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- <div class="section" id="implementing">
- <h1><a class="toc-backref" href="./dimensional-analysis.html#id45" name="implementing">Implementing Multiplication</a></h1>
- <p>Multiplication is a bit more complicated than addition and
- subtraction. So far, the dimensions of the arguments and results have
- all been identical, but when multiplying, the result will usually
- have different dimensions from either of the arguments. For
- multiplication, the relation:</p>
- <blockquote>
- (<em>x</em><sup>a</sup>)(<em>x</em><sup>b</sup>) == <em>x</em> <sup>(a + b)</sup></blockquote>
- <!-- @litre_translator.line_offset -= 7 -->
- <p>implies that the exponents of the result dimensions should be the
- sum of corresponding exponents from the argument
- dimensions. Division is similar, except that the sum is replaced by
- a difference.</p>
- <p>To combine corresponding elements from two sequences, we'll use
- MPL's <tt class="literal"><span class="pre">transform</span></tt> algorithm. <tt class="literal"><span class="pre">transform</span></tt> is a metafunction
- that iterates through two input sequences in parallel, passing an
- element from each sequence to an arbitrary binary metafunction, and
- placing the result in an output sequence.</p>
- <pre class="literal-block">
- template <class Sequence1, class Sequence2, class BinaryOperation>
- struct transform; // returns a Sequence
- </pre>
- <p>The signature above should look familiar if you're acquainted with the
- STL <tt class="literal"><span class="pre">transform</span></tt> algorithm that accepts two <em>runtime</em> sequences
- as inputs:</p>
- <pre class="literal-block">
- template <
- class InputIterator1, class InputIterator2
- , class OutputIterator, class BinaryOperation
- >
- void transform(
- InputIterator1 start1, InputIterator2 finish1
- , InputIterator2 start2
- , OutputIterator result, BinaryOperation func);
- </pre>
- <!-- @ example.wrap('namespace shield{','}')
- compile() -->
- <p>Now we just need to pass a <tt class="literal"><span class="pre">BinaryOperation</span></tt> that adds or
- subtracts in order to multiply or divide dimensions with
- <tt class="literal"><span class="pre">mpl::transform</span></tt>. If you look through the <a class="reference" href="./reference-manual.html">the MPL reference manual</a>, you'll
- come across <tt class="literal"><span class="pre">plus</span></tt> and <tt class="literal"><span class="pre">minus</span></tt> metafunctions that do just what
- you'd expect:</p>
- <pre class="literal-block">
- #include <boost/static_assert.hpp>
- #include <boost/mpl/plus.hpp>
- #include <boost/mpl/int.hpp>
- namespace mpl = boost::mpl;
- BOOST_STATIC_ASSERT((
- mpl::plus<
- mpl::int_<2>
- , mpl::int_<3>
- >::type::value == 5
- ));
- </pre>
- <!-- @ compile(pop = None) -->
- <div class="sidebar">
- <p class="sidebar-title first"><tt class="literal"><span class="pre">BOOST_STATIC_ASSERT</span></tt></p>
- <p>is a macro that causes a compilation error if its argument is
- false. The double parentheses are required because the C++
- preprocessor can't parse templates: it would otherwise be
- fooled by the comma into treating the condition as two separate
- macro arguments. Unlike its runtime analogue <tt class="literal"><span class="pre">assert(...)</span></tt>,
- <tt class="literal"><span class="pre">BOOST_STATIC_ASSERT</span></tt> can also be used at class scope,
- allowing us to put assertions in our metafunctions. See
- Chapter <a class="reference" href="./resources.html">8</a> for an in-depth discussion.</p>
- </div>
- <!-- @prefix.append('#include <boost/static_assert.hpp>') -->
- <p>At this point it might seem as though we have a solution, but we're
- not quite there yet. A naive attempt to apply the <tt class="literal"><span class="pre">transform</span></tt>
- algorithm in the implementation of <tt class="literal"><span class="pre">operator*</span></tt> yields a compiler
- error:</p>
- <pre class="literal-block">
- #include <boost/mpl/transform.hpp>
- template <class T, class D1, class D2>
- quantity<
- T
- , typename mpl::transform<D1,D2,mpl::plus>::type
- >
- operator*(quantity<T,D1> x, quantity<T,D2> y) { ... }
- </pre>
- <!-- @ example.replace('{ ... }',';')
- compile('all', pop = 1, expect_error = True)
- prefix +=['#include <boost/mpl/transform.hpp>'] -->
- <!-- @litre_translator.line_offset -= 7 -->
- <p>It fails because the protocol says that metafunction arguments
- must be types, and <tt class="literal"><span class="pre">plus</span></tt> is not a type, but a class template.
