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Пересобранный и видоизмененный conky для ОСГОС

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Галанский Сергей Александрович 2025-06-09 14:13:23 +05:00
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/* This file is part of the Vc library. {{{
Copyright © 2011-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/avx/vector.h>
#include <Vc/avx/debug.h>
#include <Vc/avx/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
#ifdef Vc_IMPL_AVX2
template <>
Vc_CONST AVX2::short_v sorted<CurrentImplementation::current()>(AVX2::short_v x_)
{
// ab cd ef gh ij kl mn op
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮
// ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮
// ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮
// 01 23 01 23 01 23 01 23
// ⎮⎮ ⎮⎮ ╳ ⎮⎮ ⎮⎮ ╳
// 01 23 32 10 01 23 32 10
// ⎮⎝ ⎮⎝ ⎠⎮ ⎠⎮ ⎮⎝ ⎮⎝ ⎠⎮ ⎠⎮
// ⎮ ╲⎮ ╳ ⎮╱ ⎮ ⎮ ╲⎮ ╳ ⎮╱ ⎮
// ⎮ ╲╱ ╲╱ ⎮ ⎮ ╲╱ ╲╱ ⎮
// ⎮ ╱╲ ╱╲ ⎮ ⎮ ╱╲ ╱╲ ⎮
// ⎮ ╱⎮ ╳ ⎮╲ ⎮ ⎮ ╱⎮ ╳ ⎮╲ ⎮
// ⎮⎛ ⎮⎛ ⎞⎮ ⎞⎮ ⎮⎛ ⎮⎛ ⎞⎮ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮
// ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮
// ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮
// 01 23 01 23 01 23 01 23
// sort pairs (one min/max)
auto x = AVX::lo128(x_.data());
auto y = AVX::hi128(x_.data());
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
auto l = _mm_min_epi16(x, y);
auto h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge left & right quads (two min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "8x2 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge quads into octs (three min/max)
x = _mm_unpacklo_epi16(h, l);
y = _mm_unpackhi_epi16(l, h);
// Vc_DEBUG << "4x4 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X2, X3, X0, X1>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(h, l));
y = Mem::permuteHi<X6, X7, X4, X5>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(l, h));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h) << " done?";
// merge octs into hexa (four min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "2x8 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi64(l, h);
y = _mm_unpackhi_epi64(l, h);
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_castps_si128(Mem::permute<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(h), _mm_castsi128_ps(l))));
y = _mm_castps_si128(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(l), _mm_castsi128_ps(h)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h);
y = Mem::permuteLo<X1, X0, X3, X2>(
Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(l, h);
return AVX::concat(x, y);
}
template <>
Vc_CONST AVX2::ushort_v sorted<CurrentImplementation::current()>(AVX2::ushort_v x_)
{
// sort pairs (one min/max)
auto x = AVX::lo128(x_.data());
auto y = AVX::hi128(x_.data());
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
auto l = _mm_min_epu16(x, y);
auto h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge left & right quads (two min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "8x2 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge quads into octs (three min/max)
x = _mm_unpacklo_epi16(h, l);
y = _mm_unpackhi_epi16(l, h);
// Vc_DEBUG << "4x4 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X2, X3, X0, X1>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(h, l));
y = Mem::permuteHi<X6, X7, X4, X5>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(l, h));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h) << " done?";
// merge octs into hexa (four min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "2x8 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi64(l, h);
y = _mm_unpackhi_epi64(l, h);
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_castps_si128(Mem::permute<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(h), _mm_castsi128_ps(l))));
y = _mm_castps_si128(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(l), _mm_castsi128_ps(h)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h);
y = Mem::permuteLo<X1, X0, X3, X2>(
Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(l, h);
return AVX::concat(x, y);
}
template <> Vc_CONST AVX2::int_v sorted<CurrentImplementation::current()>(AVX2::int_v x_)
{
using namespace AVX;
const __m256i hgfedcba = x_.data();
const __m128i hgfe = hi128(hgfedcba);
const __m128i dcba = lo128(hgfedcba);
__m128i l = _mm_min_epi32(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128i h = _mm_max_epi32(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128i x = _mm_unpacklo_epi32(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128i y = _mm_unpackhi_epi32(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_epi32(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_epi32(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_epi32(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_epi32(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128i b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
__m128i a = _mm_unpackhi_epi64(x, y); // a3 >= a2 >= a1 >= a0
// _mm_extract_epi32 may return an unsigned int, breaking these comparisons.
if (Vc_IS_UNLIKELY(static_cast<int>(_mm_extract_epi32(x, 2)) >= static_cast<int>(_mm_extract_epi32(y, 1)))) {
return concat(Reg::permute<X0, X1, X2, X3>(b), a);
} else if (Vc_IS_UNLIKELY(static_cast<int>(_mm_extract_epi32(x, 0)) >= static_cast<int>(_mm_extract_epi32(y, 3)))) {
return concat(a, Reg::permute<X0, X1, X2, X3>(b));
}
// merge
l = _mm_min_epi32(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_epi32(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_epi32(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_epi32(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_epi32(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_epi32(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_epi32(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_epi32(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_epi32(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2) ↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_epi32(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2) ↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return concat(_mm_unpacklo_epi32(l, h), _mm_unpackhi_epi32(l, h));
}
template <>
Vc_CONST AVX2::uint_v sorted<CurrentImplementation::current()>(AVX2::uint_v x_)
{
using namespace AVX;
const __m256i hgfedcba = x_.data();
const __m128i hgfe = hi128(hgfedcba);
const __m128i dcba = lo128(hgfedcba);
__m128i l = _mm_min_epu32(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128i h = _mm_max_epu32(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128i x = _mm_unpacklo_epi32(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128i y = _mm_unpackhi_epi32(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_epu32(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_epu32(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_epu32(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_epu32(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128i b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
__m128i a = _mm_unpackhi_epi64(x, y); // a3 >= a2 >= a1 >= a0
if (Vc_IS_UNLIKELY(extract_epu32<2>(x) >= extract_epu32<1>(y))) {
return concat(Reg::permute<X0, X1, X2, X3>(b), a);
} else if (Vc_IS_UNLIKELY(extract_epu32<0>(x) >= extract_epu32<3>(y))) {
return concat(a, Reg::permute<X0, X1, X2, X3>(b));
}
// merge
