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/* Double-precision vector (SVE) 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 "sv_math.h"
#include "poly_sve_f64.h"
static const struct data
{
double poly[9];
double half_pi_hi, half_pi_lo, inv_half_pi, range_val, shift;
} data = {
/* Polynomial generated with FPMinimax. */
.poly = { 0x1.5555555555556p-2, 0x1.1111111110a63p-3, 0x1.ba1ba1bb46414p-5,
0x1.664f47e5b5445p-6, 0x1.226e5e5ecdfa3p-7, 0x1.d6c7ddbf87047p-9,
0x1.7ea75d05b583ep-10, 0x1.289f22964a03cp-11,
0x1.4e4fd14147622p-12, },
.half_pi_hi = 0x1.921fb54442d18p0,
.half_pi_lo = 0x1.1a62633145c07p-54,
.inv_half_pi = 0x1.45f306dc9c883p-1,
.range_val = 0x1p23,
.shift = 0x1.8p52,
};
static svfloat64_t NOINLINE
special_case (svfloat64_t x, svfloat64_t y, svbool_t special)
{
return sv_call_f64 (tan, x, y, special);
}
/* Vector approximation for double-precision tan.
Maximum measured error is 3.48 ULP:
_ZGVsMxv_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37
want -0x1.f6ccd8ecf7deap+37. */
svfloat64_t SV_NAME_D1 (tan) (svfloat64_t x, svbool_t pg)
{
const struct data *dat = ptr_barrier (&data);
/* Invert condition to catch NaNs and Infs as well as large values. */
svbool_t special = svnot_z (pg, svaclt (pg, x, dat->range_val));
/* q = nearest integer to 2 * x / pi. */
svfloat64_t shift = sv_f64 (dat->shift);
svfloat64_t q = svmla_x (pg, shift, x, dat->inv_half_pi);
q = svsub_x (pg, q, shift);
svint64_t qi = svcvt_s64_x (pg, q);
/* Use q to reduce x to r in [-pi/4, pi/4], by:
r = x - q * pi/2, in extended precision. */
svfloat64_t r = x;
svfloat64_t half_pi = svld1rq (svptrue_b64 (), &dat->half_pi_hi);
r = svmls_lane (r, q, half_pi, 0);
r = svmls_lane (r, q, half_pi, 1);
/* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle
formula. */
r = svmul_x (pg, 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)). */
svfloat64_t r2 = svmul_x (pg, r, r);
svfloat64_t r4 = svmul_x (pg, r2, r2);
svfloat64_t r8 = svmul_x (pg, r4, r4);
/* Use offset version coeff array by 1 to evaluate from C1 onwards. */
svfloat64_t p = sv_estrin_7_f64_x (pg, r2, r4, r8, dat->poly + 1);
p = svmad_x (pg, p, r2, dat->poly[0]);
p = svmla_x (pg, r, r2, svmul_x (pg, 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). */
svbool_t use_recip
= svcmpeq (pg, svand_x (pg, svreinterpret_u64 (qi), 1), 0);
svfloat64_t n = svmad_x (pg, p, p, -1);
svfloat64_t d = svmul_x (pg, p, 2);
svfloat64_t swap = n;
n = svneg_m (n, use_recip, d);
d = svsel (use_recip, swap, d);
if (__glibc_unlikely (svptest_any (pg, special)))
return special_case (x, svdiv_x (svnot_z (pg, special), n, d), special);
return svdiv_x (pg, n, d);
}
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