2, n);
r = vfmaq_f32 (r, d->neg_half_pi_3, n);
/* If x lives in an interval, where |tan(x)|
- is finite, then use a polynomial approximation of the form
tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2).
- grows to infinity then use symmetries of tangent and the identity
tan(r) = cotan(pi/2 - r) to express tan(x) as 1/tan(-r). Finally, use
the same polynomial approximation of tan as above. */
/* Invert sign of r if odd quadrant. */
float32x4_t z = vmulq_f32 (r, vbslq_f32 (pred_alt, v_f32 (-1), v_f32 (1)));
/* Evaluate polynomial approximation of tangent on [-pi/4, pi/4]. */
float32x4_t z2 = vmulq_f32 (r, r);
float32x4_t p = eval_poly (z2, d);
float32x4_t y = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
/* Compute reciprocal and apply if required. */
float32x4_t inv_y = vdivq_f32 (v_f32 (1.0f), y);
if (unlikely (v_any_u32 (special)))
return special_case (special_arg, vbslq_f32 (pred_alt, inv_y, y), special);
return vbslq_f32 (pred_alt, inv_y, y);
}
PL_SIG (V, F, 1, tan, -3.1, 3.1)
PL_TEST_ULP (V_NAME_F1 (tan), 2.96)
PL_TEST_EXPECT_FENV (V_NAME_F1 (tan), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME_F1 (tan), 0, 0x1p-31, 5000)
PL_TEST_INTERVAL (V_NAME_F1 (tan), 0x1p-31, 0x1p15, 500000)
PL_TEST_INTERVAL (V_NAME_F1 (tan), 0x1p15, inf, 5000)
PL_TEST_INTERVAL (V_NAME_F1 (tan), -0, -0x1p-31, 5000)
PL_TEST_INTERVAL (V_NAME_F1 (tan), -0x1p-31, -0x1p15, 500000)
PL_TEST_INTERVAL (V_NAME_F1 (tan), -0x1p15, -inf, 5000)
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