What are expression templates anyway? In short, expression templates are templates that you write to capture expressions so that they can be transformed and/or evaluated lazily.

An example of normal C++ expression is:

std::sqrt(3.0) + 8.0f

The compiler sees this and creates some representation of that expression inside the compiler. This is typically an abstract syntax tree (AST). The AST for the expression above might be:

This tree structure captures all the elements of the original C++ code. The expression is a plus operation whose left side is a call to std::sqrt(3.0) and whose right side is 8.0f. The call to std::sqrt(3.0) is its own expression subtree consisting of a call node and its argument node.

A Boost.YAP version of this same tree is:

The operator+() is represented by a Boost.YAP expression whose kind is yap::expr_kind::plus and the call is represented by a Boost.YAP expression whose kind is yap::expr_kind::call. Notice that the call expression has two terminals, one for the callable, and one for its single argument.

The type that holds this expression is:

boost::yap::expression<
    boost::yap::expr_kind::plus,
    boost::hana::tuple<
        boost::yap::expression<
            boost::yap::expr_kind::call,
            boost::hana::tuple<
                boost::yap::expression<
                    boost::yap::expr_kind::terminal,
                    boost::hana::tuple<double (*)(double)>
                >,
                boost::yap::expression<
                    boost::yap::expr_kind::terminal,
                    boost::hana::tuple<double>
                >
            >
        >,
        boost::yap::expression<
            boost::yap::expr_kind::terminal,
            boost::hana::tuple<float>
        >
    >
>

That looks like a big mess; let's unpack it. You might notice that the overall shape is the same as the expression tree diagram above. We have tree-like nesting of boost::yap::expression template instantiations.

Here's the top-level boost::yap::expression again with its noisy guts removed:

boost::yap::expression<
    boost::yap::expr_kind::plus,
    boost::hana::tuple<

// Left and right operand expressions ...

    >
>

It has an expr_kind of plus as its first template parameter (it's a non-type parameter); this indicates what kind of "node" it is. In this case, the top level expression is analogous to our operator+() AST node. Its operands are the elements of its boost::hana::tuple<> data member.

The left operand to the top-level plus operation is itself a Boost.YAP expression representing std::sqrt(3.0):

boost::yap::expression<
    boost::yap::expr_kind::call,
    boost::hana::tuple<
        boost::yap::expression<
            boost::yap::expr_kind::terminal,
            boost::hana::tuple<double (*)(double)>
        >,
        boost::yap::expression<
            boost::yap::expr_kind::terminal,
            boost::hana::tuple<double>
        >
    >
>,

This expression is a call expression. The first operand to the call expression is the callable entity, in this case a pointer to std::sqrt. The remaining operands are the arguments to pass to the callable; in this case, there's only one operand after the callable, 3.0.

The children of the std::sqrt(3.0) subexpression are terminals. This means that they are leaf nodes in our notional AST.

The right operand to the top-level plus operation is of course also a Boost.YAP expression. It is also a terminal:

boost::yap::expression<
    boost::yap::expr_kind::terminal,
    boost::hana::tuple<float>
>

Notice a couple of things here: 1) non-terminals (the top-level plus operation and the call opertion in our example) have tuple elements that are all Boost.YAP expressions, and 2) terminals have tuple elements, none of which are Boost.YAP expressions (they're just normal types like float and double (*)(double)).

	Note
	From here on, I'll use the terms "expression" and "node" interchangably, and I'll also use the terms "subexpression" and "child" interchangably. Even though expression templates are not identical to tree-based ASTs, they're close enough that the terminology is interchangable without loss of meaning.

Capturing an Expression

If we want to capture an expression using Boost.YAP we have to do something to let the compiler know not just to eagerly evaulate our expression, as it does when it sees std::sqrt(3.0) + 8.0f.

To do this, we create expr_kind::terminal expressions out of one or more of the terminals in the expression we want to capture and evaluate lazily. Here, I've declared a template alias to make that easier to type:

template <typename T>
using term = boost::yap::terminal<boost::yap::expression, T>;

And here is how I might use that alias to create the terminal containing std::sqrt:

boost::yap::expression<
    boost::yap::expr_kind::plus,
    boost::hana::tuple<
        boost::yap::expression<
            boost::yap::expr_kind::call,
            boost::hana::tuple<
                boost::yap::expression<
                    boost::yap::expr_kind::terminal,
                    boost::hana::tuple<double (*)(double)>
                >,
                boost::yap::expression<
                    boost::yap::expr_kind::terminal,
                    boost::hana::tuple<double>
                >
            >
        >,
        boost::yap::expression<
            boost::yap::expr_kind::terminal,
            boost::hana::tuple<float>
        >
    >
>
yap_expr = term<double (*)(double)>{{std::sqrt}}(3.0) + 8.0f;

The reason I can then just call the terminal with a 3.0 argument and add 8.0f to the result is that I'm taking a great big shortcut in this example by using Boost.YAP's built-in example expression template, expression<>. expression<> is a template with all the operator overloads defined, including the call operator. Each operator overload returns an expression<>, which is why the + in std::sqrt(3.0) + 8.0f also works.

Note

expression<> is great for example code like what you see here, and it's great for small expression template use cases that are essentially implementation details. You should write your own expression templates for anything that is to be used in any other context. The reason for this is that most of the time your expression template system will not want to support all combinations of all possible operators and function calls. For instance, code like this: (a + b) = c; is at least unusual, if not outright wrong. Where does c go? Into a, b, or into an expiring a + b temporary? What if a is a std::string and b is a FILE *? expression<> doesn't care. You probably want to design interfaces that are more carefully considered than the "everything goes" style implied by using expression<>.

Boost.YAP comes with a handy print() function. Calling it like this:

print(std::cout, yap_expr);

Gives this output:

expr<+>
    expr<()>
        term<double (*)(double)>[=1]
        term<double>[=3]
    term<float>[=8]

This is a lot more readable. I show this to you here to give you a more concise view of the AST-like structure.

(In case you're wondering why &std::sqrt is printed as the value 1, so was I. Apparently, that's just what GCC prints for that. Weird.)

Doing Something Useful With It

Now we've seen a simple expression both described as a C++ AST and captured as a Boost.YAP expression. This just introduces the expression template mechanism; what do we do with it once we have an expression template? Consider one of the examples from the intro:

std::vector<int> v1 = {/* ... */};
std::vector<int> v2 = sort(v) | unique;

The rest of the tutorial will explain in greater detail how Boost.YAP can be used in situations like this, but the brief version is this:

There are certain idioms that Boost.YAP is written to support. Before getting into the nuts and bolts of how Boost.YAP operates, let's define these idioms.

evaluate(transform())

This is the main idiom you'll see reinforced in the examples. The idea is that you capture an expression:

auto expr_0 = /* ... */ ;

then transform it one or more times:

auto expr_1 = boost::yap::transform(expr_0, my_transform_1);
auto expr_2 = boost::yap::transform(expr_1, my_transform_2);
// ...
auto expr_n = boost::yap::transform(expr_n_minus_1, my_transform_n);

and then finally you evaluate it:

auto const result = boost::yap::evaluate(expr_n);

Each call to transform() here produces a new Expression that can subsequently be transformed. This is conceptually similar to what happens inside many compilers. Capturing the expression is analogous to the compiler's parse; the transformations are analogous to optimization passes; and the evaluation is analogous to code generation.