l = _mm_min_epu32(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_epu32(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_epi32(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_epi32(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_epu32(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_epu32(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_epi32(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_epi32(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_epu32(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2) ↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_epu32(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2) ↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return concat(_mm_unpacklo_epi32(l, h), _mm_unpackhi_epi32(l, h));
}
#endif // AVX2
template <>
Vc_CONST AVX2::float_v sorted<CurrentImplementation::current()>(AVX2::float_v x_)
{
__m256 hgfedcba = x_.data();
const __m128 hgfe = AVX::hi128(hgfedcba);
const __m128 dcba = AVX::lo128(hgfedcba);
__m128 l = _mm_min_ps(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128 h = _mm_max_ps(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128 x = _mm_unpacklo_ps(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128 y = _mm_unpackhi_ps(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_ps(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_ps(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_ps(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_ps(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128 a = _mm_castpd_ps(_mm_unpackhi_pd(_mm_castps_pd(x), _mm_castps_pd(y))); // a3 >= a2 >= a1 >= a0
__m128 b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
// merge
l = _mm_min_ps(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_ps(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_ps(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_ps(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_ps(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_ps(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_ps(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_ps(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_ps(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2),↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_ps(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2),↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return AVX::concat(_mm_unpacklo_ps(l, h), _mm_unpackhi_ps(l, h));
}
#if 0
template<> void SortHelper<double>::sort(__m256d &Vc_RESTRICT x, __m256d &Vc_RESTRICT y)
{
__m256d l = _mm256_min_pd(x, y); // ↓x3y3 ↓x2y2 ↓x1y1 ↓x0y0
__m256d h = _mm256_max_pd(x, y); // ↑x3y3 ↑x2y2 ↑x1y1 ↑x0y0
x = _mm256_unpacklo_pd(l, h); // ↑x2y2 ↓x2y2 ↑x0y0 ↓x0y0
y = _mm256_unpackhi_pd(l, h); // ↑x3y3 ↓x3y3 ↑x1y1 ↓x1y1
l = _mm256_min_pd(x, y); // ↓(↑x2y2,↑x3y3) ↓x3x2y3y2 ↓(↑x0y0,↑x1y1) ↓x1x0y1y0
h = _mm256_max_pd(x, y); // ↑x3x2y3y2 ↑(↓x2y2,↓x3y3) ↑x1x0y1y0 ↑(↓x0y0,↓x1y1)
x = _mm256_unpacklo_pd(l, h); // ↑(↓x2y2,↓x3y3) ↓x3x2y3y2 ↑(↓x0y0,↓x1y1) ↓x1x0y1y0
y = _mm256_unpackhi_pd(h, l); // ↓(↑x2y2,↑x3y3) ↑x3x2y3y2 ↓(↑x0y0,↑x1y1) ↑x1x0y1y0
l = _mm256_min_pd(x, y); // ↓(↑(↓x2y2,↓x3y3) ↓(↑x2y2,↑x3y3)) ↓x3x2y3y2 ↓(↑(↓x0y0,↓x1y1) ↓(↑x0y0,↑x1y1)) ↓x1x0y1y0
h = _mm256_max_pd(x, y); // ↑(↑(↓x2y2,↓x3y3) ↓(↑x2y2,↑x3y3)) ↑x3x2y3y2 ↑(↑(↓x0y0,↓x1y1) ↓(↑x0y0,↑x1y1)) ↑x1x0y1y0
__m256d a = Reg::permute<X2, X3, X1, X0>(Reg::permute128<X0, X1>(h, h)); // h0 h1 h3 h2
__m256d b = Reg::permute<X2, X3, X1, X0>(l); // l2 l3 l1 l0
// a3 >= a2 >= b1 >= b0
// b3 <= b2 <= a1 <= a0
// merge
l = _mm256_min_pd(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm256_min_pd(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
x = _mm256_unpacklo_pd(l, h); // ↑a2b2 ↓a2b2 ↑a0b0 ↓a0b0
y = _mm256_unpackhi_pd(l, h); // ↑a3b3 ↓a3b3 ↑a1b1 ↓a1b1
l = _mm256_min_pd(x, y); // ↓(↑a2b2,↑a3b3) ↓a2b3 ↓(↑a0b0,↑a1b1) ↓a1b0
h = _mm256_min_pd(x, y); // ↑a3b2 ↑(↓a2b2,↓a3b3) ↑a0b1 ↑(↓a0b0,↓a1b1)
x = Reg::permute128<Y0, X0>(l, h); // ↑a0b1 ↑(↓a0b0,↓a1b1) ↓(↑a0b0,↑a1b1) ↓a1b0
y = Reg::permute128<Y1, X1>(l, h); // ↑a3b2 ↑(↓a2b2,↓a3b3) ↓(↑a2b2,↑a3b3) ↓a2b3
l = _mm256_min_pd(x, y); // ↓(↑a0b1,↑a3b2) ↓(↑(↓a0b0,↓a1b1) ↑(↓a2b2,↓a3b3)) ↓(↑a0b0,↑a1b1,↑a2b2,↑a3b3) ↓b0b3
h = _mm256_min_pd(x, y); // ↑a0a3 ↑(↓a0b0,↓a1b1,↓a2b2,↓a3b3) ↑(↓(↑a0b0,↑a1b1) ↓(↑a2b2,↑a3b3)) ↑(↓a1b0,↓a2b3)
x = _mm256_unpacklo_pd(l, h); // h2 l2 h0 l0
y = _mm256_unpackhi_pd(l, h); // h3 l3 h1 l1
}
#endif
template <>
Vc_CONST AVX2::double_v sorted<CurrentImplementation::current()>(AVX2::double_v x_)
{
__m256d dcba = x_.data();
/*
* to find the second largest number find
* max(min(max(ab),max(cd)), min(max(ad),max(bc)))
* or
* max(max(min(ab),min(cd)), min(max(ab),max(cd)))
*
const __m256d adcb = avx_cast<__m256d>(AVX::concat(_mm_alignr_epi8(avx_cast<__m128i>(dc), avx_cast<__m128i>(ba), 8), _mm_alignr_epi8(avx_cast<__m128i>(ba), avx_cast<__m128i>(dc), 8)));
const __m256d l = _mm256_min_pd(dcba, adcb); // min(ad cd bc ab)
const __m256d h = _mm256_max_pd(dcba, adcb); // max(ad cd bc ab)
// max(h3, h1)
// max(min(h0,h2), min(h3,h1))
// min(max(l0,l2), max(l3,l1))
// min(l3, l1)
const __m256d ll = _mm256_min_pd(h, Reg::permute128<X0, X1>(h, h)); // min(h3h1 h2h0 h1h3 h0h2)
//const __m256d hh = _mm256_max_pd(h3 ll1_3 l1 l0, h1 ll0_2 l3 l2);
const __m256d hh = _mm256_max_pd(
Reg::permute128<X1, Y0>(_mm256_unpackhi_pd(ll, h), l),
Reg::permute128<X0, Y1>(_mm256_blend_pd(h ll, 0x1), l));
_mm256_min_pd(hh0, hh1
*/
//////////////////////////////////////////////////////////////////////////////////
// max(max(ac), max(bd))
// max(max(min(ac),min(bd)), min(max(ac),max(bd)))
// min(max(min(ac),min(bd)), min(max(ac),max(bd)))
// min(min(ac), min(bd))
__m128d l = _mm_min_pd(AVX::lo128(dcba), AVX::hi128(dcba)); // min(bd) min(ac)
__m128d h = _mm_max_pd(AVX::lo128(dcba), AVX::hi128(dcba)); // max(bd) max(ac)
__m128d h0_l0 = _mm_unpacklo_pd(l, h);
__m128d h1_l1 = _mm_unpackhi_pd(l, h);
l = _mm_min_pd(h0_l0, h1_l1);
h = _mm_max_pd(h0_l0, h1_l1);
return AVX::concat(
_mm_min_pd(l, Reg::permute<X0, X0>(h)),
_mm_max_pd(h, Reg::permute<X1, X1>(l))
);
// extract: 1 cycle
// min/max: 4 cycles
// unpacklo/hi: 2 cycles
// min/max: 4 cycles
// permute: 1 cycle
// min/max: 4 cycles
// insert: 1 cycle
// ----------------------
// total: 17 cycles
/*
__m256d cdab = Reg::permute<X2, X3, X0, X1>(dcba);
__m256d l = _mm256_min_pd(dcba, cdab);
__m256d h = _mm256_max_pd(dcba, cdab);
__m256d maxmin_ba = Reg::permute128<X0, Y0>(l, h);
__m256d maxmin_dc = Reg::permute128<X1, Y1>(l, h);
l = _mm256_min_pd(maxmin_ba, maxmin_dc);
h = _mm256_max_pd(maxmin_ba, maxmin_dc);
return _mm256_blend_pd(h, l, 0x55);
*/
/*
// a b c d
// b a d c
// sort pairs
__m256d y, l, h;
__m128d l2, h2;
y = shuffle<X1, Y0, X3, Y2>(x, x);
l = _mm256_min_pd(x, y); // min[ab ab cd cd]
h = _mm256_max_pd(x, y); // max[ab ab cd cd]
// 1 of 2 is at [0]
// 1 of 4 is at [1]
// 1 of 4 is at [2]
// 1 of 2 is at [3]
// don't be fooled by unpack here. It works differently for AVX pd than for SSE ps
x = _mm256_unpacklo_pd(l, h); // l_ab h_ab l_cd h_cd
l2 = _mm_min_pd(AVX::lo128(x), AVX::hi128(x)); // l_abcd l(h_ab hcd)
h2 = _mm_max_pd(AVX::lo128(x), AVX::hi128(x)); // h(l_ab l_cd) h_abcd
// either it is:
return AVX::concat(l2, h2);
// or:
// AVX::concat(_mm_unpacklo_pd(l2, h2), _mm_unpackhi_pd(l2, h2));
// I'd like to have four useful compares
const __m128d dc = AVX::hi128(dcba);
const __m128d ba = AVX::lo128(dcba);
