forked from mpusz/mp-units
716 lines
25 KiB
Markdown
716 lines
25 KiB
Markdown
# `mp-units` - A Units Library for C++
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## Summary
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`Units` is a compile-time enabled Modern C++ library that provides compile-time dimensional
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analysis and unit/quantity manipulation. The basic idea and design heavily bases on
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`std::chrono::duration` and extends it to work properly with many dimensions.
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Here is a small example of possible operations:
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```cpp
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// simple numeric operations
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static_assert(10km / 2 == 5km);
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// unit conversions
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static_assert(1h == 3600s);
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static_assert(1km + 1m == 1001m);
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// dimension conversions
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static_assert(1km / 1s == 1000mps);
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static_assert(2kmph * 2h == 4km);
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static_assert(2km / 2kmph == 1h);
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static_assert(1000 / 1s == 1kHz);
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static_assert(10km / 5km == 2);
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```
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## Approach
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1. Safety and performance
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- strong types
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- compile-time safety
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- `constexpr` all the things
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- as fast or even faster than when working with fundamental types
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2. The best possible user experience
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- compiler errors
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- debugging
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3. No macros in the user interface
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4. Easy extensibility
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5. No external dependencies
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6. Possibility to be standardized as a freestanding part of the C++ Standard Library
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## Overview
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The library framework consists of a few concepts: quantities, units, dimensions and their exponents. From the user's
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point of view the most important is a quantity.
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Quantity is a concrete amount of a unit for a specified dimension with a specific representation:
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```cpp
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units::quantity<units::kilometre, double> d1(123);
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auto d2 = 123km; // units::quantity<units::kilometre, std::int64_t>
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```
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There are C++ concepts provided for each such quantity type:
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```cpp
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template<typename T>
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concept Length = QuantityOf<T, length>;
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```
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With that we can easily write a function template like this:
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```cpp
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constexpr units::Velocity auto avg_speed(units::Length auto d, units::Time auto t)
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{
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return d / t;
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}
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```
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## Basic Concepts
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Below UML diagram shows the most important entities in the library design and how they relate to
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each other:
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### `Dimensions`
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`units::dimension` represents a derived dimension and is implemented as a type-list like type that
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stores an ordered list of exponents of one or more base dimensions:
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```cpp
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template<Exponent... Es>
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struct dimension : downcast_base<dimension<Es...>> {};
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```
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`units::Dimension` is a concept that is satisfied by a type that is empty and publicly
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derived from `units::dimension` class template:
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```cpp
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template<typename T>
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concept Dimension =
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std::is_empty_v<T> &&
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detail::is_dimension<downcast_base_t<T>>; // exposition only
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```
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#### `Exponents`
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`units::Exponent` concept is satisfied if provided type is an instantiation of `units::exp` class
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template:
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```cpp
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template<typename T>
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concept Exponent =
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detail::is_exp<T>; // exposition only
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```
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`units::exp` provides an information about a single dimension and its (possibly fractional)
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exponent in a derived dimension.
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```cpp
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template<typename Dim, int Num, int Den = 1>
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requires BaseDimension<Dim> || Dimension<Dim>
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struct exp {
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using dimension = Dim;
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static constexpr int num = Num;
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static constexpr int den = Den;
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};
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```
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Both a base dimension and a derived dimension can be provided to `units::exp` class template.
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`units::base_dimension` represents a base dimension and has assigned a unique compile-time text
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describing the dimension name:
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```cpp
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template<typename Name, typename Symbol>
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struct base_dimension {
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using name = Name;
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using symbol = Symbol;
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};
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```
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`units::BaseDimension` is a concept to match all types derived from `base_dimension` instantiations:
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```cpp
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template<typename T>
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concept BaseDimension = std::is_empty_v<T> &&
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requires {
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typename T::name;
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typename T::symbol;
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} &&
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std::derived_from<T, base_dimension<typename T::name, typename T::symbol>>;
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```
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For example here is a list of SI base dimensions:
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```cpp
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struct base_dim_length : base_dimension<"length", "m"> {};
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struct base_dim_mass : base_dimension<"mass", "kg"> {};
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struct base_dim_time : base_dimension<"time", "s"> {};
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struct base_dim_current : base_dimension<"current", "A"> {};
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struct base_dim_temperature : base_dimension<"temperature", "K"> {};
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struct base_dim_substance : base_dimension<"substance", "mol"> {};
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struct base_dim_luminous_intensity : base_dimension<"luminous intensity", "cd"> {};
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```
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#### `derived_dimension`
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Above design of dimensions is created with the ease of use for end users in mind. Compile-time
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errors should provide as short as possible template instantiations strings that should be easy to
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understand by every engineer. Also types visible in a debugger should be easy to understand.