This keeps the meaning of your code quite clear and easy to follow. For this reason, I think you should try to use Boost.YAP in this way when you can.

transform()-as-evaluate

This is a variant of evaluate(transform()), where the evaluate() call at the end is unnecessary, because the final (or perhaps only) transform does all the evaluation we need.

For instance, here is the get_arity transform object used in the Calc3 example (don't worry too much about the implementation — we'll return to this later in the docs in much greater detail):

struct get_arity
{
    // Base case 1: Match a placeholder terminal, and return its arity as the
    // result.
    template <long long I>
    boost::hana::llong<I> operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                                      boost::yap::placeholder<I>)
    { return boost::hana::llong_c<I>; }

    // Base case 2: Match any other terminal.  Return 0; non-placeholders do
    // not contribute to arity.
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>, T &&)
    {
        using namespace boost::hana::literals;
        return 0_c;
    }

    // Recursive case: Match any expression not covered above, and return the
    // maximum of its children's arities.
    template <boost::yap::expr_kind Kind, typename... Arg>
    auto operator() (boost::yap::expr_tag<Kind>, Arg &&... arg)
    {
        return boost::hana::maximum(
            boost::hana::make_tuple(
                boost::yap::transform(
                    boost::yap::as_expr(std::forward<Arg>(arg)),
                    get_arity{}
                )...
            )
        );
    }
};

Here is how this might be used:

auto expr = 1_p * 2_p;
auto const arity = boost::yap::transform(expr, get_arity{});
static_assert(arity.value == 2, "Called with wrong number of args.");

In this case, transform() produces a non-Expression value, all by itself. We got our result without ever needing to call evaluate().

	Note
	Whether transform() returns an Expression or non-Expression is entirely up to the caller. The transform object passed as the second argument to transform() defines what transform()'s return type will be.

Boost.YAP consists of expressions and functions that operate on them. A function that takes an expression will accept any type that models the Expression concept.

For a type T to model the Expression concept, T must contain at least an expr_kind (terminal, plus-operation, etc.) and a boost::hana::tuple<> of values. That's it.

	Note
	The boost::hana::tuple<> of values is constrained, based on the kind of the expression; see the full Expression documentation for details.

Here's an example of an expression:

template <boost::yap::expr_kind Kind, typename Tuple>
struct minimal_expr
{
    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;
};

That's a template that models ExpressionTemplate. Instantiated with the proper template parameters, it produces Expressions.

Ok, so it's not that interesting by itself — minimal_expr has no operations defined for it. But we can still use it with the Boost.YAP functions that take an Expression. Let's make a Boost.YAP plus-expression manually:

auto left = boost::yap::make_terminal<minimal_expr>(1);
auto right = boost::yap::make_terminal<minimal_expr>(41);

auto expr = boost::yap::make_expression<
    minimal_expr,
    boost::yap::expr_kind::plus
>(left, right);

If we evaluate it using evaluate(), it does what you would expect:

auto result = boost::yap::evaluate(expr);

std::cout << result << "\n"; // prints "42"

One more thing. It is important to remember that Boost.YAP expressions are all-lazy, all the time. There is no auto-evaluation of a Boost.YAP expression like there is with normal C++ expressions. If you want your expressions to be evaluated, you must call evaluate(), or define non-lazy operations that force evaluation where and when you want it. This last approach is usually the right one, and there are lots of examples of how to do this in the Examples section. In particular, checkout the Lazy Vector and TArray examples.

Because Boost.YAP operates on Expressions, it is possible to mix and match Expressions that are instantiations of different templates.

Here's why that's important. Say we have two types in a library. S is a string type, and M is a matrix type. In the code here, s and m are objects of types S and M respectively. Say we also have typical operator overloads for these types, so m * m and s[0] are well-formed expressions, but m[0] and s * s are not.

To use these with Boost.YAP I might write an expression template for each:

template <...>
struct m_expr
{
    // ...
};

BOOST_YAP_USER_BINARY_OPERATOR(times, m_expr, m_expr)

template <...>
struct s_expr
{
    // ...
    BOOST_YAP_USER_SUBSCRIPT_OPERATOR(::s_expr)
};

With this, I might write a Boost.YAP expression like:

some_expr_producing_func(S("my_matrix")) * some_matrix

I can transform this expression however I like, and do not have to worry about the fact that it contains expressions instantiated from different templates.

If Boost.YAP required an expression to be instantiated from a single expression template expr<>, expr<> would have to have both operators. This means that all of a sudden s * s and m[0] would be well-formed expressions within a Boost.YAP expression, but not for the real types S and M respectively. That would be super weird.

Most of the expression kinds are the overloadable operators (operator!(), operator<<=(), etc.), See expr_kind for the full list.

There are three special kinds of expressions:

Let's see an expression template type with some operators:

template <boost::yap::expr_kind Kind, typename Tuple>
struct lazy_vector_expr
{
    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;

    // Note that this does not return an expression; it is greedily evaluated.
    auto operator[] (std::size_t n) const;
};

BOOST_YAP_USER_BINARY_OPERATOR(plus, lazy_vector_expr, lazy_vector_expr)
BOOST_YAP_USER_BINARY_OPERATOR(minus, lazy_vector_expr, lazy_vector_expr)

Those macros are used to define operator overloads that return Expressions. As shown here, that sort of operator can be mixed with normal, non-lazy ones — the operator[] is a normal eager function.

Use of the macros is not necessary (you can write your own operators that return Expressions if you like), but it is suitable 99% of the time.

Making the operators easy to define like this allows you to define custom expression templates that have only the operators defined that are appropriate for your use case.

Detailed documentation on all the available macros can be found later in the Operator Macros section.

Transformations in Boost.YAP are done using the transform() function.

Let's take another look at the example expression from the intro:

Consider a call to transform(), operating on that expression:

auto result = boost::yap::transform(expr, xform);

Boost.YAP's transform() first looks at the top level expression, which in this case is a + expression. If the transform object xform matches the + expression, transform() is done; it just returns xform(expr). If xform does not match the + expression, transform() transforms all its operands (which for operator+() is just the left and right operands), and returns a new + expression with those transformed operands. What I mean by "match" is covered in detail below.

The overall effect of this is that transform() effectively copies an expr node that does not match xform, and returns a transformed node for an expr node that does match xform.

transform() can also take multiple transform objects. If you call it with N transform objects, it will attempt to match each of the N transforms to a given expression, one at a time and in their given order. Only if no transform matches an expression does the copy-and-recurse behavior kick in.

	Note
	There's another form of transform(), transform_strict(). transform_strict() is identical to transform() except that it does not copy or recurse into an unmatched expression. Instead, a failed match is a hard error. This is useful when you have written a transform that you expect to completely transform an expression, and you want the compiler to tell you if you've made a mistake.

