BasicMath.hpp

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/*
 *  Copyright 2019-2021 Diligent Graphics LLC
 *  Copyright 2015-2019 Egor Yusov
 *  
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *  
 *      http://www.apache.org/licenses/LICENSE-2.0
 *  
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 *
 *  In no event and under no legal theory, whether in tort (including negligence), 
 *  contract, or otherwise, unless required by applicable law (such as deliberate 
 *  and grossly negligent acts) or agreed to in writing, shall any Contributor be
 *  liable for any damages, including any direct, indirect, special, incidental, 
 *  or consequential damages of any character arising as a result of this License or 
 *  out of the use or inability to use the software (including but not limited to damages 
 *  for loss of goodwill, work stoppage, computer failure or malfunction, or any and 
 *  all other commercial damages or losses), even if such Contributor has been advised 
 *  of the possibility of such damages.
 */
 
#pragma once
 
 
#include <cmath>
#include <algorithm>
#include <iostream>
 
#include "HashUtils.hpp"
 
#ifdef _MSC_VER
#    pragma warning(push)
#    pragma warning(disable : 4201) // nonstandard extension used: nameless struct/union
#endif
 
namespace Diligent
{
 
static constexpr double PI   = 3.14159265358979323846;
static constexpr float  PI_F = 3.1415927f;
 
// Template Vector & Matrix Classes
template <class T> struct Matrix2x2;
template <class T> struct Matrix3x3;
template <class T> struct Matrix4x4;
template <class T> struct Vector4;
 
template <class T> struct Vector2
{
    union
    {
        struct
        {
            T x;
            T y;
        };
        struct
        {
            T r;
            T g;
        };
        struct
        {
            T u;
            T v;
        };
    };
 
 
    Vector2 operator-(const Vector2<T>& right) const
    {
        return Vector2{x - right.x, y - right.y};
    }
 
    Vector2& operator-=(const Vector2<T>& right)
    {
        x -= right.x;
        y -= right.y;
        return *this;
    }
 
    Vector2 operator-() const
    {
        return Vector2{-x, -y};
    }
 
    Vector2 operator+(const Vector2<T>& right) const
    {
        return Vector2{x + right.x, y + right.y};
    }
 
    Vector2& operator+=(const Vector2<T>& right)
    {
        x += right.x;
        y += right.y;
        return *this;
    }
 
    Vector2 operator*(T s) const
    {
        return Vector2{x * s, y * s};
    }
 
    Vector2 operator*(const Vector2& right) const
    {
        return Vector2{x * right.x, y * right.y};
    }
 
    Vector2& operator*=(const Vector2& right)
    {
        x *= right.x;
        y *= right.y;
        return *this;
    }
 
    Vector2& operator*=(T s)
    {
        x *= s;
        y *= s;
        return *this;
    }
 
    Vector2 operator*(const Matrix2x2<T>& m) const
    {
        Vector2 out;
        out[0] = x * m[0][0] + y * m[1][0];
        out[1] = x * m[0][1] + y * m[1][1];
        return out;
    }
 
    Vector2 operator/(const Vector2& right) const
    {
        return Vector2{x / right.x, y / right.y};
    }
 
    Vector2& operator/=(const Vector2& right)
    {
        x /= right.x;
        y /= right.y;
        return *this;
    }
 
    Vector2 operator/(T s) const
    {
        return Vector2{x / s, y / s};
    }
 
    Vector2& operator/=(T s)
    {
        x /= s;
        y /= s;
        return *this;
    }
 
    bool operator==(const Vector2& right) const
    {
        return x == right.x && y == right.y;
    }
 
    bool operator!=(const Vector2& right) const
    {
        return !(*this == right);
    }
 
    Vector2 operator<(const Vector2& right) const
    {
        return Vector2{x < right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y < right.y ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector2 operator>(const Vector2& right) const
    {
        return Vector2{x > right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y > right.y ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector2 operator<=(const Vector2& right) const
    {
        return Vector2{x <= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y <= right.y ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector2 operator>=(const Vector2& right) const
    {
        return Vector2{x >= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y >= right.y ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    T* Data() { return reinterpret_cast<T*>(this); }
 
    const T* Data() const { return reinterpret_cast<const T*>(this); }
 
    T& operator[](size_t index)
    {
        return Data()[index];
    }
 
    const T& operator[](size_t index) const
    {
        return Data()[index];
    }
 
    Vector2() :
        x{0}, y{0} {}
    Vector2(T _x, T _y) :
        x{_x}, y{_y} {}
 
    template <typename Y>
    static Vector2 MakeVector(const Y& vals)
    {
        return Vector2 //
            {
                static_cast<T>(vals[0]),
                static_cast<T>(vals[1]) //
            };
    }
 
    template <typename Y>
    Vector2<Y> Recast() const
    {
        return Vector2<Y>{static_cast<Y>(x),
                          static_cast<Y>(y)};
    }
};
 
template <class T>
Vector2<T> operator*(T s, const Vector2<T>& a)
{
    return a * s;
}
 
 
template <class T> struct Vector3
{
    union
    {
        struct
        {
            T x;
            T y;
            T z;
        };
        struct
        {
            T r;
            T g;
            T b;
        };
        struct
        {
            T u;
            T v;
            T w;
        };
    };
 
 
    Vector3 operator-(const Vector3& right) const
    {
        return Vector3{x - right.x, y - right.y, z - right.z};
    }
 
    Vector3 operator-() const
    {
        return Vector3{-x, -y, -z};
    }
 
    Vector3& operator-=(const Vector3<T>& right)
    {
        x -= right.x;
        y -= right.y;
        z -= right.z;
        return *this;
    }
 
    Vector3 operator+(const Vector3& right) const
    {
        return Vector3{x + right.x, y + right.y, z + right.z};
    }
 
    Vector3& operator+=(const Vector3<T>& right)
    {
        x += right.x;
        y += right.y;
        z += right.z;
        return *this;
    }
 
    Vector3 operator*(T s) const
    {
        return Vector3{x * s, y * s, z * s};
    }
 
    Vector3& operator*=(T s)
    {
        x *= s;
        y *= s;
        z *= s;
        return *this;
    }
 
    Vector3 operator*(const Vector3& right) const
    {
        return Vector3{x * right.x, y * right.y, z * right.z};
    }
 
    Vector3 operator*(const Matrix4x4<T>& m) const
    {
        Vector4<T> out4 = Vector4<T>(x, y, z, 1) * m;
        return Vector3{out4.x / out4.w, out4.y / out4.w, out4.z / out4.w};
    }
 
    Vector3& operator*=(const Vector3& right)
    {
        x *= right.x;
        y *= right.y;
        z *= right.z;
        return *this;
    }
 
    Vector3 operator*(const Matrix3x3<T>& m) const
    {
        Vector3 out;
        out[0] = x * m[0][0] + y * m[1][0] + z * m[2][0];
        out[1] = x * m[0][1] + y * m[1][1] + z * m[2][1];
        out[2] = x * m[0][2] + y * m[1][2] + z * m[2][2];
        return out;
    }
 
    Vector3 operator/(T s) const
    {
        return Vector3{x / s, y / s, z / s};
    }
 
    Vector3& operator/=(T s)
    {
        x /= s;
        y /= s;
        z /= s;
        return *this;
    }
 
    Vector3 operator/(const Vector3& right) const
    {
        return Vector3{x / right.x, y / right.y, z / right.z};
    }
 
    Vector3& operator/=(const Vector3& right)
    {
        x /= right.x;
        y /= right.y;
        z /= right.z;
        return *this;
    }
 
    bool operator==(const Vector3& right) const
    {
        return x == right.x && y == right.y && z == right.z;
    }
 
    bool operator!=(const Vector3& right) const
    {
        return !(*this == right);
    }
 
    Vector3 operator<(const Vector3& right) const
    {
        return Vector3{x < right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y < right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z < right.z ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector3 operator>(const Vector3& right) const
    {
        return Vector3{x > right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y > right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z > right.z ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector3 operator<=(const Vector3& right) const
    {
        return Vector3{x <= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y <= right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z <= right.z ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector3 operator>=(const Vector3& right) const
    {
        return Vector3{x >= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y >= right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z >= right.z ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    T* Data() { return reinterpret_cast<T*>(this); }
 
    const T* Data() const { return reinterpret_cast<const T*>(this); }
 
    T& operator[](size_t index)
    {
        return Data()[index];
    }
 
    const T& operator[](size_t index) const
    {
        return Data()[index];
    }
 
