525 lines
16 KiB
C++
525 lines
16 KiB
C++
// Copyright 2009-2021 Intel Corporation
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// SPDX-License-Identifier: Apache-2.0
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#pragma once
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// Transcendental functions from "ispc": https://github.com/ispc/ispc/
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// Most of the transcendental implementations in ispc code come from
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// Solomon Boulos's "syrah": https://github.com/boulos/syrah/
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#include "../simd/simd.h"
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namespace embree
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{
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namespace fastapprox
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{
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template <typename T>
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__forceinline T sin(const T &v)
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{
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static const float piOverTwoVec = 1.57079637050628662109375;
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static const float twoOverPiVec = 0.636619746685028076171875;
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auto scaled = v * twoOverPiVec;
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auto kReal = floor(scaled);
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auto k = toInt(kReal);
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// Reduced range version of x
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auto x = v - kReal * piOverTwoVec;
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auto kMod4 = k & 3;
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auto sinUseCos = (kMod4 == 1) | (kMod4 == 3);
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auto flipSign = (kMod4 > 1);
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// These coefficients are from sollya with fpminimax(sin(x)/x, [|0, 2,
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// 4, 6, 8, 10|], [|single...|], [0;Pi/2]);
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static const float sinC2 = -0.16666667163372039794921875;
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static const float sinC4 = +8.333347737789154052734375e-3;
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static const float sinC6 = -1.9842604524455964565277099609375e-4;
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static const float sinC8 = +2.760012648650445044040679931640625e-6;
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static const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
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static const float cosC2 = -0.5;
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static const float cosC4 = +4.166664183139801025390625e-2;
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static const float cosC6 = -1.388833043165504932403564453125e-3;
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static const float cosC8 = +2.47562347794882953166961669921875e-5;
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static const float cosC10 = -2.59630184018533327616751194000244140625e-7;
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auto outside = select(sinUseCos, 1., x);
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auto c2 = select(sinUseCos, T(cosC2), T(sinC2));
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auto c4 = select(sinUseCos, T(cosC4), T(sinC4));
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auto c6 = select(sinUseCos, T(cosC6), T(sinC6));
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auto c8 = select(sinUseCos, T(cosC8), T(sinC8));
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auto c10 = select(sinUseCos, T(cosC10), T(sinC10));
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auto x2 = x * x;
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auto formula = x2 * c10 + c8;
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formula = x2 * formula + c6;
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formula = x2 * formula + c4;
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formula = x2 * formula + c2;
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formula = x2 * formula + 1.;
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formula *= outside;
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formula = select(flipSign, -formula, formula);
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return formula;
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}
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template <typename T>
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__forceinline T cos(const T &v)
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{
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static const float piOverTwoVec = 1.57079637050628662109375;
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static const float twoOverPiVec = 0.636619746685028076171875;
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auto scaled = v * twoOverPiVec;
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auto kReal = floor(scaled);
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auto k = toInt(kReal);
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// Reduced range version of x
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auto x = v - kReal * piOverTwoVec;
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auto kMod4 = k & 3;
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auto cosUseCos = (kMod4 == 0) | (kMod4 == 2);
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auto flipSign = (kMod4 == 1) | (kMod4 == 2);
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const float sinC2 = -0.16666667163372039794921875;
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const float sinC4 = +8.333347737789154052734375e-3;
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const float sinC6 = -1.9842604524455964565277099609375e-4;
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const float sinC8 = +2.760012648650445044040679931640625e-6;
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const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
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const float cosC2 = -0.5;
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const float cosC4 = +4.166664183139801025390625e-2;
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const float cosC6 = -1.388833043165504932403564453125e-3;
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const float cosC8 = +2.47562347794882953166961669921875e-5;
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const float cosC10 = -2.59630184018533327616751194000244140625e-7;
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auto outside = select(cosUseCos, 1., x);
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auto c2 = select(cosUseCos, T(cosC2), T(sinC2));
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auto c4 = select(cosUseCos, T(cosC4), T(sinC4));
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auto c6 = select(cosUseCos, T(cosC6), T(sinC6));
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auto c8 = select(cosUseCos, T(cosC8), T(sinC8));
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auto c10 = select(cosUseCos, T(cosC10), T(sinC10));
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auto x2 = x * x;
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auto formula = x2 * c10 + c8;
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formula = x2 * formula + c6;
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formula = x2 * formula + c4;
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formula = x2 * formula + c2;
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formula = x2 * formula + 1.;
