mp-units/doc/DESIGN.md

mp-units - A Units Library for C++

Summary

Units is a compile-time enabled Modern C++ library that provides compile-time dimensional analysis and unit/quantity manipulation. The basic idea and design heavily bases on std::chrono::duration and extends it to work properly with many dimensions.

Here is a small example of possible operations:

// simple numeric operations
static_assert(10km / 2 == 5km);

// unit conversions
static_assert(1h == 3600s);
static_assert(1km + 1m == 1001m);

// dimension conversions
static_assert(1km / 1s == 1000mps);
static_assert(2kmph * 2h == 4km);
static_assert(2km / 2kmph == 1h);

static_assert(1000 / 1s == 1kHz);

static_assert(10km / 5km == 2);

Approach

1. Safety and performance
   • strong types
   • compile-time safety
   • constexpr all the things
2. The best possible user experience
   • compiler errors
   • debugging
3. No macros in the user interface
4. Easy extensibility
5. No external dependencies
6. Possibility to be standardized as a freestanding part of the C++ Standard Library

Overview

The library framework consists of a few concepts: quantities, units, dimensions and their exponents. From the user's point of view the most important is a quantity.

Quantity is a concrete amount of a unit for a specified dimension with a specific representation:

units::quantity<units::kilometre, double> d1(123);
auto d2 = 123km;    // stde::units::quantity<units::kilometre, std::int64_t>

There are C++ concepts provided for each such quantity type:

template<typename T>
concept Length = QuantityOf<T, length>;

With that we can easily write a function template like this:

constexpr units::Velocity auto avg_speed(units::Length auto d, units::Time auto t)
{
  return d / t;
}

Basic Concepts

Dimensions

units::dimension is a type-list like type that stores an ordered list of exponents of one or more base dimensions:

template<Exponent... Es>
struct dimension : downcast_base<dimension<Es...>> {};

units::Dimension is a Concept that is satisfied by a type that is empty and publicly derived from units::dimension class template:

template<typename T>
concept Dimension =
    std::is_empty_v<T> &&
    detail::is_dimension<downcast_from<T>>; // exposition only

Exponents

units::exp provides an information about a single base dimension and its (possibly fractional) exponent in a derived dimension:

template<const base_dimension& BaseDimension, int Num, int Den = 1>
struct exp {
  static constexpr const base_dimension& dimension = BaseDimension;
  static constexpr int num = Num;
  static constexpr int den = Den;
};

where BaseDimension is a unique sortable compile-time value:

struct base_dimension {
  const char* name;
};
constexpr bool operator==(const base_dimension& lhs, const base_dimension& rhs);
constexpr bool operator<(const base_dimension& lhs, const base_dimension& rhs);

units::Exponent concept is satisfied if provided type is an instantiation of units::exp class template:

template<typename T>
concept Exponent =
    detail::is_exp<T>;  // exposition only

make_dimension

Above design of dimensions is created with the ease of use for end users in mind. Compile-time errors should provide as short as possible template instantiations strings that should be easy to understand by every engineer. Also types visible in a debugger should be easy to understand. That is why units::dimension type for derived dimensions always stores information about only those base dimensions that are used to form that derived dimension.

However, such an approach have some challenges:

constexpr Velocity auto v1 = 1_m / 1s;
constexpr Velocity auto v2 = 2 / 2s * 1m;

static_assert(std::same_as<decltype(v1), decltype(v2)>);
static_assert(v1 == v2);

Above code, no matter what is the order of the base dimensions in an expression forming our result, must produce the same Velocity type so that both values can be easily compared. In order to achieve that, dimension class templates should never be instantiated manually but through a make_dimension_t template metaprogramming factory function:

template<Exponent... Es>
struct make_dimension {
  using type = /* unspecified */;
};

template<Exponent... Es>
using make_dimension_t = make_dimension<Es...>::type;

So for example to create a velocity type we have to do:

struct velocity : make_dimension_t<exp<base_dim_length, 1>, exp<base_dim_time, -1>> {};

In order to make make_dimension_t work as expected it has to provide unique ordering for contained base dimensions. Beside providing ordering to base dimensions it also has to:

• aggregate two arguments of the same base dimension but different exponents
• eliminate two arguments of the same base dimension and with opposite equal exponents

merge_dimension

units::merge_dimension is similar to make_dimension but instead of sorting the whole list of base dimensions from scratch it assumes that provided input dimension types are already sorted as a result of make_dimension.

