base.h

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// Copyright (c) FIRST and other WPILib contributors.
// Open Source Software; you can modify and/or share it under the terms of
// the WPILib BSD license file in the root directory of this project.
 
// Copyright (c) 2016 Nic Holthaus
//
// The MIT License (MIT)
//
// 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.
//
// ATTRIBUTION:
// Parts of this work have been adapted from:
// http://stackoverflow.com/questions/35069778/create-comparison-trait-for-template-classes-whose-parameters-are-in-a-different
// http://stackoverflow.com/questions/28253399/check-traits-for-all-variadic-template-arguments/28253503
// http://stackoverflow.com/questions/36321295/rational-approximation-of-square-root-of-stdratio-at-compile-time?noredirect=1#comment60266601_36321295
//
 
/// @file   units.h
/// @brief Complete implementation of `units` - a compile-time, header-only,
///        unit conversion library built on c++14 with no dependencies.
 
#pragma once
 
#ifdef _MSC_VER
#   pragma push_macro("pascal")
#   undef pascal
#   if _MSC_VER <= 1800
#       define _ALLOW_KEYWORD_MACROS
#       pragma warning(push)
#       pragma warning(disable : 4520)
#       pragma push_macro("constexpr")
#       define constexpr /*constexpr*/
#       pragma push_macro("noexcept")
#       define noexcept throw()
#   endif // _MSC_VER < 1800
#endif // _MSC_VER
 
#if !defined(_MSC_VER) || _MSC_VER > 1800
#   define UNIT_HAS_LITERAL_SUPPORT
#endif
 
#ifndef UNIT_LIB_DEFAULT_TYPE
#   define UNIT_LIB_DEFAULT_TYPE double
#endif
 
//--------------------
//  INCLUDES
//--------------------
 
#include <chrono>
#include <ratio>
#include <type_traits>
#include <cstdint>
#include <cmath>
#include <limits>
 
#if defined(UNIT_LIB_ENABLE_IOSTREAM)
    #include <iostream>
    #include <locale>
    #include <string>
#endif
#if __has_include(<fmt/format.h>) && !defined(UNIT_LIB_DISABLE_FMT)
    #include <locale>
    #include <string>
    #include <fmt/format.h>
#endif
 
#include <gcem.hpp>
 
//------------------------------
//  STRING FORMATTER
//------------------------------
 
namespace units
{
  namespace detail
  {
    template <typename T> std::string to_string(const T& t)
    {
      std::string str{ std::to_string(t) };
      int offset{ 1 };
 
      // remove trailing decimal points for integer value units. Locale aware!
      struct lconv * lc;
      lc = localeconv();
      char decimalPoint = *lc->decimal_point;
      if (str.find_last_not_of('0') == str.find(decimalPoint)) { offset = 0; }
      str.erase(str.find_last_not_of('0') + offset, std::string::npos);
      return str;
    }
  }
}
 
namespace units
{
    template<typename T> inline constexpr const char* name(const T&);
    template<typename T> inline constexpr const char* abbreviation(const T&);
}
 
//------------------------------
//  MACROS
//------------------------------
 
/**
 * @def         UNIT_ADD_UNIT_TAGS(namespaceName,nameSingular, namePlural, abbreviation, definition)
 * @brief       Helper macro for generating the boiler-plate code generating the tags of a new unit.
 * @details     The macro generates singular, plural, and abbreviated forms
 *              of the unit definition (e.g. `meter`, `meters`, and `m`), as aliases for the
 *              unit tag.
 * @param       namespaceName namespace in which the new units will be encapsulated.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       namePlural - plural version of the unit name, e.g. 'meters'
 * @param       abbreviation - abbreviated unit name, e.g. 'm'
 * @param       definition - the variadic parameter is used for the definition of the unit
 *              (e.g. `unit<std::ratio<1>, units::category::length_unit>`)
 * @note        a variadic template is used for the definition to allow templates with
 *              commas to be easily expanded. All the variadic 'arguments' should together
 *              comprise the unit definition.
 */
#define UNIT_ADD_UNIT_TAGS(namespaceName,nameSingular, namePlural, abbreviation, /*definition*/...)\
    namespace namespaceName\
    {\
    /** @name Units (full names plural) */ /** @{ */ typedef __VA_ARGS__ namePlural; /** @} */\
    /** @name Units (full names singular) */ /** @{ */ typedef namePlural nameSingular; /** @} */\
    /** @name Units (abbreviated) */ /** @{ */ typedef namePlural abbreviation; /** @} */\
    }
 
/**
 * @def         UNIT_ADD_UNIT_DEFINITION(namespaceName,nameSingular)
 * @brief       Macro for generating the boiler-plate code for the unit_t type definition.
 * @details     The macro generates the definition of the unit container types, e.g. `meter_t`
 * @param       namespaceName namespace in which the new units will be encapsulated.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 */
#define UNIT_ADD_UNIT_DEFINITION(namespaceName,nameSingular)\
    namespace namespaceName\
    {\
        /** @name Unit Containers */ /** @{ */ typedef unit_t<nameSingular> nameSingular ## _t; /** @} */\
    }
 
/**
 * @def         UNIT_ADD_CUSTOM_TYPE_UNIT_DEFINITION(namespaceName,nameSingular,underlyingType)
 * @brief       Macro for generating the boiler-plate code for a unit_t type definition with a non-default underlying type.
 * @details     The macro generates the definition of the unit container types, e.g. `meter_t`
 * @param       namespaceName namespace in which the new units will be encapsulated.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       underlyingType the underlying type
 */
#define UNIT_ADD_CUSTOM_TYPE_UNIT_DEFINITION(namespaceName,nameSingular, underlyingType)\
    namespace namespaceName\
    {\
    /** @name Unit Containers */ /** @{ */ typedef unit_t<nameSingular,underlyingType> nameSingular ## _t; /** @} */\
    }
/**
 * @def         UNIT_ADD_IO(namespaceName,nameSingular, abbreviation)
 * @brief       Macro for generating the boiler-plate code needed for I/O for a new unit.
 * @details     The macro generates the code to insert units into an ostream. It
 *              prints both the value and abbreviation of the unit when invoked.
 * @param       namespaceName namespace in which the new units will be encapsulated.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       abbrev - abbreviated unit name, e.g. 'm'
 * @note        When UNIT_LIB_ENABLE_IOSTREAM isn't defined, the macro does not generate any code
 */
#if __has_include(<fmt/format.h>) && !defined(UNIT_LIB_DISABLE_FMT)
    #define UNIT_ADD_IO(namespaceName, nameSingular, abbrev)\
    }\
    template <>\
    struct fmt::formatter<units::namespaceName::nameSingular ## _t> \
        : fmt::formatter<double> \
    {\
        template <typename FormatContext>\
        auto format(const units::namespaceName::nameSingular ## _t& obj,\
                                FormatContext& ctx) -> decltype(ctx.out()) \
        {\
            auto out = ctx.out();\
            out = fmt::formatter<double>::format(obj(), ctx);\
            return fmt::format_to(out, " " #abbrev);\
        }\
    };\
    namespace units\
    {\
    namespace namespaceName\
    {\
        inline std::string to_string(const nameSingular ## _t& obj)\
        {\
            return units::detail::to_string(obj()) + std::string(" "#abbrev);\
        }\
    }
#endif
#if defined(UNIT_LIB_ENABLE_IOSTREAM)
    #define UNIT_ADD_IO(namespaceName, nameSingular, abbrev)\
    namespace namespaceName\
    {\
        inline std::ostream& operator<<(std::ostream& os, const nameSingular ## _t& obj) \
        {\
            os << obj() << " "#abbrev; return os; \
        }\
        inline std::string to_string(const nameSingular ## _t& obj)\
        {\
            return units::detail::to_string(obj()) + std::string(" "#abbrev);\
        }\
    }
#endif
 
 /**
  * @def        UNIT_ADD_NAME(namespaceName,nameSingular,abbreviation)
  * @brief      Macro for generating constexpr names/abbreviations for units.
  * @details    The macro generates names for units. E.g. name() of 1_m would be "meter", and
  *             abbreviation would be "m".
  * @param      namespaceName namespace in which the new units will be encapsulated. All literal values
  *             are placed in the `units::literals` namespace.
  * @param      nameSingular singular version of the unit name, e.g. 'meter'
  * @param      abbreviation - abbreviated unit name, e.g. 'm'
  */
#define UNIT_ADD_NAME(namespaceName, nameSingular, abbrev)\
template<> inline constexpr const char* name(const namespaceName::nameSingular ## _t&)\
{\
    return #nameSingular;\
}\
template<> inline constexpr const char* abbreviation(const namespaceName::nameSingular ## _t&)\
{\
    return #abbrev;\
}
 
/**
 * @def         UNIT_ADD_LITERALS(namespaceName,nameSingular,abbreviation)
 * @brief       Macro for generating user-defined literals for units.
 * @details     The macro generates user-defined literals for units. A literal suffix is created
 *              using the abbreviation (e.g. `10.0_m`).
 * @param       namespaceName namespace in which the new units will be encapsulated. All literal values
 *              are placed in the `units::literals` namespace.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       abbreviation - abbreviated unit name, e.g. 'm'
 * @note        When UNIT_HAS_LITERAL_SUPPORT is not defined, the macro does not generate any code
 */
#if defined(UNIT_HAS_LITERAL_SUPPORT)
    #define UNIT_ADD_LITERALS(namespaceName, nameSingular, abbreviation)\
    namespace literals\
    {\
        inline constexpr namespaceName::nameSingular ## _t operator""_ ## abbreviation(long double d)\
        {\
            return namespaceName::nameSingular ## _t(static_cast<namespaceName::nameSingular ## _t::underlying_type>(d));\
        }\
        inline constexpr namespaceName::nameSingular ## _t operator""_ ## abbreviation (unsigned long long d)\
        {\
            return namespaceName::nameSingular ## _t(static_cast<namespaceName::nameSingular ## _t::underlying_type>(d));\
        }\
    }
#else
    #define UNIT_ADD_LITERALS(namespaceName, nameSingular, abbreviation)
#endif
 
/**
 * @def         UNIT_ADD(namespaceName,nameSingular, namePlural, abbreviation, definition)
 * @brief       Macro for generating the boiler-plate code needed for a new unit.
 * @details     The macro generates singular, plural, and abbreviated forms
 *              of the unit definition (e.g. `meter`, `meters`, and `m`), as well as the
 *              appropriately named unit container (e.g. `meter_t`). A literal suffix is created
 *              using the abbreviation (e.g. `10.0_m`). It also defines a class-specific
 *              cout function which prints both the value and abbreviation of the unit when invoked.
 * @param       namespaceName namespace in which the new units will be encapsulated. All literal values
 *              are placed in the `units::literals` namespace.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       namePlural - plural version of the unit name, e.g. 'meters'
 * @param       abbreviation - abbreviated unit name, e.g. 'm'
 * @param       definition - the variadic parameter is used for the definition of the unit
 *              (e.g. `unit<std::ratio<1>, units::category::length_unit>`)
 * @note        a variadic template is used for the definition to allow templates with
 *              commas to be easily expanded. All the variadic 'arguments' should together
 *              comprise the unit definition.
 */
#define UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
    UNIT_ADD_UNIT_TAGS(namespaceName,nameSingular, namePlural, abbreviation, __VA_ARGS__)\
    UNIT_ADD_UNIT_DEFINITION(namespaceName,nameSingular)\
    UNIT_ADD_NAME(namespaceName,nameSingular, abbreviation)\
    UNIT_ADD_IO(namespaceName,nameSingular, abbreviation)\
    UNIT_ADD_LITERALS(namespaceName,nameSingular, abbreviation)
 
/**
 * @def         UNIT_ADD_WITH_CUSTOM_TYPE(namespaceName,nameSingular, namePlural, abbreviation, underlyingType, definition)
 * @brief       Macro for generating the boiler-plate code needed for a new unit with a non-default underlying type.
 * @details     The macro generates singular, plural, and abbreviated forms
 *              of the unit definition (e.g. `meter`, `meters`, and `m`), as well as the
 *              appropriately named unit container (e.g. `meter_t`). A literal suffix is created
 *              using the abbreviation (e.g. `10.0_m`). It also defines a class-specific
 *              cout function which prints both the value and abbreviation of the unit when invoked.
 * @param       namespaceName namespace in which the new units will be encapsulated. All literal values
 *              are placed in the `units::literals` namespace.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       namePlural - plural version of the unit name, e.g. 'meters'
 * @param       abbreviation - abbreviated unit name, e.g. 'm'
 * @param       underlyingType - the underlying type, e.g. 'int' or 'float'
 * @param       definition - the variadic parameter is used for the definition of the unit
 *              (e.g. `unit<std::ratio<1>, units::category::length_unit>`)
 * @note        a variadic template is used for the definition to allow templates with
 *              commas to be easily expanded. All the variadic 'arguments' should together
 *              comprise the unit definition.
 */
#define UNIT_ADD_WITH_CUSTOM_TYPE(namespaceName, nameSingular, namePlural, abbreviation, underlyingType, /*definition*/...)\
    UNIT_ADD_UNIT_TAGS(namespaceName,nameSingular, namePlural, abbreviation, __VA_ARGS__)\
    UNIT_ADD_CUSTOM_TYPE_UNIT_DEFINITION(namespaceName,nameSingular,underlyingType)\
    UNIT_ADD_IO(namespaceName,nameSingular, abbreviation)\
    UNIT_ADD_LITERALS(namespaceName,nameSingular, abbreviation)
 
/**
 * @def         UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)
 * @brief       Macro to create decibel container and literals for an existing unit type.
 * @details     This macro generates the decibel unit container, cout overload, and literal definitions.
 * @param       namespaceName namespace in which the new units will be encapsulated. All literal values
 *              are placed in the `units::literals` namespace.
 * @param       nameSingular singular version of the base unit name, e.g. 'watt'
 * @param       abbreviation - abbreviated decibel unit name, e.g. 'dBW'
 */
#define UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)\
    namespace namespaceName\
    {\
        /** @name Unit Containers */ /** @{ */ typedef unit_t<nameSingular, UNIT_LIB_DEFAULT_TYPE, units::decibel_scale> abbreviation ## _t; /** @} */\
    }\
    UNIT_ADD_IO(namespaceName, abbreviation, abbreviation)\
    UNIT_ADD_LITERALS(namespaceName, abbreviation, abbreviation)
 
/**
 * @def         UNIT_ADD_CATEGORY_TRAIT(unitCategory, baseUnit)
 * @brief       Macro to create the `is_category_unit` type trait.
 * @details     This trait allows users to test whether a given type matches
 *              an intended category. This macro comprises all the boiler-plate
 *              code necessary to do so.
 * @param       unitCategory The name of the category of unit, e.g. length or mass.
 */
 
#define UNIT_ADD_CATEGORY_TRAIT_DETAIL(unitCategory)\
    namespace traits\
    {\
        /** @cond */\
        namespace detail\
        {\
            template<typename T> struct is_ ## unitCategory ## _unit_impl : std::false_type {};\
            template<typename C, typename U, typename P, typename T>\
            struct is_ ## unitCategory ## _unit_impl<units::unit<C, U, P, T>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_traits<units::unit<C, U, P, T>>::base_unit_type>, units::category::unitCategory ## _unit>::type {};\
            template<typename U, typename S, template<typename> class N>\
            struct is_ ## unitCategory ## _unit_impl<units::unit_t<U, S, N>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_t_traits<units::unit_t<U, S, N>>::unit_type>, units::category::unitCategory ## _unit>::type {};\
        }\
        /** @endcond */\
    }
 
#define UNIT_ADD_IS_UNIT_CATEGORY_TRAIT(unitCategory)\
    namespace traits\
    {\
        template<typename... T> struct is_ ## unitCategory ## _unit : std::integral_constant<bool, units::all_true<units::traits::detail::is_ ## unitCategory ## _unit_impl<std::decay_t<T>>::value...>::value> {};\
        template<typename... T> inline constexpr bool is_ ## unitCategory ## _unit_v = is_ ## unitCategory ## _unit<T...>::value;\
    }\
    template <typename T>\
    concept unitCategory ## _unit = traits::is_ ## unitCategory ## _unit_v<T>;
 
#define UNIT_ADD_CATEGORY_TRAIT(unitCategory)\
    UNIT_ADD_CATEGORY_TRAIT_DETAIL(unitCategory)\
    /** @ingroup    TypeTraits*/\
    /** @brief      Trait which tests whether a type represents a unit of unitCategory*/\
    /** @details    Inherits from `std::true_type` or `std::false_type`. Use `is_ ## unitCategory ## _unit<T>::value` to test the unit represents a unitCategory quantity.*/\
    /** @tparam     T   one or more types to test*/\
    UNIT_ADD_IS_UNIT_CATEGORY_TRAIT(unitCategory)
 