One common result of calling transform() is that you create a copy of expr, with a few matching nodes transformed. But this does not have to be the result of calling transform(), because a Boost.YAP transformation is free-form; it must return a value, but may do just about anything else. It can transform an expression into anything — a new expression of any kind, or even a non-expression value (effectively evaluating the expression). As before, here is the get_arity transform from the Calc3 example. It returns a value, not an Expression:

struct get_arity
{
    // Base case 1: Match a placeholder terminal, and return its arity as the
    // result.
    template <long long I>
    boost::hana::llong<I> operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                                      boost::yap::placeholder<I>)
    { return boost::hana::llong_c<I>; }

    // Base case 2: Match any other terminal.  Return 0; non-placeholders do
    // not contribute to arity.
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>, T &&)
    {
        using namespace boost::hana::literals;
        return 0_c;
    }

    // Recursive case: Match any expression not covered above, and return the
    // maximum of its children's arities.
    template <boost::yap::expr_kind Kind, typename... Arg>
    auto operator() (boost::yap::expr_tag<Kind>, Arg &&... arg)
    {
        return boost::hana::maximum(
            boost::hana::make_tuple(
                boost::yap::transform(
                    boost::yap::as_expr(std::forward<Arg>(arg)),
                    get_arity{}
                )...
            )
        );
    }
};

Also, note that in this case the transform is stateless, but you could also give your transform objects data members containing contextual state:

struct take_nth
{
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                     std::vector<T> const & vec)
    { return boost::yap::make_terminal(vec[n]); }

    std::size_t n;
};

	Tip
	Often when you create an expression, you will want to evaluate it in different contexts, changing its evaluation — or even entire meaning — in each context. evaluate() is wrong for this task, since it only takes values for substitution into placeholders. In these situations, you should instead use multiple transforms that evaluate your expression in different ways.

When transform() Recurses

As described above, transform() only recurses when it does not find a match. This means that if you want to transform a nonterminal, say an expr_kind::call expression we'll call C, and also C's subexpressions, you must explicitly call transform() yourself in your transform that matches C. You can see this kind of explicit transform() call in the recursive case of get_arity in the example code above.

Note

The code you write with Boost.YAP is likely going to be very generic, especially when you're writing a transform. transform() requires an Expression as its first parameter. In situations when you want to make sure that the first parameter you pass to transform() is always a Boost.YAP expression, use the as_expr() function. This is commonly needed when writing a transform in which you manually recurse by calling transform() inside one of your transform overloads.

Transform Matching

In Boost.YAP a Transform is a Callable that has zero or more overloads that model the ExpressionTransform or TagTransform concepts.

An ExpressionTransform overload takes a single parameter whose type is the expression to be transformed. Here's one from a transform object in the Future Group example:

// Transform left || right -> transform(left).
template <typename T, typename U>
auto operator() (
    future_expr<
        boost::yap::expr_kind::logical_or,
        boost::hana::tuple<T, U>
    > const & or_expr
) {
    // Recursively transform the left side, and return the result.
    // Without the recursion, we might return a terminal expression here
    // insead of a tuple.
    return boost::yap::transform(boost::yap::left(or_expr), *this);
}

ExpressionTransforms are most useful when you want to transform a narrow set of expression types (perhaps only one). In particular, you can distinguish between const and non-const, reference and non-reference, etc., in the expression and its operands in a way that you have less control over with the other kind of transform.

A TagTransform overload takes a tag that indicates the expr_kind of the expression to be transformed, and then (loosely) the value of each operand of the expression to be transformed. This looseness prevents you from needing to write out the full type of the matched expression. Here's one from the Pipable Algorithms example:

template<typename Range>
auto operator()(
    boost::yap::expr_tag<boost::yap::expr_kind::call>,
    algorithm_t a,
    Range & r)
{
    return call_algorithm(a, r);
}

TagTransforms are most useful when the transform needs to match an expression without regard to whether its operands are expr_kind::expr_ref expressions, or — if they are terminals — whether they contain or refer to their values. TagTransforms tend to be far more concise.

A More Rigorous Description of TagTransform Parameters

That "(loosely)" before probably bothered you, right? Me too. Each non-tag parameter is passed to a TagTransform by calling an operand accessor appropriate to expr's kind, and then calling a terminal-specific version of value() (terminal_value()) on the result. For example, consider a plus expression expr. The TagTransform on a transform object xform would be called like this:

xform(plus_tag, terminal_value(left(expr)), terminal_value(right(expr)))

The operand accessors (left() and right() in this example) all dereference expr_kind::expr_ref expressions before operating on them, and terminal_value() does the same.

terminal_value() works much like value(), except that it does not take the value of a nonterminal unary expression; it just forwards a nonterminal through. It still takes values out of terminals and unwraps expr_kind::expr_ref expressions, though.

The auto-unwrapping of terminals means that you can effectively ignore the presence of expr_kind::expr_ref expressions when writing a TagTransform. You can also just deal with the values inside terminals, and not the terminals themselves. Also, you can match all terminal value qualifiers (const or not, lvalue or rvalue) uniformly with a T const & parameter. Finally, you can write TagTransform parameter types that can catch conversions; for instance, you can match any negation expression containing a terminal, or a reference to one, containing a value convertible to double like this:

struct xform
{
    auto operator() (boost::yap::negate_tag, double x)
    { return /* ... */; }
}

That will match a negation of a terminal containing an unsigned int, unsigned int &, int const &, float &&, etc. It will also match a negation of a reference to such a terminal.

Mixing the Two Kinds of Transforms

You can have two overloads in your transform that match an expression, one an ExpressionTransform and one a TagTransform, and there will not be any ambiguity. The TagTransform is matched first, and the ExpressionTransform is matched only if the TagTransform did not. You don't have to worry about ambiguity, but save yourself some confusion and mix the two kinds of overloads as little as possible.

	Note
	The above only applies when you have an ExpressionTransform and a TagTransform that match the same kind of expression. Having unrelated ExpressionTransforms and TagTransforms within the same transform object is often quite useful.

Multiple Transform Objects

In the case that multiple transform objects are being used in transform(), the above logic applies to each one independently before the next one is used. In other words, in the call boost::yap::transform(expr, a, b), transform() tries to match any TagTransform from a to an expression first, then any ExpressionTransform from a, then any TagTransform from b, and finally any ExpressionTransform from b. Only the first matching overload in this sequence is used; all overloads later in the sequence or in later transforms, whether they match or not, are simply ignored.

YAP-Supplied Transforms

Boost.YAP comes with a couple of functions that return ready-made transforms, replacements() and evaluation().

The transforms returned by replacements() replace only placeholder terminals. Placeholder I is replaced by the I-1-th argument passed to replacements(). Placeholders are 1-based for consistency with other Boost and std placeholders.

There are also a couple of specialty transform functions, replace_placeholders() and evaluate(). These are convenience functions that just call transform() on an expression using replacements() or evaluation() as the transform, respectively.

The behavior of evaluation() is covered in the next section, Evaluating Expressions.

Boost.YAP expressions are evaluated explicitly, by calling the evaluate() function or calling transform() using a transform object returned from evaluation(). The former is a convenince function that does the latter.

evaluate() simply removes all the Boost.YAP machinery from an expression and evaluates it exactly as it would have been if Boost.YAP were not used. This means that functions are called, operators evaluated, etc. all as normal. To illustrate this, take a look at the implementation of operator,() used in evaluate():

template<typename T, typename U>
constexpr decltype(auto) operator()(expr_tag<expr_kind::comma>, T && t, U && u) const
{
    return transform(
               as_expr<minimal_expr>(static_cast<T &&>(t)), *this),
           transform(
               as_expr<minimal_expr>(static_cast<U &&>(u)), *this);
}

What this transformation does is transform the left and right expressions, and then use the built-in operator,() on the result. The evaluation transformations for the other operators do the same thing — evaluate the operands, then return the result of applying the built-in operator to the operands.