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That is why `units::dimension` type for derived dimensions always stores information about only
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those base dimensions that are used to form that derived dimension.
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However, such an approach have some challenges:
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```cpp
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constexpr Velocity auto v1 = 1_m / 1s;
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constexpr Velocity auto v2 = 2 / 2s * 1m;
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static_assert(std::same_as<decltype(v1), decltype(v2)>);
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static_assert(v1 == v2);
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```
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Above code, no matter what is the order of the base dimensions in an expression forming our result,
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must produce the same `Velocity` type so that both values can be easily compared. In order to achieve
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that, `dimension` class templates should never be instantiated manually but through a `derived_dimension`
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helper:
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```cpp
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template<typename Child, Exponent... Es>
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struct derived_dimension;
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```
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`Child` class template parameter is a part of a CRTP idiom and is used to provide a downcasting facility
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described later in this document.
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So for example to create a `velocity` type we have to do:
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```cpp
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struct velocity : derived_dimension<velocity, exp<base_dim_length, 1>, exp<base_dim_time, -1>> {};
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```
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In order to make `derived_dimension` work as expected it has to provide unique ordering for
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contained base dimensions. Beside providing ordering to base dimensions it also has to:
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- aggregate two arguments of the same base dimension but different exponents
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- eliminate two arguments of the same base dimension and with opposite equal exponents
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`derived_dimension` is also able to form a dimension type based not only on base dimensions but
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it can take other derived dimensions as well. So for some more complex dimensions user can
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type either:
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```cpp
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struct pressure : derived_dimension<pressure, exp<base_dim_mass, 1>, exp<base_dim_length, -1>, exp<base_dim_time, -2>> {};
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```
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or
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```cpp
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struct pressure : derived_dimension<pressure, exp<force, 1>, exp<area, -1>> {};
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```
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In the second case `derived_dimension` will extract all derived dimensions into the list of
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exponents of base dimensions. Thanks to that both cases will result with exactly the same base
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class formed only from the exponents of base units.
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#### `merge_dimension`
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`units::merge_dimension` is a type alias that works similarly to `derived_dimension` but instead
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of sorting the whole list of base dimensions from scratch it assumes that provided input `dimension`
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types are already sorted as a result of `derived_dimension`. Also contrary to `derived_dimension`
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it works only with exponents of bas dimensions (no derived dimensions allowed).
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Typical use case for `merge_dimension` is to produce final `dimension` return type of multiplying
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two different dimensions:
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```cpp
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template<Dimension D1, Dimension D2>
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struct dimension_multiply;
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template<Exponent... E1, Exponent... E2>
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struct dimension_multiply<dimension<E1...>, dimension<E2...>> {
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using type = downcast_traits_t<merge_dimension_t<dimension<E1...>, dimension<E2...>>>;
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};
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template<Dimension D1, Dimension D2>
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using dimension_multiply = dimension_multiply<typename D1::base_type, typename D2::base_type>::type;
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```
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### `Units`
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`units::unit` is a class template that expresses the unit of a specific physical dimension:
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```cpp
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template<Dimension D, Ratio R>
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requires (R::num * R::den > 0)
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struct unit : downcast_base<unit<D, R>> {
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using dimension = D;
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using ratio = R;
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};
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```
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All units are created with a `derived_unit` helper:
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```cpp
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template<typename Child, fixed_string Symbol, typename...>
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struct derived_unit;
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```
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For example to define the base unit of `length`:
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```cpp
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struct metre : derived_unit<metre, "m", length> {};
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```
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Again, similarly to `derived_dimension`, the first class template parameter is a CRTP idiom used
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to provide downcasting facility (described below).