Typical use case for merge_dimension is to produce final dimension return type of multiplying two different dimensions:

template<Dimension D1, Dimension D2>
struct dimension_multiply;

template<Exponent... E1, Exponent... E2>
struct dimension_multiply<dimension<E1...>, dimension<E2...>> {
  using type = downcasting_traits_t<merge_dimension_t<dimension<E1...>, dimension<E2...>>>;
};

template<Dimension D1, Dimension D2>
using dimension_multiply_t = dimension_multiply<typename D1::base_type, typename D2::base_type>::type;

Example implementation of merge_dimension may look like:

template<Dimension D1, Dimension D2>
struct merge_dimension {
  using type = detail::dim_consolidate_t<mp::type_list_merge_sorted_t<D1, D2, exp_dim_id_less>>;
};

Units

units::unit is a class template that expresses the unit of a specific physical dimension:

template<Dimension D, Ratio R>
  requires (R::num * R::den > 0)
struct unit : downcast_base<unit<D, R>> {
  using dimension = D;
  using ratio = R;
};

For example to define the base unit of length:

struct metre : unit<length> {};

Also there are few alias templates provided as convenience helpers to simplify Ratio handling:

• units with prefixes

struct kilometre : kilo<metre> {};

• derived units

struct kilometre_per_hour : derived_unit<velocity, kilometre, hour> {};

units::Unit is a Concept that is satisfied by a type that is empty and publicly derived from units::unit class template:

template<typename T>
concept Unit =
    std::is_empty_v<T> &&
    detail::is_unit<downcast_from<T>>;  // exposition only

Quantities

units::quantity is a class template that expresses the quantity/amount of a specific dimension expressed in a specific unit of that dimension:

template<Unit U, Scalar Rep>
class quantity;

units::Quantity is a Concept that is satisfied by a type that is an instantiation of units::quantity class template:

template<typename T>
concept Quantity =
    detail::is_quantity<T>;  // exposition only

units::quantity provides the interface really similar to std::chrono::duration. The difference is that it uses double as a default representation and has a few additional member types and functions as below:

template<Unit U, Scalar Rep = double>
class quantity {
public:
  using unit = U;
  using rep = Rep;
  using dimension = U::dimension;

  [[nodiscard]] static constexpr quantity one() noexcept { return quantity(quantity_values<Rep>::one()); }

  template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
      requires std::same_as<typename U1::dimension, dim_invert_t<typename U2::dimension>>
  [[nodiscard]] constexpr Scalar operator*(const quantity<U1, Rep1>& lhs,
                                           const quantity<U2, Rep2>& rhs);

  template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
      requires (!std::same_as<typename U1::dimension, dim_invert_t<typename U2::dimension>>) &&
               (treat_as_floating_point<decltype(lhs.count() * rhs.count())> ||
                (std::ratio_multiply<typename U1::ratio, typename U2::ratio>::den == 1))
  [[nodiscard]] constexpr Quantity operator*(const quantity<U1, Rep1>& lhs,
                                             const quantity<U2, Rep2>& rhs);

  template<Scalar Rep1, typename U, typename Rep2>
  [[nodiscard]] constexpr Quantity operator/(const Rep1& v,
                                             const quantity<U, Rep2>& q);

  template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
    requires std::same_as<typename U1::dimension, typename U2::dimension>
  [[nodiscard]] constexpr Scalar operator/(const quantity<U1, Rep1>& lhs,
                                           const quantity<U2, Rep2>& rhs);

  template<Unit U1, Scalar Rep1, Unit U2, Scalar Rep2>
    requires (!std::same_as<typename U1::dimension, typename U2::dimension>) &&
             (treat_as_floating_point<decltype(lhs.count() / rhs.count())> ||
              (ratio_divide<typename U1::ratio, typename U2::ratio>::den == 1))
  [[nodiscard]] constexpr Quantity operator/(const quantity<U1, Rep1>& lhs,
                                             const quantity<U2, Rep2>& rhs);