Function calls are done in a similar way, except that the callable is also a subexpression that needs to be evaluated before being called:

template<typename Callable, typename... Args>
constexpr decltype(auto) operator()(
    expr_tag<expr_kind::call>, Callable && callable, Args &&... args) const
{
    return transform(as_expr<minimal_expr>(static_cast<Callable &&>(callable)), *this)(
        transform(as_expr<minimal_expr>(static_cast<Args &&>(args)), *this)...
    );
}

If you got here without reading the Operators section, go read that first. Here are the operator macros and their uses:

Some operators may only be defined as member functions, and so are not covered by general-purpose the unary and binary operator macros above:

	Note
	Operands are handled in a uniform way across all functions defined by all the macros listed here. See How Expression Operands Are Treated for details.

For any expression<> operator overload, or any function defined using one of the function definition macros, operands are treated in a uniform way.

The guiding design principle here is that an expression built using Boost.YAP should match the semantics of a builtin C++ expression as closely as possible. This implies that an rvalue be treated as if it were a temporary (as it may in fact have initially been) throughout the building and transformation of an expression, and that an lvalue should retain its connection to the underlying named entity to which it refers.

For example, if you see

auto expr = a + 1;

you should expect that a will be an lvalue reference to some object of type decltype(a), regardless of whether a is a Boost.YAP Expression or a builtin type. Similarly, you should expect the 1 to be an rvalue, whether wrapped in a terminal or not.

Let's take a quick look at make_terminal(). If you call it with a T rvalue, the terminal's value type is a T, and the rvalue gets moved into it. If you call it with a T [const] lvalue, the value type is T [const] &, and the reference refers to the lvalue (read [const] as "possibly const-qualified"). This is important because you might write through the terminal later in an assignment operation. You don't want to lose the ability to do this, or be forced to write some Baroque pile of code to do so — it should be natural and easy.

And it is:

int i = 0;
auto expr = boost::yap::make_terminal(i) = 42;
evaluate(expr);
std::cout << i << "\n"; // Prints 42.

Now, there is a wrinkle. Boost.YAP's lazy expressions can be built piecemeal:

auto subexpr = boost::yap::make_terminal(1) + 2;
// This is fine, and acts more-or-less as if you wrote "1 / (1 + 2)".
auto expr = 1 / subexpr;

whereas C++'s eager builtin expressions cannot:

auto subexpr = 1 + 2;    // Same as "int subexpr = 3;".  Hm.
auto expr = 1 / subexpr; // Same as "int expr = 0;" Arg.

Ok, so since you can build these lazy Boost.YAP expressions up from subexpressions, how do we treat the subexpressions? We treat them in exactly the same way as make_terminal() treats its parameter. Rvalues are moved in, and lvalues are captured by (possibly const) reference.

	Note
	If you want to subvert the capture-by-reference semantics of using subexpressions, just std::move() them. That will force a move — or copy of values for which move is not defined.

The capture-by-reference behavior is implemented via a special expr_kind, expr_kind::expr_ref. An expr_ref expression has a single data element: a (possibly const (Can I stop saying that every time? You get it, right? Ok, good.)) reference to an expression. This additional level of indirection causes some complications at times, as you can see in the examples. Fortunately, the complications are not overly cumbersome.

So, given the rules above, here is a comprehensive breakdown of what happens when an operand is passed to a Boost.YAP operator. In this table, expr_tmpl is an ExpressionTemplate, and T is a non-Expression type. E refers to any non-expr_ref Expression. Boost.YAP does a partial decay on non-Expression operands, in which cv and reference qualifiers are left unchanged, but arrays are decayed to pointers and functions are decayed to function pointers. PARTIAL_DECAY(T) indicates such a partial decay of T.

The partial decay of non-Expression operands is another example of how Boost.YAP attempts to create expression trees that are as semantically close to builtin expressions as possible.

Boost.YAP has a convenient print() function, that prints an expression tree to a stream. It is not intended for production work (for instance, it has no formatting options), but it is excellent for debugging and instrumentation.

Since it is only a debugging aid, print() is found in a separate header not included when you include Boost.YAP with

#include <boost/yap/yap.hpp>

You must include <boost/yap/print.hpp> explicitly.

print() handles several patterns of expression specially, to allow a concise representation of a given expression tree. For example, given this definition:

struct thing {};

and this expression:

using namespace boost::yap::literals;

auto const const_lvalue_terminal_containing_rvalue = boost::yap::make_terminal("lvalue terminal");

double const d = 1.0;
auto rvalue_terminal_containing_lvalue = boost::yap::make_terminal(d);

auto thing_terminal = boost::yap::make_terminal(thing{});

auto expr =
    4_p +
    std::move(rvalue_terminal_containing_lvalue) * thing_terminal -
    const_lvalue_terminal_containing_rvalue;

print() produces this output:

expr<->
    expr<+>
        term<boost::yap::placeholder<4ll>>[=4]
        expr<*>
            term<double &>[=1]
            term<thing>[=<<unprintable-value>>] &
    term<char const*>[=lvalue terminal] const &

As you can see, print() shows one node per line, and represents the tree structure with indentation. It abbreviates non-terminal nodes in the tree expr<op>, where op is an operator symbol. Terminal nodes are abbreviated term<T>, where T is the type of value contained in the terminal; this may be a reference type or a value.

A term node may not be a terminal node at all, but an expr_kind::expr_ref expression containing a terminal. Such a expr_kind::expr_ref node has a & or const & suffix, to indicate that it is a mutable or const reference, respectively.

Each term node has a bracketed value near the end. The format is [=X] where X is the value the terminal contains. If the terminal contains a value for which no operator<<(std::ostream &, ...) overload exists (such as the thing type above), X will be <<unprintable-value>>.

Most of these examples are patterned after the examples from Boost.Proto. In part, this was done to underscore where Boost.YAP can do what Proto can, and where it cannot.

Where possible, a Proto-derived example uses syntax in main() identical to that in the original Proto example.

If you don't know anything about Proto, don't worry. The examples are useful on their own.

#include <boost/yap/expression.hpp>

#include <iostream>


int main ()
{
    evaluate(boost::yap::make_terminal(std::cout) << "Hello" << ',' << " world!\n");

    return 0;
}

#include <boost/yap/algorithm.hpp>

#include <iostream>


template <boost::yap::expr_kind Kind, typename Tuple>
struct stream_expr
{
    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;

    template <typename T>
    decltype(auto) operator<< (T && x)
    { return boost::yap::value(*this) << std::forward<T &&>(x); }
};


int main ()
{
    auto cout = boost::yap::make_terminal<stream_expr>(std::cout);
    cout << "Hello" << ',' << " world!\n";

    return 0;
}

Minimal

minimal_expr below models ExpressionTemplate; since it has no operators, an expression must be built manually.

First, the template itself:

template <boost::yap::expr_kind Kind, typename Tuple>
struct minimal_expr
{
    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;
};

This can be used to make a minimal_expr plus expression:

auto left = boost::yap::make_terminal<minimal_expr>(1);
auto right = boost::yap::make_terminal<minimal_expr>(41);

auto expr = boost::yap::make_expression<
    minimal_expr,
    boost::yap::expr_kind::plus
>(left, right);

You can evaluate, transform, or otherwise operate on minimal_expr expressions using the functions in Boost.YAP that accept an Expression:

auto result = boost::yap::evaluate(expr);

std::cout << result << "\n"; // prints "42"

	Note
	Don't use Boost.YAP this way. Use the operator macros instead. This is an example contrived only to show you the minimum requirements on a Boost.YAP-compatible template.

