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From Chapter 3 of C++ Template Metaprogramming by David Abrahams and Aleksey Gurtovoy. ISBN: 0321227255. Copyright (c) 2005 by Pearson Education, Inc. Reprinted with permission.
With the foundation laid so far, we're ready to explore one of the most basic uses of template metaprogramming techniques: adding static type checking to traditionally unchecked operations. We'll look at a practical example from science and engineering that can find applications in almost any numerical code. Along the way you'll learn some important new concepts and get a taste of metaprogramming at a high level using the MPL.
The first rule of doing physical calculations on paper is that the numbers being manipulated don't stand alone: most quantities have attached dimensions, to be ignored at our peril. As computations become more complex, keeping track of dimensions is what keeps us from inadvertently assigning a mass to what should be a length or adding acceleration to velocity—it establishes a type system for numbers.
Manual checking of types is tedious, and as a result, it's also errorprone. When human beings become bored, their attention wanders and they tend to make mistakes. Doesn't type checking seem like the sort of job a computer might be good at, though? If we could establish a framework of C++ types for dimensions and quantities, we might be able to catch errors in formulae before they cause serious problems in the real world.
Preventing quantities with different dimensions from interoperating isn't hard; we could simply represent dimensions as classes that only work with dimensions of the same type. What makes this problem interesting is that different dimensions can be combined, via multiplication or division, to produce arbitrarily complex new dimensions. For example, take Newton's law, which relates force to mass and acceleration:
F = ma
Since mass and acceleration have different dimensions, the dimensions of force must somehow capture their combination. In fact, the dimensions of acceleration are already just such a composite, a change in velocity over time:
dv/dt
Since velocity is just change in distance (l) over time (t), the fundamental dimensions of acceleration are:
(l/t)/t = l/t^{2}
And indeed, acceleration is commonly measured in "meters per second squared." It follows that the dimensions of force must be:
ml/t^{2}
and force is commonly measured in kg(m/s^{2}), or "kilogrammeters per second squared." When multiplying quantities of mass and acceleration, we multiply their dimensions as well and carry the result along, which helps us to ensure that the result is meaningful. The formal name for this bookkeeping is dimensional analysis, and our next task will be to implement its rules in the C++ type system. John Barton and Lee Nackman were the first to show how to do this in their seminal book, Scientific and Engineering C++ [BNack94]. We will recast their approach here in metaprogramming terms.
[BNack94]  John J. Barton and Lee R. Nackman, Scientific and Engineering C++: an introduction with advanced techniques and examples ISBN 0201533936, Addison Wesley, Reading Massachusetts, 1994 
An international standard called Système International d'Unites (SI), breaks every quantity down into a combination of the dimensions mass, length (or position), time, charge, temperature, intensity, and angle. To be reasonably general, our system would have to be able to represent seven or more fundamental dimensions. It also needs the ability to represent composite dimensions which, like force, are built through multiplication or division of the fundamental ones.
In general, a composite dimension is the product of powers of
fundamental dimensions.[1] If we were going to represent
these powers for manipulation at runtime, we could use an array of
seven int
s, with each position in the array holding the power
of a different fundamental dimension:
typedef int dimension[7]; // m l t ... dimension const mass = {1, 0, 0, 0, 0, 0, 0}; dimension const length = {0, 1, 0, 0, 0, 0, 0}; dimension const time = {0, 0, 1, 0, 0, 0, 0}; ...
In that representation, force would be:
dimension const force = {1, 1, 2, 0, 0, 0, 0};
that is, mlt^{2}. However, if we want to get dimensions into the type system, these arrays won't do the trick: they're all the same type! Instead, we need types that themselves represent sequences of numbers, so that two masses have the same type and a mass is a different type from a length.