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formula *= outside;
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formula = select(flipSign, -formula, formula);
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return formula;
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}
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template <typename T>
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__forceinline void sincos(const T &v, T &sinResult, T &cosResult)
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{
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const float piOverTwoVec = 1.57079637050628662109375;
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const float twoOverPiVec = 0.636619746685028076171875;
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auto scaled = v * twoOverPiVec;
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auto kReal = floor(scaled);
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auto k = toInt(kReal);
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// Reduced range version of x
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auto x = v - kReal * piOverTwoVec;
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auto kMod4 = k & 3;
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auto cosUseCos = ((kMod4 == 0) | (kMod4 == 2));
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auto sinUseCos = ((kMod4 == 1) | (kMod4 == 3));
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auto sinFlipSign = (kMod4 > 1);
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auto cosFlipSign = ((kMod4 == 1) | (kMod4 == 2));
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const float oneVec = +1.;
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const float sinC2 = -0.16666667163372039794921875;
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const float sinC4 = +8.333347737789154052734375e-3;
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const float sinC6 = -1.9842604524455964565277099609375e-4;
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const float sinC8 = +2.760012648650445044040679931640625e-6;
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const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
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const float cosC2 = -0.5;
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const float cosC4 = +4.166664183139801025390625e-2;
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const float cosC6 = -1.388833043165504932403564453125e-3;
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const float cosC8 = +2.47562347794882953166961669921875e-5;
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const float cosC10 = -2.59630184018533327616751194000244140625e-7;
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auto x2 = x * x;
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auto sinFormula = x2 * sinC10 + sinC8;
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auto cosFormula = x2 * cosC10 + cosC8;
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sinFormula = x2 * sinFormula + sinC6;
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cosFormula = x2 * cosFormula + cosC6;
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sinFormula = x2 * sinFormula + sinC4;
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cosFormula = x2 * cosFormula + cosC4;
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sinFormula = x2 * sinFormula + sinC2;
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cosFormula = x2 * cosFormula + cosC2;
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sinFormula = x2 * sinFormula + oneVec;
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cosFormula = x2 * cosFormula + oneVec;
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sinFormula *= x;
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sinResult = select(sinUseCos, cosFormula, sinFormula);
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cosResult = select(cosUseCos, cosFormula, sinFormula);
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sinResult = select(sinFlipSign, -sinResult, sinResult);
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cosResult = select(cosFlipSign, -cosResult, cosResult);
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}
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template <typename T>
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__forceinline T tan(const T &v)
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{
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const float piOverFourVec = 0.785398185253143310546875;
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const float fourOverPiVec = 1.27323949337005615234375;
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auto xLt0 = v < 0.;
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auto y = select(xLt0, -v, v);
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auto scaled = y * fourOverPiVec;
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auto kReal = floor(scaled);
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auto k = toInt(kReal);
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auto x = y - kReal * piOverFourVec;
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// If k & 1, x -= Pi/4
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auto needOffset = (k & 1) != 0;
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x = select(needOffset, x - piOverFourVec, x);
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// If k & 3 == (0 or 3) let z = tan_In...(y) otherwise z = -cot_In0To...
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auto kMod4 = k & 3;
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auto useCotan = (kMod4 == 1) | (kMod4 == 2);
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const float oneVec = 1.0;
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const float tanC2 = +0.33333075046539306640625;
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const float tanC4 = +0.13339905440807342529296875;
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const float tanC6 = +5.3348250687122344970703125e-2;
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const float tanC8 = +2.46033705770969390869140625e-2;
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const float tanC10 = +2.892402000725269317626953125e-3;
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const float tanC12 = +9.500005282461643218994140625e-3;
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const float cotC2 = -0.3333333432674407958984375;
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const float cotC4 = -2.222204394638538360595703125e-2;
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const float cotC6 = -2.11752182804048061370849609375e-3;
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const float cotC8 = -2.0846328698098659515380859375e-4;
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const float cotC10 = -2.548247357481159269809722900390625e-5;
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const float cotC12 = -3.5257363606433500535786151885986328125e-7;
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auto x2 = x * x;
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T z;
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if (any(useCotan))
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{
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auto cotVal = x2 * cotC12 + cotC10;
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cotVal = x2 * cotVal + cotC8;
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cotVal = x2 * cotVal + cotC6;
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cotVal = x2 * cotVal + cotC4;
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cotVal = x2 * cotVal + cotC2;
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cotVal = x2 * cotVal + oneVec;
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// The equation is for x * cot(x) but we need -x * cot(x) for the tan part.
