text: #ifndef Magnum_Math_TypeTraits_h
  #define Magnum_Math_TypeTraits_h
  /*
      This file is part of Magnum.
  
      Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
                  2020, 2021, 2022 Vladimír Vondruš <mosra@centrum.cz>
  
      Permission is hereby granted, free of charge, to any person obtaining a
      copy of this software and associated documentation files (the "Software"),
      to deal in the Software without restriction, including without limitation
      the rights to use, copy, modify, merge, publish, distribute, sublicense,
      and/or sell copies of the Software, and to permit persons to whom the
      Software is furnished to do so, subject to the following conditions:
  
      The above copyright notice and this permission notice shall be included
      in all copies or substantial portions of the Software.
  
      THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
      IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
      FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
      THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
      LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
      FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
      DEALINGS IN THE SOFTWARE.
  */
  
  /** @file
   * @brief Type traits
   */
  
  #include <type_traits>
  #include <Corrade/Utility/StlMath.h>
  
  #include "Magnum/Math/Math.h"
  #include "Magnum/Types.h"
  
  #ifdef MAGNUM_BUILD_DEPRECATED
  #include <Corrade/Utility/Macros.h>
  #endif
  
  #ifdef MAGNUM_BUILD_DEPRECATED
  /**
  @brief Precision when testing floats for equality
  @m_deprecated_since{2020,06} Use @ref Magnum::Math::TypeTraits::epsilon()
      instead.
  
  They have "at least" 6 significant digits of precision, taking one digit less
  for more headroom.
  */
  #ifndef FLOAT_EQUALITY_PRECISION
  #define FLOAT_EQUALITY_PRECISION \
      CORRADE_DEPRECATED_MACRO(FLOAT_EQUALITY_PRECISION, "use Math::TypeTraits instead") 1.0e-5f
  #endif
  
  /**
  @brief Precision when testing doubles for equality
  @m_deprecated_since{2020,06} Use @ref Magnum::Math::TypeTraits::epsilon()
      instead.
  
  They have "at least" 15 significant digits of precision, taking one digit less
  for more headroom.
  */
  #ifndef DOUBLE_EQUALITY_PRECISION
  #define DOUBLE_EQUALITY_PRECISION \
      CORRADE_DEPRECATED_MACRO(DOUBLE_EQUALITY_PRECISION, "use Math::TypeTraits instead") 1.0e-14
  #endif
  
  /**
  @brief Precision when testing long doubles for equality
  @m_deprecated_since{2020,06} Use @ref Magnum::Math::TypeTraits::epsilon()
      instead.
  
  They have "at least" 18 significant digits of precision on platforms where it
  is 80-bit, and 15 on platforms @ref CORRADE_LONG_DOUBLE_SAME_AS_DOUBLE "where it is 64-bit",
  taking one digit less for more headroom.
  */
  #ifndef LONG_DOUBLE_EQUALITY_PRECISION
  #ifndef CORRADE_LONG_DOUBLE_SAME_AS_DOUBLE
  #define LONG_DOUBLE_EQUALITY_PRECISION \
      CORRADE_DEPRECATED_MACRO(LONG_DOUBLE_EQUALITY_PRECISION, "use Math::TypeTraits instead") 1.0e-17l
  #else
  #define LONG_DOUBLE_EQUALITY_PRECISION \
      CORRADE_DEPRECATED_MACRO(LONG_DOUBLE_EQUALITY_PRECISION, "use Math::TypeTraits instead") 1.0e-14l
  #endif
  #endif
  #endif
  
  namespace Magnum { namespace Math {
  
  /**
  @brief Whether @p T is an arithmetic scalar type
  @m_since{2019,10}
  
  Equivalent to @ref std::true_type for all builtin scalar integer and
  floating-point types and in addition also @ref Half, @ref Deg and @ref Rad;
  equivalent to @ref std::false_type otherwise.
  
  Note that this is *different* from @ref std::is_scalar, which is @cpp true @ce
  also for enums or pointers --- it's rather closer to @ref std::is_arithmetic,
  except that it doesn't give @ref std::true_type for @cpp bool @ce. The name is
  chosen particularly for the @ref IsVector / @ref IsScalar distinction.
  @see @ref IsFloatingPoint, @ref IsIntegral, @ref UnderlyingTypeOf
  */
  template<class T> struct IsScalar
      #ifndef DOXYGEN_GENERATING_OUTPUT
      : std::false_type
      #endif
      {};
  