    Vector3() :
        x{0}, y{0}, z{0} {}
    Vector3(T _x, T _y, T _z) :
        x{_x}, y{_y}, z{_z} {}
 
    template <typename Y>
    static Vector3 MakeVector(const Y& vals)
    {
        return Vector3 //
            {
                static_cast<T>(vals[0]),
                static_cast<T>(vals[1]),
                static_cast<T>(vals[2]) //
            };
    }
 
    template <typename Y>
    Vector3<Y> Recast() const
    {
        return Vector3<Y>{static_cast<Y>(x),
                          static_cast<Y>(y),
                          static_cast<Y>(z)};
    }
 
    operator Vector2<T>() const { return Vector2<T>(x, y); }
};
 
template <class T>
Vector3<T> operator*(T s, const Vector3<T>& a)
{
    return a * s;
}
 
 
template <class T> struct Vector4
{
    union
    {
        struct
        {
            T x;
            T y;
            T z;
            T w;
        };
        struct
        {
            T r;
            T g;
            T b;
            T a;
        };
    };
 
    Vector4 operator-(const Vector4& right) const
    {
        return Vector4{x - right.x, y - right.y, z - right.z, w - right.w};
    }
 
    Vector4 operator-() const
    {
        return Vector4{-x, -y, -z, -w};
    }
 
    Vector4& operator-=(const Vector4<T>& right)
    {
        x -= right.x;
        y -= right.y;
        z -= right.z;
        w -= right.w;
        return *this;
    }
 
    Vector4 operator+(const Vector4& right) const
    {
        return Vector4{x + right.x, y + right.y, z + right.z, w + right.w};
    }
 
    Vector4& operator+=(const Vector4<T>& right)
    {
        x += right.x;
        y += right.y;
        z += right.z;
        w += right.w;
        return *this;
    }
 
    Vector4 operator*(T s) const
    {
        return Vector4{x * s, y * s, z * s, w * s};
    }
 
    Vector4& operator*=(T s)
    {
        x *= s;
        y *= s;
        z *= s;
        w *= s;
        return *this;
    }
 
    Vector4 operator*(const Vector4& right) const
    {
        return Vector4{x * right.x, y * right.y, z * right.z, w * right.w};
    }
 
    Vector4& operator*=(const Vector4& right)
    {
        x *= right.x;
        y *= right.y;
        z *= right.z;
        w *= right.w;
        return *this;
    }
 
    Vector4 operator/(T s) const
    {
        return Vector4{x / s, y / s, z / s, w / s};
    }
 
    Vector4& operator/=(T s)
    {
        x /= s;
        y /= s;
        z /= s;
        w /= s;
        return *this;
    }
 
    Vector4 operator/(const Vector4& right) const
    {
        return Vector4{x / right.x, y / right.y, z / right.z, w / right.w};
    }
 
    Vector4& operator/=(const Vector4& right)
    {
        x /= right.x;
        y /= right.y;
        z /= right.z;
        w /= right.w;
        return *this;
    }
 
    bool operator==(const Vector4& right) const
    {
        return x == right.x && y == right.y && z == right.z && w == right.w;
    }
 
    bool operator!=(const Vector4& right) const
    {
        return !(*this == right);
    }
 
    Vector4 operator*(const Matrix4x4<T>& m) const
    {
        Vector4 out;
        out[0] = x * m[0][0] + y * m[1][0] + z * m[2][0] + w * m[3][0];
        out[1] = x * m[0][1] + y * m[1][1] + z * m[2][1] + w * m[3][1];
        out[2] = x * m[0][2] + y * m[1][2] + z * m[2][2] + w * m[3][2];
        out[3] = x * m[0][3] + y * m[1][3] + z * m[2][3] + w * m[3][3];
        return out;
    }
 
    Vector4& operator=(const Vector3<T>& v3)
    {
        x = v3.x;
        y = v3.y;
        z = v3.z;
        w = 1;
        return *this;
    }
    Vector4& operator=(const Vector4&) = default;
 
    Vector4 operator<(const Vector4& right) const
    {
        return Vector4{x < right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y < right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z < right.z ? static_cast<T>(1) : static_cast<T>(0),
                       w < right.w ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector4 operator>(const Vector4& right) const
    {
        return Vector4{x > right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y > right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z > right.z ? static_cast<T>(1) : static_cast<T>(0),
                       w > right.w ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector4 operator<=(const Vector4& right) const
    {
        return Vector4{x <= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y <= right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z <= right.z ? static_cast<T>(1) : static_cast<T>(0),
                       w <= right.w ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    Vector4 operator>=(const Vector4& right) const
    {
        return Vector4{x >= right.x ? static_cast<T>(1) : static_cast<T>(0),
                       y >= right.y ? static_cast<T>(1) : static_cast<T>(0),
                       z >= right.z ? static_cast<T>(1) : static_cast<T>(0),
                       w >= right.w ? static_cast<T>(1) : static_cast<T>(0)};
    }
 
    T* Data() { return reinterpret_cast<T*>(this); }
 
    const T* Data() const { return reinterpret_cast<const T*>(this); }
 
    T& operator[](size_t index)
    {
        return Data()[index];
    }
 
    const T& operator[](size_t index) const
    {
        return Data()[index];
    }
 
    Vector4() :
        x{0}, y{0}, z{0}, w{0} {}
    Vector4(T _x, T _y, T _z, T _w) :
        x{_x}, y{_y}, z{_z}, w{_w} {}
    Vector4(const Vector3<T>& v3, T _w) :
        x{v3.x}, y{v3.y}, z{v3.z}, w{_w} {}
 
    template <typename Y>
    static Vector4 MakeVector(const Y& vals)
    {
        return Vector4 //
            {
                static_cast<T>(vals[0]),
                static_cast<T>(vals[1]),
                static_cast<T>(vals[2]),
                static_cast<T>(vals[3]) //
            };
    }
 
    template <typename Y>
    Vector4<Y> Recast() const
    {
        return Vector4<Y>{static_cast<Y>(x),
                          static_cast<Y>(y),
                          static_cast<Y>(z),
                          static_cast<Y>(w)};
    }
 
    operator Vector3<T>() const
    {
        return Vector3<T>(x, y, z);
    }
};
 
 
template <class T>
Vector4<T> operator*(T s, const Vector4<T>& a)
{
    return a * s;
}
 
 
template <class T> struct Matrix2x2
{
    union
    {
        struct
        {
            T _11;
            T _12;
            T _21;
            T _22;
        };
        struct
        {
            T m00;
            T m01;
            T m10;
            T m11;
        };
        T m[2][2];
    };
 
    explicit Matrix2x2(T value) :
        // clang-format off
        _11{value}, _12{value},
        _21{value}, _22{value}
    // clang-format on
    {
    }
 
    Matrix2x2() :
        Matrix2x2{0} {}
 
    // clang-format off
    Matrix2x2(T i11, T i12,
              T i21, T i22) :
        _11{i11}, _12{i12},
        _21{i21}, _22{i22}
    // clang-format on
    {
    }
 
    template <typename Y>
    static Matrix2x2 MakeMatrix(const Y& vals)
    {
        return Matrix2x2 //
            {
                static_cast<T>(vals[0]), static_cast<T>(vals[1]),
                static_cast<T>(vals[2]), static_cast<T>(vals[3]) //
            };
    }
 
    bool operator==(const Matrix2x2& r) const
    {
        for (int i = 0; i < 2; ++i)
            for (int j = 0; j < 2; ++j)
                if ((*this)[i][j] != r[i][j])
                    return false;
 
        return true;
    }
 
    bool operator!=(const Matrix2x2& r) const
    {
        return !(*this == r);
    }
 
    T* operator[](size_t row)
    {
        return m[row];
    }
 
    const T* operator[](size_t row) const
    {
        return m[row];
    }
 
    T* Data() { return (*this)[0]; }
 
    const T* Data() const { return (*this)[0]; }
 
 
    Matrix2x2& operator*=(T s)
    {
        for (int i = 0; i < 4; ++i)
            (reinterpret_cast<T*>(this))[i] *= s;
 
        return *this;
    }
 
    Matrix2x2& operator*=(const Matrix2x2& right)
    {
        *this = Mul(*this, right);
        return *this;
    }
 