#include <boost/yap/expression.hpp>

#include <iostream>


int main ()
{
    using namespace boost::yap::literals;

    // Displays "5"
    std::cout << evaluate( 1_p + 2.0, 3.0 ) << std::endl;

    // Displays "6"
    std::cout << evaluate( 1_p * 2_p, 3.0, 2.0 ) << std::endl;

    // Displays "0.5"
    std::cout << evaluate( (1_p - 2_p) / 2_p, 3.0, 2.0 ) << std::endl;

    return 0;
}

#include <boost/yap/expression.hpp>

#include <iostream>


int main ()
{
    using namespace boost::yap::literals;

    auto expr_1 = 1_p + 2.0;

    auto expr_1_fn = [expr_1](auto &&... args) {
        return evaluate(expr_1, args...);
    };

    auto expr_2 = 1_p * 2_p;

    auto expr_2_fn = [expr_2](auto &&... args) {
        return evaluate(expr_2, args...);
    };

    auto expr_3 = (1_p - 2_p) / 2_p;

    auto expr_3_fn = [expr_3](auto &&... args) {
        return evaluate(expr_3, args...);
    };

    // Displays "5"
    std::cout << expr_1_fn(3.0) << std::endl;

    // Displays "6"
    std::cout << expr_2_fn(3.0, 2.0) << std::endl;

    // Displays "0.5"
    std::cout << expr_3_fn(3.0, 2.0) << std::endl;

    return 0;
}

#include <boost/yap/expression.hpp>

#include <iostream>


int main ()
{
    using namespace boost::yap::literals;

    // Displays "5"
    std::cout << make_expression_function(1_p + 2.0)(3.0) << std::endl;

    // Displays "6"
    std::cout << make_expression_function(1_p * 2_p)(3.0, 2.0) << std::endl;

    // Displays "0.5"
    std::cout << make_expression_function((1_p - 2_p) / 2_p)(3.0, 2.0) << std::endl;

    return 0;
}

Calc3

Here, we introduce a Transform used to calculate expression arity, and static_assert() that the number of parameters passed by the caller matches the arity.

	Note
	The get_arity Transform doesn't produce an Expression, and it does not have to. Transforms may produce Expressions or arbitrary values. They may also have arbitrary side effects, and may be stateful.

#include <boost/yap/expression.hpp>

#include <boost/hana/maximum.hpp>

#include <iostream>


// Look! A transform!  This one transforms the expression tree into the arity
// of the expression, based on its placeholders.
struct get_arity
{
    // Base case 1: Match a placeholder terminal, and return its arity as the
    // result.
    template <long long I>
    boost::hana::llong<I> operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                                      boost::yap::placeholder<I>)
    { return boost::hana::llong_c<I>; }

    // Base case 2: Match any other terminal.  Return 0; non-placeholders do
    // not contribute to arity.
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>, T &&)
    {
        using namespace boost::hana::literals;
        return 0_c;
    }

    // Recursive case: Match any expression not covered above, and return the
    // maximum of its children's arities.
    template <boost::yap::expr_kind Kind, typename... Arg>
    auto operator() (boost::yap::expr_tag<Kind>, Arg &&... arg)
    {
        return boost::hana::maximum(
            boost::hana::make_tuple(
                boost::yap::transform(
                    boost::yap::as_expr(std::forward<Arg>(arg)),
                    get_arity{}
                )...
            )
        );
    }
};

int main ()
{
    using namespace boost::yap::literals;

    // These lambdas wrap our expressions as callables, and allow us to check
    // the arity of each as we call it.

    auto expr_1 = 1_p + 2.0;

    auto expr_1_fn = [expr_1](auto &&... args) {
        auto const arity = boost::yap::transform(expr_1, get_arity{});
        static_assert(arity.value == sizeof...(args), "Called with wrong number of args.");
        return evaluate(expr_1, args...);
    };

    auto expr_2 = 1_p * 2_p;

    auto expr_2_fn = [expr_2](auto &&... args) {
        auto const arity = boost::yap::transform(expr_2, get_arity{});
        static_assert(arity.value == sizeof...(args), "Called with wrong number of args.");
        return evaluate(expr_2, args...);
    };

    auto expr_3 = (1_p - 2_p) / 2_p;

    auto expr_3_fn = [expr_3](auto &&... args) {
        auto const arity = boost::yap::transform(expr_3, get_arity{});
        static_assert(arity.value == sizeof...(args), "Called with wrong number of args.");
        return evaluate(expr_3, args...);
    };

    // Displays "5"
    std::cout << expr_1_fn(3.0) << std::endl;

    // Displays "6"
    std::cout << expr_2_fn(3.0, 2.0) << std::endl;

    // Displays "0.5"
    std::cout << expr_3_fn(3.0, 2.0) << std::endl;

    // Static-asserts with "Called with wrong number of args."
    //std::cout << expr_3_fn(3.0) << std::endl;

    // Static-asserts with "Called with wrong number of args."
    //std::cout << expr_3_fn(3.0, 2.0, 1.0) << std::endl;

    return 0;
}

Lazy Vector

Finally, it starts to get interesting! This example shows how you can add plus and other operations to sequences of data without creating temporaries and allocating memory.

	Note
	In this example, we see a terminal type that owns the storage of its value, a std::vector<double>. See the Vector example later on to see a terminal type that does not.

#include <boost/yap/expression.hpp>

#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>


template <boost::yap::expr_kind Kind, typename Tuple>
struct lazy_vector_expr;


// This transform turns a terminal of std::vector<double> into a terminal
// containing the nth double in that vector.  Think of it as turning our
// expression of vectors into an expression of scalars.
struct take_nth
{
    boost::yap::terminal<lazy_vector_expr, double>
    operator() (boost::yap::terminal<lazy_vector_expr, std::vector<double>> const & expr);

    std::size_t n;
};

// A custom expression template that defines lazy + and - operators that
// produce expressions, and an eager [] operator that returns the nth element
// of the expression.
template <boost::yap::expr_kind Kind, typename Tuple>
struct lazy_vector_expr
{
    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;

    // Note that this does not return an expression; it is greedily evaluated.
    auto operator[] (std::size_t n) const;
};

BOOST_YAP_USER_BINARY_OPERATOR(plus, lazy_vector_expr, lazy_vector_expr)
BOOST_YAP_USER_BINARY_OPERATOR(minus, lazy_vector_expr, lazy_vector_expr)

template <boost::yap::expr_kind Kind, typename Tuple>
auto lazy_vector_expr<Kind, Tuple>::operator[] (std::size_t n) const
{ return boost::yap::evaluate(boost::yap::transform(*this, take_nth{n})); }

boost::yap::terminal<lazy_vector_expr, double>
take_nth::operator() (boost::yap::terminal<lazy_vector_expr, std::vector<double>> const & expr)
{
    double x = boost::yap::value(expr)[n];
    // This move is something of a hack; we're forcing Yap to take a copy of x
    // by using std::move().  The move indicates that the terminal should keep
    // the value of x (since, being an rvalue, it may be a temporary), rather
    // than a reference to x.  See the "How Expression Operands Are Treated"
    // section of the tutorial for details.
    return boost::yap::make_terminal<lazy_vector_expr, double>(std::move(x));
}

