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91 lines
3.4 KiB
C
91 lines
3.4 KiB
C
/* Double-precision SVE inverse tan
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Copyright (C) 2023-2024 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<https://www.gnu.org/licenses/>. */
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#include "sv_math.h"
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#include "poly_sve_f64.h"
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static const struct data
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{
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float64_t poly[20];
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float64_t pi_over_2;
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} data = {
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/* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
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[2**-1022, 1.0]. */
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.poly = { -0x1.5555555555555p-2, 0x1.99999999996c1p-3, -0x1.2492492478f88p-3,
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0x1.c71c71bc3951cp-4, -0x1.745d160a7e368p-4, 0x1.3b139b6a88ba1p-4,
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-0x1.11100ee084227p-4, 0x1.e1d0f9696f63bp-5, -0x1.aebfe7b418581p-5,
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0x1.842dbe9b0d916p-5, -0x1.5d30140ae5e99p-5, 0x1.338e31eb2fbbcp-5,
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-0x1.00e6eece7de8p-5, 0x1.860897b29e5efp-6, -0x1.0051381722a59p-6,
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0x1.14e9dc19a4a4ep-7, -0x1.d0062b42fe3bfp-9, 0x1.17739e210171ap-10,
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-0x1.ab24da7be7402p-13, 0x1.358851160a528p-16, },
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.pi_over_2 = 0x1.921fb54442d18p+0,
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};
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/* Useful constants. */
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#define SignMask (0x8000000000000000)
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/* Fast implementation of SVE atan.
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Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
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z=1/x and shift = pi/2. Largest errors are close to 1. The maximum observed
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error is 2.27 ulps:
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_ZGVsMxv_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
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want 0x1.9225645bdd7c3p-1. */
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svfloat64_t SV_NAME_D1 (atan) (svfloat64_t x, const svbool_t pg)
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{
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const struct data *d = ptr_barrier (&data);
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/* No need to trigger special case. Small cases, infs and nans
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are supported by our approximation technique. */
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svuint64_t ix = svreinterpret_u64 (x);
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svuint64_t sign = svand_x (pg, ix, SignMask);
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/* Argument reduction:
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y := arctan(x) for x < 1
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y := pi/2 + arctan(-1/x) for x > 1
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Hence, use z=-1/a if x>=1, otherwise z=a. */
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svbool_t red = svacgt (pg, x, 1.0);
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/* Avoid dependency in abs(x) in division (and comparison). */
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svfloat64_t z = svsel (red, svdivr_x (pg, x, 1.0), x);
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/* Use absolute value only when needed (odd powers of z). */
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svfloat64_t az = svabs_x (pg, z);
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az = svneg_m (az, red, az);
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/* Use split Estrin scheme for P(z^2) with deg(P)=19. */
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svfloat64_t z2 = svmul_x (pg, z, z);
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svfloat64_t x2 = svmul_x (pg, z2, z2);
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svfloat64_t x4 = svmul_x (pg, x2, x2);
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svfloat64_t x8 = svmul_x (pg, x4, x4);
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svfloat64_t y
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= svmla_x (pg, sv_estrin_7_f64_x (pg, z2, x2, x4, d->poly),
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sv_estrin_11_f64_x (pg, z2, x2, x4, x8, d->poly + 8), x8);
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/* y = shift + z + z^3 * P(z^2). */
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svfloat64_t z3 = svmul_x (pg, z2, az);
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y = svmla_x (pg, az, z3, y);
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/* Apply shift as indicated by `red` predicate. */
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y = svadd_m (red, y, d->pi_over_2);
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/* y = atan(x) if x>0, -atan(-x) otherwise. */
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y = svreinterpret_f64 (sveor_x (pg, svreinterpret_u64 (y), sign));
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return y;
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}
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