    Matrix2x2 Transpose() const
    {
        return Matrix2x2{
            _11, _21,
            _12, _22};
    }
 
    static Matrix2x2 Identity()
    {
        return Matrix2x2{
            1, 0,
            0, 1};
    }
 
    static Matrix2x2 Mul(const Matrix2x2& m1, const Matrix2x2& m2)
    {
        Matrix2x2 mOut;
        for (int i = 0; i < 2; i++)
        {
            for (int j = 0; j < 2; j++)
            {
                for (int k = 0; k < 2; k++)
                {
                    mOut.m[i][j] += m1.m[i][k] * m2.m[k][j];
                }
            }
        }
        return mOut;
    }
 
    static Matrix2x2 Rotation(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix2x2 //
            {
                c, s,
                -s, c //
            };
    }
 
    T Determinant() const
    {
        return _11 * _22 - _12 * _21;
    }
 
    Matrix2x2 Inverse() const
    {
        Matrix2x2 Inv //
            {
                +_22, -_12,
                -_21, +_11 //
            };
 
        Inv *= static_cast<T>(1) / Determinant();
        return Inv;
    }
};
 
 
template <class T> struct Matrix3x3
{
    union
    {
        struct
        {
            T _11;
            T _12;
            T _13;
            T _21;
            T _22;
            T _23;
            T _31;
            T _32;
            T _33;
        };
        struct
        {
            T m00;
            T m01;
            T m02;
            T m10;
            T m11;
            T m12;
            T m20;
            T m21;
            T m22;
        };
        T m[3][3];
    };
 
    explicit Matrix3x3(T value) :
        // clang-format off
        _11{value}, _12{value}, _13{value},
        _21{value}, _22{value}, _23{value},
        _31{value}, _32{value}, _33{value}
    // clang-format on
    {
    }
 
    Matrix3x3() :
        Matrix3x3{0} {}
 
    // clang-format off
    Matrix3x3(T i11, T i12, T i13,
              T i21, T i22, T i23,
              T i31, T i32, T i33) :
        _11{i11}, _12{i12}, _13{i13}, 
        _21{i21}, _22{i22}, _23{i23}, 
        _31{i31}, _32{i32}, _33{i33}
    // clang-format on
    {
    }
 
    template <typename Y>
    static Matrix3x3 MakeMatrix(const Y& vals)
    {
        return Matrix3x3 //
            {
                static_cast<T>(vals[0]), static_cast<T>(vals[1]), static_cast<T>(vals[2]),
                static_cast<T>(vals[3]), static_cast<T>(vals[4]), static_cast<T>(vals[5]),
                static_cast<T>(vals[6]), static_cast<T>(vals[7]), static_cast<T>(vals[8]) //
            };
    }
 
    bool operator==(const Matrix3x3& r) const
    {
        for (int i = 0; i < 3; ++i)
            for (int j = 0; j < 3; ++j)
                if ((*this)[i][j] != r[i][j])
                    return false;
 
        return true;
    }
 
    bool operator!=(const Matrix3x3& r) const
    {
        return !(*this == r);
    }
 
    T* operator[](size_t row)
    {
        return m[row];
    }
 
    const T* operator[](size_t row) const
    {
        return m[row];
    }
 
    T* Data() { return (*this)[0]; }
 
    const T* Data() const { return (*this)[0]; }
 
    Matrix3x3& operator*=(T s)
    {
        for (int i = 0; i < 9; ++i)
            (reinterpret_cast<T*>(this))[i] *= s;
 
        return *this;
    }
 
    Matrix3x3& operator*=(const Matrix3x3& right)
    {
        *this = Mul(*this, right);
        return *this;
    }
 
    Matrix3x3 Transpose() const
    {
        return Matrix3x3 //
            {
                _11, _21, _31,
                _12, _22, _32,
                _13, _23, _33 //
            };
    }
 
    static Matrix3x3 Identity()
    {
        return Matrix3x3 //
            {
                1, 0, 0,
                0, 1, 0,
                0, 0, 1 //
            };
    }
 
    static Matrix3x3 Scale(T x, T y, T z)
    {
        return Matrix3x3 //
            {
                x, 0, 0,
                0, y, 0,
                0, 0, z //
            };
    }
 
    // D3D-style left-handed matrix that rotates a point around the x axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z') = (x y z) * RotationX
    static Matrix3x3 RotationX(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix3x3 // clang-format off
            {
                1,  0,  0,
                0,  c,  s,
                0, -s,  c // clang-format on
            };
    }
 
    // D3D-style left-handed matrix that rotates a point around the y axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z' 1) = (x y z 1) * RotationY
    static Matrix3x3 RotationY(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix3x3 // clang-format off
            {
                c,  0, -s,
                0,  1,  0,
                s,  0,  c // clang-format on
            };
    }
 
    // D3D-style left-handed matrix that rotates a point around the z axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z' 1) = (x y z 1) * RotationZ
    static Matrix3x3 RotationZ(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix3x3 // clang-format off
            {
                 c,  s,  0,
                -s,  c,  0,
                 0,  0,  1 // clang-format on
            };
    }
 
    static Matrix3x3 Mul(const Matrix3x3& m1, const Matrix3x3& m2)
    {
        Matrix3x3 mOut;
        for (int i = 0; i < 3; i++)
        {
            for (int j = 0; j < 3; j++)
            {
                for (int k = 0; k < 3; k++)
                {
                    mOut.m[i][j] += m1.m[i][k] * m2.m[k][j];
                }
            }
        }
 
        return mOut;
    }
 
    T Determinant() const
    {
        T det = 0;
        det += _11 * (_22 * _33 - _32 * _23);
        det -= _12 * (_21 * _33 - _31 * _23);
        det += _13 * (_21 * _32 - _31 * _22);
        return det;
    }
};
 
template <class T> struct Matrix4x4
{
    union
    {
        struct
        {
            T _11;
            T _12;
            T _13;
            T _14;
            T _21;
            T _22;
            T _23;
            T _24;
            T _31;
            T _32;
            T _33;
            T _34;
            T _41;
            T _42;
            T _43;
            T _44;
        };
        struct
        {
            T m00;
            T m01;
            T m02;
            T m03;
            T m10;
            T m11;
            T m12;
            T m13;
            T m20;
            T m21;
            T m22;
            T m23;
            T m30;
            T m31;
            T m32;
            T m33;
        };
        T m[4][4];
    };
 
    explicit Matrix4x4(T value) :
        // clang-format off
        _11{value}, _12{value}, _13{value}, _14{value},
        _21{value}, _22{value}, _23{value}, _24{value},
        _31{value}, _32{value}, _33{value}, _34{value},
        _41{value}, _42{value}, _43{value}, _44{value}
    // clang-format on
    {
    }
 
    Matrix4x4() :
        Matrix4x4{0} {}
 
    // clang-format off
    Matrix4x4(T i11, T i12, T i13, T i14,
              T i21, T i22, T i23, T i24,
              T i31, T i32, T i33, T i34,
              T i41, T i42, T i43, T i44) :
        _11{i11}, _12{i12}, _13{i13}, _14{i14}, 
        _21{i21}, _22{i22}, _23{i23}, _24{i24}, 
        _31{i31}, _32{i32}, _33{i33}, _34{i34}, 
        _41{i41}, _42{i42}, _43{i43}, _44{i44}
    {
    }
    // clang-format on
 