// In order to define the += operator with the semantics we want, it's
// convenient to derive a terminal type from a terminal instantiation of
// lazy_vector_expr.  Note that we could have written a template
// specialization here instead -- either one would work.  That would of course
// have required more typing.
struct lazy_vector :
    lazy_vector_expr<
        boost::yap::expr_kind::terminal,
        boost::hana::tuple<std::vector<double>>
    >
{
    lazy_vector () {}

    explicit lazy_vector (std::vector<double> && vec)
    { elements = boost::hana::tuple<std::vector<double>>(std::move(vec)); }

    template <boost::yap::expr_kind Kind, typename Tuple>
    lazy_vector & operator+= (lazy_vector_expr<Kind, Tuple> const & rhs)
    {
        std::vector<double> & this_vec = boost::yap::value(*this);
        for (int i = 0, size = (int)this_vec.size(); i < size; ++i) {
            this_vec[i] += rhs[i];
        }
        return *this;
    }
};

int main ()
{
    lazy_vector v1{std::vector<double>(4, 1.0)};
    lazy_vector v2{std::vector<double>(4, 2.0)};
    lazy_vector v3{std::vector<double>(4, 3.0)};

    double d1 = (v2 + v3)[2];
    std::cout << d1 << "\n";

    v1 += v2 - v3;
    std::cout << '{' << v1[0] << ',' << v1[1]
              << ',' << v1[2] << ',' << v1[3] << '}' << "\n";

    // This expression is disallowed because it does not conform to the
    // implicit grammar.  operator+= is only defined on terminals, not
    // arbitrary expressions.
    // (v2 + v3) += v1;

    return 0;
}

#include <boost/yap/expression.hpp>

#include <boost/optional.hpp>
#include <boost/hana/fold.hpp>
#include <boost/hana/maximum.hpp>

#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>


// A super-basic matrix type, and a few associated operations.
struct matrix
{
    matrix() : values_(), rows_(0), cols_(0) {}

    matrix(int rows, int cols) : values_(rows * cols), rows_(rows), cols_(cols)
    {
        assert(0 < rows);
        assert(0 < cols);
    }

    int rows() const { return rows_; }
    int cols() const { return cols_; }

    double operator()(int r, int c) const
    { return values_[r * cols_ + c]; }
    double & operator()(int r, int c)
    { return values_[r * cols_ + c]; }

private:
    std::vector<double> values_;
    int rows_;
    int cols_;
};

matrix operator*(matrix const & lhs, double x)
{
    matrix retval = lhs;
    for (int i = 0; i < retval.rows(); ++i) {
        for (int j = 0; j < retval.cols(); ++j) {
            retval(i, j) *= x;
        }
    }
    return retval;
}
matrix operator*(double x, matrix const & lhs) { return lhs * x; }

matrix operator+(matrix const & lhs, matrix const & rhs)
{
    assert(lhs.rows() == rhs.rows());
    assert(lhs.cols() == rhs.cols());
    matrix retval = lhs;
    for (int i = 0; i < retval.rows(); ++i) {
        for (int j = 0; j < retval.cols(); ++j) {
            retval(i, j) += rhs(i, j);
        }
    }
    return retval;
}

// daxpy() means Double-precision AX Plus Y.  This crazy name comes from BLAS.
// It is more efficient than a naive implementation, because it does not
// create temporaries.  The covnention of using Y as an out-parameter comes
// from FORTRAN BLAS.
matrix & daxpy(double a, matrix const & x, matrix & y)
{
    assert(x.rows() == y.rows());
    assert(x.cols() == y.cols());
    for (int i = 0; i < y.rows(); ++i) {
        for (int j = 0; j < y.cols(); ++j) {
            y(i, j) += a * x(i, j);
        }
    }
    return y;
}

template <boost::yap::expr_kind Kind, typename Tuple>
struct self_evaluating_expr;

template <boost::yap::expr_kind Kind, typename Tuple>
auto evaluate_matrix_expr(self_evaluating_expr<Kind, Tuple> const & expr);

// This is the primary template for our expression template.  If you assign a
// self_evaluating_expr to a matrix, its conversion operator transforms and
// evaluates the expression with a call to evaluate_matrix_expr().
template <boost::yap::expr_kind Kind, typename Tuple>
struct self_evaluating_expr
{
    operator auto() const;

    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;
};

// This is a specialization of our expression template for assignment
// expressions.  The destructor transforms and evaluates via a call to
// evaluate_matrix_expr(), and then assigns the result to the variable on the
// left side of the assignment.
//
// In a production implementation, you'd need to have specializations for
// plus_assign, minus_assign, etc.
template <typename Tuple>
struct self_evaluating_expr<boost::yap::expr_kind::assign, Tuple>
{
    ~self_evaluating_expr();

    static const boost::yap::expr_kind kind = boost::yap::expr_kind::assign;

    Tuple elements;
};

struct use_daxpy
{
    // A plus-expression, which may be of the form double * matrix + matrix,
    // or may be something else.  Since our daxpy() above requires a mutable
    // "y", we only need to match a mutable lvalue matrix reference here.
    template <typename Tuple>
    auto operator()(
        boost::yap::expr_tag<boost::yap::expr_kind::plus>,
        self_evaluating_expr<boost::yap::expr_kind::multiplies, Tuple> const & expr,
        matrix & m)
    {
        // Here, we transform the left-hand side into a pair if it's the
        // double * matrix operation we're looking for.  Otherwise, we just
        // get a copy of the left side expression.
        //
        // Note that this is a bit of a cheat, done for clarity.  If we pass a
        // larger expression that happens to contain a double * matrix
        // subexpression, that subexpression will be transformed into a tuple!
        // In production code, this transform should probably only be
        // performed on an expression with all terminal members.
        auto lhs = boost::yap::transform(
            expr,
            [](boost::yap::expr_tag<boost::yap::expr_kind::multiplies>,
               double d,
               matrix const & m) {
                return std::pair<double, matrix const &>(d, m);
            });

        // If we got back a copy of expr above, just re-construct the
        // expression this function mathes; in other words, do not effectively
        // transform anything.  Otherwise, replace the expression matched by
        // this function with a call to daxpy().
        if constexpr (boost::yap::is_expr<decltype(lhs)>::value) {
            return expr + m;
        } else {
            return boost::yap::make_terminal(daxpy)(lhs.first, lhs.second, m);
        }
    }
};


// This is the heart of what self_evaluating_expr does.  If we had other
// optimizations/transformations we wanted to do, we'd put them in this
// function, either before or after the use_daxpy transformation.
template <boost::yap::expr_kind Kind, typename Tuple>
auto evaluate_matrix_expr(self_evaluating_expr<Kind, Tuple> const & expr)
{
    auto daxpy_form = boost::yap::transform(expr, use_daxpy{});
    return boost::yap::evaluate(daxpy_form);
}

template<boost::yap::expr_kind Kind, typename Tuple>
self_evaluating_expr<Kind, Tuple>::operator auto() const
{
    return evaluate_matrix_expr(*this);
}

template<typename Tuple>
self_evaluating_expr<boost::yap::expr_kind::assign, Tuple>::
    ~self_evaluating_expr()
{
    using namespace boost::hana::literals;
    boost::yap::evaluate(elements[0_c]) = evaluate_matrix_expr(elements[1_c]);
}

// In order to define the = operator with the semantics we want, it's
// convenient to derive a terminal type from a terminal instantiation of
// self_evaluating_expr.  Note that we could have written a template
// specialization here instead -- either one would work.  That would of course
// have required more typing.
struct self_evaluating :
    self_evaluating_expr<
        boost::yap::expr_kind::terminal,
        boost::hana::tuple<matrix>
    >
{
    self_evaluating() {}

    explicit self_evaluating(matrix m)
    { elements = boost::hana::tuple<matrix>(std::move(m)); }