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`derived_unit` has a few partial useful specializations:
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- helper to create a base unit of a specified dimension
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```cpp
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template<typename Child, fixed_string Symbol, Dimension D>
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struct derived_unit<Child, Symbol, D>;
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```
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```cpp
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struct metre : derived_unit<metre, "m", length> {};
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```
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- helper to create other units for a specified dimension
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```cpp
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template<typename Child, fixed_string Symbol, Dimension D, Ratio R>
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struct derived_unit<Child, Symbol, D, R>;
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```
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```cpp
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struct yard : derived_unit<yard, "yd", length, ratio<9'144, 10'000>> {};
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```
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- helper to create a unit with a SI prefix
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```cpp
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template<typename Child, fixed_string Symbol, Unit U>
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struct derived_unit<Child, Symbol, U>;
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```
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```cpp
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struct kilometre : derived_unit<kilometre, "km", kilo<metre>> {};
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```
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- helper that automatically calculates ratio based on info in desired dimension and provided list
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of units of base dimensions
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```cpp
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template<typename Child, fixed_string Symbol, Dimension D, Unit U, Unit... Us>
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struct derived_unit<Child, Symbol, D, U, Us...>;
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```
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```cpp
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struct kilometre_per_hour : derived_unit<kilometre_per_hour, "km/h", velocity, kilometre, hour> {};
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```
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`units::Unit` is a concept that is satisfied by a type that is empty and publicly
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derived from `units::unit` class template:
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```cpp
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template<typename T>
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concept Unit =
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std::is_empty_v<T> &&
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detail::is_unit<downcast_base_t<T>>; // exposition only
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```
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### `Quantities`
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`units::quantity` is a class template that expresses the quantity/amount of a specific dimension
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expressed in a specific unit of that dimension:
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```cpp
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template<Unit U, Scalar Rep = double>
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class quantity;
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```
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`units::Quantity` is a concept that is satisfied by a type that is an instantiation of `units::quantity`
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class template:
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```cpp
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template<typename T>
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concept Quantity =
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detail::is_quantity<T>; // exposition only
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```
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`units::quantity` provides the interface really similar to `std::chrono::duration`. The difference is that
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it uses `double` as a default representation and has a few additional member types and functions as below:
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```cpp
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template<Unit U, Scalar Rep = double>
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class quantity {
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public:
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using unit = U;
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using rep = Rep;
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using dimension = U::dimension;
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[[nodiscard]] static constexpr quantity one() noexcept { return quantity(quantity_values<Rep>::one()); }
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template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
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requires std::same_as<typename U1::dimension, dim_invert<typename U2::dimension>>
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[[nodiscard]] constexpr Scalar operator*(const quantity<U1, Rep1>& lhs,
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const quantity<U2, Rep2>& rhs);
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template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
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requires (!std::same_as<typename U1::dimension, dim_invert<typename U2::dimension>>) &&
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(treat_as_floating_point<decltype(lhs.count() * rhs.count())> ||
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(std::ratio_multiply<typename U1::ratio, typename U2::ratio>::den == 1))
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[[nodiscard]] constexpr Quantity operator*(const quantity<U1, Rep1>& lhs,
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const quantity<U2, Rep2>& rhs);
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template<Scalar Rep1, typename U, typename Rep2>
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[[nodiscard]] constexpr Quantity operator/(const Rep1& v,
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const quantity<U, Rep2>& q);
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template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
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requires std::same_as<typename U1::dimension, typename U2::dimension>
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[[nodiscard]] constexpr Scalar operator/(const quantity<U1, Rep1>& lhs,
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const quantity<U2, Rep2>& rhs);
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template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
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requires (!std::same_as<typename U1::dimension, typename U2::dimension>) &&
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(treat_as_floating_point<decltype(lhs.count() / rhs.count())> ||
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(ratio_divide<typename U1::ratio, typename U2::ratio>::den == 1))
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[[nodiscard]] constexpr Quantity operator/(const quantity<U1, Rep1>& lhs,
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const quantity<U2, Rep2>& rhs);
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// ...
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};
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```
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Additional functions provide the support for operations that result in a different dimension type
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than those of their arguments.
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Beside adding new elements a few other changes where applied compared to the `std::chrono::duration` class:
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1. The `duration` is using `std::common_type_t<Rep1, Rep2>` to find a common representation
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for a calculation result. Such a design was reported as problematic by numerics study group members
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as sometimes we want to provide a different type in case of multiplication and different in case of
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division. `std::common_type` lacks that additional information. That is why `units::quantity` uses
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the resulting type of a concrete operator operation and provides it directly to `units::common_quantity_t`
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type trait.
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2. `operator %` is constrained with `treat_as_floating_point` type trait to limit the types to integral
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representations only. Also `operator %(Rep)` takes `Rep` as a template argument to limit implicit
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conversions.