    // clang-format off
    Matrix4x4(const Vector4<T>& Row0,
              const Vector4<T>& Row1,
              const Vector4<T>& Row2,
              const Vector4<T>& Row3) :
        _11{Row0.x}, _12{Row0.y}, _13{Row0.z}, _14{Row0.w}, 
        _21{Row1.x}, _22{Row1.y}, _23{Row1.z}, _24{Row1.w}, 
        _31{Row2.x}, _32{Row2.y}, _33{Row2.z}, _34{Row2.w}, 
        _41{Row3.x}, _42{Row3.y}, _43{Row3.z}, _44{Row3.w}
    {
    }
    // clang-format on
 
    template <typename Y>
    static Matrix4x4 MakeMatrix(const Y& vals)
    {
        // clang-format off
        return Matrix4x4
            {
                static_cast<T>(vals[ 0]), static_cast<T>(vals[ 1]), static_cast<T>(vals[ 2]), static_cast<T>(vals[ 3]),
                static_cast<T>(vals[ 4]), static_cast<T>(vals[ 5]), static_cast<T>(vals[ 6]), static_cast<T>(vals[ 7]),
                static_cast<T>(vals[ 8]), static_cast<T>(vals[ 9]), static_cast<T>(vals[10]), static_cast<T>(vals[11]),
                static_cast<T>(vals[12]), static_cast<T>(vals[13]), static_cast<T>(vals[14]), static_cast<T>(vals[15])
            };
        // clang-format on
    }
 
    bool operator==(const Matrix4x4& r) const
    {
        for (int i = 0; i < 4; ++i)
            for (int j = 0; j < 4; ++j)
                if ((*this)[i][j] != r[i][j])
                    return false;
 
        return true;
    }
 
    bool operator!=(const Matrix4x4& r) const
    {
        return !(*this == r);
    }
 
    T* operator[](size_t row)
    {
        return m[row];
    }
 
    const T* operator[](size_t row) const
    {
        return m[row];
    }
 
    T* Data() { return (*this)[0]; }
 
    const T* Data() const { return (*this)[0]; }
 
    Matrix4x4& operator*=(T s)
    {
        for (int i = 0; i < 16; ++i)
            (reinterpret_cast<T*>(this))[i] *= s;
 
        return *this;
    }
 
    Matrix4x4& operator*=(const Matrix4x4& right)
    {
        *this = Mul(*this, right);
        return *this;
    }
 
    Matrix4x4 Transpose() const
    {
        return Matrix4x4 //
            {
                _11, _21, _31, _41,
                _12, _22, _32, _42,
                _13, _23, _33, _43,
                _14, _24, _34, _44 //
            };
    }
 
    static Matrix4x4 Identity()
    {
        return Matrix4x4 //
            {
                1, 0, 0, 0,
                0, 1, 0, 0,
                0, 0, 1, 0,
                0, 0, 0, 1 //
            };
    }
 
    static Matrix4x4 Translation(T x, T y, T z)
    {
        return Matrix4x4 //
            {
                1, 0, 0, 0,
                0, 1, 0, 0,
                0, 0, 1, 0,
                x, y, z, 1 //
            };
    }
 
    static Matrix4x4 Translation(const Vector3<T>& v)
    {
        return Translation(v.x, v.y, v.z);
    }
 
    static Matrix4x4 Scale(T x, T y, T z)
    {
        return Matrix4x4 //
            {
                x, 0, 0, 0,
                0, y, 0, 0,
                0, 0, z, 0,
                0, 0, 0, 1 //
            };
    }
 
    static Matrix4x4 Scale(const Vector3<T>& v)
    {
        return Scale(v.x, v.y, v.z);
    }
 
    static Matrix4x4 Scale(T s)
    {
        return Scale(s, s, s);
    }
 
 
    // D3D-style left-handed matrix that rotates a point around the x axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z' 1) = (x y z 1) * RotationX
    static Matrix4x4 RotationX(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix4x4 // clang-format off
            {
                1,  0,  0,  0,
                0,  c,  s,  0,
                0, -s,  c,  0,
                0,  0,  0,  1 // clang-format on
            };
    }
 
    // D3D-style left-handed matrix that rotates a point around the y axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z' 1) = (x y z 1) * RotationY
    static Matrix4x4 RotationY(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix4x4 // clang-format off
            {
                c,  0, -s,  0,
                0,  1,  0,  0,
                s,  0,  c,  0,
                0,  0,  0,  1 // clang-format on
            };
    }
 
    // D3D-style left-handed matrix that rotates a point around the z axis. Angle (in radians)
    // is measured clockwise when looking along the rotation axis toward the origin:
    // (x' y' z' 1) = (x y z 1) * RotationZ
    static Matrix4x4 RotationZ(T angleInRadians)
    {
        auto s = std::sin(angleInRadians);
        auto c = std::cos(angleInRadians);
 
        return Matrix4x4 // clang-format off
            {
                 c,  s,  0,  0,
                -s,  c,  0,  0,
                 0,  0,  1,  0,
                 0,  0,  0,  1 // clang-format on
            };
    }
 
    // 3D Rotation matrix for an arbitrary axis specified by x, y and z
    static Matrix4x4 RotationArbitrary(Vector3<T> axis, T angleInRadians)
    {
        axis = normalize(axis);
 
        auto sinAngle         = std::sin(angleInRadians);
        auto cosAngle         = std::cos(angleInRadians);
        auto oneMinusCosAngle = 1 - cosAngle;
 
        Matrix4x4 mOut;
 
        mOut._11 = 1 + oneMinusCosAngle * (axis.x * axis.x - 1);
        mOut._12 = axis.z * sinAngle + oneMinusCosAngle * axis.x * axis.y;
        mOut._13 = -axis.y * sinAngle + oneMinusCosAngle * axis.x * axis.z;
        mOut._41 = 0;
 
        mOut._21 = -axis.z * sinAngle + oneMinusCosAngle * axis.y * axis.x;
        mOut._22 = 1 + oneMinusCosAngle * (axis.y * axis.y - 1);
        mOut._23 = axis.x * sinAngle + oneMinusCosAngle * axis.y * axis.z;
        mOut._24 = 0;
 
        mOut._31 = axis.y * sinAngle + oneMinusCosAngle * axis.z * axis.x;
        mOut._32 = -axis.x * sinAngle + oneMinusCosAngle * axis.z * axis.y;
        mOut._33 = 1 + oneMinusCosAngle * (axis.z * axis.z - 1);
        mOut._34 = 0;
 
        mOut._41 = 0;
        mOut._42 = 0;
        mOut._43 = 0;
        mOut._44 = 1;
 
        return mOut;
    }
 
    static Matrix4x4 ViewFromBasis(const Vector3<T>& f3X, const Vector3<T>& f3Y, const Vector3<T>& f3Z)
    {
        return Matrix4x4 // clang-format off
            {
                f3X.x, f3Y.x, f3Z.x,   0,
                f3X.y, f3Y.y, f3Z.y,   0,
                f3X.z, f3Y.z, f3Z.z,   0,
                    0,     0,     0,   1 // clang-format on
            };
    }
 
 
    void SetNearFarClipPlanes(T zNear, T zFar, T bIsGL)
    {
        if (bIsGL)
        {
            // https://www.opengl.org/sdk/docs/man2/xhtml/gluPerspective.xml
            // http://www.terathon.com/gdc07_lengyel.pdf
            // Note that OpenGL uses right-handed coordinate system, where
            // camera is looking in negative z direction:
            //   OO
            //  |__|<--------------------
            //         -z             +z
            // Consequently, OpenGL projection matrix given by these two
            // references inverts z axis.
 