  #ifndef DOXYGEN_GENERATING_OUTPUT
  /* Can't use our own typedefs because they don't cover all types (signed char
     vs char, long vs long long etc.). Funny that char != signed char but signed
     int = int */
  template<> struct IsScalar<char>: std::true_type {};
  template<> struct IsScalar<signed char>: std::true_type {};
  template<> struct IsScalar<unsigned char>: std::true_type {};
  template<> struct IsScalar<short>: std::true_type {};
  template<> struct IsScalar<unsigned short>: std::true_type {};
  template<> struct IsScalar<int>: std::true_type {};
  template<> struct IsScalar<unsigned int>: std::true_type {};
  template<> struct IsScalar<long>: std::true_type {};
  template<> struct IsScalar<unsigned long>: std::true_type {};
  template<> struct IsScalar<long long>: std::true_type {};
  template<> struct IsScalar<unsigned long long>: std::true_type {};
  template<> struct IsScalar<float>: std::true_type {};
  template<> struct IsScalar<Half>: std::true_type {};
  template<> struct IsScalar<double>: std::true_type {};
  template<> struct IsScalar<long double>: std::true_type {};
  template<template<class> class Derived, class T> struct IsScalar<Unit<Derived, T>>: std::true_type {};
  template<class T> struct IsScalar<Deg<T>>: std::true_type {};
  template<class T> struct IsScalar<Rad<T>>: std::true_type {};
  #endif
  
  /**
  @brief Whether @p T is an arithmetic vector type
  @m_since{2019,10}
  
  Equivalent to @ref std::true_type for all @ref Vector types and their
  subclasses; equivalent to @ref std::false_type otherwise. In particular, gives
  @ref std::false_type for @ref BoolVector, all matrix types, @ref Complex or
  @ref Quaternion.
  @see @ref IsScalar, @ref IsFloatingPoint, @ref IsIntegral
  */
  template<class T> struct IsVector
      #ifndef DOXYGEN_GENERATING_OUTPUT
      : std::false_type
      #endif
      {};
  
  #ifndef DOXYGEN_GENERATING_OUTPUT
  template<std::size_t size, class T> struct IsVector<Vector<size, T>>: std::true_type {};
  template<class T> struct IsVector<Vector2<T>>: std::true_type {};
  template<class T> struct IsVector<Vector3<T>>: std::true_type {};
  template<class T> struct IsVector<Vector4<T>>: std::true_type {};
  template<class T> struct IsVector<Color3<T>>: std::true_type {};
  template<class T> struct IsVector<Color4<T>>: std::true_type {};
  #endif
  
  /**
  @brief Whether @p T is integral
  @m_since{2019,10}
  
  Equivalent to @ref std::true_type for all integral scalar and vector types
  supported by Magnum math; equivalent to @ref std::false_type otherwise.
  
  Unlike @ref std::is_integral this is @ref std::false_type for @cpp bool @ce.
  @see @ref IsFloatingPoint, @ref IsScalar, @ref IsVector
  */
  template<class T> struct IsIntegral
      #ifndef DOXYGEN_GENERATING_OUTPUT
      : std::false_type
      #endif
      {};
  
  #ifndef DOXYGEN_GENERATING_OUTPUT
  /* Can't use our own typedefs because they don't cover all types (signed char
     vs char, long vs long long etc.). Funny that char != signed char but signed
     int = int */
  template<> struct IsIntegral<char>: std::true_type {};
  template<> struct IsIntegral<signed char>: std::true_type {};
  template<> struct IsIntegral<unsigned char>: std::true_type {};
  template<> struct IsIntegral<short>: std::true_type {};
  template<> struct IsIntegral<unsigned short>: std::true_type {};
  template<> struct IsIntegral<int>: std::true_type {};
  template<> struct IsIntegral<unsigned int>: std::true_type {};
  template<> struct IsIntegral<long>: std::true_type {};
  template<> struct IsIntegral<unsigned long>: std::true_type {};
  template<> struct IsIntegral<long long>: std::true_type {};
  template<> struct IsIntegral<unsigned long long>: std::true_type {};
  template<std::size_t size, class T> struct IsIntegral<Vector<size, T>>: IsIntegral<T> {};
  template<class T> struct IsIntegral<Vector2<T>>: IsIntegral<T> {};
  template<class T> struct IsIntegral<Vector3<T>>: IsIntegral<T> {};
  template<class T> struct IsIntegral<Vector4<T>>: IsIntegral<T> {};
  template<class T> struct IsIntegral<Color3<T>>: IsIntegral<T> {};
  template<class T> struct IsIntegral<Color4<T>>: IsIntegral<T> {};
  /* I don't expect Deg/Rad to ever have an integral base type */
  #endif
  