/**
 * @def         UNIT_ADD_WITH_METRIC_PREFIXES(nameSingular, namePlural, abbreviation, definition)
 * @brief       Macro for generating the boiler-plate code needed for a new unit, including its metric
 *              prefixes from femto to peta.
 * @details     See UNIT_ADD. In addition to generating the unit definition and containers '(e.g. `meters` and 'meter_t',
 *              it also creates corresponding units with metric suffixes such as `millimeters`, and `millimeter_t`), as well as the
 *              literal suffixes (e.g. `10.0_mm`).
 * @param       namespaceName namespace in which the new units will be encapsulated. All literal values
 *              are placed in the `units::literals` namespace.
 * @param       nameSingular singular version of the unit name, e.g. 'meter'
 * @param       namePlural - plural version of the unit name, e.g. 'meters'
 * @param       abbreviation - abbreviated unit name, e.g. 'm'
 * @param       definition - the variadic parameter is used for the definition of the unit
 *              (e.g. `unit<std::ratio<1>, units::category::length_unit>`)
 * @note        a variadic template is used for the definition to allow templates with
 *              commas to be easily expanded. All the variadic 'arguments' should together
 *              comprise the unit definition.
 */
#define UNIT_ADD_WITH_METRIC_PREFIXES(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
    UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, __VA_ARGS__)\
    UNIT_ADD(namespaceName, femto ## nameSingular, femto ## namePlural, f ## abbreviation, femto<namePlural>)\
    UNIT_ADD(namespaceName, pico ## nameSingular, pico ## namePlural, p ## abbreviation, pico<namePlural>)\
    UNIT_ADD(namespaceName, nano ## nameSingular, nano ## namePlural, n ## abbreviation, nano<namePlural>)\
    UNIT_ADD(namespaceName, micro ## nameSingular, micro ## namePlural, u ## abbreviation, micro<namePlural>)\
    UNIT_ADD(namespaceName, milli ## nameSingular, milli ## namePlural, m ## abbreviation, milli<namePlural>)\
    UNIT_ADD(namespaceName, centi ## nameSingular, centi ## namePlural, c ## abbreviation, centi<namePlural>)\
    UNIT_ADD(namespaceName, deci ## nameSingular, deci ## namePlural, d ## abbreviation, deci<namePlural>)\
    UNIT_ADD(namespaceName, deca ## nameSingular, deca ## namePlural, da ## abbreviation, deca<namePlural>)\
    UNIT_ADD(namespaceName, hecto ## nameSingular, hecto ## namePlural, h ## abbreviation, hecto<namePlural>)\
    UNIT_ADD(namespaceName, kilo ## nameSingular, kilo ## namePlural, k ## abbreviation, kilo<namePlural>)\
    UNIT_ADD(namespaceName, mega ## nameSingular, mega ## namePlural, M ## abbreviation, mega<namePlural>)\
    UNIT_ADD(namespaceName, giga ## nameSingular, giga ## namePlural, G ## abbreviation, giga<namePlural>)\
    UNIT_ADD(namespaceName, tera ## nameSingular, tera ## namePlural, T ## abbreviation, tera<namePlural>)\
    UNIT_ADD(namespaceName, peta ## nameSingular, peta ## namePlural, P ## abbreviation, peta<namePlural>)\
 
 /**
  * @def        UNIT_ADD_WITH_METRIC_AND_BINARY_PREFIXES(nameSingular, namePlural, abbreviation, definition)
  * @brief      Macro for generating the boiler-plate code needed for a new unit, including its metric
  *             prefixes from femto to peta, and binary prefixes from kibi to exbi.
  * @details    See UNIT_ADD. In addition to generating the unit definition and containers '(e.g. `bytes` and 'byte_t',
  *             it also creates corresponding units with metric suffixes such as `millimeters`, and `millimeter_t`), as well as the
  *             literal suffixes (e.g. `10.0_B`).
  * @param      namespaceName namespace in which the new units will be encapsulated. All literal values
  *             are placed in the `units::literals` namespace.
  * @param      nameSingular singular version of the unit name, e.g. 'byte'
  * @param      namePlural - plural version of the unit name, e.g. 'bytes'
  * @param      abbreviation - abbreviated unit name, e.g. 'B'
  * @param      definition - the variadic parameter is used for the definition of the unit
  *             (e.g. `unit<std::ratio<1>, units::category::data_unit>`)
  * @note       a variadic template is used for the definition to allow templates with
  *             commas to be easily expanded. All the variadic 'arguments' should together
  *             comprise the unit definition.
  */
#define UNIT_ADD_WITH_METRIC_AND_BINARY_PREFIXES(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
    UNIT_ADD_WITH_METRIC_PREFIXES(namespaceName, nameSingular, namePlural, abbreviation, __VA_ARGS__)\
    UNIT_ADD(namespaceName, kibi ## nameSingular, kibi ## namePlural, Ki ## abbreviation, kibi<namePlural>)\
    UNIT_ADD(namespaceName, mebi ## nameSingular, mebi ## namePlural, Mi ## abbreviation, mebi<namePlural>)\
    UNIT_ADD(namespaceName, gibi ## nameSingular, gibi ## namePlural, Gi ## abbreviation, gibi<namePlural>)\
    UNIT_ADD(namespaceName, tebi ## nameSingular, tebi ## namePlural, Ti ## abbreviation, tebi<namePlural>)\
    UNIT_ADD(namespaceName, pebi ## nameSingular, pebi ## namePlural, Pi ## abbreviation, pebi<namePlural>)\
    UNIT_ADD(namespaceName, exbi ## nameSingular, exbi ## namePlural, Ei ## abbreviation, exbi<namePlural>)
 
//--------------------
//  UNITS NAMESPACE
//--------------------
 
/**
 * @namespace units
 * @brief Unit Conversion Library namespace
 */
namespace units
{
    //----------------------------------
    //  DOXYGEN
    //----------------------------------
 
    /**
     * @defgroup    Units Unit API
    */
 
    /**
     * @defgroup    UnitContainers Unit Containers
     * @ingroup     Units
     * @brief       Defines a series of classes which contain dimensioned values. Unit containers
     *              store a value, and support various arithmetic operations.
     */
 
    /**
     * @defgroup    UnitTypes Unit Types
     * @ingroup     Units
     * @brief       Defines a series of classes which represent units. These types are tags used by
     *              the conversion function, to create compound units, or to create `unit_t` types.
     *              By themselves, they are not containers and have no stored value.
     */
 
    /**
     * @defgroup    UnitManipulators Unit Manipulators
     * @ingroup     Units
     * @brief       Defines a series of classes used to manipulate unit types, such as `inverse<>`, `squared<>`, and metric prefixes.
     *              Unit manipulators can be chained together, e.g. `inverse<squared<pico<time::seconds>>>` to
     *              represent picoseconds^-2.
     */
 
     /**
      * @defgroup   CompileTimeUnitManipulators Compile-time Unit Manipulators
      * @ingroup    Units
      * @brief      Defines a series of classes used to manipulate `unit_value_t` types at compile-time, such as `unit_value_add<>`, `unit_value_sqrt<>`, etc.
      *             Compile-time manipulators can be chained together, e.g. `unit_value_sqrt<unit_value_add<unit_value_power<a, 2>, unit_value_power<b, 2>>>` to
      *             represent `c = sqrt(a^2 + b^2).
      */
 
     /**
     * @defgroup    UnitMath Unit Math
     * @ingroup     Units
     * @brief       Defines a collection of unit-enabled, strongly-typed versions of `<cmath>` functions.
     * @details     Includes most c++11 extensions.
     */
 
    /**
     * @defgroup    Conversion Explicit Conversion
     * @ingroup     Units
     * @brief       Functions used to convert values of one logical type to another.
     */
 
    /**
     * @defgroup    TypeTraits Type Traits
     * @ingroup     Units
     * @brief       Defines a series of classes to obtain unit type information at compile-time.
     */
 
    //------------------------------
    //  FORWARD DECLARATIONS
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace constants
    {
        namespace detail
        {
            static constexpr const UNIT_LIB_DEFAULT_TYPE PI_VAL = 3.14159265358979323846264338327950288419716939937510;
        }
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    //------------------------------
    //  RATIO TRAITS
    //------------------------------
 
    /**
     * @ingroup TypeTraits
     * @{
     */
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /// has_num implementation.
        template<class T>
        struct has_num_impl
        {
            template<class U>
            static constexpr auto test(U*)->std::is_integral<decltype(U::num)> {return std::is_integral<decltype(U::num)>{}; }
            template<typename>
            static constexpr std::false_type test(...) { return std::false_type{}; }
 
            using type = decltype(test<T>(0));
        };
    }
 
    /**
     * @brief       Trait which checks for the existence of a static numerator.
     * @details     Inherits from `std::true_type` or `std::false_type`. Use `has_num<T>::value` to test
     *              whether `class T` has a numerator static member.
     */
    template<class T>
    struct has_num : units::detail::has_num_impl<T>::type {};
 
    namespace detail
    {
        /// has_den implementation.
        template<class T>
        struct has_den_impl
        {
            template<class U>
            static constexpr auto test(U*)->std::is_integral<decltype(U::den)> { return std::is_integral<decltype(U::den)>{}; }
            template<typename>
            static constexpr std::false_type test(...) { return std::false_type{}; }
 
            using type = decltype(test<T>(0));
        };
    }
 
    /**
     * @brief       Trait which checks for the existence of a static denominator.
     * @details     Inherits from `std::true_type` or `std::false_type`. Use `has_den<T>::value` to test
     *              whether `class T` has a denominator static member.
     */
    template<class T>
    struct has_den : units::detail::has_den_impl<T>::type {};
 
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        /**
         * @brief       Trait that tests whether a type represents a std::ratio.
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_ratio<T>::value` to test
         *              whether `class T` implements a std::ratio.
         */
        template<class T>
        struct is_ratio : std::integral_constant<bool,
            has_num<T>::value &&
            has_den<T>::value>
        {};
        template<class T>
        inline constexpr bool is_ratio_v = is_ratio<T>::value;
    }
 
    //------------------------------
    //  UNIT TRAITS
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    /**
     * @brief       void type.
     * @details     Helper class for creating type traits.
     */
    template<class ...>
    struct void_t { typedef void type; };
 
    /**
     * @brief       parameter pack for boolean arguments.
     */
    template<bool...> struct bool_pack {};
 
    /**
     * @brief       Trait which tests that a set of other traits are all true.
     */
    template<bool... Args>
    struct all_true : std::is_same<units::bool_pack<true, Args...>, units::bool_pack<Args..., true>> {};
    template<bool... Args>
    inline constexpr bool all_true_t_v = all_true<Args...>::type::value;
    /** @endcond */ // DOXYGEN IGNORE
 
    /**
     * @brief namespace representing type traits which can access the properties of types provided by the units library.
     */
    namespace traits
    {
#ifdef FOR_DOXYGEN_PURPOSES_ONLY
        /**
         * @ingroup     TypeTraits
         * @brief       Traits class defining the properties of units.
         * @details     The units library determines certain properties of the units passed to
         *              them and what they represent by using the members of the corresponding
         *              unit_traits instantiation.
         */
        template<class T>
        struct unit_traits
        {
            typedef typename T::base_unit_type base_unit_type;                                          ///< Unit type that the unit was derived from. May be a `base_unit` or another `unit`. Use the `base_unit_of` trait to find the SI base unit type. This will be `void` if type `T` is not a unit.
            typedef typename T::conversion_ratio conversion_ratio;                                      ///< `std::ratio` representing the conversion factor to the `base_unit_type`. This will be `void` if type `T` is not a unit.
            typedef typename T::pi_exponent_ratio pi_exponent_ratio;                                    ///< `std::ratio` representing the exponent of pi to be used in the conversion. This will be `void` if type `T` is not a unit.
            typedef typename T::translation_ratio translation_ratio;                                    ///< `std::ratio` representing a datum translation to the base unit (i.e. degrees C to degrees F conversion). This will be `void` if type `T` is not a unit.
        };
#endif
        /** @cond */    // DOXYGEN IGNORE
        /**
         * @brief       unit traits implementation for classes which are not units.
         */
        template<class T, typename = void>
        struct unit_traits
        {
            typedef void base_unit_type;
            typedef void conversion_ratio;
            typedef void pi_exponent_ratio;
            typedef void translation_ratio;
        };
 
        template<class T>
        struct unit_traits
            <T, typename void_t<
            typename T::base_unit_type,
            typename T::conversion_ratio,
            typename T::pi_exponent_ratio,
            typename T::translation_ratio>::type>
        {
            typedef typename T::base_unit_type base_unit_type;                                          ///< Unit type that the unit was derived from. May be a `base_unit` or another `unit`. Use the `base_unit_of` trait to find the SI base unit type. This will be `void` if type `T` is not a unit.
            typedef typename T::conversion_ratio conversion_ratio;                                      ///< `std::ratio` representing the conversion factor to the `base_unit_type`. This will be `void` if type `T` is not a unit.
            typedef typename T::pi_exponent_ratio pi_exponent_ratio;                                    ///< `std::ratio` representing the exponent of pi to be used in the conversion. This will be `void` if type `T` is not a unit.
            typedef typename T::translation_ratio translation_ratio;                                    ///< `std::ratio` representing a datum translation to the base unit (i.e. degrees C to degrees F conversion). This will be `void` if type `T` is not a unit.
        };
        /** @endcond */ // END DOXYGEN IGNORE
    }
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       helper type to identify base units.
         * @details     A non-templated base class for `base_unit` which enables RTTI testing.
         */
        struct _base_unit_t {};
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests if a class is a `base_unit` type.
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_base_unit<T>::value` to test
         *              whether `class T` implements a `base_unit`.
         */
        template<class T>
        struct is_base_unit : std::is_base_of<units::detail::_base_unit_t, T> {};
    }
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       helper type to identify units.
         * @details     A non-templated base class for `unit` which enables RTTI testing.
         */
        struct _unit {};
 
        template<std::intmax_t Num, std::intmax_t Den = 1>
        using meter_ratio = std::ratio<Num, Den>;
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Traits which tests if a class is a `unit`
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_unit<T>::value` to test
         *              whether `class T` implements a `unit`.
         */
        template<class T>
        struct is_unit : std::is_base_of<units::detail::_unit, T>::type {};
        template<class T>
        inline constexpr bool is_unit_v = is_unit<T>::value;
    }
 
    /** @} */ // end of TypeTraits
 
    //------------------------------
    //  BASE UNIT CLASS
    //------------------------------
 
    /**
     * @ingroup     UnitTypes
     * @brief       Class representing SI base unit types.
     * @details     Base units are represented by a combination of `std::ratio` template parameters, each
     *              describing the exponent of the type of unit they represent. Example: meters per second
     *              would be described by a +1 exponent for meters, and a -1 exponent for seconds, thus:
     *              `base_unit<std::ratio<1>, std::ratio<0>, std::ratio<-1>>`
     * @tparam      Meter       `std::ratio` representing the exponent value for meters.
     * @tparam      Kilogram    `std::ratio` representing the exponent value for kilograms.
     * @tparam      Second      `std::ratio` representing the exponent value for seconds.
     * @tparam      Radian      `std::ratio` representing the exponent value for radians. Although radians are not SI base units, they are included because radians are described by the SI as m * m^-1, which would make them indistinguishable from scalars.
     * @tparam      Ampere      `std::ratio` representing the exponent value for amperes.
     * @tparam      Kelvin      `std::ratio` representing the exponent value for Kelvin.
     * @tparam      Mole        `std::ratio` representing the exponent value for moles.
     * @tparam      Candela     `std::ratio` representing the exponent value for candelas.
     * @tparam      Byte        `std::ratio` representing the exponent value for bytes.
     * @sa          category     for type aliases for SI base_unit types.
     */
    template<class Meter = detail::meter_ratio<0>,
    class Kilogram = std::ratio<0>,
    class Second = std::ratio<0>,
    class Radian = std::ratio<0>,
    class Ampere = std::ratio<0>,
    class Kelvin = std::ratio<0>,
    class Mole = std::ratio<0>,
    class Candela = std::ratio<0>,
    class Byte = std::ratio<0>>
    struct base_unit : units::detail::_base_unit_t
    {
        static_assert(traits::is_ratio<Meter>::value, "Template parameter `Meter` must be a `std::ratio` representing the exponent of meters the unit has");
        static_assert(traits::is_ratio<Kilogram>::value, "Template parameter `Kilogram` must be a `std::ratio` representing the exponent of kilograms the unit has");
        static_assert(traits::is_ratio<Second>::value, "Template parameter `Second` must be a `std::ratio` representing the exponent of seconds the unit has");
        static_assert(traits::is_ratio<Ampere>::value, "Template parameter `Ampere` must be a `std::ratio` representing the exponent of amperes the unit has");
        static_assert(traits::is_ratio<Kelvin>::value, "Template parameter `Kelvin` must be a `std::ratio` representing the exponent of kelvin the unit has");
        static_assert(traits::is_ratio<Candela>::value, "Template parameter `Candela` must be a `std::ratio` representing the exponent of candelas the unit has");
        static_assert(traits::is_ratio<Mole>::value, "Template parameter `Mole` must be a `std::ratio` representing the exponent of moles the unit has");
        static_assert(traits::is_ratio<Radian>::value, "Template parameter `Radian` must be a `std::ratio` representing the exponent of radians the unit has");
        static_assert(traits::is_ratio<Byte>::value, "Template parameter `Byte` must be a `std::ratio` representing the exponent of bytes the unit has");
 
        typedef Meter meter_ratio;
        typedef Kilogram kilogram_ratio;
        typedef Second second_ratio;
        typedef Radian radian_ratio;
        typedef Ampere ampere_ratio;
        typedef Kelvin kelvin_ratio;
        typedef Mole mole_ratio;
        typedef Candela candela_ratio;
        typedef Byte byte_ratio;
    };
 
    //------------------------------
    //  UNIT CATEGORIES
    //------------------------------
 
    /**
     * @brief       namespace representing the implemented base and derived unit types. These will not generally be needed by library users.
     * @sa          base_unit for the definition of the category parameters.
     */
    namespace category
    {
        // SCALAR (DIMENSIONLESS) TYPES
        typedef base_unit<> scalar_unit;            ///< Represents a quantity with no dimension.
        typedef base_unit<> dimensionless_unit; ///< Represents a quantity with no dimension.
 