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cotVal /= -x;
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z = cotVal;
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}
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auto useTan = !useCotan;
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if (any(useTan))
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{
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auto tanVal = x2 * tanC12 + tanC10;
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tanVal = x2 * tanVal + tanC8;
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tanVal = x2 * tanVal + tanC6;
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tanVal = x2 * tanVal + tanC4;
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tanVal = x2 * tanVal + tanC2;
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tanVal = x2 * tanVal + oneVec;
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// Equation was for tan(x)/x
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tanVal *= x;
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z = select(useTan, tanVal, z);
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}
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return select(xLt0, -z, z);
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}
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template <typename T>
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__forceinline T asin(const T &x0)
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{
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auto isneg = (x0 < 0.f);
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auto x = abs(x0);
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auto isnan = (x > 1.f);
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// sollya
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// fpminimax(((asin(x)-pi/2)/-sqrt(1-x)), [|0,1,2,3,4,5|],[|single...|],
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// [1e-20;.9999999999999999]);
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// avg error: 1.1105439e-06, max error 1.3187528e-06
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auto v = 1.57079517841339111328125f +
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x * (-0.21450997889041900634765625f +
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x * (8.78556668758392333984375e-2f +
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x * (-4.489909112453460693359375e-2f +
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x * (1.928029954433441162109375e-2f +
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x * (-4.3095736764371395111083984375e-3f)))));
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v *= -sqrt(1.f - x);
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v = v + 1.57079637050628662109375f;
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v = select(v < 0.f, T(0.f), v);
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v = select(isneg, -v, v);
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v = select(isnan, T(cast_i2f(0x7fc00000)), v);
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return v;
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}
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template <typename T>
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__forceinline T acos(const T &v)
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{
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return 1.57079637050628662109375f - asin(v);
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}
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template <typename T>
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__forceinline T atan(const T &v)
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{
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const float piOverTwoVec = 1.57079637050628662109375;
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// atan(-x) = -atan(x) (so flip from negative to positive first)
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// If x > 1 -> atan(x) = Pi/2 - atan(1/x)
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auto xNeg = v < 0.f;
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auto xFlipped = select(xNeg, -v, v);
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auto xGt1 = xFlipped > 1.;
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auto x = select(xGt1, rcpSafe(xFlipped), xFlipped);
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// These coefficients approximate atan(x)/x
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const float atanC0 = +0.99999988079071044921875;
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const float atanC2 = -0.3333191573619842529296875;
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const float atanC4 = +0.199689209461212158203125;
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const float atanC6 = -0.14015688002109527587890625;
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const float atanC8 = +9.905083477497100830078125e-2;
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const float atanC10 = -5.93664981424808502197265625e-2;
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const float atanC12 = +2.417283318936824798583984375e-2;
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const float atanC14 = -4.6721356920897960662841796875e-3;
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auto x2 = x * x;
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auto result = x2 * atanC14 + atanC12;
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result = x2 * result + atanC10;
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result = x2 * result + atanC8;
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result = x2 * result + atanC6;
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result = x2 * result + atanC4;
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result = x2 * result + atanC2;
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result = x2 * result + atanC0;
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result *= x;
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result = select(xGt1, piOverTwoVec - result, result);
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result = select(xNeg, -result, result);
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return result;
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}
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template <typename T>
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__forceinline T atan2(const T &y, const T &x)
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{
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const float piVec = 3.1415926536;
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// atan2(y, x) =
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//
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// atan2(y > 0, x = +-0) -> Pi/2
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// atan2(y < 0, x = +-0) -> -Pi/2
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// atan2(y = +-0, x < +0) -> +-Pi
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// atan2(y = +-0, x >= +0) -> +-0
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//
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// atan2(y >= 0, x < 0) -> Pi + atan(y/x)
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// atan2(y < 0, x < 0) -> -Pi + atan(y/x)
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// atan2(y, x > 0) -> atan(y/x)
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//
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// and then a bunch of code for dealing with infinities.