const __m256d adcb = avx_cast<__m256d>(AVX::concat(_mm_alignr_epi8(avx_cast<__m128i>(dc), avx_cast<__m128i>(ba), 8), _mm_alignr_epi8(avx_cast<__m128i>(ba), avx_cast<__m128i>(dc), 8)));
const int extraCmp = _mm_movemask_pd(_mm_cmpgt_pd(dc, ba));
// 0x0: d <= b && c <= a
// 0x1: d <= b && c > a
// 0x2: d > b && c <= a
// 0x3: d > b && c > a
switch (_mm256_movemask_pd(_mm256_cmpgt_pd(dcba, adcb))) {
// impossible: 0x0, 0xf
case 0x1: // a <= b && b <= c && c <= d && d > a
// abcd
return Reg::permute<X2, X3, X0, X1>(Reg::permute<X0, X1>(dcba, dcba));
case 0x2: // a <= b && b <= c && c > d && d <= a
// dabc
return Reg::permute<X2, X3, X0, X1>(adcb);
case 0x3: // a <= b && b <= c && c > d && d > a
// a[bd]c
if (extraCmp & 2) {
// abdc
return Reg::permute<X2, X3, X1, X0>(Reg::permute<X0, X1>(dcba, dcba));
} else {
// adbc
return Reg::permute<X3, X2, X0, X1>(adcb);
}
case 0x4: // a <= b && b > c && c <= d && d <= a
// cdab;
return Reg::permute<X2, X3, X0, X1>(dcba);
case 0x5: // a <= b && b > c && c <= d && d > a
// [ac] < [bd]
switch (extraCmp) {
case 0x0: // d <= b && c <= a
// cadb
return shuffle<>(dcba, bcda);
case 0x1: // d <= b && c > a
case 0x2: // d > b && c <= a
case 0x3: // d > b && c > a
}
case 0x6: // a <= b && b > c && c > d && d <= a
// d[ac]b
case 0x7: // a <= b && b > c && c > d && d > a
// adcb;
return permute<X1, X0, X3, X2>(permute128<X1, X0>(bcda, bcda));
case 0x8: // a > b && b <= c && c <= d && d <= a
return bcda;
case 0x9: // a > b && b <= c && c <= d && d > a
// b[ac]d;
case 0xa: // a > b && b <= c && c > d && d <= a
// [ac] > [bd]
case 0xb: // a > b && b <= c && c > d && d > a
// badc;
return permute128<X1, X0>(dcba);
case 0xc: // a > b && b > c && c <= d && d <= a
// c[bd]a;
case 0xd: // a > b && b > c && c <= d && d > a
// cbad;
return permute<X1, X0, X3, X2>(bcda);
case 0xe: // a > b && b > c && c > d && d <= a
return dcba;
}
*/
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2011-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/avx/vector.h>
#include <Vc/avx/debug.h>
#include <Vc/avx/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
#ifdef Vc_IMPL_AVX2
template <>
Vc_CONST AVX2::short_v sorted<CurrentImplementation::current()>(AVX2::short_v x_)
{
// ab cd ef gh ij kl mn op
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮
// ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮
// ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮
// 01 23 01 23 01 23 01 23
// ⎮⎮ ⎮⎮ ╳ ⎮⎮ ⎮⎮ ╳
// 01 23 32 10 01 23 32 10
// ⎮⎝ ⎮⎝ ⎠⎮ ⎠⎮ ⎮⎝ ⎮⎝ ⎠⎮ ⎠⎮
// ⎮ ╲⎮ ╳ ⎮╱ ⎮ ⎮ ╲⎮ ╳ ⎮╱ ⎮
// ⎮ ╲╱ ╲╱ ⎮ ⎮ ╲╱ ╲╱ ⎮
// ⎮ ╱╲ ╱╲ ⎮ ⎮ ╱╲ ╱╲ ⎮
// ⎮ ╱⎮ ╳ ⎮╲ ⎮ ⎮ ╱⎮ ╳ ⎮╲ ⎮
// ⎮⎛ ⎮⎛ ⎞⎮ ⎞⎮ ⎮⎛ ⎮⎛ ⎞⎮ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮ ⎮ ╳ ⎮
// ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮ ⎮⎛ ⎞⎮
// <> <> <> <> <> <> <> <>
// ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑
// ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮ ⎮⎝ ⎠⎮
// ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮ ⎮ o ⎮
// ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮ ⎮↓ ↑⎮
// 01 23 01 23 01 23 01 23
// sort pairs (one min/max)
auto x = AVX::lo128(x_.data());
auto y = AVX::hi128(x_.data());
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
auto l = _mm_min_epi16(x, y);
auto h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge left & right quads (two min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "8x2 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge quads into octs (three min/max)
x = _mm_unpacklo_epi16(h, l);
y = _mm_unpackhi_epi16(l, h);
// Vc_DEBUG << "4x4 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X2, X3, X0, X1>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(h, l));
y = Mem::permuteHi<X6, X7, X4, X5>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(l, h));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h) << " done?";
// merge octs into hexa (four min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "2x8 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi64(l, h);
y = _mm_unpackhi_epi64(l, h);
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_castps_si128(Mem::permute<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(h), _mm_castsi128_ps(l))));
y = _mm_castps_si128(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(l), _mm_castsi128_ps(h)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h);
y = Mem::permuteLo<X1, X0, X3, X2>(
Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epi16(x, y);
h = _mm_max_epi16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(l, h);
return AVX::concat(x, y);
}
template <>
Vc_CONST AVX2::ushort_v sorted<CurrentImplementation::current()>(AVX2::ushort_v x_)
{
// sort pairs (one min/max)
auto x = AVX::lo128(x_.data());
auto y = AVX::hi128(x_.data());
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
auto l = _mm_min_epu16(x, y);
auto h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge left & right quads (two min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "8x2 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
// merge quads into octs (three min/max)
x = _mm_unpacklo_epi16(h, l);
y = _mm_unpackhi_epi16(l, h);
// Vc_DEBUG << "4x4 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteLo<X2, X3, X0, X1>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(h, l));
y = Mem::permuteHi<X6, X7, X4, X5>(Mem::blend<X0, X1, Y2, Y3, X4, X5, Y6, Y7>(l, h));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h));
y = Mem::permuteLo<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h) << " done?";
// merge octs into hexa (four min/max)
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(h, l);
// Vc_DEBUG << "2x8 sorted xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi64(l, h);
y = _mm_unpackhi_epi64(l, h);
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_castps_si128(Mem::permute<X1, X0, X3, X2>(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(h), _mm_castsi128_ps(l))));
y = _mm_castps_si128(Mem::blend<X0, Y1, X2, Y3>(_mm_castsi128_ps(l), _mm_castsi128_ps(h)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(l, h);
y = Mem::permuteLo<X1, X0, X3, X2>(
Mem::permuteHi<X5, X4, X7, X6>(Mem::blend<X0, Y1, X2, Y3, X4, Y5, X6, Y7>(h, l)));
// Vc_DEBUG << "xy: " << AVX::addType<short>(x) << AVX::addType<short>(y);
l = _mm_min_epu16(x, y);
h = _mm_max_epu16(x, y);
// Vc_DEBUG << "lh: " << AVX::addType<short>(l) << AVX::addType<short>(h);
x = _mm_unpacklo_epi16(l, h);
y = _mm_unpackhi_epi16(l, h);
return AVX::concat(x, y);
}
template <> Vc_CONST AVX2::int_v sorted<CurrentImplementation::current()>(AVX2::int_v x_)
{
using namespace AVX;
const __m256i hgfedcba = x_.data();
const __m128i hgfe = hi128(hgfedcba);
const __m128i dcba = lo128(hgfedcba);
__m128i l = _mm_min_epi32(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128i h = _mm_max_epi32(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128i x = _mm_unpacklo_epi32(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128i y = _mm_unpackhi_epi32(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_epi32(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_epi32(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_epi32(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_epi32(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128i b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
__m128i a = _mm_unpackhi_epi64(x, y); // a3 >= a2 >= a1 >= a0
// _mm_extract_epi32 may return an unsigned int, breaking these comparisons.