        // SI BASE UNIT TYPES
        //                  METERS          KILOGRAMS       SECONDS         RADIANS         AMPERES         KELVIN          MOLE            CANDELA         BYTE        ---     CATEGORY
        typedef base_unit<detail::meter_ratio<1>>                                                                                                                                       length_unit;                    ///< Represents an SI base unit of length
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<1>>                                                                                                                      mass_unit;                      ///< Represents an SI base unit of mass
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<1>>                                                                                                      time_unit;                      ///< Represents an SI base unit of time
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                                                                                      angle_unit;                     ///< Represents an SI base unit of angle
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                                                                      current_unit;                   ///< Represents an SI base unit of current
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                                                      temperature_unit;               ///< Represents an SI base unit of temperature
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                                      substance_unit;                 ///< Represents an SI base unit of amount of substance
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                      luminous_intensity_unit;        ///< Represents an SI base unit of luminous intensity
 
        // SI DERIVED UNIT TYPES
        //                  METERS          KILOGRAMS       SECONDS         RADIANS         AMPERES         KELVIN          MOLE            CANDELA         BYTE        ---     CATEGORY
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<2>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>>                      solid_angle_unit;               ///< Represents an SI derived unit of solid angle
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-1>>                                                                                                     frequency_unit;                 ///< Represents an SI derived unit of frequency
        typedef base_unit<detail::meter_ratio<1>,   std::ratio<0>,  std::ratio<-1>>                                                                                                     velocity_unit;                  ///< Represents an SI derived unit of velocity
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-1>, std::ratio<1>>                                                                                      angular_velocity_unit;          ///< Represents an SI derived unit of angular velocity
        typedef base_unit<detail::meter_ratio<1>,   std::ratio<0>,  std::ratio<-2>>                                                                                                     acceleration_unit;              ///< Represents an SI derived unit of acceleration
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-2>, std::ratio<1>>                                                                                      angular_acceleration_unit;          ///< Represents an SI derived unit of angular acceleration
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-3>, std::ratio<1>>                                                                                      angular_jerk_unit;              ///< Represents an SI derived unit of angular jerk
        typedef base_unit<detail::meter_ratio<1>,   std::ratio<1>,  std::ratio<-2>>                                                                                                     force_unit;                     ///< Represents an SI derived unit of force
        typedef base_unit<detail::meter_ratio<-1>,  std::ratio<1>,  std::ratio<-2>>                                                                                                     pressure_unit;                  ///< Represents an SI derived unit of pressure
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<1>,  std::ratio<0>,  std::ratio<1>>                                                                      charge_unit;                    ///< Represents an SI derived unit of charge
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-2>>                                                                                                     energy_unit;                    ///< Represents an SI derived unit of energy
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-3>>                                                                                                     power_unit;                     ///< Represents an SI derived unit of power
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-3>, std::ratio<0>,  std::ratio<-1>>                                                                     voltage_unit;                   ///< Represents an SI derived unit of voltage
        typedef base_unit<detail::meter_ratio<-2>,  std::ratio<-1>, std::ratio<4>,  std::ratio<0>,  std::ratio<2>>                                                                      capacitance_unit;               ///< Represents an SI derived unit of capacitance
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-3>, std::ratio<0>,  std::ratio<-2>>                                                                     impedance_unit;                 ///< Represents an SI derived unit of impedance
        typedef base_unit<detail::meter_ratio<-2>,  std::ratio<-1>, std::ratio<3>,  std::ratio<0>,  std::ratio<2>>                                                                      conductance_unit;               ///< Represents an SI derived unit of conductance
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-2>, std::ratio<0>,  std::ratio<-1>>                                                                     magnetic_flux_unit;             ///< Represents an SI derived unit of magnetic flux
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<1>,  std::ratio<-2>, std::ratio<0>,  std::ratio<-1>>                                                                     magnetic_field_strength_unit;   ///< Represents an SI derived unit of magnetic field strength
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-2>, std::ratio<0>,  std::ratio<-2>>                                                                     inductance_unit;                ///< Represents an SI derived unit of inductance
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<2>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                      luminous_flux_unit;             ///< Represents an SI derived unit of luminous flux
        typedef base_unit<detail::meter_ratio<-2>,  std::ratio<0>,  std::ratio<0>,  std::ratio<2>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>                      illuminance_unit;               ///< Represents an SI derived unit of illuminance
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-1>>                                                                                                     radioactivity_unit;             ///< Represents an SI derived unit of radioactivity
 
        // OTHER UNIT TYPES
        //                  METERS          KILOGRAMS       SECONDS         RADIANS         AMPERES         KELVIN          MOLE            CANDELA         BYTE        ---     CATEGORY
        typedef base_unit<detail::meter_ratio<2>,   std::ratio<1>,  std::ratio<-2>>                                                                                                     torque_unit;                    ///< Represents an SI derived unit of torque
        typedef base_unit<detail::meter_ratio<2>>                                                                                                                                       area_unit;                      ///< Represents an SI derived unit of area
        typedef base_unit<detail::meter_ratio<3>>                                                                                                                                       volume_unit;                    ///< Represents an SI derived unit of volume
        typedef base_unit<detail::meter_ratio<-3>,  std::ratio<1>>                                                                                                                      density_unit;                   ///< Represents an SI derived unit of density
        typedef base_unit<>                                                                                                                                                     concentration_unit;             ///< Represents a unit of concentration
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>      data_unit;                      ///< Represents a unit of data size
        typedef base_unit<detail::meter_ratio<0>,   std::ratio<0>,  std::ratio<-1>, std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<0>,  std::ratio<1>>      data_transfer_rate_unit;        ///< Represents a unit of data transfer rate
    }
 
    //------------------------------
    //  UNIT CLASSES
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    /**
     * @brief       unit type template specialization for units derived from base units.
     */
    template <class, class, class, class> struct unit;
    template<class Conversion, class... Exponents, class PiExponent, class Translation>
    struct unit<Conversion, base_unit<Exponents...>, PiExponent, Translation> : units::detail::_unit
    {
        static_assert(traits::is_ratio<Conversion>::value, "Template parameter `Conversion` must be a `std::ratio` representing the conversion factor to `BaseUnit`.");
        static_assert(traits::is_ratio<PiExponent>::value, "Template parameter `PiExponent` must be a `std::ratio` representing the exponents of Pi the unit has.");
        static_assert(traits::is_ratio<Translation>::value, "Template parameter `Translation` must be a `std::ratio` representing an additive translation required by the unit conversion.");
 
        typedef typename units::base_unit<Exponents...> base_unit_type;
        typedef Conversion conversion_ratio;
        typedef Translation translation_ratio;
        typedef PiExponent pi_exponent_ratio;
    };
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @brief       Type representing an arbitrary unit.
     * @ingroup     UnitTypes
     * @details     `unit` types are used as tags for the `conversion` function. They are *not* containers
     *              (see `unit_t` for a  container class). Each unit is defined by:
     *
     *              - A `std::ratio` defining the conversion factor to the base unit type. (e.g. `std::ratio<1,12>` for inches to feet)
     *              - A base unit that the unit is derived from (or a unit category. Must be of type `unit` or `base_unit`)
     *              - An exponent representing factors of PI required by the conversion. (e.g. `std::ratio<-1>` for a radians to degrees conversion)
     *              - a ratio representing a datum translation required for the conversion (e.g. `std::ratio<32>` for a fahrenheit to celsius conversion)
     *
     *              Typically, a specific unit, like `meters`, would be implemented as a type alias
     *              of `unit`, i.e. `using meters = unit<std::ratio<1>, units::category::length_unit`, or
     *              `using inches = unit<std::ratio<1,12>, feet>`.
     * @tparam      Conversion  std::ratio representing scalar multiplication factor.
     * @tparam      BaseUnit    Unit type which this unit is derived from. May be a `base_unit`, or another `unit`.
     * @tparam      PiExponent  std::ratio representing the exponent of pi required by the conversion.
     * @tparam      Translation std::ratio representing any datum translation required by the conversion.
     */
    template<class Conversion, class BaseUnit, class PiExponent = std::ratio<0>, class Translation = std::ratio<0>>
    struct unit : units::detail::_unit
    {
        static_assert(traits::is_unit<BaseUnit>::value, "Template parameter `BaseUnit` must be a `unit` type.");
        static_assert(traits::is_ratio<Conversion>::value, "Template parameter `Conversion` must be a `std::ratio` representing the conversion factor to `BaseUnit`.");
        static_assert(traits::is_ratio<PiExponent>::value, "Template parameter `PiExponent` must be a `std::ratio` representing the exponents of Pi the unit has.");
 
        typedef typename units::traits::unit_traits<BaseUnit>::base_unit_type base_unit_type;
        typedef typename std::ratio_multiply<typename BaseUnit::conversion_ratio, Conversion> conversion_ratio;
        typedef typename std::ratio_add<typename BaseUnit::pi_exponent_ratio, PiExponent> pi_exponent_ratio;
        typedef typename std::ratio_add<std::ratio_multiply<typename BaseUnit::conversion_ratio, Translation>, typename BaseUnit::translation_ratio> translation_ratio;
    };
 
    //------------------------------
    //  BASE UNIT MANIPULATORS
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       base_unit_of trait implementation
         * @details     recursively seeks base_unit type that a unit is derived from. Since units can be
         *              derived from other units, the `base_unit_type` typedef may not represent this value.
         */
        template<class> struct base_unit_of_impl;
        template<class Conversion, class BaseUnit, class PiExponent, class Translation>
        struct base_unit_of_impl<unit<Conversion, BaseUnit, PiExponent, Translation>> : base_unit_of_impl<BaseUnit> {};
        template<class... Exponents>
        struct base_unit_of_impl<base_unit<Exponents...>>
        {
            typedef base_unit<Exponents...> type;
        };
        template<>
        struct base_unit_of_impl<void>
        {
            typedef void type;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        /**
         * @brief       Trait which returns the `base_unit` type that a unit is originally derived from.
         * @details     Since units can be derived from other `unit` types in addition to `base_unit` types,
         *              the `base_unit_type` typedef will not always be a `base_unit` (or unit category).
         *              Since compatible
         */
        template<class U>
        using base_unit_of = typename units::detail::base_unit_of_impl<U>::type;
    }
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       implementation of base_unit_multiply
         * @details     'multiples' (adds exponent ratios of) two base unit types. Base units can be found
         *              using `base_unit_of`.
         */
        template<class, class> struct base_unit_multiply_impl;
        template<class... Exponents1, class... Exponents2>
        struct base_unit_multiply_impl<base_unit<Exponents1...>, base_unit<Exponents2...>> {
            using type = base_unit<std::ratio_add<Exponents1, Exponents2>...>;
        };
 
        /**
         * @brief       represents type of two base units multiplied together
         */
        template<class U1, class U2>
        using base_unit_multiply = typename base_unit_multiply_impl<U1, U2>::type;
 
        /**
         * @brief       implementation of base_unit_divide
         * @details     'dived' (subtracts exponent ratios of) two base unit types. Base units can be found
         *              using `base_unit_of`.
         */
        template<class, class> struct base_unit_divide_impl;
        template<class... Exponents1, class... Exponents2>
        struct base_unit_divide_impl<base_unit<Exponents1...>, base_unit<Exponents2...>> {
            using type = base_unit<std::ratio_subtract<Exponents1, Exponents2>...>;
        };
 
        /**
         * @brief       represents the resulting type of `base_unit` U1 divided by U2.
         */
        template<class U1, class U2>
        using base_unit_divide = typename base_unit_divide_impl<U1, U2>::type;
 
        /**
         * @brief       implementation of inverse_base
         * @details     multiplies all `base_unit` exponent ratios by -1. The resulting type represents
         *              the inverse base unit of the given `base_unit` type.
         */
        template<class> struct inverse_base_impl;
 
        template<class... Exponents>
        struct inverse_base_impl<base_unit<Exponents...>> {
            using type = base_unit<std::ratio_multiply<Exponents, std::ratio<-1>>...>;
        };
 
        /**
         * @brief       represent the inverse type of `class U`
         * @details     E.g. if `U` is `length_unit`, then `inverse<U>` will represent `length_unit^-1`.
         */
        template<class U> using inverse_base = typename inverse_base_impl<U>::type;
 
        /**
         * @brief       implementation of `squared_base`
         * @details     multiplies all the exponent ratios of the given class by 2. The resulting type is
         *              equivalent to the given type squared.
         */
        template<class U> struct squared_base_impl;
        template<class... Exponents>
        struct squared_base_impl<base_unit<Exponents...>> {
            using type = base_unit<std::ratio_multiply<Exponents, std::ratio<2>>...>;
        };
 
        /**
         * @brief       represents the type of a `base_unit` squared.
         * @details     E.g. `squared<length_unit>` will represent `length_unit^2`.
         */
        template<class U> using squared_base = typename squared_base_impl<U>::type;
 
        /**
         * @brief       implementation of `cubed_base`
         * @details     multiplies all the exponent ratios of the given class by 3. The resulting type is
         *              equivalent to the given type cubed.
         */
        template<class U> struct cubed_base_impl;
        template<class... Exponents>
        struct cubed_base_impl<base_unit<Exponents...>> {
            using type = base_unit<std::ratio_multiply<Exponents, std::ratio<3>>...>;
        };
 
        /**
         * @brief       represents the type of a `base_unit` cubed.
         * @details     E.g. `cubed<length_unit>` will represent `length_unit^3`.
         */
        template<class U> using cubed_base = typename cubed_base_impl<U>::type;
 
        /**
         * @brief       implementation of `sqrt_base`
         * @details     divides all the exponent ratios of the given class by 2. The resulting type is
         *              equivalent to the square root of the given type.
         */
        template<class U> struct sqrt_base_impl;
        template<class... Exponents>
        struct sqrt_base_impl<base_unit<Exponents...>> {
            using type = base_unit<std::ratio_divide<Exponents, std::ratio<2>>...>;
        };
 
        /**
         * @brief       represents the square-root type of a `base_unit`.
         * @details     E.g. `sqrt<length_unit>` will represent `length_unit^(1/2)`.
         */
        template<class U> using sqrt_base = typename sqrt_base_impl<U>::type;
 
        /**
         * @brief       implementation of `cbrt_base`
         * @details     divides all the exponent ratios of the given class by 3. The resulting type is
         *              equivalent to the given type's cube-root.
         */
        template<class U> struct cbrt_base_impl;
        template<class... Exponents>
        struct cbrt_base_impl<base_unit<Exponents...>> {
            using type = base_unit<std::ratio_divide<Exponents, std::ratio<3>>...>;
        };
 
        /**
         * @brief       represents the cube-root type of a `base_unit` .
         * @details     E.g. `cbrt<length_unit>` will represent `length_unit^(1/3)`.
         */
        template<class U> using cbrt_base = typename cbrt_base_impl<U>::type;
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    //------------------------------
    //  UNIT MANIPULATORS
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       implementation of `unit_multiply`.
         * @details     multiplies two units. The base unit becomes the base units of each with their exponents
         *              added together. The conversion factors of each are multiplied by each other. Pi exponent ratios
         *              are added, and datum translations are removed.
         */
        template<class Unit1, class Unit2>
        struct unit_multiply_impl
        {
            using type = unit < std::ratio_multiply<typename Unit1::conversion_ratio, typename Unit2::conversion_ratio>,
                base_unit_multiply <traits::base_unit_of<typename Unit1::base_unit_type>, traits::base_unit_of<typename Unit2::base_unit_type>>,
                std::ratio_add<typename Unit1::pi_exponent_ratio, typename Unit2::pi_exponent_ratio>,
                std::ratio < 0 >> ;
        };
 
        /**
         * @brief       represents the type of two units multiplied together.
         * @details     recalculates conversion and exponent ratios at compile-time.
         */
        template<class U1, class U2>
        using unit_multiply = typename unit_multiply_impl<U1, U2>::type;
 
        /**
         * @brief       implementation of `unit_divide`.
         * @details     divides two units. The base unit becomes the base units of each with their exponents
         *              subtracted from each other. The conversion factors of each are divided by each other. Pi exponent ratios
         *              are subtracted, and datum translations are removed.
         */
        template<class Unit1, class Unit2>
        struct unit_divide_impl
        {
            using type = unit < std::ratio_divide<typename Unit1::conversion_ratio, typename Unit2::conversion_ratio>,
                base_unit_divide<traits::base_unit_of<typename Unit1::base_unit_type>, traits::base_unit_of<typename Unit2::base_unit_type>>,
                std::ratio_subtract<typename Unit1::pi_exponent_ratio, typename Unit2::pi_exponent_ratio>,
                std::ratio < 0 >> ;
        };
 
        /**
         * @brief       represents the type of two units divided by each other.
         * @details     recalculates conversion and exponent ratios at compile-time.
         */
        template<class U1, class U2>
        using unit_divide = typename unit_divide_impl<U1, U2>::type;
 
        /**
         * @brief       implementation of `inverse`
         * @details     inverts a unit (equivalent to 1/unit). The `base_unit` and pi exponents are all multiplied by
         *              -1. The conversion ratio numerator and denominator are swapped. Datum translation
         *              ratios are removed.
         */
        template<class Unit>
        struct inverse_impl
        {
            using type = unit < std::ratio<Unit::conversion_ratio::den, Unit::conversion_ratio::num>,
                inverse_base<traits::base_unit_of<typename units::traits::unit_traits<Unit>::base_unit_type>>,
                std::ratio_multiply<typename units::traits::unit_traits<Unit>::pi_exponent_ratio, std::ratio<-1>>,
                std::ratio < 0 >> ; // inverses are rates or change, the translation factor goes away.
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @brief       represents the inverse unit type of `class U`.
     * @ingroup     UnitManipulators
     * @tparam      U   `unit` type to invert.
     * @details     E.g. `inverse<meters>` will represent meters^-1 (i.e. 1/meters).
     */
    template<class U> using inverse = typename units::detail::inverse_impl<U>::type;
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       implementation of `squared`
         * @details     Squares the conversion ratio, `base_unit` exponents, pi exponents, and removes
         *              datum translation ratios.
         */
        template<class Unit>
        struct squared_impl
        {
            static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
            using Conversion = typename Unit::conversion_ratio;
            using type = unit < std::ratio_multiply<Conversion, Conversion>,
                squared_base<traits::base_unit_of<typename Unit::base_unit_type>>,
                std::ratio_multiply<typename Unit::pi_exponent_ratio, std::ratio<2>>,
                typename Unit::translation_ratio
            > ;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @brief       represents the unit type of `class U` squared
     * @ingroup     UnitManipulators
     * @tparam      U   `unit` type to square.
     * @details     E.g. `square<meters>` will represent meters^2.
     */
    template<class U>
    using squared = typename units::detail::squared_impl<U>::type;
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
             * @brief       implementation of `cubed`
             * @details     Cubes the conversion ratio, `base_unit` exponents, pi exponents, and removes
             *              datum translation ratios.
             */
        template<class Unit>
        struct cubed_impl
        {
            static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
            using Conversion = typename Unit::conversion_ratio;
            using type = unit < std::ratio_multiply<Conversion, std::ratio_multiply<Conversion, Conversion>>,
                cubed_base<traits::base_unit_of<typename Unit::base_unit_type>>,
                std::ratio_multiply<typename Unit::pi_exponent_ratio, std::ratio<3>>,
                typename Unit::translation_ratio> ;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @brief       represents the type of `class U` cubed.
     * @ingroup     UnitManipulators
     * @tparam      U   `unit` type to cube.
     * @details     E.g. `cubed<meters>` will represent meters^3.
     */
    template<class U>
    using cubed = typename units::detail::cubed_impl<U>::type;
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        //----------------------------------
        //  RATIO_SQRT IMPLEMENTATION
        //----------------------------------
 
        using Zero = std::ratio<0>;
        using One = std::ratio<1>;
        template <typename R> using Square = std::ratio_multiply<R, R>;
 