  // ...
};

Additional functions provide the support for operations that result in a different dimension type than those of their arguments.

Another change comparing to std::chrono::duration is that the duration is using std::common_type_t<Rep1, Rep2> to find a common representation for a calculation result. Such a design was reported as problematic by numerics study group members as sometimes we want to provide a different type in case of multiplication and different in case of division. std::common_type lacks that additional information. That is why units::quantity uses the resulting type of a concrete operator operation and provides it directly to units::common_quantity_t type trait.

quantity_cast

To explicitly force truncating conversions quantity_cast function is provided which is a direct counterpart of std::chrono::duration_cast. As a template argument user can provide here either a quantity type or only its template parameters (Unit, Rep):

template<Quantity To, typename U, typename Rep>
  requires std::same_as<typename To::dimension, typename U::dimension>
constexpr To quantity_cast(const quantity<U, Rep>& q);

template<Unit ToU, Scalar ToRep = double, typename U, typename Rep>
constexpr quantity<ToU, ToRep> quantity_cast(const quantity<U, Rep>& q);

Strong types instead of aliases, and type downcasting capability

Most of the important design decisions in the library are dictated by the requirement of providing the best user experience as possible.

For example with template aliases usage the following code:

const Velocity auto t = 20s;

could generate a following compile time error:

<path>\example\example.cpp:39:22: error: deduced initializer does not satisfy placeholder constraints
   const Velocity auto t = 20s;
                           ^~~~
In file included from <path>\example\example.cpp:23:
<path>/src/include/units/si/velocity.h:41:16: note: within 'template<class T> concept const bool stde::units::Velocity<T> [with T = stde::units::quantity<units::unit<units::dimension<units::exp<units::base_dim_time, 1> >, std::ratio<1> >, long long int>]'
   concept Velocity = Quantity<T> && std::same_as<typename T::dimension, velocity>;
           ^~~~~~~~
In file included from <path>/src/include/units/bits/tools.h:25,
                 from <path>/src/include/units/dimension.h:25,
                 from <path>/src/include/units/si/base_dimensions.h:25,
                 from <path>/src/include/units/si/velocity.h:25,
                 from <path>\example\example.cpp:23:
<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 = stde::units::dimension<units::exp<units::base_dim_time, 1> >; U = stde::units::dimension<units::exp<units::base_dim_length, 1>,stde::units::exp<units::base_dim_time, -1> >]'
     concept same_as = std::is_same_v<T, U>;
             ^~~~
<path>/src/include/units/bits/stdconcepts.h:33:18: note: 'std::is_same_v' evaluated to false

Time and velocity are not that complicated dimensions and there are much more complicated dimensions out there, but even for those dimensions

[with T = stde::units::quantity<units::unit<units::dimension<units::exp<units::base_dim_time, 1> >, std::ratio<1> >, long long int>]

and

[with T = stde::units::dimension<units::exp<units::base_dim_time, 1> >; U = stde::units::dimension<units::exp<units::base_dim_length, 1>,stde::units::exp<units::base_dim_time, -1> >]

starts to be really hard to analyze or debug.