            // We do not need to do this, because we use DX coordinate
            // system for the camera space. Thus we need to invert the
            // sign of the values in the third column in the matrix
            // from the references:
 
            _33 = -(-(zFar + zNear) / (zFar - zNear));
            _43 = -2 * zNear * zFar / (zFar - zNear);
            _34 = -(-1);
        }
        else
        {
            _33 = zFar / (zFar - zNear);
            _43 = -zNear * zFar / (zFar - zNear);
            _34 = 1;
        }
    }
 
    void GetNearFarClipPlanes(T& zNear, T& zFar, bool bIsGL) const
    {
        if (bIsGL)
        {
            zNear = _43 / (-1 - _33);
            zFar  = _43 / (+1 - _33);
        }
        else
        {
            zNear = -_43 / _33;
            zFar  = _33 / (_33 - 1) * zNear;
        }
    }
 
    static Matrix4x4 Projection(T fov, T aspectRatio, T zNear, T zFar, bool bIsGL) // Left-handed projection
    {
        Matrix4x4 mOut;
        auto      yScale = static_cast<T>(1) / std::tan(fov / static_cast<T>(2));
        auto      xScale = yScale / aspectRatio;
        mOut._11         = xScale;
        mOut._22         = yScale;
 
        mOut.SetNearFarClipPlanes(zNear, zFar, bIsGL);
 
        return mOut;
    }
 
    static Matrix4x4 OrthoOffCenter(T left, T right, T bottom, T top, T zNear, T zFar, bool bIsGL) // Left-handed ortho projection
    {
        auto _22 = (bIsGL ? 2 : 1) / (zFar - zNear);
        auto _32 = (bIsGL ? zNear + zFar : zNear) / (zNear - zFar);
        // clang-format off
        return Matrix4x4
            {
                         2   / (right - left),                                 0,     0,    0,
                                            0,                2 / (top - bottom),     0,    0,
                                            0,                                 0,   _22,    0,                
                (left + right)/(left - right),   (top + bottom) / (bottom - top),   _32,    1
            };
        // clang-format on
    }
 
    static Matrix4x4 Ortho(T width, T height, T zNear, T zFar, bool bIsGL) // Left-handed ortho projection
    {
        return OrthoOffCenter(
            -width * static_cast<T>(0.5),
            +width * static_cast<T>(0.5),
            -height * static_cast<T>(0.5),
            +height * static_cast<T>(0.5),
            zNear, zFar, bIsGL);
    }
 
    static Matrix4x4 Mul(const Matrix4x4& m1, const Matrix4x4& m2)
    {
        Matrix4x4 mOut;
        for (int i = 0; i < 4; i++)
        {
            for (int j = 0; j < 4; j++)
            {
                for (int k = 0; k < 4; k++)
                {
                    mOut.m[i][j] += m1.m[i][k] * m2.m[k][j];
                }
            }
        }
        return mOut;
    }
 
 
    T Determinant() const
    {
        T det = 0.f;
 
        det += _11 *
            Matrix3x3<T>(_22, _23, _24,
                         _32, _33, _34,
                         _42, _43, _44)
                .Determinant();
 
        det -= _12 *
            Matrix3x3<T>(_21, _23, _24,
                         _31, _33, _34,
                         _41, _43, _44)
                .Determinant();
 
        det += _13 *
            Matrix3x3<T>(_21, _22, _24,
                         _31, _32, _34,
                         _41, _42, _44)
                .Determinant();
 
        det -= _14 *
            Matrix3x3<T>(_21, _22, _23,
                         _31, _32, _33,
                         _41, _42, _43)
                .Determinant();
 
        return det;
    }
 
    Matrix4x4 Inverse() const
    {
        Matrix4x4 inv;
 
        // row 1
        inv._11 =
            Matrix3x3<T>(_22, _23, _24,
                         _32, _33, _34,
                         _42, _43, _44)
                .Determinant();
 
        inv._12 =
            -Matrix3x3<T>(_21, _23, _24,
                          _31, _33, _34,
                          _41, _43, _44)
                 .Determinant();
 
        inv._13 =
            Matrix3x3<T>(_21, _22, _24,
                         _31, _32, _34,
                         _41, _42, _44)
                .Determinant();
 
        inv._14 =
            -Matrix3x3<T>(_21, _22, _23,
                          _31, _32, _33,
                          _41, _42, _43)
                 .Determinant();
 
 
        // row 2
        inv._21 =
            -Matrix3x3<T>(_12, _13, _14,
                          _32, _33, _34,
                          _42, _43, _44)
                 .Determinant();
 
        inv._22 =
            Matrix3x3<T>(_11, _13, _14,
                         _31, _33, _34,
                         _41, _43, _44)
                .Determinant();
 
        inv._23 =
            -Matrix3x3<T>(_11, _12, _14,
                          _31, _32, _34,
                          _41, _42, _44)
                 .Determinant();
 
        inv._24 =
            Matrix3x3<T>(_11, _12, _13,
                         _31, _32, _33,
                         _41, _42, _43)
                .Determinant();
 
 
        // row 3
        inv._31 =
            Matrix3x3<T>(_12, _13, _14,
                         _22, _23, _24,
                         _42, _43, _44)
                .Determinant();
 
        inv._32 =
            -Matrix3x3<T>(_11, _13, _14,
                          _21, _23, _24,
                          _41, _43, _44)
                 .Determinant();
 
        inv._33 =
            Matrix3x3<T>(_11, _12, _14,
                         _21, _22, _24,
                         _41, _42, _44)
                .Determinant();
 
        inv._34 =
            -Matrix3x3<T>(_11, _12, _13,
                          _21, _22, _23,
                          _41, _42, _43)
                 .Determinant();
 
 
        // row 4
        inv._41 =
            -Matrix3x3<T>(_12, _13, _14,
                          _22, _23, _24,
                          _32, _33, _34)
                 .Determinant();
 
        inv._42 =
            Matrix3x3<T>(_11, _13, _14,
                         _21, _23, _24,
                         _31, _33, _34)
                .Determinant();
 
        inv._43 =
            -Matrix3x3<T>(_11, _12, _14,
                          _21, _22, _24,
                          _31, _32, _34)
                 .Determinant();
 
        inv._44 =
            Matrix3x3<T>(_11, _12, _13,
                         _21, _22, _23,
                         _31, _32, _33)
                .Determinant();
 
        auto det = _11 * inv._11 + _12 * inv._12 + _13 * inv._13 + _14 * inv._14;
        inv      = inv.Transpose();
        inv *= static_cast<T>(1) / det;
 
        return inv;
    }
 
    Matrix4x4 RemoveTranslation() const
    {
        return Matrix4x4 // clang-format off
            {
                _11, _12, _13, _14,
                _21, _22, _23, _24,
                _31, _32, _33, _34,
                  0,   0,   0, _44 // clang-format on
            };
    }
};
 
// Template Vector Operations
 
 
template <class T>
T dot(const Vector2<T>& a, const Vector2<T>& b)
{
    return a.x * b.x + a.y * b.y;
}
 
template <class T>
T dot(const Vector3<T>& a, const Vector3<T>& b)
{
    return a.x * b.x + a.y * b.y + a.z * b.z;
}
 
template <class T>
T dot(const Vector4<T>& a, const Vector4<T>& b)
{
    return a.x * b.x + a.y * b.y + a.z * b.z + a.w * b.w;
}
 
template <class VectorType>
auto length(const VectorType& a) -> decltype(dot(a, a))
{
    return sqrt(dot(a, a));
}
 
 
template <class T>
Vector3<T> min(const Vector3<T>& a, const Vector3<T>& b)
{
    return Vector3<T>(std::min(a.x, b.x), std::min(a.y, b.y), std::min(a.z, b.z));
}
 
template <class T>
Vector4<T> min(const Vector4<T>& a, const Vector4<T>& b)
{
    return Vector4<T>(std::min(a.x, b.x), std::min(a.y, b.y), std::min(a.z, b.z), std::min(a.w, b.w));
}
 
template <class T>
Vector3<T> max(const Vector3<T>& a, const Vector3<T>& b)
{
    return Vector3<T>(std::max(a.x, b.x), std::max(a.y, b.y), std::max(a.z, b.z));
}
 
template <class T>
Vector4<T> max(const Vector4<T>& a, const Vector4<T>& b)
{
    return Vector4<T>(std::max(a.x, b.x), std::max(a.y, b.y), std::max(a.z, b.z), std::max(a.w, b.w));
}
 
template <class T>
Vector2<T> abs(const Vector2<T>& a)
{
    // WARNING: abs() on gcc is for integers only!
    return Vector2<T>(a.x < 0 ? -a.x : a.x,
                      a.y < 0 ? -a.y : a.y);
}
 
template <class T>
Vector3<T> abs(const Vector3<T>& a)
{
    // WARNING: abs() on gcc is for integers only!
    return Vector3<T>(a.x < 0 ? -a.x : a.x,
                      a.y < 0 ? -a.y : a.y,
                      a.z < 0 ? -a.z : a.z);
}
 
template <class T>
Vector4<T> abs(const Vector4<T>& a)
{
    // WARNING: abs() on gcc is for integers only!
    return Vector4<T>(a.x < 0 ? -a.x : a.x,
                      a.y < 0 ? -a.y : a.y,
                      a.z < 0 ? -a.z : a.z,
                      a.w < 0 ? -a.w : a.w);
}
 