        // Find the largest std::integer N such that Predicate<N>::value is true.
        template <template <std::intmax_t N> class Predicate, typename enabled = void>
        struct BinarySearch {
            template <std::intmax_t N>
            struct SafeDouble_ {
                static constexpr const std::intmax_t value = 2 * N;
                static_assert(value > 0, "Overflows when computing 2 * N");
            };
 
            template <intmax_t Lower, intmax_t Upper, typename Condition1 = void, typename Condition2 = void>
            struct DoubleSidedSearch_ : DoubleSidedSearch_<Lower, Upper,
                std::integral_constant<bool, (Upper - Lower == 1)>,
                std::integral_constant<bool, ((Upper - Lower>1 && Predicate<Lower + (Upper - Lower) / 2>::value))>> {};
 
            template <intmax_t Lower, intmax_t Upper>
            struct DoubleSidedSearch_<Lower, Upper, std::false_type, std::false_type> : DoubleSidedSearch_<Lower, Lower + (Upper - Lower) / 2> {};
 
            template <intmax_t Lower, intmax_t Upper, typename Condition2>
            struct DoubleSidedSearch_<Lower, Upper, std::true_type, Condition2> : std::integral_constant<intmax_t, Lower>{};
 
            template <intmax_t Lower, intmax_t Upper, typename Condition1>
            struct DoubleSidedSearch_<Lower, Upper, Condition1, std::true_type> : DoubleSidedSearch_<Lower + (Upper - Lower) / 2, Upper>{};
 
            template <std::intmax_t Lower, class enabled1 = void>
            struct SingleSidedSearch_ : SingleSidedSearch_<Lower, std::integral_constant<bool, Predicate<SafeDouble_<Lower>::value>::value>>{};
 
            template <std::intmax_t Lower>
            struct SingleSidedSearch_<Lower, std::false_type> : DoubleSidedSearch_<Lower, SafeDouble_<Lower>::value> {};
 
            template <std::intmax_t Lower>
            struct SingleSidedSearch_<Lower, std::true_type> : SingleSidedSearch_<SafeDouble_<Lower>::value>{};
 
            static constexpr const std::intmax_t value = SingleSidedSearch_<1>::value;
        };
 
        template <template <std::intmax_t N> class Predicate>
        struct BinarySearch<Predicate, std::enable_if_t<!Predicate<1>::value>> : std::integral_constant<std::intmax_t, 0>{};
 
        // Find largest std::integer N such that N<=sqrt(R)
        template <typename R>
        struct Integer {
            template <std::intmax_t N> using Predicate_ = std::ratio_less_equal<std::ratio<N>, std::ratio_divide<R, std::ratio<N>>>;
            static constexpr const std::intmax_t value = BinarySearch<Predicate_>::value;
        };
 
        template <typename R>
        struct IsPerfectSquare {
            static constexpr const std::intmax_t DenSqrt_ = Integer<std::ratio<R::den>>::value;
            static constexpr const std::intmax_t NumSqrt_ = Integer<std::ratio<R::num>>::value;
            static constexpr const bool value =( DenSqrt_ * DenSqrt_ == R::den && NumSqrt_ * NumSqrt_ == R::num);
            using Sqrt = std::ratio<NumSqrt_, DenSqrt_>;
        };
 
        // Represents sqrt(P)-Q.
        template <typename Tp, typename Tq>
        struct Remainder {
            using P = Tp;
            using Q = Tq;
        };
 
        // Represents 1/R = I + Rem where R is a Remainder.
        template <typename R>
        struct Reciprocal {
            using P_ = typename R::P;
            using Q_ = typename R::Q;
            using Den_ = std::ratio_subtract<P_, Square<Q_>>;
            using A_ = std::ratio_divide<Q_, Den_>;
            using B_ = std::ratio_divide<P_, Square<Den_>>;
            static constexpr const std::intmax_t I_ = (A_::num + Integer<std::ratio_multiply<B_, Square<std::ratio<A_::den>>>>::value) / A_::den;
            using I = std::ratio<I_>;
            using Rem = Remainder<B_, std::ratio_subtract<I, A_>>;
        };
 
        // Expands sqrt(R) to continued fraction:
        // f(x)=C1+1/(C2+1/(C3+1/(...+1/(Cn+x)))) = (U*x+V)/(W*x+1) and sqrt(R)=f(Rem).
        // The error |f(Rem)-V| = |(U-W*V)x/(W*x+1)| <= |U-W*V|*Rem <= |U-W*V|/I' where
        // I' is the std::integer part of reciprocal of Rem.
        template <typename Tr, std::intmax_t N>
        struct ContinuedFraction {
            template <typename T>
            using Abs_ = std::conditional_t<std::ratio_less<T, Zero>::value, std::ratio_subtract<Zero, T>, T>;
 
            using R = Tr;
            using Last_ = ContinuedFraction<R, N - 1>;
            using Reciprocal_ = Reciprocal<typename Last_::Rem>;
            using Rem = typename Reciprocal_::Rem;
            using I_ = typename Reciprocal_::I;
            using Den_ = std::ratio_add<typename Last_::W, I_>;
            using U = std::ratio_divide<typename Last_::V, Den_>;
            using V = std::ratio_divide<std::ratio_add<typename Last_::U, std::ratio_multiply<typename Last_::V, I_>>, Den_>;
            using W = std::ratio_divide<One, Den_>;
            using Error = Abs_<std::ratio_divide<std::ratio_subtract<U, std::ratio_multiply<V, W>>, typename Reciprocal<Rem>::I>>;
        };
 
        template <typename Tr>
        struct ContinuedFraction<Tr, 1> {
            using R = Tr;
            using U = One;
            using V = std::ratio<Integer<R>::value>;
            using W = Zero;
            using Rem = Remainder<R, V>;
            using Error = std::ratio_divide<One, typename Reciprocal<Rem>::I>;
        };
 
        template <typename R, typename Eps, std::intmax_t N = 1, typename enabled = void>
        struct Sqrt_ : Sqrt_<R, Eps, N + 1> {};
 
        template <typename R, typename Eps, std::intmax_t N>
        struct Sqrt_<R, Eps, N, std::enable_if_t<std::ratio_less_equal<typename ContinuedFraction<R, N>::Error, Eps>::value>> {
            using type = typename ContinuedFraction<R, N>::V;
        };
 
        template <typename R, typename Eps, typename enabled = void>
        struct Sqrt {
            static_assert(std::ratio_greater_equal<R, Zero>::value, "R can't be negative");
        };
 
        template <typename R, typename Eps>
        struct Sqrt<R, Eps, std::enable_if_t<std::ratio_greater_equal<R, Zero>::value && IsPerfectSquare<R>::value>> {
            using type = typename IsPerfectSquare<R>::Sqrt;
        };
 
        template <typename R, typename Eps>
        struct Sqrt<R, Eps, std::enable_if_t<(std::ratio_greater_equal<R, Zero>::value && !IsPerfectSquare<R>::value)>> : Sqrt_<R, Eps>{};
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @ingroup     TypeTraits
     * @brief       Calculate square root of a ratio at compile-time
     * @details     Calculates a rational approximation of the square root of the ratio. The error
     *              in the calculation is bounded by 1/epsilon (Eps). E.g. for the default value
     *              of 10000000000, the maximum error will be a/10000000000, or 1e-8, or said another way,
     *              the error will be on the order of 10^-9. Since these calculations are done at
     *              compile time, it is advisable to set epsilon to the highest value that does not
     *              cause an integer overflow in the calculation. If you can't compile `ratio_sqrt`
     *              due to overflow errors, reducing the value of epsilon sufficiently will correct
     *              the problem.\n\n
     *              `ratio_sqrt` is guaranteed to converge for all values of `Ratio` which do not
     *              overflow.
     * @note        This function provides a rational approximation, _NOT_ an exact value.
     * @tparam      Ratio   ratio to take the square root of. This can represent any rational value,
     *                      _not_ just integers or values with integer roots.
     * @tparam      Eps     Value of epsilon, which represents the inverse of the maximum allowable
     *                      error. This value should be chosen to be as high as possible before
     *                      integer overflow errors occur in the compiler.
     */
    template<typename Ratio, std::intmax_t Eps = 10000000000>
    using ratio_sqrt = typename  units::detail::Sqrt<Ratio, std::ratio<1, Eps>>::type;
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       implementation of `sqrt`
         * @details     square roots the conversion ratio, `base_unit` exponents, pi exponents, and removes
         *              datum translation ratios.
         */
        template<class Unit, std::intmax_t Eps>
        struct sqrt_impl
        {
            static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
            using Conversion = typename Unit::conversion_ratio;
            using type = unit <ratio_sqrt<Conversion, Eps>,
                sqrt_base<traits::base_unit_of<typename Unit::base_unit_type>>,
                std::ratio_divide<typename Unit::pi_exponent_ratio, std::ratio<2>>,
                typename Unit::translation_ratio>;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @ingroup     UnitManipulators
     * @brief       represents the square root of type `class U`.
     * @details     Calculates a rational approximation of the square root of the unit. The error
     *              in the calculation is bounded by 1/epsilon (Eps). E.g. for the default value
     *              of 10000000000, the maximum error will be a/10000000000, or 1e-8, or said another way,
     *              the error will be on the order of 10^-9. Since these calculations are done at
     *              compile time, it is advisable to set epsilon to the highest value that does not
     *              cause an integer overflow in the calculation. If you can't compile `ratio_sqrt`
     *              due to overflow errors, reducing the value of epsilon sufficiently will correct
     *              the problem.\n\n
     *              `ratio_sqrt` is guaranteed to converge for all values of `Ratio` which do not
     *              overflow.
     * @tparam      U   `unit` type to take the square root of.
     * @tparam      Eps Value of epsilon, which represents the inverse of the maximum allowable
     *                  error. This value should be chosen to be as high as possible before
     *                  integer overflow errors occur in the compiler.
     * @note        USE WITH CAUTION. The is an approximate value. In general, squared<sqrt<meter>> != meter,
     *              i.e. the operation is not reversible, and it will result in propagated approximations.
     *              Use only when absolutely necessary.
     */
    template<class U, std::intmax_t Eps = 10000000000>
    using square_root = typename units::detail::sqrt_impl<U, Eps>::type;
 
    //------------------------------
    //  COMPOUND UNITS
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
             * @brief       implementation of compound_unit
             * @details     multiplies a variadic list of units together, and is inherited from the resulting
             *              type.
             */
        template<class U, class... Us> struct compound_impl;
        template<class U> struct compound_impl<U> { using type = U; };
        template<class U1, class U2, class...Us>
        struct compound_impl<U1, U2, Us...>
            : compound_impl<unit_multiply<U1, U2>, Us...> {};
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @brief       Represents a unit type made up from other units.
     * @details     Compound units are formed by multiplying the units of all the types provided in
     *              the template argument. Types provided must inherit from `unit`. A compound unit can
     *              be formed from any number of other units, and unit manipulators like `inverse` and
     *              `squared` are supported. E.g. to specify acceleration, on could create
     *              `using acceleration = compound_unit<length::meters, inverse<squared<seconds>>;`
     * @tparam      U...    units which, when multiplied together, form the desired compound unit.
     * @ingroup     UnitTypes
     */
    template<class U, class... Us>
    using compound_unit = typename units::detail::compound_impl<U, Us...>::type;
 
    //------------------------------
    //  PREFIXES
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /**
         * @brief       prefix applicator.
         * @details     creates a unit type from a prefix and a unit
         */
        template<class Ratio, class Unit>
        struct prefix
        {
            static_assert(traits::is_ratio<Ratio>::value, "Template parameter `Ratio` must be a `std::ratio`.");
            static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
            typedef typename units::unit<Ratio, Unit> type;
        };
 
        /// recursive exponential implementation
        template <int N, class U>
        struct power_of_ratio
        {
            typedef std::ratio_multiply<U, typename power_of_ratio<N - 1, U>::type> type;
        };
 
        /// End recursion
        template <class U>
        struct power_of_ratio<1, U>
        {
            typedef U type;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @ingroup UnitManipulators
     * @{
     * @ingroup Decimal Prefixes
     * @{
     */
    template<class U> using atto    = typename units::detail::prefix<std::atto,	U>::type;           ///< Represents the type of `class U` with the metric 'atto' prefix appended.   @details E.g. atto<meters> represents meters*10^-18     @tparam U unit type to apply the prefix to.
    template<class U> using femto   = typename units::detail::prefix<std::femto,U>::type;           ///< Represents the type of `class U` with the metric 'femto' prefix appended.  @details E.g. femto<meters> represents meters*10^-15    @tparam U unit type to apply the prefix to.
    template<class U> using pico    = typename units::detail::prefix<std::pico,	U>::type;           ///< Represents the type of `class U` with the metric 'pico' prefix appended.   @details E.g. pico<meters> represents meters*10^-12     @tparam U unit type to apply the prefix to.
    template<class U> using nano    = typename units::detail::prefix<std::nano,	U>::type;           ///< Represents the type of `class U` with the metric 'nano' prefix appended.   @details E.g. nano<meters> represents meters*10^-9      @tparam U unit type to apply the prefix to.
    template<class U> using micro   = typename units::detail::prefix<std::micro,U>::type;           ///< Represents the type of `class U` with the metric 'micro' prefix appended.  @details E.g. micro<meters> represents meters*10^-6     @tparam U unit type to apply the prefix to.
    template<class U> using milli   = typename units::detail::prefix<std::milli,U>::type;           ///< Represents the type of `class U` with the metric 'milli' prefix appended.  @details E.g. milli<meters> represents meters*10^-3     @tparam U unit type to apply the prefix to.
    template<class U> using centi   = typename units::detail::prefix<std::centi,U>::type;           ///< Represents the type of `class U` with the metric 'centi' prefix appended.  @details E.g. centi<meters> represents meters*10^-2     @tparam U unit type to apply the prefix to.
    template<class U> using deci    = typename units::detail::prefix<std::deci,	U>::type;           ///< Represents the type of `class U` with the metric 'deci' prefix appended.   @details E.g. deci<meters> represents meters*10^-1      @tparam U unit type to apply the prefix to.
    template<class U> using deca    = typename units::detail::prefix<std::deca,	U>::type;           ///< Represents the type of `class U` with the metric 'deca' prefix appended.   @details E.g. deca<meters> represents meters*10^1       @tparam U unit type to apply the prefix to.
    template<class U> using hecto   = typename units::detail::prefix<std::hecto,U>::type;           ///< Represents the type of `class U` with the metric 'hecto' prefix appended.  @details E.g. hecto<meters> represents meters*10^2      @tparam U unit type to apply the prefix to.
    template<class U> using kilo    = typename units::detail::prefix<std::kilo,	U>::type;           ///< Represents the type of `class U` with the metric 'kilo' prefix appended.   @details E.g. kilo<meters> represents meters*10^3       @tparam U unit type to apply the prefix to.
    template<class U> using mega    = typename units::detail::prefix<std::mega,	U>::type;           ///< Represents the type of `class U` with the metric 'mega' prefix appended.   @details E.g. mega<meters> represents meters*10^6       @tparam U unit type to apply the prefix to.
    template<class U> using giga    = typename units::detail::prefix<std::giga,	U>::type;           ///< Represents the type of `class U` with the metric 'giga' prefix appended.   @details E.g. giga<meters> represents meters*10^9       @tparam U unit type to apply the prefix to.
    template<class U> using tera    = typename units::detail::prefix<std::tera,	U>::type;           ///< Represents the type of `class U` with the metric 'tera' prefix appended.   @details E.g. tera<meters> represents meters*10^12      @tparam U unit type to apply the prefix to.
    template<class U> using peta    = typename units::detail::prefix<std::peta,	U>::type;           ///< Represents the type of `class U` with the metric 'peta' prefix appended.   @details E.g. peta<meters> represents meters*10^15      @tparam U unit type to apply the prefix to.
    template<class U> using exa     = typename units::detail::prefix<std::exa,	U>::type;            ///< Represents the type of `class U` with the metric 'exa' prefix appended.    @details E.g. exa<meters> represents meters*10^18       @tparam U unit type to apply the prefix to.
    /** @} @} */
 
