# Closure semantics¶

The xtensor library is a tensor expression library implementing numpy-style broadcasting and universal functions but in a lazy fashion.

If x and y are two tensor expressions with compatible shapes, the result of x + y is not a tensor but an expression that does not hold any value. Values of x + y are computed upon access or when the result is assigned to a container such as xt::xtensor or xt::xarray. The same holds for most functions in xtensor, views, broadcasting views, etc.

In order to be able to perform the differed computation of x + y, the returned expression must hold references, const references or copies of the members x and y, depending on how arguments were passed to operator+. The actual types held by the expressions are the closure types.

The concept of closure type is key in the implementation of xtensor and appears in all the expressions defined in xtensor, and the utility functions and metafunctions complement the tools of the standard library for the move semantics.

## Basic rules for determining closure types¶

The two main requirements are the following:

• when an argument passed to the function returning an expression (here, operator+) is an rvalue, the closure type is always a value and the rvalue is moved.

• when an argument passed to the function returning an expression is an lvalue reference, the closure type is a reference of the same type.

It is important for the closure type not to be a reference when the passed argument is an rvalue, which can result in dangling references.

Following the conventions of the C++ standard library for naming type traits, we provide two type traits classes providing an implementation of these rules in the xutils.hpp header, closure_type, and const_closure_type. The latter adds the const qualifier to the reference even when the provided argument is not const.

template <class S>
struct closure_type
{
using underlying_type = std::conditional_t<
std::is_const<std::remove_reference_t<S>>::value,
const std::decay_t<S>,
std::decay_t<S>>;
using type = typename std::conditional<
std::is_lvalue_reference<S>::value,
underlying_type&,
underlying_type>::type;
};

template <class S>
using closure_type_t = typename closure_type<S>::type;


The implementation for const_closure_type is slightly shorter.

template <class S>
struct const_closure_type
{
using underlying_type = std::decay_t<S>;
using type = typename std::conditional<
std::is_lvalue_reference<S>::value,
underlying_type>::type;
};

template <class S>
using const_closure_type_t = typename const_closure_type<S>::type;


Using this mechanism, we were able to

• avoid dangling references in nested expressions,

• hold references whenever possible,

• take advantage of the move semantics when holding references is not possible.

## Closure types and scalar wrappers¶

A requirement for xtensor is the ability to mix scalars and tensors in tensor expressions. In order to do so, scalar values are wrapped into the xscalar wrapper, which is a cheap 0-D tensor expression holding a single scalar value.

For the xscalar to be a proper proxy on the scalar value, if actually holds a closure type on the scalar value.

The logic for this is encoded into xtensor’s xclosure type trait.

template <class E, class EN = void>
struct xclosure
{
using type = closure_t<E>;
};

template <class E>
struct xclosure<E, disable_xexpression<std::decay_t<E>>>
{
using type = xscalar<closure_t<E>>;
};

template <class E>
using xclosure_t = typename xclosure<E>::type;


In doing so, we ensure const-correctness, we avoid dangling reference, and ensure that lvalues remain lvalues. The const_xclosure follows the same scheme:

template <class E, class EN = void>
struct const_xclosure
{
using type = const_closure_type_t<E>;
};

template <class E>
struct const_xclosure<E, disable_xexpression<std::decay_t<E>>>
{
using type = xscalar<std::decay_t<E>>;
};

template <class E>
using const_xclosure_t = typename const_xclosure<E>::type;


## Writing functions that return expressions¶

xtensor closure semantics are not meant to prevent users from doing mistakes, since it would also prevent them from doing something clever.

This section covers cases where understanding C++ move semantics and xtensor closure semantics helps writing better code with xtensor.

### Returning evaluated or unevaluated expressions¶

A key feature of xtensor is that a function returning e.g. x + y / z where x, y and z are xtensor expressions does not actually perform any computation. It is only evaluated upon access or assignment. The returned expression holds values or references for x, y and z depending on the lvalue-ness of the variables passed to the expression, using the closure semantics described earlier. This may result in dangling references when using local variables of a function in an unevaluated expression unless one properly forwards / move the variables.

Note

The following rule of thumbs prevents dangling references in the xtensor closure semantics:

• If the laziness is not important for your use case, returning xt::eval(x + y / z) will return an evaluated container and avoid these complications.

