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Interface Introduction

New style of definitions

The mp-units library decided to use a rather unusual pattern to define entities. Here is how we define metre and second SI base units:

inline constexpr struct metre final : named_unit<"m", kind_of<isq::length>> {} metre;
inline constexpr struct second final : named_unit<"s", kind_of<isq::time>> {} second;

Please note that the above reuses the same identifier for a type and its value. The rationale behind this is that:

  • Users always work with values and never have to spell such a type name.
  • The types appear in the compilation errors and during debugging.


To improve compiler errors' readability and make it easier to correlate them with a user's written code, a new idiom in the library is to use the same identifier for a type and its instance.

Also, to prevent possible issues in compile-time logic, all of the library's entities must be marked final. This prevents the users to derive own strong types from them, which would prevent expression template simplification of equivalent entities.

Strong types instead of aliases

Let's look again at the above units definitions. Another important point to notice is that all the types describing entities in the library are short, nicely named identifiers that derive from longer, more verbose class template instantiations. This is really important to improve the user experience while debugging the program or analyzing the compilation error.


Such a practice is rare in the industry. Some popular C++ physical units libraries generate enormously long error messages where even only the first line failed to fit on a slide with a tiny font.

Entities composability

Many physical units libraries (in C++ or any other programming language) assign strong types to library entities (e.g., derived units). While metre_per_second as a type may not look too scary, consider, for example, units of angular momentum. If we followed this path, its coherent unit would look like kilogram_metre_sq_per_second. Now, consider how many scaled versions of this unit you would predefine in the library to ensure that all users are happy with your choice? How expensive would it be from the implementation point of view? What about potential future standardization efforts?

This is why in mp-units, we put a strong requirement to make everything as composable as possible. For example, to create a quantity with a unit of speed, one may write:

quantity<si::metre / si::second> q;

In case we use such a unit often and would prefer to have a handy helper for it, we can always do something like this:

constexpr auto metre_per_second = si::metre / si::second;
quantity<metre_per_second> q;

or choose any shorter identifier of our choice.

Coming back to the angular momentum case, thanks to the composability of units, a user can create such a quantity in the following way:

using namespace mp_units::si::unit_symbols;
auto q = la_vector{1, 2, 3} * isq::angular_momentum[kg * m2 / s];

It is a much better solution. It is terse and easy to understand. Please also notice how easy it is to obtain any scaled version of such a unit (e.g., mg * square(mm) / min) without having to introduce hundreds of types to predefine them.

Value-based equations

The mp-units library is based on C++20, significantly improving user experience. One of such improvements is the usage of value-based equations.

As we have learned above, the entities are being used as values in the code, and they compose. Moreover, derived entities can be defined in the library using such value-based equations. This is a huge improvement compared to what we can find in other physical units libraries or what we have to deal with when we want to write some equations for std::ratio.

For example, below are a few definitions of the SI derived units showing the power of C++20 extensions to Non-Type Template Parameters, which allow us to directly pass a result of the value-based unit equation to a class template definition:

inline constexpr struct newton final : named_unit<"N", kilogram * metre / square(second)> {} newton;
inline constexpr struct pascal final : named_unit<"Pa", newton / square(metre)> {} pascal;
inline constexpr struct joule  final : named_unit<"J", newton * metre> {} joule;

Expression templates

The previous chapter provided a rationale for not having predefined types for derived entities. In many libraries, such an approach results in long and unreadable compilation errors, as framework-generated types are typically far from being easy to read and understand.

The mp-units library greatly improves the user experience by extensively using expression templates. Such expressions are used consistently throughout the entire library to describe the results of:

  • dimension equation - the result is put into the derived_dimension<> class template
  • quantity equation - the result is put into the derived_quantity_spec<> class template
  • unit equation - the result is put into the derived_unit<> class template

For example, if we take the above-defined base units and put the results of their division into the quantity class template like this:

quantity<metre / second> q;

we will observe the following type in the debugger

(gdb) ptype q
type = class mp_units::quantity<mp_units::derived_unit<metre, mp_units::per<second>>(), double> [with Rep = double] {

The same type identifier will be visible in the compilation error (in case it happens).