    /**
     * @ingroup UnitManipulators
     * @{
     * @ingroup Binary Prefixes
     * @{
     */
    template<class U> using kibi    = typename units::detail::prefix<std::ratio<1024>,                  U>::type;   ///< Represents the type of `class U` with the binary 'kibi' prefix appended.   @details E.g. kibi<bytes> represents bytes*2^10 @tparam U unit type to apply the prefix to.
    template<class U> using mebi    = typename units::detail::prefix<std::ratio<1048576>,               U>::type;   ///< Represents the type of `class U` with the binary 'mibi' prefix appended.   @details E.g. mebi<bytes> represents bytes*2^20 @tparam U unit type to apply the prefix to.
    template<class U> using gibi    = typename units::detail::prefix<std::ratio<1073741824>,            U>::type;   ///< Represents the type of `class U` with the binary 'gibi' prefix appended.   @details E.g. gibi<bytes> represents bytes*2^30 @tparam U unit type to apply the prefix to.
    template<class U> using tebi    = typename units::detail::prefix<std::ratio<1099511627776>,         U>::type;   ///< Represents the type of `class U` with the binary 'tebi' prefix appended.   @details E.g. tebi<bytes> represents bytes*2^40 @tparam U unit type to apply the prefix to.
    template<class U> using pebi    = typename units::detail::prefix<std::ratio<1125899906842624>,      U>::type;   ///< Represents the type of `class U` with the binary 'pebi' prefix appended.   @details E.g. pebi<bytes> represents bytes*2^50 @tparam U unit type to apply the prefix to.
    template<class U> using exbi    = typename units::detail::prefix<std::ratio<1152921504606846976>,   U>::type;   ///< Represents the type of `class U` with the binary 'exbi' prefix appended.   @details E.g. exbi<bytes> represents bytes*2^60 @tparam U unit type to apply the prefix to.
    /** @} @} */
 
    //------------------------------
    //  CONVERSION TRAITS
    //------------------------------
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which checks whether two units can be converted to each other
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_convertible_unit<U1, U2>::value` to test
         *              whether `class U1` is convertible to `class U2`. Note: convertible has both the semantic meaning,
         *              (i.e. meters can be converted to feet), and the c++ meaning of conversion (type meters can be
         *              converted to type feet). Conversion is always symmetric, so if U1 is convertible to U2, then
         *              U2 will be convertible to U1.
         * @tparam      U1 Unit to convert from.
         * @tparam      U2 Unit to convert to.
         * @sa          is_convertible_unit_t
         */
        template<class U1, class U2>
        struct is_convertible_unit : std::is_same <traits::base_unit_of<typename units::traits::unit_traits<U1>::base_unit_type>,
            base_unit_of<typename units::traits::unit_traits<U2>::base_unit_type >> {};
        template<class U1, class U2>
        inline constexpr bool is_convertible_unit_v = is_convertible_unit<U1, U2>::value;
    }
 
    //------------------------------
    //  CONVERSION FUNCTION
    //------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        constexpr inline UNIT_LIB_DEFAULT_TYPE pow(UNIT_LIB_DEFAULT_TYPE x, unsigned long long y)
        {
            return y == 0 ? 1.0 : x * pow(x, y - 1);
        }
 
        constexpr inline UNIT_LIB_DEFAULT_TYPE abs(UNIT_LIB_DEFAULT_TYPE x)
        {
            return x < 0 ? -x : x;
        }
 
        /// convert dispatch for units which are both the same
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::true_type, std::false_type, std::false_type) noexcept
        {
            return value;
        }
 
        /// convert dispatch for units which are both the same
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::true_type, std::false_type, std::true_type) noexcept
        {
            return value;
        }
 
        /// convert dispatch for units which are both the same
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::true_type, std::true_type, std::false_type) noexcept
        {
            return value;
        }
 
        /// convert dispatch for units which are both the same
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::true_type, std::true_type, std::true_type) noexcept
        {
            return value;
        }
 
        /// convert dispatch for units of different types w/ no translation and no PI
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::false_type, std::false_type, std::false_type) noexcept
        {
            return ((value * Ratio::num) / Ratio::den);
        }
 
        /// convert dispatch for units of different types w/ no translation, but has PI in numerator
        // constepxr with PI in numerator
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr
        std::enable_if_t<(PiRatio::num / PiRatio::den >= 1 && PiRatio::num % PiRatio::den == 0), T>
        convert(const T& value, std::false_type, std::true_type, std::false_type) noexcept
        {
            return ((value * pow(constants::detail::PI_VAL, PiRatio::num / PiRatio::den) * Ratio::num) / Ratio::den);
        }
 
        /// convert dispatch for units of different types w/ no translation, but has PI in denominator
        // constexpr with PI in denominator
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr
        std::enable_if_t<(PiRatio::num / PiRatio::den <= -1 && PiRatio::num % PiRatio::den == 0), T>
        convert(const T& value, std::false_type, std::true_type, std::false_type) noexcept
        {
            return (value * Ratio::num) / (Ratio::den * pow(constants::detail::PI_VAL, -PiRatio::num / PiRatio::den));
        }
 
        /// convert dispatch for units of different types w/ no translation, but has PI in numerator
        // Not constexpr - uses std::pow
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline // sorry, this can't be constexpr!
        std::enable_if_t<(PiRatio::num / PiRatio::den < 1 && PiRatio::num / PiRatio::den > -1), T>
        convert(const T& value, std::false_type, std::true_type, std::false_type) noexcept
        {
            return ((value * std::pow(constants::detail::PI_VAL, PiRatio::num / PiRatio::den)  * Ratio::num) / Ratio::den);
        }
 
        /// convert dispatch for units of different types with a translation, but no PI
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, std::false_type, std::false_type, std::true_type) noexcept
        {
            return ((value * Ratio::num) / Ratio::den) + (static_cast<UNIT_LIB_DEFAULT_TYPE>(Translation::num) / Translation::den);
        }
 
        /// convert dispatch for units of different types with a translation AND PI
        template<class UnitFrom, class UnitTo, class Ratio, class PiRatio, class Translation, typename T>
        static inline constexpr T convert(const T& value, const std::false_type, const std::true_type, const std::true_type) noexcept
        {
            return ((value * std::pow(constants::detail::PI_VAL, PiRatio::num / PiRatio::den) * Ratio::num) / Ratio::den) + (static_cast<UNIT_LIB_DEFAULT_TYPE>(Translation::num) / Translation::den);
        }
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @ingroup     Conversion
     * @brief       converts a <i>value</i> from one type to another.
     * @details     Converts a <i>value</i> of a built-in arithmetic type to another unit. This does not change
     *              the type of <i>value</i>, only what it contains. E.g. @code double result = convert<length::meters, length::feet>(1.0); // result == 3.28084 @endcode
     * @sa          unit_t  for implicit conversion of unit containers.
     * @tparam      UnitFrom unit tag to convert <i>value</i> from. Must be a `unit` type (i.e. is_unit<UnitFrom>::value == true),
     *              and must be convertible to `UnitTo` (i.e. is_convertible_unit<UnitFrom, UnitTo>::value == true).
     * @tparam      UnitTo unit tag to convert <i>value</i> to. Must be a `unit` type (i.e. is_unit<UnitTo>::value == true),
     *              and must be convertible from `UnitFrom` (i.e. is_convertible_unit<UnitFrom, UnitTo>::value == true).
     * @tparam      T type of <i>value</i>. It is inferred from <i>value</i>, and is expected to be a built-in arithmetic type.
     * @param[in]   value Arithmetic value to convert from `UnitFrom` to `UnitTo`. The value should represent
     *              a quantity in units of `UnitFrom`.
     * @returns     value, converted from units of `UnitFrom` to `UnitTo`.
     */
    template<class UnitFrom, class UnitTo, typename T = UNIT_LIB_DEFAULT_TYPE>
    static inline constexpr T convert(const T& value) noexcept
    {
        static_assert(traits::is_unit<UnitFrom>::value, "Template parameter `UnitFrom` must be a `unit` type.");
        static_assert(traits::is_unit<UnitTo>::value, "Template parameter `UnitTo` must be a `unit` type.");
        static_assert(traits::is_convertible_unit<UnitFrom, UnitTo>::value, "Units are not compatible.");
 
        using Ratio = std::ratio_divide<typename UnitFrom::conversion_ratio, typename UnitTo::conversion_ratio>;
        using PiRatio = std::ratio_subtract<typename UnitFrom::pi_exponent_ratio, typename UnitTo::pi_exponent_ratio>;
        using Translation = std::ratio_divide<std::ratio_subtract<typename UnitFrom::translation_ratio, typename UnitTo::translation_ratio>, typename UnitTo::conversion_ratio>;
 
        using isSame = typename std::is_same<std::decay_t<UnitFrom>, std::decay_t<UnitTo>>::type;
        using piRequired = std::integral_constant<bool, !(std::is_same<std::ratio<0>, PiRatio>::value)>;
        using translationRequired = std::integral_constant<bool, !(std::is_same<std::ratio<0>, Translation>::value)>;
 
        return units::detail::convert<UnitFrom, UnitTo, Ratio, PiRatio, Translation, T>
            (value, isSame{}, piRequired{}, translationRequired{});
    }
 
    //----------------------------------
    //  NON-LINEAR SCALE TRAITS
    //----------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace traits
    {
        namespace detail
        {
            /**
            * @brief        implementation of has_operator_parenthesis
            * @details      checks that operator() returns the same type as `Ret`
            */
            template<class T, class Ret>
            struct has_operator_parenthesis_impl
            {
                template<class U>
                static constexpr auto test(U*) -> decltype(std::declval<U>()()) { return decltype(std::declval<U>()()){}; }
                template<typename>
                static constexpr std::false_type test(...) { return std::false_type{}; }
 
                using type = typename std::is_same<Ret, decltype(test<T>(0))>::type;
            };
        }
 
        /**
         * @brief       checks that `class T` has an `operator()` member which returns `Ret`
         * @details     used as part of the linear_scale concept.
         */
        template<class T, class Ret>
        struct has_operator_parenthesis : traits::detail::has_operator_parenthesis_impl<T, Ret>::type {};
    }
 
    namespace traits
    {
        namespace detail
        {
            /**
            * @brief        implementation of has_value_member
            * @details      checks for a member named `m_member` with type `Ret`
            */
            template<class T, class Ret>
            struct has_value_member_impl
            {
                template<class U>
                static constexpr auto test(U* p) -> decltype(p->m_value) { return p->m_value; }
                template<typename>
                static constexpr auto test(...)->std::false_type { return std::false_type{}; }
 
                using type = typename std::is_same<std::decay_t<Ret>, std::decay_t<decltype(test<T>(0))>>::type;
            };
        }
 
        /**
         * @brief       checks for a member named `m_member` with type `Ret`
         * @details     used as part of the linear_scale concept checker.
         */
        template<class T, class Ret>
        struct has_value_member : traits::detail::has_value_member_impl<T, Ret>::type {};
        template<class T, class Ret>
        inline constexpr bool has_value_member_v = has_value_member<T, Ret>::value;
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests that `class T` meets the requirements for a non-linear scale
         * @details     A non-linear scale must:
         *              - be default constructible
         *              - have an `operator()` member which returns the non-linear value stored in the scale
         *              - have an accessible `m_value` member type which stores the linearized value in the scale.
         *
         *              Linear/nonlinear scales are used by `units::unit` to store values and scale them
         *              if they represent things like dB.
         */
        template<class T, class Ret>
        struct is_nonlinear_scale : std::integral_constant<bool,
            std::is_default_constructible<T>::value &&
            has_operator_parenthesis<T, Ret>::value &&
            has_value_member<T, Ret>::value &&
            std::is_trivial<T>::value>
        {};
    }
 
    //------------------------------
    //  UNIT_T TYPE TRAITS
    //------------------------------
 
    namespace traits
    {
#ifdef FOR_DOXYGEN_PURPOSOES_ONLY
        /**
        * @ingroup      TypeTraits
        * @brief        Trait for accessing the publicly defined types of `units::unit_t`
        * @details      The units library determines certain properties of the unit_t types passed to them
        *               and what they represent by using the members of the corresponding unit_t_traits instantiation.
        */
        template<typename T>
        struct unit_t_traits
        {
            typedef typename T::non_linear_scale_type non_linear_scale_type;    ///< Type of the unit_t non_linear_scale (e.g. linear_scale, decibel_scale). This property is used to enable the proper linear or logarithmic arithmetic functions.
            typedef typename T::underlying_type underlying_type;                ///< Underlying storage type of the `unit_t`, e.g. `double`.
            typedef typename T::value_type value_type;                          ///< Synonym for underlying type. May be removed in future versions. Prefer underlying_type.
            typedef typename T::unit_type unit_type;                            ///< Type of unit the `unit_t` represents, e.g. `meters`
        };
#endif
 
        /** @cond */    // DOXYGEN IGNORE
        /**
         * @brief       unit_t_traits specialization for things which are not unit_t
         * @details
         */
        template<typename T, typename = void>
        struct unit_t_traits
        {
            typedef void non_linear_scale_type;
            typedef void underlying_type;
            typedef void value_type;
            typedef void unit_type;
        };
 
        /**
         * @ingroup     TypeTraits
         * @brief       Trait for accessing the publicly defined types of `units::unit_t`
         * @details
         */
        template<typename T>
        struct unit_t_traits <T, typename void_t<
            typename T::non_linear_scale_type,
            typename T::underlying_type,
            typename T::value_type,
            typename T::unit_type>::type>
        {
            typedef typename T::non_linear_scale_type non_linear_scale_type;
            typedef typename T::underlying_type underlying_type;
            typedef typename T::value_type value_type;
            typedef typename T::unit_type unit_type;
        };
        /** @endcond */ // END DOXYGEN IGNORE
    }
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether two container types derived from `unit_t` are convertible to each other
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_convertible_unit_t<U1, U2>::value` to test
         *              whether `class U1` is convertible to `class U2`. Note: convertible has both the semantic meaning,
         *              (i.e. meters can be converted to feet), and the c++ meaning of conversion (type meters can be
         *              converted to type feet). Conversion is always symmetric, so if U1 is convertible to U2, then
         *              U2 will be convertible to U1.
         * @tparam      U1 Unit to convert from.
         * @tparam      U2 Unit to convert to.
         * @sa          is_convertible_unit
         */
        template<class U1, class U2>
        struct is_convertible_unit_t : std::integral_constant<bool,
            is_convertible_unit<typename units::traits::unit_t_traits<U1>::unit_type, typename units::traits::unit_t_traits<U2>::unit_type>::value>
        {};
    }
 
    //----------------------------------
    //  UNIT TYPE
    //----------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    // forward declaration
    template<typename T> struct linear_scale;
    template<typename T> struct decibel_scale;
 
    namespace detail
    {
        /**
        * @brief        helper type to identify units.
        * @details      A non-templated base class for `unit` which enables RTTI testing.
        */
        struct _unit_t {};
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
        // forward declaration
        #if !defined(_MSC_VER) || _MSC_VER > 1800 // bug in VS2013 prevents this from working
        template<typename... T> struct is_dimensionless_unit;
        #else
        template<typename T1, typename T2 = T1, typename T3 = T1> struct is_dimensionless_unit;
        #endif
 
        /**
         * @ingroup     TypeTraits
         * @brief       Traits which tests if a class is a `unit`
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_unit<T>::value` to test
         *              whether `class T` implements a `unit`.
         */
        template<class T>
        struct is_unit_t : std::is_base_of<units::detail::_unit_t, T>::type {};
        template<class T>
        inline constexpr bool is_unit_t_v = is_unit_t<T>::value;
    }
 
    /**
     * @ingroup     UnitContainers
     * @brief       Container for values which represent quantities of a given unit.
     * @details     Stores a value which represents a quantity in the given units. Unit containers
     *              (except scalar values) are *not* convertible to built-in c++ types, in order to
     *              provide type safety in dimensional analysis. Unit containers *are* implicitly
     *              convertible to other compatible unit container types. Unit containers support
     *              various types of arithmetic operations, depending on their scale type.
     *
     *              The value of a `unit_t` can only be changed on construction, or by assignment
     *              from another `unit_t` type. If necessary, the underlying value can be accessed
     *              using `operator()`: @code
     *              meter_t m(5.0);
     *              double val = m(); // val == 5.0 @endcode.
     * @tparam      Units unit tag for which type of units the `unit_t` represents (e.g. meters)
     * @tparam      T underlying type of the storage. Defaults to double.
     * @tparam      NonLinearScale optional scale class for the units. Defaults to linear (i.e. does
     *              not scale the unit value). Examples of non-linear scales could be logarithmic,
     *              decibel, or richter scales. Non-linear scales must adhere to the non-linear-scale
     *              concept, i.e. `is_nonlinear_scale<...>::value` must be `true`.
     * @sa
     *              - \ref lengthContainers "length unit containers"
     *              - \ref massContainers "mass unit containers"
     *              - \ref timeContainers "time unit containers"
     *              - \ref angleContainers "angle unit containers"
     *              - \ref currentContainers "current unit containers"
     *              - \ref temperatureContainers "temperature unit containers"
     *              - \ref substanceContainers "substance unit containers"
     *              - \ref luminousIntensityContainers "luminous intensity unit containers"
     *              - \ref solidAngleContainers "solid angle unit containers"
     *              - \ref frequencyContainers "frequency unit containers"
     *              - \ref velocityContainers "velocity unit containers"
     *              - \ref angularVelocityContainers "angular velocity unit containers"
     *              - \ref accelerationContainers "acceleration unit containers"
     *              - \ref forceContainers "force unit containers"
     *              - \ref pressureContainers "pressure unit containers"
     *              - \ref chargeContainers "charge unit containers"
     *              - \ref energyContainers "energy unit containers"
     *              - \ref powerContainers "power unit containers"
     *              - \ref voltageContainers "voltage unit containers"
     *              - \ref capacitanceContainers "capacitance unit containers"
     *              - \ref impedanceContainers "impedance unit containers"
     *              - \ref magneticFluxContainers "magnetic flux unit containers"
     *              - \ref magneticFieldStrengthContainers "magnetic field strength unit containers"
     *              - \ref inductanceContainers "inductance unit containers"
     *              - \ref luminousFluxContainers "luminous flux unit containers"
     *              - \ref illuminanceContainers "illuminance unit containers"
     *              - \ref radiationContainers "radiation unit containers"
     *              - \ref torqueContainers "torque unit containers"
     *              - \ref areaContainers "area unit containers"
     *              - \ref volumeContainers "volume unit containers"
     *              - \ref densityContainers "density unit containers"
     *              - \ref concentrationContainers "concentration unit containers"
     *              - \ref constantContainers "constant unit containers"
     */
    template<class Units, typename T = UNIT_LIB_DEFAULT_TYPE, template<typename> class NonLinearScale = linear_scale>
    class unit_t : public NonLinearScale<T>, units::detail::_unit_t
    {
        static_assert(traits::is_unit<Units>::value, "Template parameter `Units` must be a unit tag. Check that you aren't using a unit type (_t).");
        static_assert(traits::is_nonlinear_scale<NonLinearScale<T>, T>::value, "Template parameter `NonLinearScale` does not conform to the `is_nonlinear_scale` concept.");
 
    protected:
 
        using nls = NonLinearScale<T>;
        using nls::m_value;
 
    public:
 
        typedef NonLinearScale<T> non_linear_scale_type;                                            ///< Type of the non-linear scale of the unit_t (e.g. linear_scale)
        typedef T underlying_type;                                                                  ///< Type of the underlying storage of the unit_t (e.g. double)
        typedef T value_type;                                                                       ///< Synonym for underlying type. May be removed in future versions. Prefer underlying_type.
        typedef Units unit_type;                                                                    ///< Type of `unit` the `unit_t` represents (e.g. meters)
 