if (Vc_IS_UNLIKELY(static_cast<int>(_mm_extract_epi32(x, 2)) >= static_cast<int>(_mm_extract_epi32(y, 1)))) {
return concat(Reg::permute<X0, X1, X2, X3>(b), a);
} else if (Vc_IS_UNLIKELY(static_cast<int>(_mm_extract_epi32(x, 0)) >= static_cast<int>(_mm_extract_epi32(y, 3)))) {
return concat(a, Reg::permute<X0, X1, X2, X3>(b));
}
// merge
l = _mm_min_epi32(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_epi32(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_epi32(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_epi32(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_epi32(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_epi32(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_epi32(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_epi32(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_epi32(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2) ↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_epi32(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2) ↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return concat(_mm_unpacklo_epi32(l, h), _mm_unpackhi_epi32(l, h));
}
template <>
Vc_CONST AVX2::uint_v sorted<CurrentImplementation::current()>(AVX2::uint_v x_)
{
using namespace AVX;
const __m256i hgfedcba = x_.data();
const __m128i hgfe = hi128(hgfedcba);
const __m128i dcba = lo128(hgfedcba);
__m128i l = _mm_min_epu32(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128i h = _mm_max_epu32(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128i x = _mm_unpacklo_epi32(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128i y = _mm_unpackhi_epi32(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_epu32(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_epu32(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_epu32(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_epu32(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128i b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
__m128i a = _mm_unpackhi_epi64(x, y); // a3 >= a2 >= a1 >= a0
if (Vc_IS_UNLIKELY(extract_epu32<2>(x) >= extract_epu32<1>(y))) {
return concat(Reg::permute<X0, X1, X2, X3>(b), a);
} else if (Vc_IS_UNLIKELY(extract_epu32<0>(x) >= extract_epu32<3>(y))) {
return concat(a, Reg::permute<X0, X1, X2, X3>(b));
}
// merge
l = _mm_min_epu32(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_epu32(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_epi32(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_epi32(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_epu32(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_epu32(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_epi32(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_epi32(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_epu32(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2) ↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_epu32(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2) ↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return concat(_mm_unpacklo_epi32(l, h), _mm_unpackhi_epi32(l, h));
}
#endif // AVX2
template <>
Vc_CONST AVX2::float_v sorted<CurrentImplementation::current()>(AVX2::float_v x_)
{
__m256 hgfedcba = x_.data();
const __m128 hgfe = AVX::hi128(hgfedcba);
const __m128 dcba = AVX::lo128(hgfedcba);
__m128 l = _mm_min_ps(hgfe, dcba); // ↓hd ↓gc ↓fb ↓ea
__m128 h = _mm_max_ps(hgfe, dcba); // ↑hd ↑gc ↑fb ↑ea
__m128 x = _mm_unpacklo_ps(l, h); // ↑fb ↓fb ↑ea ↓ea
__m128 y = _mm_unpackhi_ps(l, h); // ↑hd ↓hd ↑gc ↓gc
l = _mm_min_ps(x, y); // ↓(↑fb,↑hd) ↓hfdb ↓(↑ea,↑gc) ↓geca
h = _mm_max_ps(x, y); // ↑hfdb ↑(↓fb,↓hd) ↑geca ↑(↓ea,↓gc)
x = _mm_min_ps(l, Reg::permute<X2, X2, X0, X0>(h)); // 2(hfdb) 1(hfdb) 2(geca) 1(geca)
y = _mm_max_ps(h, Reg::permute<X3, X3, X1, X1>(l)); // 4(hfdb) 3(hfdb) 4(geca) 3(geca)
__m128 a = _mm_castpd_ps(_mm_unpackhi_pd(_mm_castps_pd(x), _mm_castps_pd(y))); // a3 >= a2 >= a1 >= a0
__m128 b = Reg::shuffle<Y0, Y1, X0, X1>(y, x); // b3 <= b2 <= b1 <= b0
// merge
l = _mm_min_ps(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm_max_ps(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
a = _mm_unpacklo_ps(l, h); // ↑a1b1 ↓a1b1 ↑a0b0 ↓a0b0
b = _mm_unpackhi_ps(l, h); // ↑a3b3 ↓a3b3 ↑a2b2 ↓a2b2
l = _mm_min_ps(a, b); // ↓(↑a1b1,↑a3b3) ↓a1b3 ↓(↑a0b0,↑a2b2) ↓a0b2
h = _mm_max_ps(a, b); // ↑a3b1 ↑(↓a1b1,↓a3b3) ↑a2b0 ↑(↓a0b0,↓a2b2)
a = _mm_unpacklo_ps(l, h); // ↑a2b0 ↓(↑a0b0,↑a2b2) ↑(↓a0b0,↓a2b2) ↓a0b2
b = _mm_unpackhi_ps(l, h); // ↑a3b1 ↓(↑a1b1,↑a3b3) ↑(↓a1b1,↓a3b3) ↓a1b3
l = _mm_min_ps(a, b); // ↓(↑a2b0,↑a3b1) ↓(↑a0b0,↑a2b2,↑a1b1,↑a3b3) ↓(↑(↓a0b0,↓a2b2),↑(↓a1b1,↓a3b3)) ↓a0b3
h = _mm_max_ps(a, b); // ↑a3b0 ↑(↓(↑a0b0,↑a2b2),↓(↑a1b1,↑a3b3)) ↑(↓a0b0,↓a2b2,↓a1b1,↓a3b3) ↑(↓a0b2,↓a1b3)
return AVX::concat(_mm_unpacklo_ps(l, h), _mm_unpackhi_ps(l, h));
}
#if 0
template<> void SortHelper<double>::sort(__m256d &Vc_RESTRICT x, __m256d &Vc_RESTRICT y)
{
__m256d l = _mm256_min_pd(x, y); // ↓x3y3 ↓x2y2 ↓x1y1 ↓x0y0
__m256d h = _mm256_max_pd(x, y); // ↑x3y3 ↑x2y2 ↑x1y1 ↑x0y0
x = _mm256_unpacklo_pd(l, h); // ↑x2y2 ↓x2y2 ↑x0y0 ↓x0y0
y = _mm256_unpackhi_pd(l, h); // ↑x3y3 ↓x3y3 ↑x1y1 ↓x1y1
l = _mm256_min_pd(x, y); // ↓(↑x2y2,↑x3y3) ↓x3x2y3y2 ↓(↑x0y0,↑x1y1) ↓x1x0y1y0
h = _mm256_max_pd(x, y); // ↑x3x2y3y2 ↑(↓x2y2,↓x3y3) ↑x1x0y1y0 ↑(↓x0y0,↓x1y1)
x = _mm256_unpacklo_pd(l, h); // ↑(↓x2y2,↓x3y3) ↓x3x2y3y2 ↑(↓x0y0,↓x1y1) ↓x1x0y1y0
y = _mm256_unpackhi_pd(h, l); // ↓(↑x2y2,↑x3y3) ↑x3x2y3y2 ↓(↑x0y0,↑x1y1) ↑x1x0y1y0
l = _mm256_min_pd(x, y); // ↓(↑(↓x2y2,↓x3y3) ↓(↑x2y2,↑x3y3)) ↓x3x2y3y2 ↓(↑(↓x0y0,↓x1y1) ↓(↑x0y0,↑x1y1)) ↓x1x0y1y0
h = _mm256_max_pd(x, y); // ↑(↑(↓x2y2,↓x3y3) ↓(↑x2y2,↑x3y3)) ↑x3x2y3y2 ↑(↑(↓x0y0,↓x1y1) ↓(↑x0y0,↑x1y1)) ↑x1x0y1y0
__m256d a = Reg::permute<X2, X3, X1, X0>(Reg::permute128<X0, X1>(h, h)); // h0 h1 h3 h2
__m256d b = Reg::permute<X2, X3, X1, X0>(l); // l2 l3 l1 l0
// a3 >= a2 >= b1 >= b0
// b3 <= b2 <= a1 <= a0
// merge
l = _mm256_min_pd(a, b); // ↓a3b3 ↓a2b2 ↓a1b1 ↓a0b0
h = _mm256_min_pd(a, b); // ↑a3b3 ↑a2b2 ↑a1b1 ↑a0b0
x = _mm256_unpacklo_pd(l, h); // ↑a2b2 ↓a2b2 ↑a0b0 ↓a0b0
y = _mm256_unpackhi_pd(l, h); // ↑a3b3 ↓a3b3 ↑a1b1 ↓a1b1
l = _mm256_min_pd(x, y); // ↓(↑a2b2,↑a3b3) ↓a2b3 ↓(↑a0b0,↑a1b1) ↓a1b0
h = _mm256_min_pd(x, y); // ↑a3b2 ↑(↓a2b2,↓a3b3) ↑a0b1 ↑(↓a0b0,↓a1b1)
x = Reg::permute128<Y0, X0>(l, h); // ↑a0b1 ↑(↓a0b0,↓a1b1) ↓(↑a0b0,↑a1b1) ↓a1b0
y = Reg::permute128<Y1, X1>(l, h); // ↑a3b2 ↑(↓a2b2,↓a3b3) ↓(↑a2b2,↑a3b3) ↓a2b3
l = _mm256_min_pd(x, y); // ↓(↑a0b1,↑a3b2) ↓(↑(↓a0b0,↓a1b1) ↑(↓a2b2,↓a3b3)) ↓(↑a0b0,↑a1b1,↑a2b2,↑a3b3) ↓b0b3
h = _mm256_min_pd(x, y); // ↑a0a3 ↑(↓a0b0,↓a1b1,↓a2b2,↓a3b3) ↑(↓(↑a0b0,↑a1b1) ↓(↑a2b2,↑a3b3)) ↑(↓a1b0,↓a2b3)
x = _mm256_unpacklo_pd(l, h); // h2 l2 h0 l0
y = _mm256_unpackhi_pd(l, h); // h3 l3 h1 l1
}
#endif
template <>
Vc_CONST AVX2::double_v sorted<CurrentImplementation::current()>(AVX2::double_v x_)
{
__m256d dcba = x_.data();
/*
* to find the second largest number find
* max(min(max(ab),max(cd)), min(max(ad),max(bc)))
* or
* max(max(min(ab),min(cd)), min(max(ab),max(cd)))
*
const __m256d adcb = avx_cast<__m256d>(AVX::concat(_mm_alignr_epi8(avx_cast<__m128i>(dc), avx_cast<__m128i>(ba), 8), _mm_alignr_epi8(avx_cast<__m128i>(ba), avx_cast<__m128i>(dc), 8)));
const __m256d l = _mm256_min_pd(dcba, adcb); // min(ad cd bc ab)
const __m256d h = _mm256_max_pd(dcba, adcb); // max(ad cd bc ab)
// max(h3, h1)
// max(min(h0,h2), min(h3,h1))
// min(max(l0,l2), max(l3,l1))
// min(l3, l1)
const __m256d ll = _mm256_min_pd(h, Reg::permute128<X0, X1>(h, h)); // min(h3h1 h2h0 h1h3 h0h2)
//const __m256d hh = _mm256_max_pd(h3 ll1_3 l1 l0, h1 ll0_2 l3 l2);
const __m256d hh = _mm256_max_pd(
Reg::permute128<X1, Y0>(_mm256_unpackhi_pd(ll, h), l),
Reg::permute128<X0, Y1>(_mm256_blend_pd(h ll, 0x1), l));
_mm256_min_pd(hh0, hh1
