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<div class="section" id="implementing-division">
<h1><a class="toc-backref" href="./dimensional-analysis.html#id46" name="implementing-division">Implementing Division</a></h1>
<p>Division is similar to multiplication, but instead of adding
exponents, we must subtract them.  Rather than writing out a near
duplicate of <tt class="literal"><span class="pre">plus_f</span></tt>, we can use the following trick to make
<tt class="literal"><span class="pre">minus_f</span></tt> much simpler:</p>
<pre class="literal-block">
struct minus_f
{
    template &lt;class T1, class T2&gt;
    struct apply
      : mpl::minus&lt;T1,T2&gt; {};
};
</pre>
<!-- @ # The following is OK because we showed how to get at mpl_plus
prefix.append('#include <boost/mpl/minus.hpp>')
compile(1)  -->
<p>Here <tt class="literal"><span class="pre">minus_f::apply</span></tt> uses inheritance to expose the nested
<tt class="literal"><span class="pre">type</span></tt> of its base class, <tt class="literal"><span class="pre">mpl::minus</span></tt>, so we don't have to
write:</p>
<pre class="literal-block">
typedef typename ...::type type
</pre>
<!-- @ignore() -->
<p>We don't have to write
<tt class="literal"><span class="pre">typename</span></tt> here (in fact, it would be illegal), because the
compiler knows that dependent names in <tt class="literal"><span class="pre">apply</span></tt>'s initializer
list must be base classes. <a class="footnote-reference" href="#plus-too" id="id7" name="id7">[2]</a>  This powerful
simplification is known as <strong>metafunction forwarding</strong>; we'll apply
it often as the book goes on. <a class="footnote-reference" href="#edg" id="id8" name="id8">[3]</a></p>
<table class="footnote" frame="void" id="plus-too" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id7" name="plus-too">[2]</a></td><td>In case you're wondering, the same approach could
have been applied to <tt class="literal"><span class="pre">plus_f</span></tt>, but since it's a little subtle,
we introduced the straightforward but verbose formulation
first.</td></tr>
</tbody>
</table>
<table class="footnote" frame="void" id="edg" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id8" name="edg">[3]</a></td><td>Users of EDG-based compilers should consult <a class="reference" href="./resources.html">the book's</a> Appendix C
for a caveat about metafunction forwarding.  You can tell whether
you have an EDG compiler by checking the preprocessor symbol
<tt class="literal"><span class="pre">__EDG_VERSION__</span></tt>, which is defined by all EDG-based compilers.</td></tr>
</tbody>
</table>
<p>Syntactic tricks notwithstanding, writing trivial classes to wrap
existing metafunctions is going to get boring pretty quickly.  Even
though the definition of <tt class="literal"><span class="pre">minus_f</span></tt> was far less verbose than that
of <tt class="literal"><span class="pre">plus_f</span></tt>, it's still an awful lot to type.  Fortunately, MPL gives
us a <em>much</em> simpler way to pass metafunctions around.  Instead of
building a whole metafunction class, we can invoke <tt class="literal"><span class="pre">transform</span></tt>
this way:</p>
<pre class="literal-block">
typename mpl::transform&lt;D1,D2, <strong>mpl::minus&lt;_1,_2&gt;</strong> &gt;::type
</pre>
<!-- @# Make it harmless but legit C++ so we can syntax check later
example.wrap('template <class D1,class D2>', 'fff(D1,D2);')

# We explain placeholders below, so we can henceforth use them
# without qualification -->
<p>Those funny looking arguments (<tt class="literal"><span class="pre">_1</span></tt> and <tt class="literal"><span class="pre">_2</span></tt>) are known as
<strong>placeholders</strong>, and they signify that when the <tt class="literal"><span class="pre">transform</span></tt>'s
<tt class="literal"><span class="pre">BinaryOperation</span></tt> is invoked, its first and second arguments will
be passed on to <tt class="literal"><span class="pre">minus</span></tt> in the positions indicated by <tt class="literal"><span class="pre">_1</span></tt> and
<tt class="literal"><span class="pre">_2</span></tt>, respectively.  The whole type <tt class="literal"><span class="pre">mpl::minus&lt;_1,_2&gt;</span></tt> is
known as a <strong>placeholder expression</strong>.</p>
<div class="note">
<p class="admonition-title first">Note</p>
<p>MPL's placeholders are in the <tt class="literal"><span class="pre">mpl::placeholders</span></tt>
namespace and defined in <tt class="literal"><span class="pre">boost/mpl/placeholders.hpp</span></tt>.  In
this book we will usually assume that you have written:</p>
<pre class="literal-block">
#include&lt;boost/mpl/placeholders.hpp&gt;
using namespace mpl::placeholders;
</pre>
<p>so that they can be accessed without qualification.</p>
</div>
<!-- @ prefix.append(str(example)) # move to common prefix
ignore() -->
<p>Here's our division operator written using placeholder
expressions:</p>
<pre class="literal-block">
template &lt;class T, class D1, class D2&gt;
quantity&lt; 
    T
  , typename mpl::transform&lt;D1,D2,<strong>mpl::minus&lt;_1,_2&gt;</strong> &gt;::type
&gt;
operator/(quantity&lt;T,D1&gt; x, quantity&lt;T,D2&gt; y)
{
   typedef typename 
     mpl::transform&lt;D1,D2,<strong>mpl::minus&lt;_1,_2&gt;</strong> &gt;::type dim;

   return quantity&lt;T,dim&gt;( x.value() / y.value() );
}
</pre>
<!-- @compile('all', pop = 1) -->
<p>This code is considerably simpler.  We can simplify it even further
by factoring the code that calculates the new dimensions into its
own metafunction:</p>
<pre class="literal-block">
template &lt;class D1, class D2&gt;
struct <strong>divide_dimensions</strong>
  : mpl::transform&lt;D1,D2,mpl::minus&lt;_1,_2&gt; &gt; // forwarding again
{};

template &lt;class T, class D1, class D2&gt;
quantity&lt;T, typename <strong>divide_dimensions&lt;D1,D2&gt;</strong>::type&gt;
operator/(quantity&lt;T,D1&gt; x, quantity&lt;T,D2&gt; y)
{
   return quantity&lt;T, typename <strong>divide_dimensions&lt;D1,D2&gt;</strong>::type&gt;(
      x.value() / y.value());
}
</pre>
<!-- @compile('all', pop = None) -->
<p>Now we can verify our &quot;force-on-a-laptop&quot; computation by reversing
it, as follows:</p>
<pre class="literal-block">
quantity&lt;float,mass&gt; m2 = f/a;
float rounding_error = std::abs((m2 - m).value());
</pre>
<!-- @example.wrap('''
#include <cassert>
#include <cmath>
int main()
{
    quantity<float,mass> m(5.0f);
    quantity<float,acceleration> a(9.8f);
    quantity<float,force> f = m * a;
''','''
    assert(rounding_error < .001);
}''')

dimensional_analysis = stack[:-1] # save for later

run('all') -->
<p>If we got everything right, <tt class="literal"><span class="pre">rounding_error</span></tt> should be very close
to zero.  These are boring calculations, but they're just the sort
of thing that could ruin a whole program (or worse) if you got them
wrong.  If we had written <tt class="literal"><span class="pre">a/f</span></tt> instead of <tt class="literal"><span class="pre">f/a</span></tt>, there would have
been a compilation error, preventing a mistake from propagating
throughout our program.</p>
</div>

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