        /**
         * @ingroup     Constructors
         * @brief       default constructor.
         */
        constexpr unit_t() = default;
 
        /**
         * @brief       constructor
         * @details     constructs a new unit_t using the non-linear scale's constructor.
         * @param[in]   value   unit value magnitude.
         * @param[in]   args    additional constructor arguments are forwarded to the non-linear scale constructor. Which
         *                      args are required depends on which scale is used. For the default (linear) scale,
         *                      no additional args are necessary.
         */
        template<class... Args>
        inline explicit constexpr unit_t(const T value, const Args&... args) noexcept : nls(value, args...)
        {
 
        }
 
        /**
         * @brief       constructor
         * @details     enable implicit conversions from T types ONLY for linear scalar units
         * @param[in]   value value of the unit_t
         */
        template<class Ty, class = typename std::enable_if<traits::is_dimensionless_unit<Units>::value && std::is_arithmetic<Ty>::value>::type>
        inline constexpr unit_t(const Ty value) noexcept : nls(value)
        {
 
        }
 
        /**
         * @brief       chrono constructor
         * @details     enable implicit conversions from std::chrono::duration types ONLY for time units
         * @param[in]   value value of the unit_t
         */
        template<class Rep, class Period, class = std::enable_if_t<std::is_arithmetic<Rep>::value && traits::is_ratio<Period>::value>>
        inline constexpr unit_t(const std::chrono::duration<Rep, Period>& value) noexcept :
        nls(units::convert<unit<std::ratio<1,1000000000>, category::time_unit>, Units>(static_cast<T>(std::chrono::duration_cast<std::chrono::nanoseconds>(value).count())))
        {
 
        }
 
        /**
         * @brief       copy constructor (converting)
         * @details     performs implicit unit conversions if required.
         * @param[in]   rhs unit to copy.
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr unit_t(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) noexcept :
        nls(units::convert<UnitsRhs, Units, T>(rhs.m_value), std::true_type() /*store linear value*/)
        {
 
        }
 
        /**
         * @brief       assignment
         * @details     performs implicit unit conversions if required
         * @param[in]   rhs unit to copy.
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline unit_t& operator=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) noexcept
        {
            nls::m_value = units::convert<UnitsRhs, Units, T>(rhs.m_value);
            return *this;
        }
 
        /**
        * @brief        assignment
        * @details      performs implicit conversions from built-in types ONLY for scalar units
        * @param[in]    rhs value to copy.
        */
        template<class Ty, class = std::enable_if_t<traits::is_dimensionless_unit<Units>::value && std::is_arithmetic<Ty>::value>>
        inline unit_t& operator=(const Ty& rhs) noexcept
        {
            nls::m_value = rhs;
            return *this;
        }
 
        /**
         * @brief       less-than
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` is less than the value of `rhs`
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr bool operator<(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return (nls::m_value < units::convert<UnitsRhs, Units>(rhs.m_value));
        }
 
        /**
         * @brief       less-than or equal
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` is less than or equal to the value of `rhs`
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr bool operator<=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return (nls::m_value <= units::convert<UnitsRhs, Units>(rhs.m_value));
        }
 
        /**
         * @brief       greater-than
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` is greater than the value of `rhs`
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr bool operator>(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return (nls::m_value > units::convert<UnitsRhs, Units>(rhs.m_value));
        }
 
        /**
         * @brief       greater-than or equal
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` is greater than or equal to the value of `rhs`
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr bool operator>=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return (nls::m_value >= units::convert<UnitsRhs, Units>(rhs.m_value));
        }
 
        /**
         * @brief       equality
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` exactly equal to the value of rhs.
         * @note        This may not be suitable for all applications when the underlying_type of unit_t is a double.
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs, std::enable_if_t<std::is_floating_point<T>::value || std::is_floating_point<Ty>::value, int> = 0>
        inline constexpr bool operator==(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return detail::abs(nls::m_value - units::convert<UnitsRhs, Units>(rhs.m_value)) < std::numeric_limits<T>::epsilon() *
                detail::abs(nls::m_value + units::convert<UnitsRhs, Units>(rhs.m_value)) ||
                detail::abs(nls::m_value - units::convert<UnitsRhs, Units>(rhs.m_value)) < (std::numeric_limits<T>::min)();
        }
 
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs, std::enable_if_t<std::is_integral<T>::value && std::is_integral<Ty>::value, int> = 0>
        inline constexpr bool operator==(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return nls::m_value == units::convert<UnitsRhs, Units>(rhs.m_value);
        }
 
        /**
         * @brief       inequality
         * @details     compares the linearized value of two units. Performs unit conversions if necessary.
         * @param[in]   rhs right-hand side unit for the comparison
         * @returns     true IFF the value of `this` is not equal to the value of rhs.
         * @note        This may not be suitable for all applications when the underlying_type of unit_t is a double.
         */
        template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
        inline constexpr bool operator!=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const noexcept
        {
            return !(*this == rhs);
        }
 
        /**
         * @brief       unit value
         * @returns     value of the unit in it's underlying, non-safe type.
         */
        inline constexpr underlying_type value() const noexcept
        {
            return static_cast<underlying_type>(*this);
        }
 
        /**
         * @brief       unit value
         * @returns     value of the unit converted to an arithmetic, non-safe type.
         */
        template<typename Ty, class = std::enable_if_t<std::is_arithmetic<Ty>::value>>
        inline constexpr Ty to() const noexcept
        {
            return static_cast<Ty>(*this);
        }
 
        /**
         * @brief       linearized unit value
         * @returns     linearized value of unit which has a non-linear scale. For `unit_t` types with
         *              linear scales, this is equivalent to `value`.
         */
        template<typename Ty, class = std::enable_if_t<std::is_arithmetic<Ty>::value>>
        inline constexpr Ty toLinearized() const noexcept
        {
            return static_cast<Ty>(m_value);
        }
 
        /**
         * @brief       conversion
         * @details     Converts to a different unit container. Units can be converted to other containers
         *              implicitly, but this can be used in cases where explicit notation of a conversion
         *              is beneficial, or where an r-value container is needed.
         * @tparam      U unit (not unit_t) to convert to
         * @returns     a unit container with the specified units containing the equivalent value to
         *              *this.
         */
        template<class U>
        inline constexpr unit_t<U> convert() const noexcept
        {
            static_assert(traits::is_unit<U>::value, "Template parameter `U` must be a unit type.");
            return unit_t<U>(*this);
        }
 
        /**
         * @brief       implicit type conversion.
         * @details     only enabled for scalar unit types.
         */
        template<class Ty, std::enable_if_t<traits::is_dimensionless_unit<Units>::value && std::is_arithmetic<Ty>::value, int> = 0>
        inline constexpr operator Ty() const noexcept
        {
            // this conversion also resolves any PI exponents, by converting from a non-zero PI ratio to a zero-pi ratio.
            return static_cast<Ty>(units::convert<Units, unit<std::ratio<1>, units::category::scalar_unit>>((*this)()));
        }
 
        /**
         * @brief       explicit type conversion.
         * @details     only enabled for non-dimensionless unit types.
         */
        template<class Ty, std::enable_if_t<!traits::is_dimensionless_unit<Units>::value && std::is_arithmetic<Ty>::value, int> = 0>
        inline constexpr explicit operator Ty() const noexcept
        {
            return static_cast<Ty>((*this)());
        }
 
        /**
         * @brief       chrono implicit type conversion.
         * @details     only enabled for time unit types.
         */
        template<typename U = Units, std::enable_if_t<units::traits::is_convertible_unit<U, unit<std::ratio<1>, category::time_unit>>::value, int> = 0>
        inline constexpr operator std::chrono::nanoseconds() const noexcept
        {
            return std::chrono::duration_cast<std::chrono::nanoseconds>(std::chrono::duration<double, std::nano>(units::convert<Units, unit<std::ratio<1,1000000000>, category::time_unit>>((*this)())));
        }
 
        /**
         * @brief       returns the unit name
         */
        inline constexpr const char* name() const noexcept
        {
            return units::name(*this);
        }
 
        /**
         * @brief       returns the unit abbreviation
         */
        inline constexpr const char* abbreviation() const noexcept
        {
            return units::abbreviation(*this);
        }
 
    public:
 
        template<class U, typename Ty, template<typename> class Nlt>
        friend class unit_t;
    };
 
    //------------------------------
    //  UNIT_T NON-MEMBER FUNCTIONS
    //------------------------------
 
    /**
     * @ingroup     UnitContainers
     * @brief       Constructs a unit container from an arithmetic type.
     * @details     make_unit can be used to construct a unit container from an arithmetic type, as an alternative to
     *              using the explicit constructor. Unlike the explicit constructor it forces the user to explicitly
     *              specify the units.
     * @tparam      UnitType Type to construct.
     * @tparam      Ty      Arithmetic type.
     * @param[in]   value   Arithmetic value that represents a quantity in units of `UnitType`.
     */
    template<class UnitType, typename T, class = std::enable_if_t<std::is_arithmetic<T>::value>>
    inline constexpr UnitType make_unit(const T value) noexcept
    {
        static_assert(traits::is_unit_t<UnitType>::value, "Template parameter `UnitType` must be a unit type (_t).");
 
        return UnitType(value);
    }
 
#if defined(UNIT_LIB_ENABLE_IOSTREAM)
    template<class Units, typename T, template<typename> class NonLinearScale>
    inline std::ostream& operator<<(std::ostream& os, const unit_t<Units, T, NonLinearScale>& obj) noexcept
    {
        using BaseUnits = unit<std::ratio<1>, typename traits::unit_traits<Units>::base_unit_type>;
        os << convert<Units, BaseUnits>(obj());
 
        if (traits::unit_traits<Units>::base_unit_type::meter_ratio::num != 0) { os << " m"; }
        if (traits::unit_traits<Units>::base_unit_type::meter_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::meter_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::meter_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::meter_ratio::den != 1) { os << "/"   << traits::unit_traits<Units>::base_unit_type::meter_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::kilogram_ratio::num != 0) { os << " kg"; }
        if (traits::unit_traits<Units>::base_unit_type::kilogram_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::kilogram_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::kilogram_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::kilogram_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::kilogram_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::second_ratio::num != 0) { os << " s"; }
        if (traits::unit_traits<Units>::base_unit_type::second_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::second_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::second_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::second_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::second_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::ampere_ratio::num != 0) { os << " A"; }
        if (traits::unit_traits<Units>::base_unit_type::ampere_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::ampere_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::ampere_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::ampere_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::ampere_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::kelvin_ratio::num != 0) { os << " K"; }
        if (traits::unit_traits<Units>::base_unit_type::kelvin_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::kelvin_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::kelvin_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::kelvin_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::kelvin_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::mole_ratio::num != 0) { os << " mol"; }
        if (traits::unit_traits<Units>::base_unit_type::mole_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::mole_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::mole_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::mole_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::mole_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::candela_ratio::num != 0) { os << " cd"; }
        if (traits::unit_traits<Units>::base_unit_type::candela_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::candela_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::candela_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::candela_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::candela_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::radian_ratio::num != 0) { os << " rad"; }
        if (traits::unit_traits<Units>::base_unit_type::radian_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::radian_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::radian_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::radian_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::radian_ratio::den; }
 
        if (traits::unit_traits<Units>::base_unit_type::byte_ratio::num != 0) { os << " b"; }
        if (traits::unit_traits<Units>::base_unit_type::byte_ratio::num != 0 &&
            traits::unit_traits<Units>::base_unit_type::byte_ratio::num != 1) { os << "^" << traits::unit_traits<Units>::base_unit_type::byte_ratio::num; }
        if (traits::unit_traits<Units>::base_unit_type::byte_ratio::den != 1) { os << "/" << traits::unit_traits<Units>::base_unit_type::byte_ratio::den; }
 
        return os;
    }
#endif
 
    template<class Units, typename T, template<typename> class NonLinearScale, typename RhsType>
    inline unit_t<Units, T, NonLinearScale>& operator+=(unit_t<Units, T, NonLinearScale>& lhs, const RhsType& rhs) noexcept
    {
        static_assert(traits::is_convertible_unit_t<unit_t<Units, T, NonLinearScale>, RhsType>::value ||
            (traits::is_dimensionless_unit<decltype(lhs)>::value && std::is_arithmetic<RhsType>::value),
            "parameters are not compatible units.");
 
        lhs = lhs + rhs;
        return lhs;
    }
 
    template<class Units, typename T, template<typename> class NonLinearScale, typename RhsType>
    inline unit_t<Units, T, NonLinearScale>& operator-=(unit_t<Units, T, NonLinearScale>& lhs, const RhsType& rhs) noexcept
    {
        static_assert(traits::is_convertible_unit_t<unit_t<Units, T, NonLinearScale>, RhsType>::value ||
            (traits::is_dimensionless_unit<decltype(lhs)>::value && std::is_arithmetic<RhsType>::value),
            "parameters are not compatible units.");
 
        lhs = lhs - rhs;
        return lhs;
    }
 
    template<class Units, typename T, template<typename> class NonLinearScale, typename RhsType>
    inline unit_t<Units, T, NonLinearScale>& operator*=(unit_t<Units, T, NonLinearScale>& lhs, const RhsType& rhs) noexcept
    {
        static_assert((traits::is_dimensionless_unit<RhsType>::value || std::is_arithmetic<RhsType>::value),
            "right-hand side parameter must be dimensionless.");
 
        lhs = lhs * rhs;
        return lhs;
    }
 
    template<class Units, typename T, template<typename> class NonLinearScale, typename RhsType>
    inline unit_t<Units, T, NonLinearScale>& operator/=(unit_t<Units, T, NonLinearScale>& lhs, const RhsType& rhs) noexcept
    {
        static_assert((traits::is_dimensionless_unit<RhsType>::value || std::is_arithmetic<RhsType>::value),
            "right-hand side parameter must be dimensionless.");
 
        lhs = lhs / rhs;
        return lhs;
    }
 
    //------------------------------
    //  UNIT_T UNARY OPERATORS
    //------------------------------
 
    // unary addition: +T
    template<class Units, typename T, template<typename> class NonLinearScale>
    constexpr inline unit_t<Units, T, NonLinearScale> operator+(const unit_t<Units, T, NonLinearScale>& u) noexcept
    {
        return u;
    }
 
    // prefix increment: ++T
    template<class Units, typename T, template<typename> class NonLinearScale>
    inline unit_t<Units, T, NonLinearScale>& operator++(unit_t<Units, T, NonLinearScale>& u) noexcept
    {
        u = unit_t<Units, T, NonLinearScale>(u() + 1);
        return u;
    }
 
    // postfix increment: T++
    template<class Units, typename T, template<typename> class NonLinearScale>
    inline unit_t<Units, T, NonLinearScale> operator++(unit_t<Units, T, NonLinearScale>& u, int) noexcept
    {
        auto ret = u;
        u = unit_t<Units, T, NonLinearScale>(u() + 1);
        return ret;
    }
 
    // unary addition: -T
    template<class Units, typename T, template<typename> class NonLinearScale>
    constexpr inline unit_t<Units, T, NonLinearScale> operator-(const unit_t<Units, T, NonLinearScale>& u) noexcept
    {
        return unit_t<Units, T, NonLinearScale>(-u());
    }
 
    // prefix increment: --T
    template<class Units, typename T, template<typename> class NonLinearScale>
    inline unit_t<Units, T, NonLinearScale>& operator--(unit_t<Units, T, NonLinearScale>& u) noexcept
    {
        u = unit_t<Units, T, NonLinearScale>(u() - 1);
        return u;
    }
 
    // postfix increment: T--
    template<class Units, typename T, template<typename> class NonLinearScale>
    inline unit_t<Units, T, NonLinearScale> operator--(unit_t<Units, T, NonLinearScale>& u, int) noexcept
    {
        auto ret = u;
        u = unit_t<Units, T, NonLinearScale>(u() - 1);
        return ret;
    }
 
    //------------------------------
    //  UNIT_CAST
    //------------------------------
 
    /**
     * @ingroup     Conversion
     * @brief       Casts a unit container to an arithmetic type.
     * @details     unit_cast can be used to remove the strong typing from a unit class, and convert it
     *              to a built-in arithmetic type. This may be useful for compatibility with libraries
     *              and legacy code that don't support `unit_t` types. E.g
     * @code        meter_t unitVal(5);
     *  double value = units::unit_cast<double>(unitVal);   // value = 5.0
     * @endcode
     * @tparam      T       Type to cast the unit type to. Must be a built-in arithmetic type.
     * @param       value   Unit value to cast.
     * @sa          unit_t::to
     */
    template<typename T, typename Units, class = std::enable_if_t<std::is_arithmetic<T>::value && traits::is_unit_t<Units>::value>>
    inline constexpr T unit_cast(const Units& value) noexcept
    {
        return static_cast<T>(value);
    }
 