  /**
  @brief Whether @p T is floating-point
  @m_since{2019,10}
  
  Equivalent to @ref std::true_type for all floating-point scalar and vector
  types supported by Magnum math including @ref Half, @ref Deg and @ref Rad;
  equivalent to @ref std::false_type otherwise.
  @see @ref IsIntegral, @ref IsScalar, @ref IsVector, @ref std::is_floating_point
  */
  template<class T> struct IsFloatingPoint
      #ifndef DOXYGEN_GENERATING_OUTPUT
      : std::false_type
      #endif
      {};
  
  #ifndef DOXYGEN_GENERATING_OUTPUT
  template<> struct IsFloatingPoint<Float>: std::true_type {};
  template<> struct IsFloatingPoint<Half>: std::true_type {};
  template<> struct IsFloatingPoint<Double>: std::true_type {};
  template<> struct IsFloatingPoint<long double>: std::true_type {};
  template<std::size_t size, class T> struct IsFloatingPoint<Vector<size, T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Vector2<T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Vector3<T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Vector4<T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Color3<T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Color4<T>>: IsFloatingPoint<T> {};
  /* Deg<Half> is legal but Half is not an arithmetic type (and thus not
     floating-point), so need to check the underlying type as well */
  template<template<class> class Derived, class T> struct IsFloatingPoint<Unit<Derived, T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Deg<T>>: IsFloatingPoint<T> {};
  template<class T> struct IsFloatingPoint<Rad<T>>: IsFloatingPoint<T> {};
  #endif
  
  /**
  @brief Whether @p T is a unitless type
  @m_since{2019,10}
  
  Equivalent to @ref std::true_type for scalar or vector types that have an
  unitless underlying type (i.e., not @ref Deg or @ref Rad); @ref std::false_type
  otherwise. Some math functions such as @ref sqrt() or @ref log() work only with
  unitless types because the resulting unit couldn't be expressed otherwise.
  @see @ref IsScalar, @ref IsVector
  */
  template<class T> struct IsUnitless
      #ifndef DOXYGEN_GENERATING_OUTPUT
      : std::integral_constant<bool, IsScalar<T>::value || IsVector<T>::value>
      #endif
      {};
  template<template<class> class Derived, class T> struct IsUnitless<Unit<Derived, T>>: std::false_type {};
  template<class T> struct IsUnitless<Deg<T>>: std::false_type {};
  template<class T> struct IsUnitless<Rad<T>>: std::false_type {};
  
  namespace Implementation {
      template<class T> struct UnderlyingType {
          static_assert(IsScalar<T>::value, "type is not scalar");
          typedef T Type;
      };
      template<template<class> class Derived, class T> struct UnderlyingType<Unit<Derived, T>> {
          typedef T Type;
      };
      template<class T> struct UnderlyingType<Deg<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Rad<T>> { typedef T Type; };
      template<std::size_t size, class T> struct UnderlyingType<Vector<size, T>> {
          typedef T Type;
      };
      template<class T> struct UnderlyingType<Vector2<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Vector3<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Vector4<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Color3<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Color4<T>> { typedef T Type; };
      template<std::size_t cols, std::size_t rows, class T> struct UnderlyingType<RectangularMatrix<cols, rows, T>> {
          typedef T Type;
      };
      template<std::size_t size, class T> struct UnderlyingType<Matrix<size, T>> {
          typedef T Type;
      };
      template<class T> struct UnderlyingType<Matrix3<T>> { typedef T Type; };
      template<class T> struct UnderlyingType<Matrix4<T>> { typedef T Type; };
  }
  
  /**
  @brief Underlying type of a math type
  @m_since{2019,10}
  
  For builtin scalar types returns the type itself, for wrapped types like
  @ref Deg or @ref Rad returns the underlying builtin type, for vector and matrix
  types the type of their components.
  
  For scalar types it's guaranteed that the input type is always explicitly
  convertible to the output type and the output type is usable with standard APIs
  such as @ref std::isinf().
  */
  template<class T> using UnderlyingTypeOf = typename Implementation::UnderlyingType<T>::Type;
  
  namespace Implementation {
      template<class T> struct TypeTraitsDefault {
          TypeTraitsDefault() = delete;
  
          constexpr static bool equals(T a, T b) {
              return a == b;
          }
  
          constexpr static bool equalsZero(T a, T) {
              return !a;
          }
      };
  }
  
  /**
  @brief Traits class for builtin arithmetic types
  
  Useful for detecting type features at compile time without the need for
  repeated code such as method overloading, cascaded ifs or template
  specializations for given types. All builtin arithmetic types have this class
  implemented.
  */
  template<class T> struct TypeTraits: Implementation::TypeTraitsDefault<T> {
      /*
       * The following values are implemented as inline functions, not as
       * static const variables, because the compiler will inline the return
       * values instead of referencing to static data and unlike static const
       * variables the functions can be overloaded, deleted and hidden.
       */
  