    BOOST_YAP_USER_ASSIGN_OPERATOR(self_evaluating_expr, ::self_evaluating_expr);
};

BOOST_YAP_USER_BINARY_OPERATOR(plus, self_evaluating_expr, self_evaluating_expr)
BOOST_YAP_USER_BINARY_OPERATOR(minus, self_evaluating_expr, self_evaluating_expr)
BOOST_YAP_USER_BINARY_OPERATOR(multiplies, self_evaluating_expr, self_evaluating_expr)


int main()
{
    matrix identity(2, 2);
    identity(0, 0) = 1.0;
    identity(1, 1) = 1.0;

    // These are YAP-ified terminal expressions.
    self_evaluating m1(identity);
    self_evaluating m2(identity);
    self_evaluating m3(identity);

    // This transforms the YAP expression to use daxpy(), so it creates no
    // temporaries.  The transform happens in the destructor of the
    // assignment-expression specialization of self_evaluating_expr.
    m1 = 3.0 * m2 + m3;

    // Same as above, except that it uses the matrix conversion operator on
    // the self_evaluating_expr primary template, because here we're assigning
    // a YAP expression to a non-YAP-ified matrix.
    matrix m_result_1 = 3.0 * m2 + m3;

    // Creates temporaries and does not use daxpy(), because the A * X + Y
    // pattern does not occur within the expression.
    matrix m_result_2 = 3.0 * m2;

    return 0;
}

TArray

Proto refers to this as the "mini-library for linear algebra" example. It shows how quite complicated expressions involving sequences can be evaluated elementwise, requiring no temporaries.

	Note
	The original Proto example used a terminal that contained an array of three ints; Boost.YAP cannot represent this, and so this example uses a std::array<T, 3> instead. Boost.YAP decays int[3] to int *, since that is what is done in a C++ expression. See How Expression Operands Are Treated for details.

#include <boost/yap/algorithm.hpp>
#include <boost/yap/print.hpp>

#include <array>
#include <iostream>


template <boost::yap::expr_kind Kind, typename Tuple>
struct tarray_expr;


struct take_nth
{
    boost::yap::terminal<tarray_expr, int>
    operator() (boost::yap::terminal<tarray_expr, std::array<int, 3>> const & expr);

    std::size_t n;
};

// Another custom expression template.  In this case, we static_assert() that
// it only gets instantiated with terminals with pre-approved value types.
template <boost::yap::expr_kind Kind, typename Tuple>
struct tarray_expr
{
    // Make sure that, if this expression is a terminal, its value is one we
    // want to support.  Note that the presence of expr_kind::expr_ref makes
    // life slightly more difficult; we have to account for int const & and
    // int & as well as int.
    static_assert(
        Kind != boost::yap::expr_kind::terminal ||
        std::is_same<Tuple, boost::hana::tuple<int const &>>{} ||
        std::is_same<Tuple, boost::hana::tuple<int &>>{} ||
        std::is_same<Tuple, boost::hana::tuple<int>>{} ||
        std::is_same<Tuple, boost::hana::tuple<std::array<int, 3>>>{},
        "tarray_expr instantiated with an unsupported terminal type."
    );

    static const boost::yap::expr_kind kind = Kind;

    Tuple elements;

    int operator[] (std::size_t n) const
    { return boost::yap::evaluate(boost::yap::transform(*this, take_nth{n})); }
};

// Define operators +, -, *, and /.
BOOST_YAP_USER_BINARY_OPERATOR(plus, tarray_expr, tarray_expr)
BOOST_YAP_USER_BINARY_OPERATOR(minus, tarray_expr, tarray_expr)
BOOST_YAP_USER_BINARY_OPERATOR(multiplies, tarray_expr, tarray_expr)
BOOST_YAP_USER_BINARY_OPERATOR(divides, tarray_expr, tarray_expr)


boost::yap::terminal<tarray_expr, int>
take_nth::operator() (boost::yap::terminal<tarray_expr, std::array<int, 3>> const & expr)
{
    int x = boost::yap::value(expr)[n];
    // Again, this is the move hack to get x into the resulting terminal as a
    // value instead of a reference.
    return boost::yap::make_terminal<tarray_expr>(std::move(x));
}


// Stream-out operators for the two kinds of terminals we support.

std::ostream & operator<< (std::ostream & os, boost::yap::terminal<tarray_expr, int> expr)
{ return os << '{' << boost::yap::value(expr) << '}'; }

std::ostream & operator<< (std::ostream & os, boost::yap::terminal<tarray_expr, std::array<int, 3>> expr)
{
    std::array<int, 3> const & a = boost::yap::value(expr);
    return os << '{' << a[0] << ", " << a[1] << ", " << a[2] << '}';
}

// Stream-out operators for general expressions.  Note that we have to treat
// the reference case separately; this also could have been done using
// constexpr if in a single function template.

template <typename Tuple>
std::ostream & operator<< (std::ostream & os, tarray_expr<boost::yap::expr_kind::expr_ref, Tuple> const & expr)
{ return os << boost::yap::deref(expr); }

template <boost::yap::expr_kind Kind, typename Tuple>
std::ostream & operator<< (std::ostream & os, tarray_expr<Kind, Tuple> const & expr)
{
    if (Kind == boost::yap::expr_kind::plus || Kind == boost::yap::expr_kind::minus)
        os << '(';
    os << boost::yap::left(expr) << " " << op_string(Kind) << " " << boost::yap::right(expr);
    if (Kind == boost::yap::expr_kind::plus || Kind == boost::yap::expr_kind::minus)
        os << ')';
    return os;
}


// Since we want different behavior on terminals than on other kinds of
// expressions, we create a custom type that does so.
struct tarray :
    tarray_expr<
        boost::yap::expr_kind::terminal,
        boost::hana::tuple<std::array<int, 3>>
    >
{
    explicit tarray (int i = 0, int j = 0, int k = 0)
    {
        (*this)[0] = i;
        (*this)[1] = j;
        (*this)[2] = k;
    }

    explicit tarray (std::array<int, 3> a)
    {
        (*this)[0] = a[0];
        (*this)[1] = a[1];
        (*this)[2] = a[2];
    }

    int & operator[] (std::ptrdiff_t i)
    { return boost::yap::value(*this)[i]; }

    int const & operator[] (std::ptrdiff_t i) const
    { return boost::yap::value(*this)[i]; }

    template <typename T>
    tarray & operator= (T const & t)
    {
        // We use as_expr() here to make sure that the value passed to
        // assign() is an expression.  as_expr() simply forwards expressions
        // through, and wraps non-expressions as terminals.
        return assign(boost::yap::as_expr< ::tarray_expr>(t));
    }

    template <typename Expr>
    tarray & printAssign (Expr const & expr)
    {
        *this = expr;
        std::cout << *this << " = " << expr << std::endl;
        return *this;
    }

private:
    template <typename Expr>
    tarray & assign (Expr const & expr)
    {
        (*this)[0] = expr[0];
        (*this)[1] = expr[1];
        (*this)[2] = expr[2];
        return *this;
    }
};


int main()
{
    tarray a(3,1,2);

    tarray b;

    std::cout << a << std::endl;
    std::cout << b << std::endl;

    b[0] = 7; b[1] = 33; b[2] = -99;

    tarray c(a);

    std::cout << c << std::endl;

    a = 0;

    std::cout << a << std::endl;
    std::cout << b << std::endl;
    std::cout << c << std::endl;

    a = b + c;

    std::cout << a << std::endl;

    a.printAssign(b+c*(b + 3*c));

    return 0;
}

#include <boost/yap/yap.hpp>

#include <array>
#include <iostream>


struct take_nth
{
    auto operator() (boost::yap::terminal<boost::yap::expression, std::array<int, 3>> const & expr)
    {
        int x = boost::yap::value(expr)[n];
        // The move forces the terminal to store the value of x, not a
        // reference.
        return boost::yap::make_terminal(std::move(x));
    }

    std::size_t n;
};