*/
//////////////////////////////////////////////////////////////////////////////////
// max(max(ac), max(bd))
// max(max(min(ac),min(bd)), min(max(ac),max(bd)))
// min(max(min(ac),min(bd)), min(max(ac),max(bd)))
// min(min(ac), min(bd))
__m128d l = _mm_min_pd(AVX::lo128(dcba), AVX::hi128(dcba)); // min(bd) min(ac)
__m128d h = _mm_max_pd(AVX::lo128(dcba), AVX::hi128(dcba)); // max(bd) max(ac)
__m128d h0_l0 = _mm_unpacklo_pd(l, h);
__m128d h1_l1 = _mm_unpackhi_pd(l, h);
l = _mm_min_pd(h0_l0, h1_l1);
h = _mm_max_pd(h0_l0, h1_l1);
return AVX::concat(
_mm_min_pd(l, Reg::permute<X0, X0>(h)),
_mm_max_pd(h, Reg::permute<X1, X1>(l))
);
// extract: 1 cycle
// min/max: 4 cycles
// unpacklo/hi: 2 cycles
// min/max: 4 cycles
// permute: 1 cycle
// min/max: 4 cycles
// insert: 1 cycle
// ----------------------
// total: 17 cycles
/*
__m256d cdab = Reg::permute<X2, X3, X0, X1>(dcba);
__m256d l = _mm256_min_pd(dcba, cdab);
__m256d h = _mm256_max_pd(dcba, cdab);
__m256d maxmin_ba = Reg::permute128<X0, Y0>(l, h);
__m256d maxmin_dc = Reg::permute128<X1, Y1>(l, h);
l = _mm256_min_pd(maxmin_ba, maxmin_dc);
h = _mm256_max_pd(maxmin_ba, maxmin_dc);
return _mm256_blend_pd(h, l, 0x55);
*/
/*
// a b c d
// b a d c
// sort pairs
__m256d y, l, h;
__m128d l2, h2;
y = shuffle<X1, Y0, X3, Y2>(x, x);
l = _mm256_min_pd(x, y); // min[ab ab cd cd]
h = _mm256_max_pd(x, y); // max[ab ab cd cd]
// 1 of 2 is at [0]
// 1 of 4 is at [1]
// 1 of 4 is at [2]
// 1 of 2 is at [3]
// don't be fooled by unpack here. It works differently for AVX pd than for SSE ps
x = _mm256_unpacklo_pd(l, h); // l_ab h_ab l_cd h_cd
l2 = _mm_min_pd(AVX::lo128(x), AVX::hi128(x)); // l_abcd l(h_ab hcd)
h2 = _mm_max_pd(AVX::lo128(x), AVX::hi128(x)); // h(l_ab l_cd) h_abcd
// either it is:
return AVX::concat(l2, h2);
// or:
// AVX::concat(_mm_unpacklo_pd(l2, h2), _mm_unpackhi_pd(l2, h2));
// I'd like to have four useful compares
const __m128d dc = AVX::hi128(dcba);
const __m128d ba = AVX::lo128(dcba);
const __m256d adcb = avx_cast<__m256d>(AVX::concat(_mm_alignr_epi8(avx_cast<__m128i>(dc), avx_cast<__m128i>(ba), 8), _mm_alignr_epi8(avx_cast<__m128i>(ba), avx_cast<__m128i>(dc), 8)));
const int extraCmp = _mm_movemask_pd(_mm_cmpgt_pd(dc, ba));
// 0x0: d <= b && c <= a
// 0x1: d <= b && c > a
// 0x2: d > b && c <= a
// 0x3: d > b && c > a
switch (_mm256_movemask_pd(_mm256_cmpgt_pd(dcba, adcb))) {
// impossible: 0x0, 0xf
case 0x1: // a <= b && b <= c && c <= d && d > a
// abcd
return Reg::permute<X2, X3, X0, X1>(Reg::permute<X0, X1>(dcba, dcba));
case 0x2: // a <= b && b <= c && c > d && d <= a
// dabc
return Reg::permute<X2, X3, X0, X1>(adcb);
case 0x3: // a <= b && b <= c && c > d && d > a
// a[bd]c
if (extraCmp & 2) {
// abdc
return Reg::permute<X2, X3, X1, X0>(Reg::permute<X0, X1>(dcba, dcba));
} else {
// adbc
return Reg::permute<X3, X2, X0, X1>(adcb);
}
case 0x4: // a <= b && b > c && c <= d && d <= a
// cdab;
return Reg::permute<X2, X3, X0, X1>(dcba);
case 0x5: // a <= b && b > c && c <= d && d > a
// [ac] < [bd]
switch (extraCmp) {
case 0x0: // d <= b && c <= a
// cadb
return shuffle<>(dcba, bcda);
case 0x1: // d <= b && c > a
case 0x2: // d > b && c <= a
case 0x3: // d > b && c > a
}
case 0x6: // a <= b && b > c && c > d && d <= a
// d[ac]b
case 0x7: // a <= b && b > c && c > d && d > a
// adcb;
return permute<X1, X0, X3, X2>(permute128<X1, X0>(bcda, bcda));
case 0x8: // a > b && b <= c && c <= d && d <= a
return bcda;
case 0x9: // a > b && b <= c && c <= d && d > a
// b[ac]d;
case 0xa: // a > b && b <= c && c > d && d <= a
// [ac] > [bd]
case 0xb: // a > b && b <= c && c > d && d > a
// badc;
return permute128<X1, X0>(dcba);
case 0xc: // a > b && b > c && c <= d && d <= a
// c[bd]a;
case 0xd: // a > b && b > c && c <= d && d > a
// cbad;
return permute<X1, X0, X3, X2>(bcda);
case 0xe: // a > b && b > c && c > d && d <= a
return dcba;
}
*/
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2009-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/sse/vector.h>
#include <Vc/sse/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
template <>
Vc_CONST SSE::short_v sorted<CurrentImplementation::current()>(SSE::short_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <>
Vc_CONST SSE::ushort_v sorted<CurrentImplementation::current()>(SSE::ushort_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <> Vc_CONST SSE::int_v sorted<CurrentImplementation::current()>(SSE::int_v x_)
{
__m128i x = x_.data();
/*
// in 16,67% of the cases the merge can be replaced by an append
// x = [a b c d]
// y = [c d a b]
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
__m128i l = SSE::min_epi32(x, y); // min[ac bd ac bd]
__m128i h = SSE::max_epi32(x, y); // max[ac bd ac bd]
if (IS_UNLIKELY(_mm_cvtsi128_si32(h) <= l[1])) { // l[0] < h[0] < l[1] < h[1]
return _mm_unpacklo_epi32(l, h);
}
// h[0] > l[1]
*/
// sort pairs
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epi32(x, y);
__m128i h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <> Vc_CONST SSE::uint_v sorted<CurrentImplementation::current()>(SSE::uint_v x_)
{
__m128i x = x_.data();
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epu32(x, y);
__m128i h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <>
Vc_CONST SSE::float_v sorted<CurrentImplementation::current()>(SSE::float_v x_)
{
__m128 x = x_.data();
__m128 y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(2, 3, 0, 1));
__m128 l = _mm_min_ps(x, y);
__m128 h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_unpackhi_ps(h, l);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_movehl_ps(x, x);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
return _mm_unpacklo_ps(l, h);
// X __m128 k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(2, 2, 0, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(1, 0, 3, 2));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(1, 0, 1, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(3, 1, 2, 0));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(0, 1, 1, 0));
// X return SSE::blendv_ps(x, y, k);
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2009-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/sse/vector.h>
#include <Vc/sse/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
template <>
Vc_CONST SSE::short_v sorted<CurrentImplementation::current()>(SSE::short_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <>
Vc_CONST SSE::ushort_v sorted<CurrentImplementation::current()>(SSE::ushort_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <> Vc_CONST SSE::int_v sorted<CurrentImplementation::current()>(SSE::int_v x_)
{
__m128i x = x_.data();
/*
// in 16,67% of the cases the merge can be replaced by an append
// x = [a b c d]
// y = [c d a b]
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
__m128i l = SSE::min_epi32(x, y); // min[ac bd ac bd]
__m128i h = SSE::max_epi32(x, y); // max[ac bd ac bd]
if (IS_UNLIKELY(_mm_cvtsi128_si32(h) <= l[1])) { // l[0] < h[0] < l[1] < h[1]
return _mm_unpacklo_epi32(l, h);
}
// h[0] > l[1]
*/
// sort pairs
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epi32(x, y);
__m128i h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <> Vc_CONST SSE::uint_v sorted<CurrentImplementation::current()>(SSE::uint_v x_)
{
__m128i x = x_.data();
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epu32(x, y);
__m128i h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <>
Vc_CONST SSE::float_v sorted<CurrentImplementation::current()>(SSE::float_v x_)
{
__m128 x = x_.data();
__m128 y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(2, 3, 0, 1));
__m128 l = _mm_min_ps(x, y);
__m128 h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_unpackhi_ps(h, l);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_movehl_ps(x, x);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
return _mm_unpacklo_ps(l, h);