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#### `quantity_cast`
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To explicitly force truncating conversions `quantity_cast` function is provided which is a direct
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counterpart of `std::chrono::duration_cast`. As a template argument user can provide here either
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a `quantity` type or only its template parameters (`Unit`, `Rep`):
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```cpp
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template<Quantity To, typename U, typename Rep>
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requires same_dim<typename To::dimension, typename U::dimension>
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[[nodiscard]] constexpr To quantity_cast(const quantity<U, Rep>& q);
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template<Unit ToU, Scalar ToRep, typename U, typename Rep>
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[[nodiscard]] constexpr quantity<ToU, ToRep> quantity_cast(const quantity<U, Rep>& q);
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template<Unit ToU, typename U, typename Rep>
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[[nodiscard]] constexpr quantity<ToU, Rep> quantity_cast(const quantity<U, Rep>& q);
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template<Scalar ToRep, typename U, typename Rep>
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[[nodiscard]] constexpr quantity<U, ToRep> quantity_cast(const quantity<U, Rep>& q);
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```
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## Strong types instead of aliases, and type downcasting facility
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Most of the important design decisions in the library are dictated by the requirement of providing
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the best user experience as possible.
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For example with template aliases usage the following code:
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```cpp
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const Velocity auto t = 20s;
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```
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could generate a following compile time error:
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```text
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<path>\example\example.cpp:39:22: error: deduced initializer does not satisfy placeholder constraints
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const Velocity auto t = 20s;
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^~~~
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In file included from <path>\example\example.cpp:23:
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<path>/src/include/units/si/velocity.h:41:16: note: within 'template<class T> concept const bool units::Velocity<T> [with T = units::quantity<units::unit<units::dimension<units::exp<units::base_dim_time, 1> >, std::ratio<1> >, long long int>]'
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concept Velocity = Quantity<T> && std::same_as<typename T::dimension, velocity>;
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^~~~~~~~
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In file included from <path>/src/include/units/bits/tools.h:25,
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from <path>/src/include/units/dimension.h:25,
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from <path>/src/include/units/si/base_dimensions.h:25,
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from <path>/src/include/units/si/velocity.h:25,
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from <path>\example\example.cpp:23:
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<path>/src/include/units/bits/stdconcepts.h:33:18: note: within 'template<class T, class U> concept const bool std::same_as<T, U> [with T = units::dimension<units::exp<units::base_dim_time, 1> >; U = units::dimension<units::exp<units::base_dim_length, 1>,units::exp<units::base_dim_time, -1> >]'
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concept same_as = std::is_same_v<T, U>;
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^~~~
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<path>/src/include/units/bits/stdconcepts.h:33:18: note: 'std::is_same_v' evaluated to false
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```
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Time and velocity are not that complicated dimensions and there are much more complicated dimensions
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out there, but even for those dimensions
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```text
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[with T = units::quantity<units::unit<units::dimension<units::exp<units::base_dim_time, 1> >, std::ratio<1> >, long long int>]
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```
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and
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```text
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[with T = units::dimension<units::exp<units::base_dim_time, 1> >; U = units::dimension<units::exp<units::base_dim_length, 1>,units::exp<units::base_dim_time, -1> >]
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```
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starts to be really hard to analyze or debug.
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That is why it was decided to provide automated downcasting capability when possible. Thanks to this feature the
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same code will result with such an error:
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```text
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<path>\example\example.cpp:40:22: error: deduced initializer does not satisfy placeholder constraints
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const Velocity t = 20s;
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^~~~
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In file included from <path>\example\example.cpp:23:
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<path>/src/include/units/si/velocity.h:48:16: note: within 'template<class T> concept const bool units::Velocity<T> [with T = units::quantity<units::second, long long int>]'
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concept Velocity = Quantity<T> && std::same_as<typename T::dimension, velocity>;
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^~~~~~~~
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In file included from <path>/src/include/units/bits/tools.h:25,
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from <path>/src/include/units/dimension.h:25,
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from <path>/src/include/units/si/base_dimensions.h:25,
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from <path>/src/include/units/si/velocity.h:25,
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from <path>\example\example.cpp:23:
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<path>/src/include/units/bits/stdconcepts.h:33:18: note: within 'template<class T, class U> concept const bool std::same_as<T, U> [with T = units::time; U = units::velocity]'
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concept same_as = std::is_same_v<T, U>;
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^~~~
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<path>/src/include/units/bits/stdconcepts.h:33:18: note: 'std::is_same_v' evaluated to false
|
||
```
|
||
|
||
Now
|
||
|
||
```text
|
||
[with T = units::quantity<units::second, long long int>]
|
||
```
|
||
|
||
and
|
||
|
||
```text
|
||
[with T = units::time; U = units::velocity]
|
||
```
|
||
|
||
are not arguably much easier to understand thus provide better user experience.