Expressions templates are extensively used throughout the library to improve the readability of the resulting types.


As mentioned above, equations can be performed on dimensions, quantities, and units. Each such domain must introduce an identity object that can be used in the resulting expressions. Here is the list of identities used in the library:

Domain Concept Identity
Dimension dimension_one
QuantitySpec dimensionless
Unit one

In the equations, a user can explicitly refer to an identity object. For example:

constexpr auto my_unit = one / second;


Another way to achieve the same result is to call an inverse() function:

constexpr auto my_unit = inverse(second);

Both cases will result in the same expression template being generated and put into the wrapper class template.

Supported operations and their results

There are only a few operations that one can do on such entities, and the result of each of them has its unique representation in the library:

Operation Resulting template expression arguments
A * B A, B
B * A A, B
A * A power<A, 2>
{identity} * A A
A * {identity} A
A / B A, per<B>
A / A {identity}
A / {identity} A
{identity} / A {identity}, per<A>
pow<2>(A) power<A, 2>
pow<2>({identity}) {identity}
sqrt(A) or pow<1, 2>(A) power<A, 1, 2>
sqrt({identity}) or pow<1, 2>({identity}) {identity}

Simplifying the resulting expression templates

To limit the length and improve the readability of generated types, there are many rules to simplify the resulting expression template.

  1. Ordering

    The resulting comma-separated arguments of multiplication are always sorted according to a specific predicate. This is why:

    static_assert(A * B == B * A);
    static_assert(std::is_same_v<decltype(A * B), decltype(B * A)>);

    This is probably the most important of all the steps, as it allows comparing types and enables the rest of the simplification rules.

  2. Aggregation

    In case two of the same identifiers are found next to each other on the argument list they will be aggregated in one entry:

    Before After
    A, A power<A, 2>
    A, power<A, 2> power<A, 3>
    power<A, 1, 2>, power<A, 2> power<A, 5, 2>
    power<A, 1, 2>, power<A, 1, 2> A
  3. Simplification

    In case two of the same identifiers are found in the numerator and denominator argument lists; they are being simplified into one entry:

    Before After
    A, per<A> {identity}
    power<A, 2>, per<A> A
    power<A, 3>, per<A> power<A, 2>
    A, per<power<A, 2>> {identity}, per<A>

    It is important to notice here that only the elements with exactly the same type are being simplified. This means that, for example, m/m results in one, but km/m will not be simplified. The resulting derived unit will preserve both symbols and their relative magnitude. This allows us to properly print symbols of some units or constants that require such behavior. For example, the Hubble constant is expressed in km⋅s⁻¹⋅Mpc⁻¹, where both km and Mpc are units of length.

    Also, to prevent possible issues in compile-time logic, all of the library's entities must be marked final. This prevents the users to derive own strong types from them, which would prevent expression template simplification of equivalent entities.

  4. Repacking

    In case an expression uses two results of other operations, the components of its arguments are repacked into one resulting type and simplified there.

    For example, assuming:

    constexpr auto X = A / B;


    Operation Resulting template expression arguments
    X * B A
    X * A power<A, 2>, per<B>
    X * X power<A, 2>, per<power<B, 2>>
    X / X {identity}
    X / A {identity}, per<B>
    X / B A, per<power<B, 2>>


Thanks to all of the features described above, a user may write the code like this one:

using namespace mp_units::si::unit_symbols;
quantity speed = 60. * isq::speed[km / h];
quantity duration = 8 * s;
quantity acceleration = speed / duration;
std::cout << "acceleration: " << acceleration << " (" << / s2) << ")\n";

The acceleration quantity, being the result of the above code, has the following type (after stripping the mp_units namespace for brevity):

quantity<reference<derived_quantity_spec<isq::speed, per<isq::time>>{}, derived_unit<si::kilo_<si::metre{}>, per<non_si::hour, si::second>>{}>{}, int>

and the text output presents:

acceleration: 7.5 km h⁻¹ s⁻¹ (2.08333 m/s²)