// X __m128 k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(2, 2, 0, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(1, 0, 3, 2));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(1, 0, 1, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(3, 1, 2, 0));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(0, 1, 1, 0));
// X return SSE::blendv_ps(x, y, k);
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2009-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/sse/vector.h>
#include <Vc/sse/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
template <>
Vc_CONST SSE::short_v sorted<CurrentImplementation::current()>(SSE::short_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <>
Vc_CONST SSE::ushort_v sorted<CurrentImplementation::current()>(SSE::ushort_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <> Vc_CONST SSE::int_v sorted<CurrentImplementation::current()>(SSE::int_v x_)
{
__m128i x = x_.data();
/*
// in 16,67% of the cases the merge can be replaced by an append
// x = [a b c d]
// y = [c d a b]
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
__m128i l = SSE::min_epi32(x, y); // min[ac bd ac bd]
__m128i h = SSE::max_epi32(x, y); // max[ac bd ac bd]
if (IS_UNLIKELY(_mm_cvtsi128_si32(h) <= l[1])) { // l[0] < h[0] < l[1] < h[1]
return _mm_unpacklo_epi32(l, h);
}
// h[0] > l[1]
*/
// sort pairs
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epi32(x, y);
__m128i h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <> Vc_CONST SSE::uint_v sorted<CurrentImplementation::current()>(SSE::uint_v x_)
{
__m128i x = x_.data();
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epu32(x, y);
__m128i h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <>
Vc_CONST SSE::float_v sorted<CurrentImplementation::current()>(SSE::float_v x_)
{
__m128 x = x_.data();
__m128 y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(2, 3, 0, 1));
__m128 l = _mm_min_ps(x, y);
__m128 h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_unpackhi_ps(h, l);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_movehl_ps(x, x);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
return _mm_unpacklo_ps(l, h);
// X __m128 k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(2, 2, 0, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(1, 0, 3, 2));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(1, 0, 1, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(3, 1, 2, 0));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(0, 1, 1, 0));
// X return SSE::blendv_ps(x, y, k);
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2009-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
#include <Vc/sse/vector.h>
#include <Vc/sse/macros.h>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Detail
{
template <>
Vc_CONST SSE::short_v sorted<CurrentImplementation::current()>(SSE::short_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = _mm_min_epi16(x, y);
hi = _mm_max_epi16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <>
Vc_CONST SSE::ushort_v sorted<CurrentImplementation::current()>(SSE::ushort_v x_)
{
__m128i lo, hi, y, x = x_.data();
// sort pairs
y = Mem::permute<X1, X0, X3, X2, X5, X4, X7, X6>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, hi);
// merge left and right quads
y = Mem::permute<X3, X2, X1, X0, X7, X6, X5, X4>(x);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xcc>(lo, hi);
y = _mm_srli_si128(x, 2);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = SSE::blend_epi16<0xaa>(lo, _mm_slli_si128(hi, 2));
// merge quads into octs
y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
y = _mm_shufflelo_epi16(y, _MM_SHUFFLE(0, 1, 2, 3));
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
x = _mm_unpacklo_epi16(lo, hi);
y = _mm_srli_si128(x, 8);
lo = SSE::min_epu16(x, y);
hi = SSE::max_epu16(x, y);
return _mm_unpacklo_epi16(lo, hi);
}
template <> Vc_CONST SSE::int_v sorted<CurrentImplementation::current()>(SSE::int_v x_)
{
__m128i x = x_.data();
/*
// in 16,67% of the cases the merge can be replaced by an append
// x = [a b c d]
// y = [c d a b]
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(1, 0, 3, 2));
__m128i l = SSE::min_epi32(x, y); // min[ac bd ac bd]
__m128i h = SSE::max_epi32(x, y); // max[ac bd ac bd]
if (IS_UNLIKELY(_mm_cvtsi128_si32(h) <= l[1])) { // l[0] < h[0] < l[1] < h[1]
return _mm_unpacklo_epi32(l, h);
}
// h[0] > l[1]
*/
// sort pairs
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epi32(x, y);
__m128i h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epi32(x, y);
h = SSE::max_epi32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <> Vc_CONST SSE::uint_v sorted<CurrentImplementation::current()>(SSE::uint_v x_)
{
__m128i x = x_.data();
__m128i y = _mm_shuffle_epi32(x, _MM_SHUFFLE(2, 3, 0, 1));
__m128i l = SSE::min_epu32(x, y);
__m128i h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi32(h, l);
// sort quads
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
x = _mm_unpacklo_epi32(l, h);
y = _mm_unpackhi_epi64(x, x);
l = SSE::min_epu32(x, y);
h = SSE::max_epu32(x, y);
return _mm_unpacklo_epi32(l, h);
}
template <>
Vc_CONST SSE::float_v sorted<CurrentImplementation::current()>(SSE::float_v x_)
{
__m128 x = x_.data();
__m128 y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(2, 3, 0, 1));
__m128 l = _mm_min_ps(x, y);
__m128 h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_unpackhi_ps(h, l);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
x = _mm_unpacklo_ps(l, h);
y = _mm_movehl_ps(x, x);
l = _mm_min_ps(x, y);
h = _mm_max_ps(x, y);
return _mm_unpacklo_ps(l, h);
// X __m128 k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(2, 2, 0, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(1, 0, 3, 2));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(1, 0, 1, 0));
// X x = SSE::blendv_ps(x, y, k);
// X y = _mm_shuffle_ps(x, x, _MM_SHUFFLE(3, 1, 2, 0));
// X k = _mm_cmpgt_ps(x, y);
// X k = _mm_shuffle_ps(k, k, _MM_SHUFFLE(0, 1, 1, 0));
// X return SSE::blendv_ps(x, y, k);
}
} // namespace Detail
} // namespace Vc
// vim: foldmethod=marker

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif

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/* This file is part of the Vc library. {{{
Copyright © 2012-2015 Matthias Kretz <kretz@kde.org>
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
* Neither the names of contributing organizations nor the
names of its contributors may be used to endorse or promote products
derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
}}}*/
// enable bit operators for easier portable bit manipulation on floats
#define Vc_ENABLE_FLOAT_BIT_OPERATORS 1
#include <Vc/vector.h>
#if defined(Vc_IMPL_SSE) || defined(Vc_IMPL_AVX)
#include <Vc/common/macros.h>
//#include <Vc/IO>
namespace Vc_VERSIONED_NAMESPACE
{
namespace Common
{
namespace
{
using Vc::Vector;
template <typename T, typename Abi>
using Const = typename std::conditional<std::is_same<Abi, VectorAbi::Avx>::value,
AVX::Const<T>, SSE::Const<T>>::type;
template <typename V>
using best_int_v_for =
typename std::conditional<(V::size() <= Vector<int, VectorAbi::Best<int>>::size()),
Vector<int, VectorAbi::Best<int>>,
SimdArray<int, V::size()>>::type;
template <typename Abi> using float_int_v = best_int_v_for<Vector<float, Abi>>;
template <typename Abi> using double_int_v = best_int_v_for<Vector<double, Abi>>;
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> cosSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant< 1, 0x500000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x360800, -10>(); // -1/6!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x2AAAAB, -5>(); // 1/4!