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auto yOverX = y * rcpSafe(x);
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auto atanArg = atan(yOverX);
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auto xLt0 = x < 0.f;
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auto yLt0 = y < 0.f;
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auto offset = select(xLt0,
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select(yLt0, T(-piVec), T(piVec)), 0.f);
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return offset + atanArg;
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}
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template <typename T>
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__forceinline T exp(const T &v)
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{
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const float ln2Part1 = 0.6931457519;
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const float ln2Part2 = 1.4286067653e-6;
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const float oneOverLn2 = 1.44269502162933349609375;
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auto scaled = v * oneOverLn2;
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auto kReal = floor(scaled);
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auto k = toInt(kReal);
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// Reduced range version of x
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auto x = v - kReal * ln2Part1;
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x -= kReal * ln2Part2;
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// These coefficients are for e^x in [0, ln(2)]
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const float one = 1.;
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const float c2 = 0.4999999105930328369140625;
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const float c3 = 0.166668415069580078125;
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const float c4 = 4.16539050638675689697265625e-2;
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const float c5 = 8.378830738365650177001953125e-3;
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const float c6 = 1.304379315115511417388916015625e-3;
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const float c7 = 2.7555381529964506626129150390625e-4;
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auto result = x * c7 + c6;
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result = x * result + c5;
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result = x * result + c4;
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result = x * result + c3;
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result = x * result + c2;
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result = x * result + one;
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result = x * result + one;
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// Compute 2^k (should differ for float and double, but I'll avoid
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// it for now and just do floats)
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const int fpbias = 127;
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auto biasedN = k + fpbias;
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auto overflow = kReal > fpbias;
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// Minimum exponent is -126, so if k is <= -127 (k + 127 <= 0)
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// we've got underflow. -127 * ln(2) -> -88.02. So the most
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// negative float input that doesn't result in zero is like -88.
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auto underflow = kReal <= -fpbias;
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const int infBits = 0x7f800000;
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biasedN <<= 23;
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// Reinterpret this thing as float
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auto twoToTheN = asFloat(biasedN);
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// Handle both doubles and floats (hopefully eliding the copy for float)
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auto elemtype2n = twoToTheN;
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result *= elemtype2n;
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result = select(overflow, cast_i2f(infBits), result);
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result = select(underflow, 0., result);
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return result;
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}
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// Range reduction for logarithms takes log(x) -> log(2^n * y) -> n
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// * log(2) + log(y) where y is the reduced range (usually in [1/2, 1)).
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template <typename T, typename R>
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__forceinline void __rangeReduceLog(const T &input,
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T &reduced,
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R &exponent)
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{
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auto intVersion = asInt(input);
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// single precision = SEEE EEEE EMMM MMMM MMMM MMMM MMMM MMMM
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// exponent mask = 0111 1111 1000 0000 0000 0000 0000 0000
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// 0x7 0xF 0x8 0x0 0x0 0x0 0x0 0x0
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// non-exponent = 1000 0000 0111 1111 1111 1111 1111 1111
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// = 0x8 0x0 0x7 0xF 0xF 0xF 0xF 0xF
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//const int exponentMask(0x7F800000)
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static const int nonexponentMask = 0x807FFFFF;
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// We want the reduced version to have an exponent of -1 which is
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// -1 + 127 after biasing or 126
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static const int exponentNeg1 = (126l << 23);
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// NOTE(boulos): We don't need to mask anything out since we know
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// the sign bit has to be 0. If it's 1, we need to return infinity/nan
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// anyway (log(x), x = +-0 -> infinity, x < 0 -> NaN).