      #ifdef DOXYGEN_GENERATING_OUTPUT
      /**
       * @brief Corresponding floating-point type for normalization
       *
       * If the type is not already floating-point, defines smallest larger
       * floating-point type.
       */
      typedef U FloatingPointType;
  
      /**
       * @brief Type name
       *
       * Returns a string representation of type name, such as
       * @cpp "UnsignedInt" @ce for @relativeref{Magnum,UnsignedInt}.
       */
      constexpr static const char* name();
  
      /**
       * @brief Epsilon value for fuzzy compare
       *
       * Returns minimal difference between numbers to be considered
       * inequal. Not implemented for arbitrary types. Returns @cpp 1 @ce for
       * integer types and
       *
       * -    @cpp 1.0e-5f @ce for @cpp float @ce,
       * -    @cpp 1.0e-15 @ce for @cpp double @ce,
       * -    @cpp 1.0e-17l @ce for @cpp long double @ce on platforms where it is
       *      80-bit, and @cpp 1.0e-14l @ce on platforms
       *      @ref CORRADE_LONG_DOUBLE_SAME_AS_DOUBLE "where it is 64-bit".
       *
       * This matches fuzzy comparison precision in @ref Corrade::TestSuite and
       * is always one digit less than how @ref Corrade::Utility::Debug or
       * @ref Corrade::Utility::format() prints given type.
       */
      constexpr static T epsilon();
  
      /**
       * @brief Fuzzy compare
       *
       * Uses fuzzy compare for all floating-point types except @ref Half (using
       * the @ref epsilon() value), pure equality comparison everywhere else.
       * The @ref Half type has representable values sparse enough that no fuzzy
       * comparison needs to be done. Algorithm adapted from
       * http://floating-point-gui.de/errors/comparison/.
       * @see @ref Math::equal(T, T), @ref Math::notEqual(T, T)
       */
      static bool equals(T a, T b);
  
      /**
       * @brief Fuzzy compare to zero with magnitude
       *
       * Uses fuzzy compare for floating-point types (using @ref epsilon()
       * value), pure equality comparison everywhere else. Use this function when
       * comparing e.g. a calculated nearly-zero difference with zero, knowing
       * the magnitude of original values so the epsilon can be properly scaled.
       * In other words, the following lines are equivalent:
       *
       * @snippet MagnumMath.cpp TypeTraits-equalsZero
       */
      static bool equalsZero(T a, T magnitude);
      #endif
  };
  
  /**
  @brief Equality comparison of scalar types
  @m_since{2019,10}
  
  Calls @ref TypeTraits<T>::equals() --- using fuzzy compare for floating-point
  types and doing equality comparison on integral types. Scalar complement to
  @ref equal(const Vector<size, T>& a, const Vector<size, T>&).
  */
  template<class T> inline typename std::enable_if<IsScalar<T>::value, bool>::type equal(T a, T b) {
      return TypeTraits<T>::equals(a, b);
  }
  
  /**
  @brief Non-equality comparison of scalar types
  @m_since{2019,10}
  
  Calls @ref TypeTraits<T>::equals() --- using fuzzy compare for floating-point
  types and doing equality comparison on integral types. Scalar complement to
  @ref notEqual(const Vector<size, T>& a, const Vector<size, T>&).
  */
  template<class T> inline typename std::enable_if<IsScalar<T>::value, bool>::type notEqual(T a, T b) {
      return !TypeTraits<T>::equals(a, b);
  }
  
  /* Integral scalar types */
  namespace Implementation {
      template<class> struct TypeTraitsName;
      #ifndef DOXYGEN_GENERATING_OUTPUT
      #define _c(type) template<> struct TypeTraitsName<type> { \
          constexpr static const char* name() { return #type; } \
      };
      _c(UnsignedByte)
      _c(Byte)
      _c(UnsignedShort)
      _c(Short)
      _c(UnsignedInt)
      _c(Int)
      _c(UnsignedLong)
      _c(Long)
      _c(Float)
      _c(Half)
      _c(Double)
      _c(long double)
      #undef _c
      #endif
  
      template<class T> struct TypeTraitsIntegral: TypeTraitsDefault<T>, TypeTraitsName<T> {
          constexpr static T epsilon() { return T(1); }
      };
  }
  
