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/* Double-precision vector (Advanced SIMD) tan function
Copyright (C) 2023 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"
#include "poly_advsimd_f64.h"
static const struct data
{
float64x2_t poly[9];
float64x2_t half_pi_hi, half_pi_lo, two_over_pi, shift;
#if !WANT_SIMD_EXCEPT
float64x2_t range_val;
#endif
} data = {
/* Coefficients generated using FPMinimax. */
.poly = { V2 (0x1.5555555555556p-2), V2 (0x1.1111111110a63p-3),
V2 (0x1.ba1ba1bb46414p-5), V2 (0x1.664f47e5b5445p-6),
V2 (0x1.226e5e5ecdfa3p-7), V2 (0x1.d6c7ddbf87047p-9),
V2 (0x1.7ea75d05b583ep-10), V2 (0x1.289f22964a03cp-11),
V2 (0x1.4e4fd14147622p-12) },
.half_pi_hi = V2 (0x1.921fb54442d18p0),
.half_pi_lo = V2 (0x1.1a62633145c07p-54),
.two_over_pi = V2 (0x1.45f306dc9c883p-1),
.shift = V2 (0x1.8p52),
#if !WANT_SIMD_EXCEPT
.range_val = V2 (0x1p23),
#endif
};
#define RangeVal 0x4160000000000000 /* asuint64(0x1p23). */
#define TinyBound 0x3e50000000000000 /* asuint64(2^-26). */
#define Thresh 0x310000000000000 /* RangeVal - TinyBound. */
/* Special cases (fall back to scalar calls). */
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t x)
{
return v_call_f64 (tan, x, x, v_u64 (-1));
}
/* Vector approximation for double-precision tan.
Maximum measured error is 3.48 ULP:
__v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37
want -0x1.f6ccd8ecf7deap+37. */
float64x2_t VPCS_ATTR V_NAME_D1 (tan) (float64x2_t x)
{
const struct data *dat = ptr_barrier (&data);
/* Our argument reduction cannot calculate q with sufficient accuracy for very
large inputs. Fall back to scalar routine for all lanes if any are too
large, or Inf/NaN. If fenv exceptions are expected, also fall back for tiny
input to avoid underflow. */
#if WANT_SIMD_EXCEPT
uint64x2_t iax = vreinterpretq_u64_f64 (vabsq_f64 (x));
/* iax - tiny_bound > range_val - tiny_bound. */
uint64x2_t special
= vcgtq_u64 (vsubq_u64 (iax, v_u64 (TinyBound)), v_u64 (Thresh));
if (__glibc_unlikely (v_any_u64 (special)))
return special_case (x);
#endif
/* q = nearest integer to 2 * x / pi. */
float64x2_t q
= vsubq_f64 (vfmaq_f64 (dat->shift, x, dat->two_over_pi), dat->shift);
int64x2_t qi = vcvtq_s64_f64 (q);
/* Use q to reduce x to r in [-pi/4, pi/4], by:
r = x - q * pi/2, in extended precision. */
float64x2_t r = x;
r = vfmsq_f64 (r, q, dat->half_pi_hi);
r = vfmsq_f64 (r, q, dat->half_pi_lo);
/* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle
formula. */
r = vmulq_n_f64 (r, 0.5);
/* Approximate tan(r) using order 8 polynomial.
tan(x) is odd, so polynomial has the form:
tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ...
Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ...
Then compute the approximation by:
tan(r) ~= r + r^3 * (C0 + r^2 * P(r)). */
float64x2_t r2 = vmulq_f64 (r, r), r4 = vmulq_f64 (r2, r2),
r8 = vmulq_f64 (r4, r4);
/* Offset coefficients to evaluate from C1 onwards. */
float64x2_t p = v_estrin_7_f64 (r2, r4, r8, dat->poly + 1);
p = vfmaq_f64 (dat->poly[0], p, r2);
p = vfmaq_f64 (r, r2, vmulq_f64 (p, r));
/* Recombination uses double-angle formula:
tan(2x) = 2 * tan(x) / (1 - (tan(x))^2)
and reciprocity around pi/2:
tan(x) = 1 / (tan(pi/2 - x))
to assemble result using change-of-sign and conditional selection of
numerator/denominator, dependent on odd/even-ness of q (hence quadrant). */
float64x2_t n = vfmaq_f64 (v_f64 (-1), p, p);
float64x2_t d = vaddq_f64 (p, p);
uint64x2_t no_recip = vtstq_u64 (vreinterpretq_u64_s64 (qi), v_u64 (1));
#if !WANT_SIMD_EXCEPT
uint64x2_t special = vceqzq_u64 (vcaleq_f64 (x, dat->range_val));
if (__glibc_unlikely (v_any_u64 (special)))
return special_case (x);
#endif
return vdivq_f64 (vbslq_f64 (no_recip, n, vnegq_f64 (d)),
vbslq_f64 (no_recip, d, n));
}
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