That is why it was decided to provide automated downcasting capability when possible. With that the same code will result with such an error:

<path>\example\example.cpp:40:22: error: deduced initializer does not satisfy placeholder constraints
   const Velocity t = 20s;
                      ^~~~
In file included from <path>\example\example.cpp:23:
<path>/src/include/units/si/velocity.h:48:16: note: within 'template<class T> concept const bool stde::units::Velocity<T> [with T = stde::units::quantity<units::second, long long int>]'
   concept Velocity = Quantity<T> && std::same_as<typename T::dimension, velocity>;
           ^~~~~~~~
In file included from <path>/src/include/units/bits/tools.h:25,
                 from <path>/src/include/units/dimension.h:25,
                 from <path>/src/include/units/si/base_dimensions.h:25,
                 from <path>/src/include/units/si/velocity.h:25,
                 from <path>\example\example.cpp:23:
<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 = stde::units::time; U = stde::units::velocity]'
     concept same_as = std::is_same_v<T, U>;
             ^~~~
<path>/src/include/units/bits/stdconcepts.h:33:18: note: 'std::is_same_v' evaluated to false

Now

[with T = stde::units::quantity<units::second, long long int>]

and

[with T = stde::units::time; U = stde::units::velocity]

are not arguably much easier to understand thus provide better user experience.

Downcasting capability is provided through dedicated downcasting_traits, concept, a few helper aliases and by base_type member type in downcast_base class template.

template<typename BaseType>
struct downcast_base {
  using base_type = BaseType;
};

template<typename T>
concept bool Downcastable =
    requires {
      typename T::base_type;
    } &&
    std::derived_from<T, downcast_base<typename T::base_type>>;

template<Downcastable T>
using downcast_from = T::base_type;

template<Downcastable T>
using downcast_to = std::type_identity<T>;

template<Downcastable T>
struct downcasting_traits : downcast_to<T> {};

template<Downcastable T>
using downcasting_traits_t = downcasting_traits<T>::type;

With that the downcasting functionality is enabled by:

struct length : make_dimension_t<exp<base_dim_length, 1>> {};
template<> struct downcasting_traits<downcast_from<length>> : downcast_to<length> {};

struct kilometre : unit<length, std::kilo> {};
template<> struct downcasting_traits<downcast_from<kilometre>> : downcast_to<kilometre> {};

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:

   inline constexpr units::base_dimension base_dim_digital_information{"digital information"};
   

2. Create a new dimension type with the recipe of how to construct it from base dimensions and provide downcasting trait for it:

   struct digital_information : units::make_dimension_t<units::exp<base_dim_digital_information, 1>> {};
   template<>
   struct units::downcasting_traits<units::downcast_from<digital_information>> : units::downcast_to<digital_information> {};
   

3. Define a concept that will match a new dimension:

   template<typename T>
   concept DigitalInformation = units::QuantityOf<T, digital_information>;
   

4. Define units and provide downcasting traits for them:

   struct bit : units::unit<digital_information> {};
   template<> struct units::downcasting_traits<units::downcast_from<bit>> : units::downcast_to<bit> {};
   
   struct byte : units::unit<digital_information, units::ratio<8>> {};
   template<> struct units::downcasting_traits<units::downcast_from<byte>> : units::downcast_to<byte> {};
   

5. Provide user-defined literals for the most important units:

   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 make_dimension? How to do it?

2. What to do with time which is ambiguous (conflict wit ANSI C)?

3. What to do with std::chrono::duration?

4. Should we provide seconds<int> or stay with quantity<second, int>?

5. 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?

6. Do we need non-linear scale?

7. Should we provide cmath-like functions for quantities?

8. What should be the resulting type of auto d = 1km + 1ft;?

9. Should we require explicit casts (i.e. quantity_cast) between different systems of measurement?

10. Should we support integral representations?

11. Provide ostream overloads to print quantity units (use std::format)?

12. 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.

13. Should we standardize accompany tools (downcasting_traits, type_list operations, common_ratio, etc)?

14. Do we need to support fractional exponents (i.e. dimension<exp<"length", 2, 3>> as 2/3)?

15. 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.

16. J imaginary constants are a GCC extension