• Otherwise, the key is to move lvalues that become invalid when leaving the current scope.

• If you would need to move more than once, take a look at the Reusing expressions / sharing expressions.

Example: moving local variables and forwarding universal references

Let us first consider the following implementation of the mean function in xtensor:

template <class E>
inline auto mean(E&& e) noexcept
{
using value_type = typename std::decay_t<E>::value_type;
auto size = e.size();
auto s = sum(std::forward<E>(e));
return std::move(s) / value_type(size);
}


The first thing to take into account is that the result of the final division is an expression, which performs the actual computation upon access or assignment.

• In order to perform the division, the expression must hold the values or references on the numerator and denominator.

• Since s is a local variable, it will be destroyed upon leaving the scope of the function, and more importantly, it is an lvalue.

• A consequence of s being an lvalue and a local variable, is that the s / value_type(size) would end up holding a dangling const reference on s.

• Hence we must call return std::move(s) / value_type(size).

The other place in this example where the C++ move semantics is used is the line s = sum(std::forward<E>(e)). The goal is to have the unevaluated s expression hold a const reference or a value for e depending on the lvalue-ness of the parameter passed to the function.

### Reusing expressions / sharing expressions¶

Sometimes it is necessary to use a xexpression in two separate places in another xexpression. For example, when computing something like sin(A) + cos(A) we can see A being referenced twice. This works fine if we can guarantee that A has a long enough lifetime. However, when writing generic interfaces that accept rvalues we cannot always guarantee that A will live long enough. Another scenario is the creation of a temporary, which needs to be used at more than one place in the resulting expression. We can only std::move(…) the temporary once into the expression to hand lifetime management to the expression.

In order to solve this problem, xtensor offers two solutions: the first involves ad-hoc lambda construction and the second utilizes shared pointers wrapped in a xshared_expression.

We can rewrite the sin(A) + cos(A) function as a lambda that we use to create a vectorized xfunction, and xtensor has a simple utility to achieve this:

template <class E>
inline auto sin_plus_cos(E&& e) noexcept
{
auto func = [](auto x) -> decltype(sin(x) + cos(x)) {
return sin(x) + cos(x);
};
return detail::make_lambda_function(std::move(func), std::forward<E>(e));
}


Note: writing a lambda is just sugar for writing a functor. Also, using auto x as the function argument enables automatic xsimd acceleration.

As the data flow through the lambda is entirely transparent to the compiler, using this construct is generally faster than using xshared_expressions. The usage of xshared_expression also requires the creation of a shared_ptr which dynamically allocates some memory and is therefore slow(ish). But under certain circumstances it might be required, e.g. to implement a fully lazy average:

template <class E, class W>
inline auto average(E&& e, W&& weights, std::ptrdiff_t axis) noexcept
{
auto shared_weights = xt::make_xshared(std::move(weights));
auto expr = xt::sum(e * shared_weights , {axis}) / xt::sum(shared_weights);
// the following line prints how often shared_weights is used
std::cout << shared_weights.use_count() << std::endl; // ==> 4
return expr;
}


We can see that, before returning from the function, four copies of shared_weights exist: two in the two xt::sum functions, and one is the temporary. The last one lies in weights itself, it is a technical requirement for the share syntax. After returning from the function, only two copies of the xshared_expression will exist. As discussed before, xt::make_xshared has the same overhead as creating a std::shared_ptr which is used internally by the shared expression.

Another syntax can be used if you don’t want to have a temporary variable for the shared expression:

template <class E, class W>
inline auto average(E&& e, W&& weights, std::ptrdiff_t axis) noexcept
{
auto expr = xt::sum(e * xt::share(weights) , {axis}) / xt::sum(xt::share(weights));
// the following line prints how often shared_weights is used
std::cout << shared_weights.use_count() << std::endl; // ==> 3
return expr;
}


In that case only three copies of the shared weights exist. Notice that contrary to make_xshare, share also accepts lvalues; this is to avoid the required std::move, however share will turn its argument into an rvalue and will move it into the shared expression. Thus share invalidates its argument, and the only thing that can be done with an expression upon which share has been called is another call to share. Therefore share should be called on rvalue references or temporary expressions only.