|
||
|
||
Downcasting facility provides a type substitution mechanism. It connects a specific primary template
|
||
class specialization with a strong type assigned to it by the user. A simplified mental model of the
|
||
facility may be represented as:
|
||
|
||
```cpp
|
||
struct metre : unit<dimension<exp<base_dim_length, 1>>, std::ratio<1, 1, 0>>;
|
||
```
|
||
|
||
In the above example `metre` is a downcasting target (child class) and a specific `unit` class
|
||
template specialization is a downcasting source (base class). The downcasting facility provides
|
||
1 to 1 tpe substitution mechanism. Only one child class can be created for a specific base class
|
||
template instantiation.
|
||
|
||
Downcasting facility is provided through 2 dedicated types, a concept, and a few helper template aliases.
|
||
|
||
```cpp
|
||
template<typename BaseType>
|
||
struct downcast_base {
|
||
using base_type = BaseType;
|
||
friend auto downcast_guide(downcast_base);
|
||
};
|
||
```
|
||
|
||
`units::downcast_base` is a class that implements CRTP idiom, marks the base of downcasting
|
||
facility with a `base_type` member type, and provides a declaration of downcasting ADL friendly
|
||
(Hidden Friend) entry point member function `downcast_guide`. An important design point is that
|
||
this function does not return any specific type in its declaration. This non-member function
|
||
is going to be defined in a child class template `downcast_helper` and will return a target
|
||
type of the downcasting operation there.
|
||
|
||
```cpp
|
||
template<typename T>
|
||
concept Downcastable =
|
||
requires {
|
||
typename T::base_type;
|
||
} &&
|
||
std::derived_from<T, downcast_base<typename T::base_type>>;
|
||
```
|
||
|
||
`units::Downcastable` is a concepts that verifies if a type implements and can be used in a downcasting
|
||
facility.
|
||
|
||
```cpp
|
||
template<typename Target, Downcastable T>
|
||
struct downcast_helper : T {
|
||
friend auto downcast_guide(typename downcast_helper::downcast_base) { return Target(); }
|
||
};
|
||
```
|
||
|
||
`units::downcast_helper` is another CRTP class template that provides the implementation of a
|
||
non-member friend function of the `downcast_base` class template which defines the target
|
||
type of a downcasting operation. It is used in the following way to define `dimension` and
|
||
`unit` types in the library:
|
||
|
||
```cpp
|
||
template<typename Child, Exponent... Es>
|
||
struct derived_dimension : downcast_helper<Child, detail::make_dimension_t<Es...>> {};
|
||
```
|
||
|
||
```cpp
|
||
template<typename Child, fixed_string Symbol, Dimension D>
|
||
struct derived_unit<Child, Symbol, D, R> : downcast_helper<Child, unit<D, ratio<1>>> {};
|
||
```
|
||
|
||
With such CRTP types the only thing the user has to do to register a new type to the downcasting
|
||
facility is to publicly derive from one of those CRTP types and provide its new child type as
|
||
the first template parameter of the CRTP type.
|
||
|
||
```cpp
|
||
struct metre : derived_unit<metre, "m", length> {};
|
||
```
|
||
|
||
Above types are used to define base and target of a downcasting operation. To perform the actual
|
||
downcasting operation a dedicated template alias is provided:
|
||
|
||
```cpp
|
||
template<Downcastable T>
|
||
using downcast_target = decltype(detail::downcast_target_impl<T>());
|
||
```
|
||
|
||
`units::downcast_target` is used to obtain the target type of the downcasting operation registered
|
||
for a given specialization in a base type.
|
||
|
||
For example to determine a downcasted type of a quantity multiply operation the following can be done:
|
||
|
||
```cpp
|
||
using dim = dimension_multiply<typename U1::dimension, typename U2::dimension>;
|
||
using common_rep = decltype(lhs.count() * rhs.count());
|
||
using ret = quantity<downcast_target<unit<dim, ratio_multiply<typename U1::ratio, typename U2::ratio>>>, common_rep>;
|
||
```
|
||
|
||
`detail::downcast_target_impl` checks if a downcasting target is registered for the specific base class.
