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112 lines
4.1 KiB
C
112 lines
4.1 KiB
C
/* Single-precision AdvSIMD 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 "v_math.h"
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#include "poly_advsimd_f32.h"
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static const struct data
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{
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float32x4_t poly[8];
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float32x4_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**-128, 1.0].
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Generated using fpminimax between FLT_MIN and 1. */
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.poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
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V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
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V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
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.pi_over_2 = V4 (0x1.921fb6p+0f),
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};
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#define SignMask v_u32 (0x80000000)
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#define P(i) d->poly[i]
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#define TinyBound 0x30800000 /* asuint(0x1p-30). */
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#define BigBound 0x4e800000 /* asuint(0x1p30). */
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#if WANT_SIMD_EXCEPT
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static float32x4_t VPCS_ATTR NOINLINE
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special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
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{
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return v_call_f32 (atanf, x, y, special);
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}
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#endif
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/* Fast implementation of vector atanf based on
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atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
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using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps:
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_ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */
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float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x)
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{
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const struct data *d = ptr_barrier (&data);
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/* Small cases, infs and nans are supported by our approximation technique,
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but do not set fenv flags correctly. Only trigger special case if we need
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fenv. */
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uint32x4_t ix = vreinterpretq_u32_f32 (x);
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uint32x4_t sign = vandq_u32 (ix, SignMask);
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#if WANT_SIMD_EXCEPT
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uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000));
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uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)),
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v_u32 (BigBound - TinyBound));
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/* If any lane is special, fall back to the scalar routine for all lanes. */
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if (__glibc_unlikely (v_any_u32 (special)))
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return special_case (x, x, v_u32 (-1));
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#endif
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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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uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0));
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/* Avoid dependency in abs(x) in division (and comparison). */
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float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x);
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float32x4_t shift = vreinterpretq_f32_u32 (
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vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2)));
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/* Use absolute value only when needed (odd powers of z). */
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float32x4_t az = vbslq_f32 (
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SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z);
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/* Calculate the polynomial approximation.
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Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
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a standard implementation using z8 creates spurious underflow
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in the very last fma (when z^8 is small enough).
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Therefore, we split the last fma into a mul and an fma.
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Horner and single-level Estrin have higher errors that exceed
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threshold. */
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float32x4_t z2 = vmulq_f32 (z, z);
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float32x4_t z4 = vmulq_f32 (z2, z2);
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float32x4_t y = vfmaq_f32 (
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v_pairwise_poly_3_f32 (z2, z4, d->poly), z4,
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vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4)));
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/* y = shift + z * P(z^2). */
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y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift);
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/* y = atan(x) if x>0, -atan(-x) otherwise. */
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y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign));
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return y;
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}
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libmvec_hidden_def (V_NAME_F1 (atan))
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HALF_WIDTH_ALIAS_F1 (atan)
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