- Somehow we need to make metafunctions like <tt class="literal"><span class="pre">plus</span></tt> fit the
- metadata mold.</p>
- <p>One natural way to introduce polymorphism between metafunctions and
- metadata is to employ the wrapper idiom that gave us polymorphism
- between types and integral constants. Instead of a nested integral
- constant, we can use a class template nested within a
- <strong>metafunction class</strong>:</p>
- <pre class="literal-block">
- struct plus_f
- {
- template <class T1, class T2>
- struct apply
- {
- typedef typename mpl::plus<T1,T2>::type type;
- };
- };
- </pre>
- <div class="admonition-definition admonition">
- <p class="admonition-title first">Definition</p>
- <p>A <strong>Metafunction Class</strong> is a class with a publicly accessible
- nested metafunction called <tt class="literal"><span class="pre">apply</span></tt>.</p>
- </div>
- <p>Whereas a metafunction is a template but not a type, a
- metafunction class wraps that template within an ordinary
- non-templated class, which <em>is</em> a type. Since metafunctions
- operate on and return types, a metafunction class can be passed as
- an argument to, or returned from, another metafunction.</p>
- <p>Finally, we have a <tt class="literal"><span class="pre">BinaryOperation</span></tt> type that we can pass to
- <tt class="literal"><span class="pre">transform</span></tt> without causing a compilation error:</p>
- <pre class="literal-block">
- template <class T, class D1, class D2>
- quantity<
- T
- , typename mpl::transform<D1,D2,<strong>plus_f</strong>>::type // new dimensions
- >
- operator*(quantity<T,D1> x, quantity<T,D2> y)
- {
- typedef typename mpl::transform<D1,D2,<strong>plus_f</strong>>::type dim;
- return quantity<T,dim>( x.value() * y.value() );
- }
- </pre>
- <p>Now, if we want to compute the force exterted by gravity on a 5 kilogram
- laptop computer, that's just the acceleration due to gravity (9.8
- m/sec<sup>2</sup>) times the mass of the laptop:</p>
- <pre class="literal-block">
- quantity<float,mass> m(5.0f);
- quantity<float,acceleration> a(9.8f);
- std::cout << "force = " << (m * a).value();
- </pre>
- <!-- @example.wrap('#include <iostream>\nvoid ff() {', '}')
- compile('all', pop = 1) -->
- <p>Our <tt class="literal"><span class="pre">operator*</span></tt> multiplies the runtime values (resulting in
- 6.0f), and our metaprogram code uses <tt class="literal"><span class="pre">transform</span></tt> to sum the
- meta-sequences of fundamental dimension exponents, so that the
- result type contains a representation of a new list of exponents,
- something like:</p>
- <pre class="literal-block">
- mpl::vector_c<int,1,1,-2,0,0,0,0>
- </pre>
- <!-- @example.wrap('''
- #include <boost/mpl/vector_c.hpp>
- typedef''', 'xxxx;')
- compile() -->
- <!-- @litre_translator.line_offset -= 7 -->
- <p>However, if we try to write:</p>
- <pre class="literal-block">
- quantity<float,force> f = m * a;
- </pre>
- <!-- @ ma_function_args = '(quantity<float,mass> m, quantity<float,acceleration> a)'
- example.wrap('void bogus%s {' % ma_function_args, '}')
- compile('all', pop = 1, expect_error = True) -->
- <!-- @litre_translator.line_offset -= 7 -->
- <p>we'll run into a little problem. Although the result of
- <tt class="literal"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">a</span></tt> does indeed represent a force with exponents of mass,
- length, and time 1, 1, and -2 respectively, the type returned by
- <tt class="literal"><span class="pre">transform</span></tt> isn't a specialization of <tt class="literal"><span class="pre">vector_c</span></tt>. Instead,
- <tt class="literal"><span class="pre">transform</span></tt> works generically on the elements of its inputs and
- builds a new sequence with the appropriate elements: a type with
- many of the same sequence properties as
- <tt class="literal"><span class="pre">mpl::vector_c<int,1,1,-2,0,0,0,0></span></tt>, but with a different C++ type
- altogether. If you want to see the type's full name, you can try
- to compile the example yourself and look at the error message, but
- the exact details aren't important. The point is that
- <tt class="literal"><span class="pre">force</span></tt> names a different type, so the assignment above will fail.</p>
- <p>In order to resolve the problem, we can add an implicit conversion
- from the multiplication's result type to <tt class="literal"><span class="pre">quantity<float,force></span></tt>.
- Since we can't predict the exact types of the dimensions involved
- in any computation, this conversion will have to be templated,
- something like:</p>
- <pre class="literal-block">
- template <class T, class Dimensions>
- struct quantity
- {
- // converting constructor
- template <class OtherDimensions>
- quantity(quantity<T,OtherDimensions> const& rhs)
- : m_value(rhs.value())
- {
- }
- ...
- </pre>
- <!-- @ example.append("""
- explicit quantity(T x)
- : m_value(x)
- {}
- T value() const { return m_value; }
- private:
- T m_value;
- };""")
- stack[quantity_declaration] = example
- ignore() -->
- <p>Unfortunately, such a general conversion undermines our whole
- purpose, allowing nonsense such as:</p>
- <pre class="literal-block">
- // Should yield a force, not a mass!
- quantity<float,mass> bogus = m * a;
- </pre>
- <!-- @ example.wrap('void bogus2%s {' % ma_function_args, '}')
- bogus_example = example
- compile('all', pop = 1) -->
- <p>We can correct that problem using another MPL algorithm,
- <tt class="literal"><span class="pre">equal</span></tt>, which tests that two sequences have the same elements:</p>
- <pre class="literal-block">
- template <class OtherDimensions>
- quantity(quantity<T,OtherDimensions> const& rhs)
- : m_value(rhs.value())
- {
- BOOST_STATIC_ASSERT((
- mpl::equal<Dimensions,OtherDimensions>::type::value
- ));
- }
- </pre>
- <!-- @ example.wrap('''
- #include <boost/mpl/equal.hpp>
- template <class T, class Dimensions>
- struct quantity
- {
- explicit quantity(T x)
- : m_value(x)
- {}
- T value() const { return m_value; }
- ''','''
- private:
- T m_value;
- };''')
- stack[quantity_declaration] = example
- stack[-1] = bogus_example
- compile('all', pop = 1, expect_error = True) -->
- <p>Now, if the dimensions of the two quantities fail to match, the
- assertion will cause a compilation error.</p>
- </div>
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