 
template <typename T>
T clamp(T val, T _min, T _max)
{
    return val < _min ? _min : (val > _max ? _max : val);
}
 
template <class T>
Vector2<T> clamp(const Vector2<T>& a, const Vector2<T>& _min, const Vector2<T>& _max)
{
    return Vector2<T>(clamp(a.x, _min.x, _max.x),
                      clamp(a.y, _min.y, _max.y));
}
 
template <class T>
Vector3<T> clamp(const Vector3<T>& a, const Vector3<T>& _min, const Vector3<T>& _max)
{
    return Vector3<T>(clamp(a.x, _min.x, _max.x),
                      clamp(a.y, _min.y, _max.y),
                      clamp(a.z, _min.z, _max.z));
}
 
template <class T>
Vector4<T> clamp(const Vector4<T>& a, const Vector4<T>& _min, const Vector4<T>& _max)
{
    return Vector4<T>(clamp(a.x, _min.x, _max.x),
                      clamp(a.y, _min.y, _max.y),
                      clamp(a.z, _min.z, _max.z),
                      clamp(a.w, _min.w, _max.w));
}
 
 
template <class T>
Vector3<T> cross(const Vector3<T>& a, const Vector3<T>& b)
{
    // |   i    j    k   |
    // |  a.x  a.y  a.z  |
    // |  b.x  b.y  b.z  |
    return Vector3<T>((a.y * b.z) - (a.z * b.y), (a.z * b.x) - (a.x * b.z), (a.x * b.y) - (a.y * b.x));
}
 
template <class T, class Y>
Vector3<T> cross(const Vector3<T>& a, const Vector3<T>& b)
{
    // |   i    j    k   |
    // |  a.x  a.y  a.z  |
    // |  b.x  b.y  b.z  |
    return Vector3<T> //
        {
            static_cast<T>(Y{a.y} * Y{b.z} - Y{a.z} * Y{b.y}),
            static_cast<T>(Y{a.z} * Y{b.x} - Y{a.x} * Y{b.z}),
            static_cast<T>(Y{a.x} * Y{b.y} - Y{a.y} * Y{b.x}) //
        };
}
 
inline Vector3<float> high_precision_cross(const Vector3<float>& a, const Vector3<float>& b)
{
    return cross<float, double>(a, b);
}
 
inline Vector3<int32_t> high_precision_cross(const Vector3<int32_t>& a, const Vector3<int32_t>& b)
{
    return cross<int32_t, int64_t>(a, b);
}
 
template <class VectorType>
VectorType normalize(const VectorType& a)
{
    auto len = length(a);
    return a / len;
}
 
 
// Template Matrix-Matrix multiplications
 
template <class T>
Matrix4x4<T> operator*(const Matrix4x4<T>& m1, const Matrix4x4<T>& m2)
{
    return Matrix4x4<T>::Mul(m1, m2);
}
 
template <class T>
Matrix3x3<T> operator*(const Matrix3x3<T>& m1, const Matrix3x3<T>& m2)
{
    return Matrix3x3<T>::Mul(m1, m2);
}
 
template <class T>
Matrix2x2<T> operator*(const Matrix2x2<T>& m1, const Matrix2x2<T>& m2)
{
    return Matrix2x2<T>::Mul(m1, m2);
}
 
 
// Template Matrix-Vector multiplications
 
template <class T>
Vector4<T> operator*(const Matrix4x4<T>& m, const Vector4<T>& v)
{
    Vector4<T> out;
    out[0] = m[0][0] * v.x + m[0][1] * v.y + m[0][2] * v.z + m[0][3] * v.w;
    out[1] = m[1][0] * v.x + m[1][1] * v.y + m[1][2] * v.z + m[1][3] * v.w;
    out[2] = m[2][0] * v.x + m[2][1] * v.y + m[2][2] * v.z + m[2][3] * v.w;
    out[3] = m[3][0] * v.x + m[3][1] * v.y + m[3][2] * v.z + m[3][3] * v.w;
    return out;
}
 
template <class T>
Vector3<T> operator*(const Matrix3x3<T>& m, Vector3<T>& v)
{
    Vector3<T> out;
    out[0] = m[0][0] * v.x + m[0][1] * v.y + m[0][2] * v.z;
    out[1] = m[1][0] * v.x + m[1][1] * v.y + m[1][2] * v.z;
    out[2] = m[2][0] * v.x + m[2][1] * v.y + m[2][2] * v.z;
    return out;
}
 
template <class T>
Vector2<T> operator*(const Matrix2x2<T>& m, const Vector2<T>& v)
{
    Vector2<T> out;
    out[0] = m[0][0] * v.x + m[0][1] * v.y;
    out[1] = m[1][0] * v.x + m[1][1] * v.y;
    return out;
}
 
// Common HLSL-compatible vector typedefs
 
using uint  = uint32_t;
using uint2 = Vector2<uint>;
using uint3 = Vector3<uint>;
using uint4 = Vector4<uint>;
 
using int2 = Vector2<int32_t>;
using int3 = Vector3<int32_t>;
using int4 = Vector4<int32_t>;
 
using float2 = Vector2<float>;
using float3 = Vector3<float>;
using float4 = Vector4<float>;
 
using double2 = Vector2<double>;
using double3 = Vector3<double>;
using double4 = Vector4<double>;
 
using float4x4 = Matrix4x4<float>;
using float3x3 = Matrix3x3<float>;
using float2x2 = Matrix2x2<float>;
 
using double4x4 = Matrix4x4<double>;
using double3x3 = Matrix3x3<double>;
using double2x2 = Matrix2x2<double>;
 
 
struct Quaternion
{
    float4 q;
 
    Quaternion(const float4& _q) noexcept :
        q{_q}
    {}
    Quaternion(float x, float y, float z, float w) noexcept :
        q{x, y, z, w}
    {
    }
    Quaternion() noexcept {}
 
    bool operator==(const Quaternion& right) const
    {
        return q == right.q;
    }
 
    template <typename Y>
    static Quaternion MakeQuaternion(const Y& vals)
    {
        return Quaternion{float4::MakeVector(vals)};
    }
 
    static Quaternion RotationFromAxisAngle(const float3& axis, float angle)
    {
        Quaternion out{0, 0, 0, 1};
        float      norm = length(axis);
        if (norm != 0)
        {
            float sina2 = sin(0.5f * angle);
            out.q[0]    = sina2 * axis[0] / norm;
            out.q[1]    = sina2 * axis[1] / norm;
            out.q[2]    = sina2 * axis[2] / norm;
            out.q[3]    = cos(0.5f * angle);
        }
        return out;
    }
 
    void GetAxisAngle(float3& outAxis, float& outAngle) const
    {
        float sina2 = sqrt(q[0] * q[0] + q[1] * q[1] + q[2] * q[2]);
        outAngle    = 2.0f * atan2(sina2, q[3]);
        float r     = (sina2 > 0) ? (1.0f / sina2) : 0;
        outAxis[0]  = r * q[0];
        outAxis[1]  = r * q[1];
        outAxis[2]  = r * q[2];
    }
 
    float4x4 ToMatrix() const
    {
        float4x4 out;
        float    yy2 = 2.0f * q[1] * q[1];
        float    xy2 = 2.0f * q[0] * q[1];
        float    xz2 = 2.0f * q[0] * q[2];
        float    yz2 = 2.0f * q[1] * q[2];
        float    zz2 = 2.0f * q[2] * q[2];
        float    wz2 = 2.0f * q[3] * q[2];
        float    wy2 = 2.0f * q[3] * q[1];
        float    wx2 = 2.0f * q[3] * q[0];
        float    xx2 = 2.0f * q[0] * q[0];
        out[0][0]    = -yy2 - zz2 + 1.0f;
        out[0][1]    = xy2 + wz2;
        out[0][2]    = xz2 - wy2;
        out[0][3]    = 0;
        out[1][0]    = xy2 - wz2;
        out[1][1]    = -xx2 - zz2 + 1.0f;
        out[1][2]    = yz2 + wx2;
        out[1][3]    = 0;
        out[2][0]    = xz2 + wy2;
        out[2][1]    = yz2 - wx2;
        out[2][2]    = -xx2 - yy2 + 1.0f;
        out[2][3]    = 0;
        out[3][0] = out[3][1] = out[3][2] = 0;
        out[3][3]                         = 1;
        return out;
    }
 
    static Quaternion Mul(const Quaternion& q1, const Quaternion& q2)
    {
        Quaternion q1_q2;
        q1_q2.q.x = +q1.q.x * q2.q.w + q1.q.y * q2.q.z - q1.q.z * q2.q.y + q1.q.w * q2.q.x;
        q1_q2.q.y = -q1.q.x * q2.q.z + q1.q.y * q2.q.w + q1.q.z * q2.q.x + q1.q.w * q2.q.y;
        q1_q2.q.z = +q1.q.x * q2.q.y - q1.q.y * q2.q.x + q1.q.z * q2.q.w + q1.q.w * q2.q.z;
        q1_q2.q.w = -q1.q.x * q2.q.x - q1.q.y * q2.q.y - q1.q.z * q2.q.z + q1.q.w * q2.q.w;
        return q1_q2;
    }
 