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auto biasedExponent = intVersion >> 23; // This number is [0, 255] but it means [-127, 128]
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auto offsetExponent = biasedExponent + 1; // Treat the number as if it were 2^{e+1} * (1.m)/2
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exponent = offsetExponent - 127; // get the real value
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// Blend the offset_exponent with the original input (do this in
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// int for now, until I decide if float can have & and ¬)
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auto blended = (intVersion & nonexponentMask) | (exponentNeg1);
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reduced = asFloat(blended);
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}
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template <typename T> struct ExponentType { };
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template <int N> struct ExponentType<vfloat_impl<N>> { typedef vint<N> Ty; };
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template <> struct ExponentType<float> { typedef int Ty; };
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template <typename T>
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__forceinline T log(const T &v)
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{
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T reduced;
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typename ExponentType<T>::Ty exponent;
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|
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const int nanBits = 0x7fc00000;
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const int negInfBits = 0xFF800000;
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const float nan = cast_i2f(nanBits);
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const float negInf = cast_i2f(negInfBits);
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auto useNan = v < 0.;
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auto useInf = v == 0.;
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auto exceptional = useNan | useInf;
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const float one = 1.0;
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|
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auto patched = select(exceptional, one, v);
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__rangeReduceLog(patched, reduced, exponent);
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|
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const float ln2 = 0.693147182464599609375;
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|
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auto x1 = one - reduced;
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const float c1 = +0.50000095367431640625;
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const float c2 = +0.33326041698455810546875;
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const float c3 = +0.2519190013408660888671875;
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const float c4 = +0.17541764676570892333984375;
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const float c5 = +0.3424419462680816650390625;
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const float c6 = -0.599632322788238525390625;
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const float c7 = +1.98442304134368896484375;
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const float c8 = -2.4899270534515380859375;
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const float c9 = +1.7491014003753662109375;
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|
|
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auto result = x1 * c9 + c8;
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result = x1 * result + c7;
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|
result = x1 * result + c6;
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|
result = x1 * result + c5;
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|
result = x1 * result + c4;
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|
result = x1 * result + c3;
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result = x1 * result + c2;
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result = x1 * result + c1;
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|
result = x1 * result + one;
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|
|
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// Equation was for -(ln(red)/(1-red))
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result *= -x1;
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|
result += toFloat(exponent) * ln2;
|
|
|
|
return select(exceptional,
|
|
select(useNan, T(nan), T(negInf)),
|
|
result);
|
|
}
|
|
|
|
template <typename T>
|
|
__forceinline T pow(const T &x, const T &y)
|
|
{
|
|
auto x1 = abs(x);
|
|
auto z = exp(y * log(x1));
|
|
|
|
// Handle special cases
|
|
const float twoOver23 = 8388608.0f;
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|
auto yInt = y == round(y);
|
|
auto yOddInt = select(yInt, asInt(abs(y) + twoOver23) << 31, 0); // set sign bit
|
|
|
|
// x == 0
|
|
z = select(x == 0.0f,
|
|
select(y < 0.0f, T(inf) | signmsk(x),
|
|
select(y == 0.0f, T(1.0f), asFloat(yOddInt) & x)), z);
|
|
|
|
// x < 0
|
|
auto xNegative = x < 0.0f;
|
|
if (any(xNegative))
|
|
{
|
|
auto z1 = z | asFloat(yOddInt);
|
|
z1 = select(yInt, z1, std::numeric_limits<float>::quiet_NaN());
|
|
z = select(xNegative, z1, z);
|
|
}
|
|
|
|
auto xFinite = isfinite(x);
|
|
auto yFinite = isfinite(y);
|
|
if (all(xFinite & yFinite))
|
|
return z;
|
|
|
|
// x finite and y infinite
|
|
z = select(andn(xFinite, yFinite),
|
|
select(x1 == 1.0f, 1.0f,
|
|
select((x1 > 1.0f) ^ (y < 0.0f), inf, T(0.0f))), z);
|
|
|
|
// x infinite
|
|
z = select(xFinite, z,
|
|
select(y == 0.0f, 1.0f,
|
|
select(y < 0.0f, T(0.0f), inf) | (asFloat(yOddInt) & x)));
|
|
|
|
return z;
|
|
}
|
|
|
|
template <typename T>
|
|
__forceinline T pow(const T &x, float y)
|
|
{
|
|
return pow(x, T(y));
|
|
}
|
|
|
|
} // namespace fastapprox
|
|
|
|
} // namespace embree
|