return y * (x2 * x2) - .5f * x2 + 1.f;
// alternative (appears neither faster nor more precise):
// return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> cosSeries(const Vector<double, Abi> &x)
{
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant< 1, 0xAC00000000000, -45>(); // 1/16!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x9394000000000, -37>(); // -1/14!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1EED8C0000000, -29>(); // 1/12!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x27E4FB7400000, -22>(); // -1/10!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0xA01A01A018000, -16>(); // 1/8!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x6C16C16C16C00, -10>(); // -1/6!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x5555555555554, -5>(); // 1/4!
return (y * x2 - .5f) * x2 + 1.f;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<float, Abi> sinSeries(const Vector<float, Abi> &x)
{
const Vector<float, Abi> x2 = x * x;
Vector<float, Abi> y;
y = Vc::Detail::floatConstant<-1, 0x4E6000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::floatConstant< 1, 0x088880, -7>(); // 1/5!
y = y * x2 + Vc::Detail::floatConstant<-1, 0x2AAAAB, -3>(); // -1/3!
return y * (x2 * x) + x;
}
template <typename Abi>
static Vc_ALWAYS_INLINE Vector<double, Abi> sinSeries(const Vector<double, Abi> &x)
{
// x = [0, 0.7854 = pi/4]
// x² = [0, 0.6169 = pi²/8]
const Vector<double, Abi> x2 = x * x;
Vector<double, Abi> y;
y = Vc::Detail::doubleConstant<-1, 0xACF0000000000, -41>(); // -1/15!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x6124400000000, -33>(); // 1/13!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xAE64567000000, -26>(); // -1/11!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x71DE3A5540000, -19>(); // 1/9!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0xA01A01A01A000, -13>(); // -1/7!
y = y * x2 + Vc::Detail::doubleConstant< 1, 0x1111111111110, -7>(); // 1/5!
y = y * x2 + Vc::Detail::doubleConstant<-1, 0x5555555555555, -3>(); // -1/3!
return y * (x2 * x) + x;
}
/**\internal
* Fold \p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using a bitmask,
* `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the truncation
* expression. The `+= fmod` effect can the be achieved by using rounding instead of
* truncation:
* `y = round(x / ½π) * 2`.
* If precision allows, `2/π * x` is better (faster).
*/
template <class T, class Abi> struct folded {
Vector<T, Abi> x, quadrant;
};
template <typename Abi>
static Vc_ALWAYS_INLINE folded<float, Abi> foldInput(const Vector<float, Abi> &x)
{
using V = Vector<float, Abi>;
using IV = best_int_v_for<V>;
using C = Const<float, Abi>;
folded<float, Abi> r;
r.x = abs(x);
if (Vc_IS_UNLIKELY(all_of(r.x < C::_pi_4()))) {
r.quadrant = 0;
} else if (Vc_IS_LIKELY(all_of(r.x < 33 * C::_pi_4()))) {
const float _2_over_pi = Vc::Detail::floatConstant<1, 0x22F983, -1>(); // ½π
V y = round(r.x * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // y mod 4
r.x -= y * Vc::Detail::floatConstant<1, 0x490fe0, 0>(); // pi/2
r.x -= y * Vc::Detail::floatConstant<-1, 0x2bbbd3, -21>(); // pi/2 remainder
} else {
typedef SimdArray<double, V::size()> VD;
VD xd = simd_cast<VD>(abs(r.x));
constexpr double _2_over_pi =
Vc::Detail::doubleConstant<1, 0x45F306DC9C883, -1>();
VD y = round(xd * _2_over_pi);
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3); // = y mod 4
constexpr double pi_over_2 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, 0>();
r.x = simd_cast<V>(xd - y * pi_over_2);
}
//std::cout << std::hexfloat << r.x << ' ' << r.quadrant << std::defaultfloat << '\n';
return r;
/*
More alternatives to avoid double precision calculation follow. Their main problem
is unfortunate rounding on intermediate values of x (after subtraction):
if (all_of(y <= 0x1000)) {
// 0 <= x <= 0x1.921fb6p12
// π/4 - 12 bits precision (12 zero bits):
//const auto _pi_4_1 = Vc::Detail::floatConstant< 1, 0x491000, -1>();
//const auto _pi_4_2 = Vc::Detail::floatConstant<-1, 0x157000, -19>();
//const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x6F4B9F, -32>();
x -= y * C::_pi_4_hi();
x -= y * C::_pi_4_rem1();
x -= y * C::_pi_4_rem2();
} else { //if (all_of(y <= 0x10000)) {
// consider y = 0x1.fffep+16 / x = 0x1.921e24p+16
// critical input: 0x1.921fb6p+12 <= x <= 0x1.921fb6p+16
// π/4 - 8 bits precision (16 zero bits):
const auto _pi_4_1 = Vc::Detail::floatConstant<+1, 0x490000, -1>();
const auto _pi_4_2 = Vc::Detail::floatConstant<+1, 0x7E0000, -13>();
const auto _pi_4_3 = Vc::Detail::floatConstant<-1, 0x2C0000, -22>();
const auto _pi_4_4 = Vc::Detail::floatConstant<+1, 0x085A31, -31>();
x -= y * _pi_4_1; // (x -= 0x1.91fe6ep+16) -> 0x1.fb6000p+4
x -= y * _pi_4_2; // (x -= 0x1.fbfe04p+4) -> -0x1.3c0800p-6
x -= y * _pi_4_3; // (x -= 0x1.57fea8p-5) -> 0x1.bf5a80p-9
x -= y * _pi_4_4; // (x -= 0x1.10b352p-14) -> 0x1.b6e4e6p-9
x.setQnan(y > 0x10000);
}
*/
}
template <typename Abi>
static Vc_ALWAYS_INLINE folded<double, Abi> foldInput(const Vector<double, Abi> &x)
{
using V = Vector<double, Abi>;
using IV = best_int_v_for<V>;
using C = Const<double, Abi> ;
folded<double, Abi> r;
r.x = abs(x);
constexpr double pi_over_4 = Vc::Detail::doubleConstant<1, 0x921FB54442D18, -1>();
if (Vc_IS_UNLIKELY(all_of(r.x < pi_over_4))) {
r.quadrant = 0;
return r;
}
const V y = round(r.x / (2 * pi_over_4));
r.quadrant = simd_cast<V>(simd_cast<IV>(y) & 3);
if (Vc_IS_LIKELY(all_of(r.x < 1025 * pi_over_4))) {
// x - y * pi/2, y uses no more than 11 mantissa bits
r.x -= y * Vc::Detail::doubleConstant< 1, 0x921FB54443000, 0>();
r.x -= y * Vc::Detail::doubleConstant<-1, 0x73DCB3B39A000, -43>();
r.x -= y * Vc::Detail::doubleConstant< 1, 0x45C06E0E68948, -86>();
} else if (Vc_IS_LIKELY(all_of(y <= Vc::Detail::doubleConstant<1, 0, 30>()))) {
// x - y * pi/2, y uses no more than 29 mantissa bits
r.x -= y * Vc::Detail::doubleConstant<1, 0x921FB40000000, 0>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x4442D00000000, -24>();
r.x -= y * Vc::Detail::doubleConstant<1, 0x8469898CC5170, -48>();
} else {
// x - y * pi/2, y may require all mantissa bits
const V y_hi = y & C::highMask(26);
const V y_lo = y - y_hi;
const auto _pi_2_1 = Vc::Detail::doubleConstant<1, 0x921FB50000000, 0>();
const auto _pi_2_2 = Vc::Detail::doubleConstant<1, 0x110B460000000, -26>();
const auto _pi_2_3 = Vc::Detail::doubleConstant<1, 0x1A62630000000, -54>();
const auto _pi_2_4 = Vc::Detail::doubleConstant<1, 0x8A2E03707344A, -81>();
/*
std::cout << std::hexfloat;
std::cout
<< y_hi << '\n'
<< y_lo << '\n'
<< _pi_2_1 << '\n'
<< _pi_2_2 << '\n'
<< _pi_2_3 << '\n'
<< y_hi * _pi_2_1 << '\n'
<< y_hi * _pi_2_2 << '\n'
<< y_hi * _pi_2_3 << '\n'
<< y * _pi_2_4 << '\n'
<< y_lo * _pi_2_1 << '\n'
<< y_lo * _pi_2_2 << '\n'
<< y_lo * _pi_2_3 << '\n';
std::cout << std::defaultfloat;
*/
r.x = r.x
- y_hi * _pi_2_1
- max(y_hi * _pi_2_2, y_lo * _pi_2_1)
- min(y_hi * _pi_2_2, y_lo * _pi_2_1)
- max(y_hi * _pi_2_3, y_lo * _pi_2_2)
- min(y_hi * _pi_2_3, y_lo * _pi_2_2)
- max(y * _pi_2_4, y_lo * _pi_2_3)
- min(y * _pi_2_4, y_lo * _pi_2_3);
}
return r;
}
constexpr double signmask = -0.;
constexpr float signmaskf = -0.f;
} // anonymous namespace
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section the input is
* in.