    Quaternion& operator=(const Quaternion& rhs)
    {
        q = rhs.q;
        return *this;
    }
 
    Quaternion& operator*=(const Quaternion& rhs)
    {
        *this = Mul(*this, rhs);
        return *this;
    }
 
    float3 RotateVector(const float3& v) const
    {
        const float3 axis(q.x, q.y, q.z);
        return v + 2.f * cross(axis, cross(axis, v) + q.w * v);
    }
};
 
inline Quaternion operator*(const Quaternion& q1, const Quaternion& q2)
{
    return Quaternion::Mul(q1, q2);
}
 
inline Quaternion normalize(const Quaternion& q)
{
    return Quaternion{normalize(q.q)};
}
 
// https://en.wikipedia.org/wiki/Slerp
inline Quaternion slerp(Quaternion v0, Quaternion v1, float t, bool DoNotNormalize = false)
{
    // Only unit quaternions are valid rotations.
    // Normalize to avoid undefined behavior.
    if (!DoNotNormalize)
    {
        v0 = normalize(v0);
        v1 = normalize(v1);
    }
 
    // Compute the cosine of the angle between the two vectors.
    auto dp = dot(v0.q, v1.q);
 
    // If the dot product is negative, slerp won't take
    // the shorter path. Note that v1 and -v1 are equivalent when
    // the negation is applied to all four components. Fix by
    // reversing one quaternion.
    if (dp < 0)
    {
        v1.q = -v1.q;
        dp   = -dp;
    }
 
    const double DOT_THRESHOLD = 0.9995;
    if (dp > DOT_THRESHOLD)
    {
        // If the inputs are too close for comfort, linearly interpolate
        // and normalize the result.
 
        Quaternion result = Quaternion{v0.q + t * (v1.q - v0.q)};
        result.q          = normalize(result.q);
        return result;
    }
 
    // Since dot is in range [0, DOT_THRESHOLD], acos is safe
    auto theta_0     = std::acos(dp);     // theta_0 = angle between input vectors
    auto theta       = theta_0 * t;       // theta = angle between v0 and result
    auto sin_theta   = std::sin(theta);   // compute this value only once
    auto sin_theta_0 = std::sin(theta_0); // compute this value only once
 
    auto s0 = cos(theta) - dp * sin_theta / sin_theta_0; // == sin(theta_0 - theta) / sin(theta_0)
    auto s1 = sin_theta / sin_theta_0;
 
    auto v = Quaternion{v0.q * s0 + v1.q * s1};
    if (!DoNotNormalize)
    {
        v = normalize(v);
    }
    return v;
}
 
 
template <typename T>
T lerp(const T& Left, const T& Right, float w)
{
    return Left * (1.f - w) + Right * w;
}
 
template <typename T>
T SmoothStep(T Left, T Right, T w)
{
    auto t = clamp((w - Left) / (Right - Left), static_cast<T>(0), static_cast<T>(1));
    return t * t * (static_cast<T>(3) - static_cast<T>(2) * t);
}
 
template <typename T>
T max3(const T& x, const T& y, const T& z)
{
    return std::max(std::max(x, y), z);
}
 
template <typename T>
T min3(const T& x, const T& y, const T& z)
{
    return std::min(std::min(x, y), z);
}
 
inline float4 RGBA8Unorm_To_F4Color(Uint32 RGBA8)
{
    // clang-format off
    return float4
    {
            static_cast<float>((RGBA8 >>  0u) & 0xFF) / 255.f,
            static_cast<float>((RGBA8 >>  8u) & 0xFF) / 255.f,
            static_cast<float>((RGBA8 >> 16u) & 0xFF) / 255.f,
            static_cast<float>((RGBA8 >> 24u) & 0xFF) / 255.f
    };
    // clang-format on
}
 
inline Uint32 F4Color_To_RGBA8Unorm(const float4& f4Color)
{
    Uint32 RGBA8U = 0;
    RGBA8U |= static_cast<Uint32>(clamp(f4Color.r, 0.f, 1.f) * 255.f) << 0u;
    RGBA8U |= static_cast<Uint32>(clamp(f4Color.g, 0.f, 1.f) * 255.f) << 8u;
    RGBA8U |= static_cast<Uint32>(clamp(f4Color.b, 0.f, 1.f) * 255.f) << 16u;
    RGBA8U |= static_cast<Uint32>(clamp(f4Color.a, 0.f, 1.f) * 255.f) << 24u;
    return RGBA8U;
}
 
template <typename T>
struct _FastFloatIntermediateType
{
};
 
template <>
struct _FastFloatIntermediateType<float>
{
    // All floats that have fractional part are representable as 32-bit int
    using Type = Int32;
};
 
template <>
struct _FastFloatIntermediateType<double>
{
    // All doubles that have fractional part are representable as 64-bit int
    using Type = Int64;
};
 
// At least on MSVC std::floor is an actual function call into ucrtbase.dll.
// All floats/doubles that have fractional parts also fit into integer
// representable range, so we can do much better.
template <typename T>
T FastFloor(T x)
{
    auto i   = static_cast<typename _FastFloatIntermediateType<T>::Type>(x);
    auto flr = static_cast<T>(i);
    //   x         flr    floor(x)  flr <= x
    // +1.0   ->   1.0      1.0       true
    // +0.5   ->   0.0      0.0       true
    //  0.0   ->   0.0      0.0       true
    // -0.5   ->   0.0     -1.0      false
    // -1.0   ->  -1.0     -1.0       true
 
    return flr <= x ? flr : flr - 1;
}
 
template <typename T>
T FastCeil(T x)
{
    return -FastFloor(-x);
}
 
 
template <typename T>
Diligent::Vector2<T> FastFloor(const Diligent::Vector2<T>& vec)
{
    return Diligent::Vector2<T>{
        FastFloor(vec.x),
        FastFloor(vec.y)};
}
 
template <typename T>
Diligent::Vector3<T> FastFloor(const Diligent::Vector3<T>& vec)
{
    return Diligent::Vector3<T>{
        FastFloor(vec.x),
        FastFloor(vec.y),
        FastFloor(vec.z)};
}
 
template <typename T>
Diligent::Vector4<T> FastFloor(const Diligent::Vector4<T>& vec)
{
    return Diligent::Vector4<T>{
        FastFloor(vec.x),
        FastFloor(vec.y),
        FastFloor(vec.z),
        FastFloor(vec.w)};
}
 
 
template <typename T>
Diligent::Vector2<T> FastCeil(const Diligent::Vector2<T>& vec)
{
    return Diligent::Vector2<T>{
        FastCeil(vec.x),
        FastCeil(vec.y)};
}
 
template <typename T>
Diligent::Vector3<T> FastCeil(const Diligent::Vector3<T>& vec)
{
    return Diligent::Vector3<T>{
        FastCeil(vec.x),
        FastCeil(vec.y),
        FastCeil(vec.z)};
}
 
template <typename T>
Diligent::Vector4<T> FastCeil(const Diligent::Vector4<T>& vec)
{
    return Diligent::Vector4<T>{
        FastCeil(vec.x),
        FastCeil(vec.y),
        FastCeil(vec.z),
        FastCeil(vec.w)};
}
 
inline Uint32 BitInterleave16(Uint16 _x, Uint16 _y)
{
    // https://graphics.stanford.edu/~seander/bithacks.html#InterleaveBMN
 
