godot/thirdparty/embree/common/math/transcendental.h

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