* sine ≈ x + x³
* cosine ≈ 1 - x²
*
* sine:
* Map -x to x and invert the output
* Extend precision of x - n * π/4 by calculating
* ((x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sin(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 527449))) {
return simd_cast<V>(sin(simd_cast<Vc::double_v, 0>(x)),
sin(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
}
template <>
template <>
Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::double_v &x)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::cos(const Vc::float_v &x)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
return simd_cast<V>(cos(simd_cast<Vc::double_v, 0>(x)),
cos(simd_cast<Vc::double_v, 1>(x)));
}
const auto f = foldInput(x);
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
return cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::double_v &x, Vc::double_v *s,
Vc::double_v *c)
{
using V = Vc::double_v;
const auto f = foldInput(x);
// quadrant | cosine | sine
// 0 | cosSeries, + | sinSeries
// 1 | sinSeries, - | cosSeries
// 2 | cosSeries, - | sinSeries, sign flip
// 3 | sinSeries, + | cosSeries, sign flip
const V sin_sign = (x ^ (1 - f.quadrant)) & signmask;
const V cos_sign = ((0.5 - f.quadrant) & (f.quadrant - 2.5)) & signmask;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
void Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::sincos(const Vc::float_v &x, Vc::float_v *s,
Vc::float_v *c)
{
using V = Vc::float_v;
if (Vc_IS_UNLIKELY(any_of(abs(x) >= 393382))) {
Vc::double_v s0, s1, c0, c1;
sincos(simd_cast<Vc::double_v, 0>(x), &s0, &c0);
sincos(simd_cast<Vc::double_v, 1>(x), &s1, &c1);
*s = simd_cast<V>(s0, s1);
*c = simd_cast<V>(c0, c1);
return;
}
const auto f = foldInput(x);
const V sin_sign = (x ^ (1 - f.quadrant)) & signmaskf;
const V cos_sign = ((0.5f - f.quadrant) & (f.quadrant - 2.5f)) & signmaskf;
const V sin_s = sinSeries(f.x);
const V cos_s = cosSeries(f.x);
*s = sin_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, sin_s, cos_s);
*c = cos_sign ^ iif(f.quadrant == 0 || f.quadrant == 2, cos_s, sin_s);
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::asin(const Vc::float_v &_x)
{
typedef Vc::float_v V;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M &negative = _x < V::Zero();
const V &a = abs(_x);
const M outOfRange = a > V::One();
const M &small = a < C::smallAsinInput();
const M &gt_0_5 = a > C::_1_2();
V x = a;
V z = a * a;
z(gt_0_5) = (V::One() - a) * C::_1_2();
x(gt_0_5) = sqrt(z);
z = ((((C::asinCoeff0(0) * z
+ C::asinCoeff0(1)) * z
+ C::asinCoeff0(2)) * z
+ C::asinCoeff0(3)) * z
+ C::asinCoeff0(4)) * z * x
+ x;
z(gt_0_5) = C::_pi_2() - (z + z);
z(small) = a;
z(negative) = -z;
z.setQnan(outOfRange);
return z;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::asin (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M negative = _x < V::Zero();
const V a = abs(_x);
const M outOfRange = a > V::One();
const M small = a < C::smallAsinInput();
const M large = a > C::largeAsinInput();
V zz = V::One() - a;
const V r = (((C::asinCoeff0(0) * zz + C::asinCoeff0(1)) * zz + C::asinCoeff0(2)) * zz +
C::asinCoeff0(3)) * zz + C::asinCoeff0(4);
const V s = (((zz + C::asinCoeff1(0)) * zz + C::asinCoeff1(1)) * zz +
C::asinCoeff1(2)) * zz + C::asinCoeff1(3);
V sqrtzz = sqrt(zz + zz);
V z = C::_pi_4() - sqrtzz;
z -= sqrtzz * (zz * r / s) - C::_pi_2_rem();
z += C::_pi_4();
V a2 = a * a;
const V p = ((((C::asinCoeff2(0) * a2 + C::asinCoeff2(1)) * a2 + C::asinCoeff2(2)) * a2 +
C::asinCoeff2(3)) * a2 + C::asinCoeff2(4)) * a2 + C::asinCoeff2(5);
const V q = ((((a2 + C::asinCoeff3(0)) * a2 + C::asinCoeff3(1)) * a2 +
C::asinCoeff3(2)) * a2 + C::asinCoeff3(3)) * a2 + C::asinCoeff3(4);
z(!large) = a * (a2 * p / q) + a;
z(negative) = -z;
z(small) = _x;
z.setQnan(outOfRange);
return z;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan(const Vc::float_v &_x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
V x = abs(_x);
const M &gt_tan_3pi_8 = x > C::atanThrsHi();
const M &gt_tan_pi_8 = x > C::atanThrsLo() && !gt_tan_3pi_8;
V y = V::Zero();
y(gt_tan_3pi_8) = C::_pi_2();
y(gt_tan_pi_8) = C::_pi_4();
x(gt_tan_3pi_8) = -V::One() / x;
x(gt_tan_pi_8) = (x - V::One()) / (x + V::One());
const V &x2 = x * x;
y += (((C::atanP(0) * x2
- C::atanP(1)) * x2
+ C::atanP(2)) * x2
- C::atanP(3)) * x2 * x
+ x;
y(_x < V::Zero()) = -y;
y.setQnan(isnan(_x));
return y;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan (const Vc::double_v &_x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
M sign = _x < V::Zero();
V x = abs(_x);
M finite = isfinite(_x);
V ret = C::_pi_2();
V y = V::Zero();
const M large = x > C::atanThrsHi();
const M gt_06 = x > C::atanThrsLo();
V tmp = (x - V::One()) / (x + V::One());
tmp(large) = -V::One() / x;
x(gt_06) = tmp;
y(gt_06) = C::_pi_4();
y(large) = C::_pi_2();
V z = x * x;
const V p = (((C::atanP(0) * z + C::atanP(1)) * z + C::atanP(2)) * z + C::atanP(3)) * z + C::atanP(4);
const V q = ((((z + C::atanQ(0)) * z + C::atanQ(1)) * z + C::atanQ(2)) * z + C::atanQ(3)) * z + C::atanQ(4);
z = z * p / q;
z = x * z + x;
V morebits = C::_pi_2_rem();
morebits(!large) *= C::_1_2();
z(gt_06) += morebits;
ret(finite) = y + z;
ret(sign) = -ret;
ret.setQnan(isnan(_x));
return ret;
}
template <>
template <>
Vc::float_v Trigonometric<Vc::Detail::TrigonometricImplementation<
Vc::CurrentImplementation::current()>>::atan2(const Vc::float_v &y,
const Vc::float_v &x)
{
using V = Vc::float_v;
typedef Const<float, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(C::_pi(), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
template<> template<> Vc::double_v Trigonometric<Vc::Detail::TrigonometricImplementation<Vc::CurrentImplementation::current()>>::atan2 (const Vc::double_v &y, const Vc::double_v &x) {
typedef Vc::double_v V;
typedef Const<double, V::abi> C;
typedef V::Mask M;
const M xZero = x == V::Zero();
const M yZero = y == V::Zero();
const M xMinusZero = xZero && isnegative(x);
const M yNeg = y < V::Zero();
const M xInf = !isfinite(x);
const M yInf = !isfinite(y);
V a = copysign(V(C::_pi()), y);
a.setZero(x >= V::Zero());
// setting x to any finite value will have atan(y/x) return sign(y/x)*pi/2, just in case x is inf
V _x = x;
_x(yInf) = copysign(V::One(), x);
a += atan(y / _x);
// if x is +0 and y is +/-0 the result is +0
a.setZero(xZero && yZero);
// for x = -0 we add/subtract pi to get the correct result
a(xMinusZero) += copysign(C::_pi(), y);
// atan2(-Y, +/-0) = -pi/2
a(xZero && yNeg) = -C::_pi_2();
// if both inputs are inf the output is +/- (3)pi/4
a(xInf && yInf) += copysign(C::_pi_4(), x ^ ~y);
// correct the sign of y if the result is 0
a(a == V::Zero()) = copysign(a, y);
// any NaN input will lead to NaN output
a.setQnan(isnan(y) || isnan(x));
return a;
}
}
}
#endif