// Since this example doesn't constrain the operators defined on its
// expressions, we can just use boost::yap::expression<> as the expression
// template.
using vec3_terminal = boost::yap::expression<
    boost::yap::expr_kind::terminal,
    boost::hana::tuple<std::array<int, 3>>
>;

// Customize the terminal type we use by adding index and assignment
// operations.
struct vec3 : vec3_terminal
{
    explicit vec3 (int i = 0, int j = 0, int k = 0)
    {
        (*this)[0] = i;
        (*this)[1] = j;
        (*this)[2] = k;
    }

    explicit vec3 (std::array<int, 3> a)
    {
        (*this)[0] = a[0];
        (*this)[1] = a[1];
        (*this)[2] = a[2];
    }

    int & operator[] (std::ptrdiff_t i)
    { return boost::yap::value(*this)[i]; }

    int const & operator[] (std::ptrdiff_t i) const
    { return boost::yap::value(*this)[i]; }

    template <typename T>
    vec3 & operator= (T const & t)
    {
        decltype(auto) expr = boost::yap::as_expr(t);
        (*this)[0] = boost::yap::evaluate(boost::yap::transform(expr, take_nth{0}));
        (*this)[1] = boost::yap::evaluate(boost::yap::transform(expr, take_nth{1}));
        (*this)[2] = boost::yap::evaluate(boost::yap::transform(expr, take_nth{2}));
        return *this;
    }

    void print() const
    {
        std::cout << '{' << (*this)[0]
                  << ", " << (*this)[1]
                  << ", " << (*this)[2]
                  << '}' << std::endl;
    }
};

// This is a stateful transform that keeps a running count of the terminals it
// has seen.
struct count_leaves_impl
{
    auto operator() (vec3_terminal const & expr)
    {
        value += 1;
        return expr;
    }

    int value = 0;
};

template <typename Expr>
int count_leaves (Expr const & expr)
{
    count_leaves_impl impl;
    boost::yap::transform(boost::yap::as_expr(expr), impl);
    return impl.value;
}


int main()
{
    vec3 a, b, c;

    c = 4;

    b[0] = -1;
    b[1] = -2;
    b[2] = -3;

    a = b + c;

    a.print();

    vec3 d;
    auto expr1 = b + c;
    d = expr1;
    d.print();

    int num = count_leaves(expr1);
    std::cout << num << std::endl;

    num = count_leaves(b + 3 * c);
    std::cout << num << std::endl;

    num = count_leaves(b + c * d);
    std::cout << num << std::endl;

    return 0;
}

#include <boost/yap/yap.hpp>

#include <vector>
#include <iostream>


struct take_nth
{
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                     std::vector<T> const & vec)
    { return boost::yap::make_terminal(vec[n]); }

    std::size_t n;
};

// A stateful transform that records whether all the std::vector<> terminals
// it has seen are equal to the given size.
struct equal_sizes_impl
{
    template <typename T>
    auto operator() (boost::yap::expr_tag<boost::yap::expr_kind::terminal>,
                     std::vector<T> const & vec)
    {
        auto const expr_size = vec.size();
        if (expr_size != size)
            value = false;
        return 0;
    }

    std::size_t const size;
    bool value;
};

template <typename Expr>
bool equal_sizes (std::size_t size, Expr const & expr)
{
    equal_sizes_impl impl{size, true};
    boost::yap::transform(boost::yap::as_expr(expr), impl);
    return impl.value;
}


// Assigns some expression e to the given vector by evaluating e elementwise,
// to avoid temporaries and allocations.
template <typename T, typename Expr>
std::vector<T> & assign (std::vector<T> & vec, Expr const & e)
{
    decltype(auto) expr = boost::yap::as_expr(e);
    assert(equal_sizes(vec.size(), expr));
    for (std::size_t i = 0, size = vec.size(); i < size; ++i) {
        vec[i] = boost::yap::evaluate(
            boost::yap::transform(boost::yap::as_expr(expr), take_nth{i}));
    }
    return vec;
}

// As assign() above, just using +=.
template <typename T, typename Expr>
std::vector<T> & operator+= (std::vector<T> & vec, Expr const & e)
{
    decltype(auto) expr = boost::yap::as_expr(e);
    assert(equal_sizes(vec.size(), expr));
    for (std::size_t i = 0, size = vec.size(); i < size; ++i) {
        vec[i] += boost::yap::evaluate(
            boost::yap::transform(boost::yap::as_expr(expr), take_nth{i}));
    }
    return vec;
}

// Define a type trait that identifies std::vectors.
template <typename T>
struct is_vector : std::false_type {};

template <typename T, typename A>
struct is_vector<std::vector<T, A>> : std::true_type {};

// Define all the expression-returning numeric operators we need.  Each will
// accept any std::vector<> as any of its arguments, and then any value in the
// remaining argument, if any -- some of the operators below are unary.
BOOST_YAP_USER_UDT_UNARY_OPERATOR(negate, boost::yap::expression, is_vector); // -
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(multiplies, boost::yap::expression, is_vector); // *
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(divides, boost::yap::expression, is_vector); // /
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(modulus, boost::yap::expression, is_vector); // %
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(plus, boost::yap::expression, is_vector); // +
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(minus, boost::yap::expression, is_vector); // -
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(less, boost::yap::expression, is_vector); // <
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(greater, boost::yap::expression, is_vector); // >
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(less_equal, boost::yap::expression, is_vector); // <=
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(greater_equal, boost::yap::expression, is_vector); // >=
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(equal_to, boost::yap::expression, is_vector); // ==
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(not_equal_to, boost::yap::expression, is_vector); // !=
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(logical_or, boost::yap::expression, is_vector); // ||
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(logical_and, boost::yap::expression, is_vector); // &&
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(bitwise_and, boost::yap::expression, is_vector); // &
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(bitwise_or, boost::yap::expression, is_vector); // |
BOOST_YAP_USER_UDT_ANY_BINARY_OPERATOR(bitwise_xor, boost::yap::expression, is_vector); // ^

int main()
{
    int i;
    int const n = 10;
    std::vector<int> a,b,c,d;
    std::vector<double> e(n);

    for (i = 0; i < n; ++i)
    {
        a.push_back(i);
        b.push_back(2*i);
        c.push_back(3*i);
        d.push_back(i);
    }

    // After this point, no allocations occur.

    assign(b, 2);
    assign(d, a + b * c);

    a += if_else(d < 30, b, c);

    assign(e, c);
    e += e - 4 / (c + 1);

    for (i = 0; i < n; ++i)
    {
        std::cout
            << " a(" << i << ") = " << a[i]
            << " b(" << i << ") = " << b[i]
            << " c(" << i << ") = " << c[i]
            << " d(" << i << ") = " << d[i]
            << " e(" << i << ") = " << e[i]
            << std::endl;
    }

    return 0;
}