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<td class="header-group page-location"><a href="../index.html" class="navigation-link">Front Page</a> / <a href="./tutorial-metafunctions.html" class="navigation-link">Tutorial: Metafunctions and Higher-Order Metaprogramming</a> / <a href="./higher-order.html" class="navigation-link">Higher-Order Metafunctions</a></td>
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<div class="section" id="higher-order">
<h1><a class="toc-backref" href="./tutorial-metafunctions.html#id47" name="higher-order">Higher-Order Metafunctions</a></h1>
<p>In the previous section we used two different forms —
metafunction classes and placeholder expressions — 
to pass and return metafunctions just like any other metadata.
Bundling metafunctions into &quot;first class metadata&quot; allows
<tt class="literal"><span class="pre">transform</span></tt> to perform an infinite variety of different
operations: in our case, multiplication and division of dimensions.
Though the idea of using functions to manipulate other functions
may seem simple, its great power and flexibility <a class="citation-reference" href="#hudak89" id="id9" name="id9">[Hudak89]</a> has
earned it a fancy title: <strong>higher-order functional programming</strong>.
A function that operates on another function is known as a 
<strong>higher-order function</strong>.  It follows that <tt class="literal"><span class="pre">transform</span></tt> is a
higher-order 
metafunction: a metafunction that operates on another metafunction.</p>
<table class="citation" frame="void" id="hudak89" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id9" name="hudak89">[Hudak89]</a></td><td>Paul Hudak. &quot;Conception, Evolution, and Application of
Functional Programming Languages,&quot; ACM Computing Surveys 21,
no. 3 Pages: 359 - 411. New York: ACM Press. 1989.
ISSN:0360-0300. http://doi.acm.org/10.1145/72551.72554.</td></tr>
</tbody>
</table>
<p>Now that we've seen the power of higher-order metafunctions at
work, it would be good to be able to create new ones.  In order to
explore the basic mechanisms, let's try a simple example.  Our task
is to write a metafunction called <tt class="literal"><span class="pre">twice</span></tt>, which — given a unary
metafunction <em>f</em> and arbitrary metadata <em>x</em> — computes:</p>
<blockquote>
<em>twice</em>(<em>f</em>, <em>x</em>) := <em>f</em>(<em>f</em>(<em>x</em>))</blockquote>
<p>This might seem like a trivial example, and in fact it is.  You
won't find much use for <tt class="literal"><span class="pre">twice</span></tt> in real code.  We hope you'll
bear with us anyway:  Because it doesn't do much more than accept
and invoke a metafunction, <tt class="literal"><span class="pre">twice</span></tt> captures all the essential
elements of &quot;higher-orderness&quot; without any distracting details.</p>
<p>If <em>f</em> is a metafunction class, the definition of <tt class="literal"><span class="pre">twice</span></tt> is
straightforward:</p>
<pre class="literal-block">
template &lt;class F, class X&gt;
struct twice
{
    typedef typename F::template apply&lt;X&gt;::type once;    // f(x)
    typedef typename F::template apply&lt;once&gt;::type type; // f(f(x))
};
</pre>
<!-- @ prefix.append(
 '''#include <boost/type_traits/add_pointer.hpp>
    #include <boost/static_assert.hpp>
    #include <boost/type_traits/is_same.hpp>''')

twice_test = '''
#include <boost/mpl/assert.hpp>
struct add_pointer_f
{ 
    template <class T> struct apply : boost::add_pointer<T>
    {};
};
BOOST_MPL_ASSERT((boost::is_same<twice<add_pointer_f,int>::type,int**>));
'''
example.append(twice_test)
compile() -->
<!-- @litre_translator.line_offset -= 7 -->
<p>Or, applying metafunction forwarding:</p>
<pre class="literal-block">
template &lt;class F, class X&gt;
struct twice
  : F::template apply&lt;
       typename F::template apply&lt;X&gt;::type
    &gt;
{};
</pre>
<!-- @ example.append(twice_test)
compile() -->
<div class="admonition-c-language-note admonition">
<p class="admonition-title first">C++ Language Note</p>
<p>The C++ standard requires the <tt class="literal"><span class="pre">template</span></tt> keyword when we use a
<strong>dependent name</strong> that refers to a member template.
<tt class="literal"><span class="pre">F::apply</span></tt> may or may not name a template, <em>depending</em> on the
particular <tt class="literal"><span class="pre">F</span></tt> that is passed.  See <a class="reference" href="./resources.html">the book's</a> Appendix B for more
information about <tt class="literal"><span class="pre">template</span></tt>.</p>
</div>
<p>Given the need to sprinkle our code with the <tt class="literal"><span class="pre">template</span></tt> keyword,
it would be nice to reduce the syntactic burden of invoking
metafunction classes.  As usual, the solution is to factor the
pattern into a metafunction:</p>
<pre class="literal-block">
template &lt;class UnaryMetaFunctionClass, class Arg&gt;
struct apply1
  : UnaryMetaFunctionClass::template apply&lt;Arg&gt;
{};
</pre>
<p>Now <tt class="literal"><span class="pre">twice</span></tt> is just:</p>
<pre class="literal-block">
template &lt;class F, class X&gt;
struct twice
  : apply1&lt;F, typename apply1&lt;F,X&gt;::type&gt;
{};
</pre>
<p>To see <tt class="literal"><span class="pre">twice</span></tt> at work, we can apply it to a little metafunction
class built around the <tt class="literal"><span class="pre">add_pointer</span></tt> metafunction:</p>
<pre class="literal-block">
struct add_pointer_f
{
    template &lt;class T&gt;
    struct apply : boost::add_pointer&lt;T&gt; {};
};
</pre>
<!-- @litre_translator.line_offset -= 7 -->
<p>Now we can use <tt class="literal"><span class="pre">twice</span></tt> with <tt class="literal"><span class="pre">add_pointer_f</span></tt> to build
pointers-to-pointers:</p>
<pre class="literal-block">
BOOST_STATIC_ASSERT((
    boost::is_same&lt;
         twice&lt;add_pointer_f, int&gt;::type
       , int**
    &gt;::value
));
</pre>
<!-- @ apply1 = stack[-4]
add_pointer_f = stack[-2]
compile('all', pop = 0) -->
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