  #ifndef DOXYGEN_GENERATING_OUTPUT
  template<> struct TypeTraits<UnsignedByte>: Implementation::TypeTraitsIntegral<UnsignedByte> {
      typedef Float FloatingPointType;
  };
  template<> struct TypeTraits<Byte>: Implementation::TypeTraitsIntegral<Byte> {
      typedef Float FloatingPointType;
  };
  template<> struct TypeTraits<UnsignedShort>: Implementation::TypeTraitsIntegral<UnsignedShort> {
      typedef Float FloatingPointType;
  };
  template<> struct TypeTraits<Short>: Implementation::TypeTraitsIntegral<Short> {
      typedef Float FloatingPointType;
  };
  template<> struct TypeTraits<UnsignedInt>: Implementation::TypeTraitsIntegral<UnsignedInt> {
      typedef Double FloatingPointType;
  };
  template<> struct TypeTraits<Int>: Implementation::TypeTraitsIntegral<Int> {
      typedef Double FloatingPointType;
  };
  template<> struct TypeTraits<UnsignedLong>: Implementation::TypeTraitsIntegral<UnsignedLong> {
      typedef long double FloatingPointType;
  };
  template<> struct TypeTraits<Long>: Implementation::TypeTraitsIntegral<Long> {
      typedef long double FloatingPointType;
  };
  
  /* Floating-point scalar types */
  namespace Implementation {
  
  template<class T> struct TypeTraitsFloatingPoint: TypeTraitsName<T> {
      TypeTraitsFloatingPoint() = delete;
  
      static bool equals(T a, T b);
      static bool equalsZero(T a, T epsilon);
  };
  
  template<class T> bool TypeTraitsFloatingPoint<T>::equals(const T a, const T b) {
      /* Shortcut for binary equality (also infinites) */
      if(a == b) return true;
  
      const T absA = std::abs(a);
      const T absB = std::abs(b);
      const T difference = std::abs(a - b);
  
      /* One of the numbers is zero or both are extremely close to it, relative
         error is meaningless */
      if(a == T{} || b == T{} || difference < TypeTraits<T>::epsilon())
          return difference < TypeTraits<T>::epsilon();
  
      /* Relative error */
      return difference/(absA + absB) < TypeTraits<T>::epsilon();
  }
  
  template<class T> bool TypeTraitsFloatingPoint<T>::equalsZero(const T a, const T magnitude) {
      /* Shortcut for binary equality */
      if(a == T(0.0)) return true;
  
      const T absA = std::abs(a);
  
      /* The value is extremely close to zero, relative error is meaningless */
      if(absA < TypeTraits<T>::epsilon())
          return absA < TypeTraits<T>::epsilon();
  
      /* Relative error */
      return absA*T(0.5)/magnitude < TypeTraits<T>::epsilon();
  }
  
  }
  
  template<> struct TypeTraits<Float>: Implementation::TypeTraitsFloatingPoint<Float> {
      typedef Float FloatingPointType;
  
      constexpr static Float epsilon() { return 1.0e-5f; }
  };
  /* A bit special -- using integer comparison for equality but presenting itself
     as a floating-point type so Color's fullChannel() works correctly */
  template<> struct TypeTraits<Half>: Implementation::TypeTraitsName<Half>, Implementation::TypeTraitsDefault<Half> {
      typedef Half FloatingPointType;
  };
  template<> struct TypeTraits<Double>: Implementation::TypeTraitsFloatingPoint<Double> {
      typedef Double FloatingPointType;
  
      constexpr static Double epsilon() { return 1.0e-14; }
  };
  template<> struct TypeTraits<long double>: Implementation::TypeTraitsFloatingPoint<long double> {
      typedef long double FloatingPointType;
  
      #ifndef CORRADE_LONG_DOUBLE_SAME_AS_DOUBLE
      constexpr static long double epsilon() { return 1.0e-17l; }
      #else
      constexpr static long double epsilon() { return 1.0e-14l; }
      #endif
  };
  
  namespace Implementation {
  
  /* Comparing squared length to 1 is not sufficient to compare within range
     [1 - epsilon, 1 + epsilon], as e.g. Quaternion with dot() = 1 + 1e-7 when
     converted to matrix has column vectors with dot() = 1 + 1e-6, which is just
     above 1 + epsilon. Thus it's needed to compare sqrt(dot()) in range
     [1 - epsilon, 1 + epsilon] or dot() in range [1 - 2*epsilon + epsilon^2,
     1 + 2*epsilon + epsilon^2]. Because epsilon^2 is way off machine precision,
     it's omitted. */
  template<class T> inline bool isNormalizedSquared(T lengthSquared) {
      return std::abs(lengthSquared - T(1)) < T(2)*TypeTraits<T>::epsilon();
  }
  
  }
  #endif
  
  }}
  
  #endif
  