    // Interleave lower 16 bits of x and y, so the bits of x
    // are in the even positions and bits from y in the odd;
    // x | (y << 1) gets the resulting 32-bit Morton Number.
    // x and y must initially be less than 65536.
    Uint32 x = _x;
    Uint32 y = _y;
 
    x = (x | (x << 8u)) & 0x00FF00FFu;
    x = (x | (x << 4u)) & 0x0F0F0F0Fu;
    x = (x | (x << 2u)) & 0x33333333u;
    x = (x | (x << 1u)) & 0x55555555u;
 
    y = (y | (y << 8u)) & 0x00FF00FFu;
    y = (y | (y << 4u)) & 0x0F0F0F0Fu;
    y = (y | (y << 2u)) & 0x33333333u;
    y = (y | (y << 1u)) & 0x55555555u;
 
    return x | (y << 1u);
}
 
template <typename T>
typename std::enable_if<std::is_integral<T>::value, T>::type ExtractLSB(T& bits)
{
    if (bits == T{0})
        return 0;
 
    const T bit = bits & ~(bits - T{1});
    bits &= ~bit;
 
    return bit;
}
 
template <typename T>
typename std::enable_if<std::is_enum<T>::value, T>::type ExtractLSB(T& bits)
{
    return static_cast<T>(ExtractLSB(reinterpret_cast<typename std::underlying_type<T>::type&>(bits)));
}
 
 
inline std::ostream& operator<<(std::ostream& os, const float4& vec)
{
    return os << "float4(" << vec.x << ", " << vec.y << ", " << vec.z << ", " << vec.w << ')';
}
inline std::ostream& operator<<(std::ostream& os, const float3& vec)
{
    return os << "float3(" << vec.x << ", " << vec.y << ", " << vec.z << ')';
}
inline std::ostream& operator<<(std::ostream& os, const float2& vec)
{
    return os << "float2(" << vec.x << ", " << vec.y << ')';
}
 
inline std::ostream& operator<<(std::ostream& os, const int4& vec)
{
    return os << "int4(" << vec.x << ", " << vec.y << ", " << vec.z << ", " << vec.w << ')';
}
inline std::ostream& operator<<(std::ostream& os, const int3& vec)
{
    return os << "int3(" << vec.x << ", " << vec.y << ", " << vec.z << ')';
}
inline std::ostream& operator<<(std::ostream& os, const int2& vec)
{
    return os << "int2(" << vec.x << ", " << vec.y << ')';
}
 
 
inline std::ostream& operator<<(std::ostream& os, const uint4& vec)
{
    return os << "uint4(" << vec.x << ", " << vec.y << ", " << vec.z << ", " << vec.w << ')';
}
inline std::ostream& operator<<(std::ostream& os, const uint3& vec)
{
    return os << "uint3(" << vec.x << ", " << vec.y << ", " << vec.z << ')';
}
inline std::ostream& operator<<(std::ostream& os, const uint2& vec)
{
    return os << "uint2(" << vec.x << ", " << vec.y << ')';
}
 
} // namespace Diligent
 
 
 
namespace std
{
template <typename T>
Diligent::Vector2<T> max(const Diligent::Vector2<T>& Left, const Diligent::Vector2<T>& Right)
{
    return Diligent::Vector2<T>(
        std::max(Left.x, Right.x),
        std::max(Left.y, Right.y));
}
 
template <typename T>
Diligent::Vector3<T> max(const Diligent::Vector3<T>& Left, const Diligent::Vector3<T>& Right)
{
    return Diligent::Vector3<T>(
        std::max(Left.x, Right.x),
        std::max(Left.y, Right.y),
        std::max(Left.z, Right.z));
}
 
template <typename T>
Diligent::Vector4<T> max(const Diligent::Vector4<T>& Left, const Diligent::Vector4<T>& Right)
{
    return Diligent::Vector4<T>(
        std::max(Left.x, Right.x),
        std::max(Left.y, Right.y),
        std::max(Left.z, Right.z),
        std::max(Left.w, Right.w));
}
 
 
template <typename T>
Diligent::Vector2<T> min(const Diligent::Vector2<T>& Left, const Diligent::Vector2<T>& Right)
{
    return Diligent::Vector2<T>(
        std::min(Left.x, Right.x),
        std::min(Left.y, Right.y));
}
 
template <typename T>
Diligent::Vector3<T> min(const Diligent::Vector3<T>& Left, const Diligent::Vector3<T>& Right)
{
    return Diligent::Vector3<T>(
        std::min(Left.x, Right.x),
        std::min(Left.y, Right.y),
        std::min(Left.z, Right.z));
}
 
template <typename T>
Diligent::Vector4<T> min(const Diligent::Vector4<T>& Left, const Diligent::Vector4<T>& Right)
{
    return Diligent::Vector4<T>(
        std::min(Left.x, Right.x),
        std::min(Left.y, Right.y),
        std::min(Left.z, Right.z),
        std::min(Left.w, Right.w));
}
 
template <typename T>
Diligent::Vector2<T> floor(const Diligent::Vector2<T>& vec)
{
    return Diligent::Vector2<T>(
        std::floor(vec.x),
        std::floor(vec.y));
}
 
template <typename T>
Diligent::Vector3<T> floor(const Diligent::Vector3<T>& vec)
{
    return Diligent::Vector3<T>(
        std::floor(vec.x),
        std::floor(vec.y),
        std::floor(vec.z));
}
 
template <typename T>
Diligent::Vector4<T> floor(const Diligent::Vector4<T>& vec)
{
    return Diligent::Vector4<T>(
        std::floor(vec.x),
        std::floor(vec.y),
        std::floor(vec.z),
        std::floor(vec.w));
}
 
 
template <typename T>
Diligent::Vector2<T> ceil(const Diligent::Vector2<T>& vec)
{
    return Diligent::Vector2<T>(
        std::ceil(vec.x),
        std::ceil(vec.y));
}
 
template <typename T>
Diligent::Vector3<T> ceil(const Diligent::Vector3<T>& vec)
{
    return Diligent::Vector3<T>(
        std::ceil(vec.x),
        std::ceil(vec.y),
        std::ceil(vec.z));
}
 
template <typename T>
Diligent::Vector4<T> ceil(const Diligent::Vector4<T>& vec)
{
    return Diligent::Vector4<T>(
        std::ceil(vec.x),
        std::ceil(vec.y),
        std::ceil(vec.z),
        std::ceil(vec.w));
}
 
 
template <typename T>
struct hash<Diligent::Vector2<T>>
{
    size_t operator()(const Diligent::Vector2<T>& v2) const
    {
        return Diligent::ComputeHash(v2.x, v2.y);
    }
};
 
template <typename T>
struct hash<Diligent::Vector3<T>>
{
    size_t operator()(const Diligent::Vector3<T>& v3) const
    {
        return Diligent::ComputeHash(v3.x, v3.y, v3.z);
    }
};
 
template <typename T>
struct hash<Diligent::Vector4<T>>
{
    size_t operator()(const Diligent::Vector4<T>& v4) const
    {
        return Diligent::ComputeHash(v4.x, v4.y, v4.z, v4.w);
    }
};
 
template <typename T>
struct hash<Diligent::Matrix2x2<T>>
{
    size_t operator()(const Diligent::Matrix2x2<T>& m) const
    {
        return Diligent::ComputeHash(
            m.m00, m.m01,
            m.m10, m.m11);
    }
};
 
template <typename T>
struct hash<Diligent::Matrix3x3<T>>
{
    size_t operator()(const Diligent::Matrix3x3<T>& m) const
    {
        return Diligent::ComputeHash(
            m.m00, m.m01, m.m02,
            m.m10, m.m11, m.m12,
            m.m20, m.m21, m.m22);
    }
};
 
template <typename T>
struct hash<Diligent::Matrix4x4<T>>
{
    size_t operator()(const Diligent::Matrix4x4<T>& m) const
    {
        return Diligent::ComputeHash(
            m.m00, m.m01, m.m02, m.m03,
            m.m10, m.m11, m.m12, m.m13,
            m.m20, m.m21, m.m22, m.m23,
            m.m30, m.m31, m.m32, m.m33);
    }
};
} // namespace std
 
#ifdef _MSC_VER
#    pragma warning(pop)
#endif