|
||
If yes, it returns the registered type, otherwise it works like a regular identity type returning
|
||
a provided base class.
|
||
|
||
```cpp
|
||
namespace detail {
|
||
|
||
template<typename T>
|
||
concept has_downcast = requires {
|
||
downcast_guide(std::declval<downcast_base<T>>());
|
||
};
|
||
|
||
template<typename T>
|
||
constexpr auto downcast_target_impl()
|
||
{
|
||
if constexpr(has_downcast<T>)
|
||
return decltype(downcast_guide(std::declval<downcast_base<T>>()))();
|
||
else
|
||
return T();
|
||
}
|
||
|
||
}
|
||
```
|
||
|
||
Additionally there is on more simple helper alias provided that is used in the internal
|
||
library implementation:
|
||
|
||
```cpp
|
||
template<Downcastable T>
|
||
using downcast_base_t = T::base_type;
|
||
```
|
||
|
||
|
||
## Adding custom dimensions and units
|
||
|
||
In order to extend the library with custom dimensions the user has to:
|
||
1. Create a new base dimension if the predefined ones are not enough to form a new derived dimension:
|
||
|
||
```cpp
|
||
struct base_dim_digital_information : units::base_dimension<"digital information", "b"> {};
|
||
```
|
||
|
||
2. Create a new dimension type with the recipe of how to construct it from base dimensions and
|
||
register it for a downcasting facility:
|
||
|
||
```cpp
|
||
struct digital_information : units::derived_dimension<digital_information, units::exp<base_dim_digital_information, 1>> {};
|
||
```
|
||
|
||
3. Define a concept that will match a new dimension:
|
||
|
||
```cpp
|
||
template<typename T>
|
||
concept DigitalInformation = units::QuantityOf<T, digital_information>;
|
||
```
|
||
|
||
4. Define units and register them to a downcasting facility:
|
||
|
||
```cpp
|
||
struct bit : units::derived_unit<bit, "b", digital_information> {};
|
||
struct byte : units::derived_unit<byte, "B", digital_information, units::ratio<8>> {};
|
||
```
|
||
|
||
5. Provide user-defined literals for the most important units:
|
||
|
||
```cpp
|
||
inline namespace literals {
|
||
constexpr auto operator""_b(unsigned long long l) { return units::quantity<bit, std::int64_t>(l); }
|
||
constexpr auto operator""_b(long double l) { return units::quantity<bit, long double>(l); }
|
||
|
||
constexpr auto operator""_B(unsigned long long l) { return units::quantity<byte, std::int64_t>(l); }
|
||
constexpr auto operator""_B(long double l) { return units::quantity<byte, long double>(l); }
|
||
}
|
||
```
|
||
|
||
|
||
## Open questions
|
||
|
||
1. Should we ensure that dimension is always a result of `derived_dimension` and unit is a result
|
||
of `derived_unit`? How to do it?
|
||
|
||
2. What to do with `std::chrono::duration`?
|
||
|
||
3. Should we provide `seconds<int>` or stay with `quantity<second, int>`?
|
||
|
||
4. What is the best way to add support for temperatures?
|
||
|
||
Temperature absolute values not only require `std::ratio` but also should be adjusted/shifted
|
||
by some constant values (i.e. [°C] = [K] − 273.15). Relative temperatures does need an offset.
|
||
Users will most probably have problems with differentiating those two. Maybe the best solution
|
||
is to provide only `K` support in quantity and provide non-member helper conversion functions
|
||
with verbose names to convert to `°C` and `°C`?
|
||
|
||
5. Do we need a non-linear scale?
|
||
|
||
6. Should we provide integral UDLs or just leave floating point ones?
|
||
|
||
7. Should we provide support for dimensionless quantities?
|
||
|
||
Because dimensionless quantities have no associated units, they behave as normal scalars,
|
||
and allow implicit conversion to and from the underlying value type or types that are
|
||
convertible to/from that value type.
|
||
|
||
8. Should we standardize accompany tools (downcasting facility, `type_list` operations, `common_ratio`, etc)?
|
||
|
||
9. `k`, `K`, `W`, `F` UDLs conflict with gcc GNU extensions (<https://gcc.gnu.org/onlinedocs/gcc-4.3.0/gcc/Fixed_002dPoint.html>)
|
||
for floating point types.
|
||
|
||
10. `J` imaginary constants are a GCC extension
|
||
|
||
11. Do we need custom/multiple systems?
|