    //------------------------------
    //  NON-LINEAR SCALE TRAITS
    //------------------------------
 
    // forward declaration
    template<typename T> struct decibel_scale;
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether a type is inherited from a linear scale.
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `has_linear_scale<U1 [, U2, ...]>::value` to test
         *              one or more types to see if they represent unit_t's whose scale is linear.
         * @tparam      T   one or more types to test.
         */
#if !defined(_MSC_VER) || _MSC_VER > 1800   // bug in VS2013 prevents this from working
        template<typename... T>
        struct has_linear_scale : std::integral_constant<bool, units::all_true<std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T>::underlying_type>, T>::value...>::value > {};
        template<typename... T>
        inline constexpr bool has_linear_scale_v = has_linear_scale<T...>::value;
#else
        template<typename T1, typename T2 = T1, typename T3 = T1>
        struct has_linear_scale : std::integral_constant<bool,
            std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T1>::underlying_type>, T1>::value &&
            std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T2>::value &&
            std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T3>::underlying_type>, T3>::value> {};
        template<typename T1, typename T2 = T1, typename T3 = T1>
        inline constexpr bool has_linear_scale_v = has_linear_scale<T1, T2, T3>::value;
#endif
 
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether a type is inherited from a decibel scale.
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `has_decibel_scale<U1 [, U2, ...]>::value` to test
         *              one or more types to see if they represent unit_t's whose scale is in decibels.
         * @tparam      T   one or more types to test.
         */
#if !defined(_MSC_VER) || _MSC_VER > 1800   // bug in VS2013 prevents this from working
        template<typename... T>
        struct has_decibel_scale : std::integral_constant<bool, units::all_true<std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T>::underlying_type>, T>::value...>::value> {};
        template<typename... T>
        inline constexpr bool has_decibel_scale_v = has_decibel_scale<T...>::value;
#else
        template<typename T1, typename T2 = T1, typename T3 = T1>
        struct has_decibel_scale : std::integral_constant<bool,
            std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T1>::underlying_type>, T1>::value &&
            std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T2>::value &&
            std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T3>::value> {};
        template<typename T1, typename T2 = T1, typename T3 = T1>
        inline constexpr bool has_decibel_scale_v = has_decibel_scale<T1, T2, T3>::value;
#endif
 
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether two types has the same non-linear scale.
         * @details     Inherits from `std::true_type` or `std::false_type`. Use `is_same_scale<U1 , U2>::value` to test
         *              whether two types have the same non-linear scale.
         * @tparam      T1  left hand type.
         * @tparam      T2  right hand type
         */
        template<typename T1, typename T2>
        struct is_same_scale : std::integral_constant<bool,
            std::is_same<typename units::traits::unit_t_traits<T1>::non_linear_scale_type, typename units::traits::unit_t_traits<T2>::non_linear_scale_type>::value>
        {};
        template<typename T1, typename T2>
        inline constexpr bool is_same_scale_v = is_same_scale<T1, T2>::value;
    }
 
    //----------------------------------
    //  NON-LINEAR SCALES
    //----------------------------------
 
    // Non-linear transforms are used to pre and post scale units which are defined in terms of non-
    // linear functions of their current value. A good example of a non-linear scale would be a
    // logarithmic or decibel scale
 
    //------------------------------
    //  LINEAR SCALE
    //------------------------------
 
    /**
     * @brief       unit_t scale which is linear
     * @details     Represents units on a linear scale. This is the appropriate unit_t scale for almost
     *              all units almost all of the time.
     * @tparam      T   underlying storage type
     * @sa          unit_t
     */
    template<typename T>
    struct linear_scale
    {
        inline constexpr linear_scale() = default;                                                  ///< default constructor.
        inline constexpr linear_scale(const linear_scale&) = default;
        inline ~linear_scale() = default;
        inline linear_scale& operator=(const linear_scale&) = default;
#if defined(_MSC_VER) && (_MSC_VER > 1800)
        inline constexpr linear_scale(linear_scale&&) = default;
        inline linear_scale& operator=(linear_scale&&) = default;
#endif
        template<class... Args>
        inline constexpr linear_scale(const T& value, Args&&...) noexcept : m_value(value) {}   ///< constructor.
        inline constexpr T operator()() const noexcept { return m_value; }                          ///< returns value.
 
        T m_value;                                                                                  ///< linearized value.
    };
 
    //----------------------------------
    //  SCALAR (LINEAR) UNITS
    //----------------------------------
 
    // Scalar units are the *ONLY* units implicitly convertible to/from built-in types.
    namespace dimensionless
    {
        typedef unit<std::ratio<1>, units::category::scalar_unit> scalar;
        typedef unit<std::ratio<1>, units::category::dimensionless_unit> dimensionless;
 
        typedef unit_t<scalar> scalar_t;
        typedef scalar_t dimensionless_t;
    }
 
// ignore the redeclaration of the default template parameters
#if defined(_MSC_VER)
#   pragma warning(push)
#   pragma warning(disable : 4348)
#endif
    UNIT_ADD_CATEGORY_TRAIT(scalar)
    UNIT_ADD_CATEGORY_TRAIT(dimensionless)
#if defined(_MSC_VER)
#   pragma warning(pop)
#endif
 
    //------------------------------
    //  LINEAR ARITHMETIC
    //------------------------------
 
    template<class UnitTypeLhs, class UnitTypeRhs, std::enable_if_t<!traits::is_same_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
    constexpr inline int operator+(const UnitTypeLhs& /* lhs */, const UnitTypeRhs& /* rhs */) noexcept
    {
        static_assert(traits::is_same_scale<UnitTypeLhs, UnitTypeRhs>::value, "Cannot add units with different linear/non-linear scales.");
        return 0;
    }
 
    /// Addition operator for unit_t types with a linear_scale.
    template<class UnitTypeLhs, class UnitTypeRhs, std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
    inline constexpr UnitTypeLhs operator+(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return UnitTypeLhs(lhs() + convert<UnitsRhs, UnitsLhs>(rhs()));
    }
 
    /// Addition operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
    template<typename T, std::enable_if_t<std::is_arithmetic<T>::value, int> = 0>
    inline constexpr dimensionless::scalar_t operator+(const dimensionless::scalar_t& lhs, T rhs) noexcept
    {
        return dimensionless::scalar_t(lhs() + rhs);
    }
 
    /// Addition operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
    template<typename T, std::enable_if_t<std::is_arithmetic<T>::value, int> = 0>
    inline constexpr dimensionless::scalar_t operator+(T lhs, const dimensionless::scalar_t& rhs) noexcept
    {
        return dimensionless::scalar_t(lhs + rhs());
    }
 
    /// Subtraction operator for unit_t types with a linear_scale.
    template<class UnitTypeLhs, class UnitTypeRhs, std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
    inline constexpr UnitTypeLhs operator-(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return UnitTypeLhs(lhs() - convert<UnitsRhs, UnitsLhs>(rhs()));
    }
 
    /// Subtraction operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
    template<typename T, std::enable_if_t<std::is_arithmetic<T>::value, int> = 0>
    inline constexpr dimensionless::scalar_t operator-(const dimensionless::scalar_t& lhs, T rhs) noexcept
    {
        return dimensionless::scalar_t(lhs() - rhs);
    }
 
    /// Subtraction operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
    template<typename T, std::enable_if_t<std::is_arithmetic<T>::value, int> = 0>
    inline constexpr dimensionless::scalar_t operator-(T lhs, const dimensionless::scalar_t& rhs) noexcept
    {
        return dimensionless::scalar_t(lhs - rhs());
    }
 
    /// Multiplication type for convertible unit_t types with a linear scale. @returns the multiplied value, with the same type as left-hand side unit.
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
        inline constexpr auto operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>>
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return  unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>>
            (lhs() * convert<UnitsRhs, UnitsLhs>(rhs()));
    }
 
    /// Multiplication type for non-convertible unit_t types with a linear scale. @returns the multiplied value, whose type is a compound unit of the left and right hand side values.
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<!traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr auto operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return unit_t<compound_unit<UnitsLhs, UnitsRhs>>
            (lhs() * rhs());
    }
 
    /// Multiplication by a dimensionless unit for unit_t types with a linear scale.
    template<class UnitTypeLhs, typename UnitTypeRhs,
        std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value && traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr UnitTypeLhs operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        // the cast makes sure factors of PI are handled as expected
        return UnitTypeLhs(lhs() * static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs));
    }
 
    /// Multiplication by a dimensionless unit for unit_t types with a linear scale.
    template<class UnitTypeLhs, typename UnitTypeRhs,
        std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && traits::is_dimensionless_unit<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr UnitTypeRhs operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        // the cast makes sure factors of PI are handled as expected
        return UnitTypeRhs(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) * rhs());
    }
 
    /// Multiplication by a scalar for unit_t types with a linear scale.
    template<class UnitTypeLhs, typename T,
        std::enable_if_t<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeLhs>::value, int> = 0>
        inline constexpr UnitTypeLhs operator*(const UnitTypeLhs& lhs, T rhs) noexcept
    {
        return UnitTypeLhs(lhs() * rhs);
    }
 
    /// Multiplication by a scalar for unit_t types with a linear scale.
    template<class UnitTypeRhs, typename T,
        std::enable_if_t<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeRhs>::value, int> = 0>
        inline constexpr UnitTypeRhs operator*(T lhs, const UnitTypeRhs& rhs) noexcept
    {
        return UnitTypeRhs(lhs * rhs());
    }
 
    /// Division for convertible unit_t types with a linear scale. @returns the lhs divided by rhs value, whose type is a scalar
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
        inline constexpr dimensionless::scalar_t operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return dimensionless::scalar_t(lhs() / convert<UnitsRhs, UnitsLhs>(rhs()));
    }
 
    /// Division for non-convertible unit_t types with a linear scale. @returns the lhs divided by the rhs, with a compound unit type of lhs/rhs
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<!traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr auto operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept ->  unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>>
    {
        using UnitsLhs = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return unit_t<compound_unit<UnitsLhs, inverse<UnitsRhs>>>
            (lhs() / rhs());
    }
 
    /// Division by a dimensionless unit for unit_t types with a linear scale
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value && traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr UnitTypeLhs operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept
    {
        return UnitTypeLhs(lhs() / static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs));
    }
 
    /// Division of a dimensionless unit  by a unit_t type with a linear scale
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value && traits::is_dimensionless_unit<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
        inline constexpr auto operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
    {
        return unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
            (static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) / rhs());
    }
 
    /// Division by a scalar for unit_t types with a linear scale
    template<class UnitTypeLhs, typename T,
        std::enable_if_t<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeLhs>::value, int> = 0>
        inline constexpr UnitTypeLhs operator/(const UnitTypeLhs& lhs, T rhs) noexcept
    {
        return UnitTypeLhs(lhs() / rhs);
    }
 
    /// Division of a scalar  by a unit_t type with a linear scale
    template<class UnitTypeRhs, typename T,
        std::enable_if_t<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeRhs>::value, int> = 0>
        inline constexpr auto operator/(T lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
    {
        using UnitsRhs = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        return unit_t<inverse<UnitsRhs>>
            (lhs / rhs());
    }
 
    //----------------------------------
    //  SCALAR COMPARISONS
    //----------------------------------
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator==(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return detail::abs(lhs - static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs)) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::epsilon() * detail::abs(lhs + static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs)) ||
            detail::abs(lhs - static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs)) < (std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::min)();
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator==(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return detail::abs(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) - rhs) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::epsilon() * detail::abs(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) + rhs) ||
            detail::abs(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) - rhs) < (std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::min)();
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator!=(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return!(lhs == static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs));
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator!=(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return !(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) == rhs);
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator>=(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return std::isgreaterequal(lhs, static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs));
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator>=(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return std::isgreaterequal(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs), rhs);
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator>(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return lhs > static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs);
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator>(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) > rhs;
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator<=(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return std::islessequal(lhs, static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs));
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator<=(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return std::islessequal(static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs), rhs);
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator<(const UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs) noexcept
    {
        return lhs < static_cast<UNIT_LIB_DEFAULT_TYPE>(rhs);
    }
 
    template<typename Units, class = std::enable_if_t<units::traits::is_dimensionless_unit<Units>::value>>
    constexpr bool operator<(const Units& lhs, const UNIT_LIB_DEFAULT_TYPE rhs) noexcept
    {
        return static_cast<UNIT_LIB_DEFAULT_TYPE>(lhs) < rhs;
    }
 
    //----------------------------------
    //  POW
    //----------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        /// recursive exponential implementation
        template <int N, class U> struct power_of_unit
        {
            typedef typename units::detail::unit_multiply<U, typename power_of_unit<N - 1, U>::type> type;
        };
 
        /// End recursion
        template <class U> struct power_of_unit<1, U>
        {
            typedef U type;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace math
    {
        /**
         * @brief       computes the value of <i>value</i> raised to the <i>power</i>
         * @details     Only implemented for linear_scale units. <i>Power</i> must be known at compile time, so the resulting unit type can be deduced.
         * @tparam      power exponential power to raise <i>value</i> by.
         * @param[in]   value `unit_t` derived type to raise to the given <i>power</i>
         * @returns     new unit_t, raised to the given exponent
         */
        template<int power, class UnitType, class = typename std::enable_if<traits::has_linear_scale<UnitType>::value, int>>
        inline constexpr auto pow(const UnitType& value) noexcept -> unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
        {
            return unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
                (gcem::pow(value(), power));
        }
 
        /**
         * @brief       computes the value of <i>value</i> raised to the <i>power</i> as a constexpr
         * @details     Only implemented for linear_scale units. <i>Power</i> must be known at compile time, so the resulting unit type can be deduced.
         *              Additionally, the power must be <i>a positive, integral, value</i>.
         * @tparam      power exponential power to raise <i>value</i> by.
         * @param[in]   value `unit_t` derived type to raise to the given <i>power</i>
         * @returns     new unit_t, raised to the given exponent
         */
        template<int power, class UnitType, class = typename std::enable_if<traits::has_linear_scale<UnitType>::value, int>>
        inline constexpr auto cpow(const UnitType& value) noexcept -> unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
        {
            static_assert(power >= 0, "cpow cannot accept negative numbers. Try units::math::pow instead.");
            return unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
                (detail::pow(value(), power));
        }
    }
 
    //------------------------------
    //  DECIBEL SCALE
    //------------------------------
 
    /**
    * @brief        unit_t scale for representing decibel values.
    * @details      internally stores linearized values. `operator()` returns the value in dB.
    * @tparam       T   underlying storage type
    * @sa           unit_t
    */
    template<typename T>
    struct decibel_scale
    {
        inline constexpr decibel_scale() = default;
        inline constexpr decibel_scale(const decibel_scale&) = default;
        inline ~decibel_scale() = default;
        inline decibel_scale& operator=(const decibel_scale&) = default;
#if defined(_MSC_VER) && (_MSC_VER > 1800)
        inline constexpr decibel_scale(decibel_scale&&) = default;
        inline decibel_scale& operator=(decibel_scale&&) = default;
#endif
        inline constexpr decibel_scale(const T value) noexcept : m_value(std::pow(10, value / 10)) {}
        template<class... Args>
        inline constexpr decibel_scale(const T value, std::true_type, Args&&...) noexcept : m_value(value) {}
        inline constexpr T operator()() const noexcept { return 10 * std::log10(m_value); }
 
        T m_value;  ///< linearized value
    };
 
    //------------------------------
    //  SCALAR (DECIBEL) UNITS
    //------------------------------
 
    /**
     * @brief       namespace for unit types and containers for units that have no dimension (scalar units)
     * @sa          See unit_t for more information on unit type containers.
     */
    namespace dimensionless
    {
        typedef unit_t<scalar, UNIT_LIB_DEFAULT_TYPE, decibel_scale> dB_t;
        typedef dB_t dBi_t;
    }
#if defined(UNIT_LIB_ENABLE_IOSTREAM)
    namespace dimensionless
    {
        inline std::ostream& operator<<(std::ostream& os, const dB_t& obj) { os << obj() << " dB"; return os; }
    }
#endif
}
#if __has_include(<fmt/format.h>) && !defined(UNIT_LIB_DISABLE_FMT)
template <>
struct fmt::formatter<units::dimensionless::dB_t> : fmt::formatter<double>
{
    template <typename FormatContext>
    auto format(const units::dimensionless::dB_t& obj,
                            FormatContext& ctx) -> decltype(ctx.out())
    {
        auto out = ctx.out();
        out = fmt::formatter<double>::format(obj(), ctx);
        return fmt::format_to(out, " dB");
    }
};
#endif
 
namespace units {
    //------------------------------
    //  DECIBEL ARITHMETIC
    //------------------------------
 
    /// Addition for convertible unit_t types with a decibel_scale
    template<class UnitTypeLhs, class UnitTypeRhs,
        std::enable_if_t<traits::has_decibel_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
    constexpr inline auto operator+(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>, typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type, decibel_scale>
    {
        using LhsUnits = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;
 
        return unit_t<compound_unit<squared<LhsUnits>>, underlying_type, decibel_scale>
            (lhs.template toLinearized<underlying_type>() * convert<RhsUnits, LhsUnits>(rhs.template toLinearized<underlying_type>()), std::true_type());
    }
 
    /// Addition between unit_t types with a decibel_scale and dimensionless dB units
    template<class UnitTypeLhs, std::enable_if_t<traits::has_decibel_scale<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value, int> = 0>
    constexpr inline UnitTypeLhs operator+(const UnitTypeLhs& lhs, const dimensionless::dB_t& rhs) noexcept
    {
        using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;
        return UnitTypeLhs(lhs.template toLinearized<underlying_type>() * rhs.template toLinearized<underlying_type>(), std::true_type());
    }
 
    /// Addition between unit_t types with a decibel_scale and dimensionless dB units
    template<class UnitTypeRhs, std::enable_if_t<traits::has_decibel_scale<UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
    constexpr inline UnitTypeRhs operator+(const dimensionless::dB_t& lhs, const UnitTypeRhs& rhs) noexcept
    {
        using underlying_type = typename units::traits::unit_t_traits<UnitTypeRhs>::underlying_type;
        return UnitTypeRhs(lhs.template toLinearized<underlying_type>() * rhs.template toLinearized<underlying_type>(), std::true_type());
    }
 
    /// Subtraction for convertible unit_t types with a decibel_scale
    template<class UnitTypeLhs, class UnitTypeRhs, std::enable_if_t<traits::has_decibel_scale<UnitTypeLhs, UnitTypeRhs>::value, int> = 0>
    constexpr inline auto operator-(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>, typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type, decibel_scale>
    {
        using LhsUnits = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
        using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;
 
        return unit_t<compound_unit<LhsUnits, inverse<RhsUnits>>, underlying_type, decibel_scale>
            (lhs.template toLinearized<underlying_type>() / convert<RhsUnits, LhsUnits>(rhs.template toLinearized<underlying_type>()), std::true_type());
    }
 
    /// Subtraction between unit_t types with a decibel_scale and dimensionless dB units
    template<class UnitTypeLhs, std::enable_if_t<traits::has_decibel_scale<UnitTypeLhs>::value && !traits::is_dimensionless_unit<UnitTypeLhs>::value, int> = 0>
    constexpr inline UnitTypeLhs operator-(const UnitTypeLhs& lhs, const dimensionless::dB_t& rhs) noexcept
    {
        using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;
        return UnitTypeLhs(lhs.template toLinearized<underlying_type>() / rhs.template toLinearized<underlying_type>(), std::true_type());
    }
 
    /// Subtraction between unit_t types with a decibel_scale and dimensionless dB units
    template<class UnitTypeRhs, std::enable_if_t<traits::has_decibel_scale<UnitTypeRhs>::value && !traits::is_dimensionless_unit<UnitTypeRhs>::value, int> = 0>
    constexpr inline auto operator-(const dimensionless::dB_t& lhs, const UnitTypeRhs& rhs) noexcept -> unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>, typename units::traits::unit_t_traits<UnitTypeRhs>::underlying_type, decibel_scale>
    {
        using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
        using underlying_type = typename units::traits::unit_t_traits<RhsUnits>::underlying_type;
 
        return unit_t<inverse<RhsUnits>, underlying_type, decibel_scale>
            (lhs.template toLinearized<underlying_type>() / rhs.template toLinearized<underlying_type>(), std::true_type());
    }
 
    //----------------------------------
    //  UNIT RATIO CLASS
    //----------------------------------
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        template<class Units>
        struct _unit_value_t {};
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    namespace traits
    {
#ifdef FOR_DOXYGEN_PURPOSES_ONLY
        /**
        * @ingroup      TypeTraits
        * @brief        Trait for accessing the publicly defined types of `units::unit_value_t_traits`
        * @details      The units library determines certain properties of the `unit_value_t` types passed to
        *               them and what they represent by using the members of the corresponding `unit_value_t_traits`
        *               instantiation.
        */
        template<typename T>
        struct unit_value_t_traits
        {
            typedef typename T::unit_type unit_type;    ///< Dimension represented by the `unit_value_t`.
            typedef typename T::ratio ratio;            ///< Quantity represented by the `unit_value_t`, expressed as arational number.
        };
#endif
 
        /** @cond */    // DOXYGEN IGNORE
        /**
         * @brief       unit_value_t_traits specialization for things which are not unit_t
         * @details
         */
        template<typename T, typename = void>
        struct unit_value_t_traits
        {
            typedef void unit_type;
            typedef void ratio;
        };
 
        /**
         * @ingroup     TypeTraits
         * @brief       Trait for accessing the publicly defined types of `units::unit_value_t_traits`
         * @details
         */
        template<typename T>
        struct unit_value_t_traits <T, typename void_t<
            typename T::unit_type,
            typename T::ratio>::type>
        {
            typedef typename T::unit_type unit_type;
            typedef typename T::ratio ratio;
        };
        /** @endcond */ // END DOXYGEN IGNORE
    }
 
    //------------------------------------------------------------------------------
    //  COMPILE-TIME UNIT VALUES AND ARITHMETIC
    //------------------------------------------------------------------------------
 
    /**
     * @ingroup     UnitContainers
     * @brief       Stores a rational unit value as a compile-time constant
     * @details     unit_value_t is useful for performing compile-time arithmetic on known
     *              unit quantities.
     * @tparam      Units   units represented by the `unit_value_t`
     * @tparam      Num     numerator of the represented value.
     * @tparam      Denom   denominator of the represented value.
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        This is intentionally identical in concept to a `std::ratio`.
     *
     */
    template<typename Units, std::uintmax_t Num, std::uintmax_t Denom = 1>
    struct unit_value_t : units::detail::_unit_value_t<Units>
    {
        typedef Units unit_type;
        typedef std::ratio<Num, Denom> ratio;
 
        static_assert(traits::is_unit<Units>::value, "Template parameter `Units` must be a unit type.");
        static constexpr const unit_t<Units> value() { return unit_t<Units>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den); }
    };
 
    namespace traits
    {
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether a type is a unit_value_t representing the given unit type.
         * @details     e.g. `is_unit_value_t<meters, myType>::value` would test that `myType` is a
         *              `unit_value_t<meters>`.
         * @tparam      Units   units that the `unit_value_t` is supposed to have.
         * @tparam      T       type to test.
         */
        template<typename T, typename Units = typename traits::unit_value_t_traits<T>::unit_type>
        struct is_unit_value_t : std::integral_constant<bool,
            std::is_base_of<units::detail::_unit_value_t<Units>, T>::value>
        {};
        template<typename T, typename Units = typename traits::unit_value_t_traits<T>::unit_type>
        inline constexpr bool is_unit_value_t_v = is_unit_value_t<T, Units>::value;
 
        /**
         * @ingroup     TypeTraits
         * @brief       Trait which tests whether type T is a unit_value_t with a unit type in the given category.
         * @details     e.g. `is_unit_value_t_category<units::category::length, unit_value_t<feet>>::value` would be true
         */
        template<typename Category, typename T>
        struct is_unit_value_t_category : std::integral_constant<bool,
            std::is_same<units::traits::base_unit_of<typename traits::unit_value_t_traits<T>::unit_type>, Category>::value>
        {
            static_assert(is_base_unit<Category>::value, "Template parameter `Category` must be a `base_unit` type.");
        };
        template<typename Category, typename T>
        inline constexpr bool is_unit_value_t_category_v = is_unit_value_t_category<Category, T>::value;
    }
 
    /** @cond */    // DOXYGEN IGNORE
    namespace detail
    {
        // base class for common arithmetic
        template<class U1, class U2>
        struct unit_value_arithmetic
        {
            static_assert(traits::is_unit_value_t<U1>::value, "Template parameter `U1` must be a `unit_value_t` type.");
            static_assert(traits::is_unit_value_t<U2>::value, "Template parameter `U2` must be a `unit_value_t` type.");
 
            using _UNIT1 = typename traits::unit_value_t_traits<U1>::unit_type;
            using _UNIT2 = typename traits::unit_value_t_traits<U2>::unit_type;
            using _CONV1 = typename units::traits::unit_traits<_UNIT1>::conversion_ratio;
            using _CONV2 = typename units::traits::unit_traits<_UNIT2>::conversion_ratio;
            using _RATIO1 = typename traits::unit_value_t_traits<U1>::ratio;
            using _RATIO2 = typename traits::unit_value_t_traits<U2>::ratio;
            using _RATIO2CONV = typename std::ratio_divide<std::ratio_multiply<_RATIO2, _CONV2>, _CONV1>;
            using _PI_EXP = std::ratio_subtract<typename units::traits::unit_traits<_UNIT2>::pi_exponent_ratio, typename units::traits::unit_traits<_UNIT1>::pi_exponent_ratio>;
        };
    }
    /** @endcond */ // END DOXYGEN IGNORE
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       adds two unit_value_t types at compile-time
     * @details     The resulting unit will the the `unit_type` of `U1`
     * @tparam      U1  left-hand `unit_value_t`
     * @tparam      U2  right-hand `unit_value_t`
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `std::ratio_add`
     */
    template<class U1, class U2>
    struct unit_value_add : units::detail::unit_value_arithmetic<U1, U2>, units::detail::_unit_value_t<typename traits::unit_value_t_traits<U1>::unit_type>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U2>;
        typedef typename Base::_UNIT1 unit_type;
        using ratio = std::ratio_add<typename Base::_RATIO1, typename Base::_RATIO2CONV>;
 
        static_assert(traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, "Unit types are not compatible.");
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of sum
         * @details     Returns the calculated value of the sum of `U1` and `U2`, in the same
         *              units as `U1`.
         * @returns     Value of the sum in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO1::num / Base::_RATIO1::den) +
            ((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO2CONV::num / Base::_RATIO2CONV::den) * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE
    };
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       subtracts two unit_value_t types at compile-time
     * @details     The resulting unit will the the `unit_type` of `U1`
     * @tparam      U1  left-hand `unit_value_t`
     * @tparam      U2  right-hand `unit_value_t`
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `std::ratio_subtract`
     */
    template<class U1, class U2>
    struct unit_value_subtract : units::detail::unit_value_arithmetic<U1, U2>, units::detail::_unit_value_t<typename traits::unit_value_t_traits<U1>::unit_type>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U2>;
 
        typedef typename Base::_UNIT1 unit_type;
        using ratio = std::ratio_subtract<typename Base::_RATIO1, typename Base::_RATIO2CONV>;
 
        static_assert(traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, "Unit types are not compatible.");
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of difference
         * @details     Returns the calculated value of the difference of `U1` and `U2`, in the same
         *              units as `U1`.
         * @returns     Value of the difference in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO1::num / Base::_RATIO1::den) - ((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO2CONV::num / Base::_RATIO2CONV::den)
                * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE   };
    };
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       multiplies two unit_value_t types at compile-time
     * @details     The resulting unit will the the `unit_type` of `U1 * U2`
     * @tparam      U1  left-hand `unit_value_t`
     * @tparam      U2  right-hand `unit_value_t`
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `std::ratio_multiply`
     */
    template<class U1, class U2>
    struct unit_value_multiply : units::detail::unit_value_arithmetic<U1, U2>,
        units::detail::_unit_value_t<typename std::conditional<traits::is_convertible_unit<typename traits::unit_value_t_traits<U1>::unit_type,
            typename traits::unit_value_t_traits<U2>::unit_type>::value, compound_unit<squared<typename traits::unit_value_t_traits<U1>::unit_type>>,
            compound_unit<typename traits::unit_value_t_traits<U1>::unit_type, typename traits::unit_value_t_traits<U2>::unit_type>>::type>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U2>;
 
        using unit_type = std::conditional_t<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, compound_unit<squared<typename Base::_UNIT1>>, compound_unit<typename Base::_UNIT1, typename Base::_UNIT2>>;
        using ratio = std::conditional_t<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, std::ratio_multiply<typename Base::_RATIO1, typename Base::_RATIO2CONV>, std::ratio_multiply<typename Base::_RATIO1, typename Base::_RATIO2>>;
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of product
         * @details     Returns the calculated value of the product of `U1` and `U2`, in units
         *              of `U1 x U2`.
         * @returns     Value of the product in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE
    };
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       divides two unit_value_t types at compile-time
     * @details     The resulting unit will the the `unit_type` of `U1`
     * @tparam      U1  left-hand `unit_value_t`
     * @tparam      U2  right-hand `unit_value_t`
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `std::ratio_divide`
     */
    template<class U1, class U2>
    struct unit_value_divide : units::detail::unit_value_arithmetic<U1, U2>,
        units::detail::_unit_value_t<typename std::conditional<traits::is_convertible_unit<typename traits::unit_value_t_traits<U1>::unit_type,
        typename traits::unit_value_t_traits<U2>::unit_type>::value, dimensionless::scalar, compound_unit<typename traits::unit_value_t_traits<U1>::unit_type,
        inverse<typename traits::unit_value_t_traits<U2>::unit_type>>>::type>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U2>;
 
        using unit_type = std::conditional_t<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, dimensionless::scalar, compound_unit<typename Base::_UNIT1, inverse<typename Base::_UNIT2>>>;
        using ratio = std::conditional_t<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, std::ratio_divide<typename Base::_RATIO1, typename Base::_RATIO2CONV>, std::ratio_divide<typename Base::_RATIO1, typename Base::_RATIO2>>;
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of quotient
         * @details     Returns the calculated value of the quotient of `U1` and `U2`, in units
         *              of `U1 x U2`.
         * @returns     Value of the quotient in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE
    };
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       raises unit_value_to a power at compile-time
     * @details     The resulting unit will the `unit_type` of `U1` squared
     * @tparam      U1  `unit_value_t` to take the exponentiation of.
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `units::math::pow`
     */
    template<class U1, int power>
    struct unit_value_power : units::detail::unit_value_arithmetic<U1, U1>, units::detail::_unit_value_t<typename units::detail::power_of_unit<power, typename traits::unit_value_t_traits<U1>::unit_type>::type>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U1>;
 
        using unit_type = typename units::detail::power_of_unit<power, typename Base::_UNIT1>::type;
        using ratio = typename units::detail::power_of_ratio<power, typename Base::_RATIO1>::type;
        using pi_exponent = std::ratio_multiply<std::ratio<power>, typename Base::_UNIT1::pi_exponent_ratio>;
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of exponentiation
         * @details     Returns the calculated value of the exponentiation of `U1`, in units
         *              of `U1^power`.
         * @returns     Value of the exponentiation in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)pi_exponent::num / pi_exponent::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE   };
    };
 
    /**
     * @ingroup     CompileTimeUnitManipulators
     * @brief       calculates square root of unit_value_t at compile-time
     * @details     The resulting unit will the square root `unit_type` of `U1`
     * @tparam      U1  `unit_value_t` to take the square root of.
     * @sa          unit_value_t_traits to access information about the properties of the class,
     *              such as it's unit type and rational value.
     * @note        very similar in concept to `units::ratio_sqrt`
     */
    template<class U1, std::intmax_t Eps = 10000000000>
    struct unit_value_sqrt : units::detail::unit_value_arithmetic<U1, U1>, units::detail::_unit_value_t<square_root<typename traits::unit_value_t_traits<U1>::unit_type, Eps>>
    {
        /** @cond */    // DOXYGEN IGNORE
        using Base = units::detail::unit_value_arithmetic<U1, U1>;
 
        using unit_type = square_root<typename Base::_UNIT1, Eps>;
        using ratio = ratio_sqrt<typename Base::_RATIO1, Eps>;
        using pi_exponent = ratio_sqrt<typename Base::_UNIT1::pi_exponent_ratio, Eps>;
        /** @endcond */ // END DOXYGEN IGNORE
 
        /**
         * @brief       Value of square root
         * @details     Returns the calculated value of the square root of `U1`, in units
         *              of `U1^1/2`.
         * @returns     Value of the square root in the appropriate units.
         */
        static constexpr const unit_t<unit_type> value() noexcept
        {
            using UsePi = std::integral_constant<bool, Base::_PI_EXP::num != 0>;
            return value(UsePi());
        }
 
        /** @cond */    // DOXYGEN IGNORE
        // value if PI isn't involved
        static constexpr const unit_t<unit_type> value(std::false_type) noexcept
        {
            return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
        }
 
        // value if PI *is* involved
        static constexpr const unit_t<unit_type> value(std::true_type) noexcept
        {
            return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::detail::PI_VAL, ((UNIT_LIB_DEFAULT_TYPE)pi_exponent::num / pi_exponent::den)));
        }
        /** @endcond */ // END DOXYGEN IGNORE
    };
 
    //----------------------------------
    //  UNIT-ENABLED CMATH FUNCTIONS
    //----------------------------------
 
    /**
     * @brief       namespace for unit-enabled versions of the `<cmath>` library
     * @details     Includes trigonometric functions, exponential/log functions, rounding functions, etc.
     * @sa          See `unit_t` for more information on unit type containers.
     */
    namespace math
    {
 
        //----------------------------------
        //  MIN/MAX FUNCTIONS
        //----------------------------------
        // XXX: min/max are defined here instead of math.h to avoid a conflict with
        // the "_min" user-defined literal in time.h.
 
        template<class UnitTypeLhs, class UnitTypeRhs>
        UnitTypeLhs (min)(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
        {
            static_assert(traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value, "Unit types are not compatible.");
            UnitTypeLhs r(rhs);
            return (lhs < r ? lhs : r);
        }
 
        template<class UnitTypeLhs, class UnitTypeRhs>
        UnitTypeLhs (max)(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
        {
            static_assert(traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value, "Unit types are not compatible.");
            UnitTypeLhs r(rhs);
            return (lhs > r ? lhs : r);
        }
    }
}
 
#ifdef _MSC_VER
#   if _MSC_VER <= 1800
#       pragma warning(pop)
#       undef constexpr
#       pragma pop_macro("constexpr")
#       undef noexcept
#       pragma pop_macro("noexcept")
#       undef _ALLOW_KEYWORD_MACROS
#   endif // _MSC_VER < 1800
#   pragma pop_macro("pascal")
#endif // _MSC_VER
 
#if defined(UNIT_HAS_LITERAL_SUPPORT)
namespace units::literals {}
using namespace units::literals;
#endif  // UNIT_HAS_LITERAL_SUPPORT
 
#if __has_include(<fmt/format.h>) && !defined(UNIT_LIB_DISABLE_FMT)
#include "units/formatter.h"
#endif