Fortunately, the MPL provides us with a collection of type sequences. For example, we can build a sequence of the builtin signed integral types this way:
#include <boost/mpl/vector.hpp> typedef boost::mpl::vector< signed char, short, int, long> signed_types;
How can we use a type sequence to represent numbers? Just as
numerical metafunctions pass and return wrapper types having a
nested ::value
, so numerical sequences are really sequences of
wrapper types (another example of polymorphism). To make this sort
of thing easier, MPL supplies the int_<N>
class template, which
presents its integral argument as a nested ::value
:
#include <boost/mpl/int.hpp> namespace mpl = boost::mpl; // namespace alias static int const five = mpl::int_<5>::value;
namespace alias = namespacename;
declares alias to be a synonym for namespacename. Many
examples in this book will use mpl::
to indicate
boost::mpl::
, but will omit the alias that makes it legal
C++.
In fact, the library contains a whole suite of integral constant
wrappers such as long_
and bool_
, each one wrapping a
different type of integral constant within a class template.
Now we can build our fundamental dimensions:
typedef mpl::vector< mpl::int_<1>, mpl::int_<0>, mpl::int_<0>, mpl::int_<0> , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> > mass; typedef mpl::vector< mpl::int_<0>, mpl::int_<1>, mpl::int_<0>, mpl::int_<0> , mpl::int_<0>, mpl::int_<0>, mpl::int_<0> > length; ...
Whew! That's going to get tiring pretty quickly. Worse, it's hard to read and verify: The essential information, the powers of each fundamental dimension, is buried in repetitive syntactic "noise." Accordingly, MPL supplies integral sequence wrappers that allow us to write:
#include <boost/mpl/vector_c.hpp> typedef mpl::vector_c<int,1,0,0,0,0,0,0> mass; typedef mpl::vector_c<int,0,1,0,0,0,0,0> length; // or position typedef mpl::vector_c<int,0,0,1,0,0,0,0> time; typedef mpl::vector_c<int,0,0,0,1,0,0,0> charge; typedef mpl::vector_c<int,0,0,0,0,1,0,0> temperature; typedef mpl::vector_c<int,0,0,0,0,0,1,0> intensity; typedef mpl::vector_c<int,0,0,0,0,0,0,1> angle;
Even though they have different types, you can think of these
mpl::vector_c
specializations as being equivalent to the more
verbose versions above that use mpl::vector
.
If we want, we can also define a few composite dimensions:
// base dimension: m l t ... typedef mpl::vector_c<int,0,1,1,0,0,0,0> velocity; // l/t typedef mpl::vector_c<int,0,1,2,0,0,0,0> acceleration; // l/(t^{2}) typedef mpl::vector_c<int,1,1,1,0,0,0,0> momentum; // ml/t typedef mpl::vector_c<int,1,1,2,0,0,0,0> force; // ml/(t^{2})
And, incidentally, the dimensions of scalars (like pi) can be described as:
typedef mpl::vector_c<int,0,0,0,0,0,0,0> scalar;
The types listed above are still pure metadata; to typecheck real
computations we'll need to somehow bind them to our runtime data.
A simple numeric value wrapper, parameterized on the datatype T
and on its dimensions, fits the bill:
template <class T, class Dimensions> struct quantity { explicit quantity(T x) : m_value(x) {} T value() const { return m_value; } private: T m_value; };
Now we have a way to represent numbers associated with dimensions. For instance, we can say:
quantity<float,length> l( 1.0f ); quantity<float,mass> m( 2.0f );
Note that Dimensions
doesn't appear anywhere in the definition
of quantity
outside the template parameter list; its only
role is to ensure that l
and m
have different types.
Because they do, we cannot make the mistake of assigning a length
to a mass:
m = l; // compiletime type error
We can now easily write the rules for addition and subtraction, since the dimensions of the arguments must always match.
template <class T, class D> quantity<T,D> operator+(quantity<T,D> x, quantity<T,D> y) { return quantity<T,D>(x.value() + y.value()); } template <class T, class D> quantity<T,D> operator(quantity<T,D> x, quantity<T,D> y) { return quantity<T,D>(x.value()  y.value()); }
These operators enable us to write code like:
quantity<float,length> len1( 1.0f ); quantity<float,length> len2( 2.0f ); len1 = len1 + len2; // OK
but prevent us from trying to add incompatible dimensions:
len1 = len2 + quantity<float,mass>( 3.7f ); // error
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:
(x^{a})(x^{b}) == x ^{(a + b)}
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.
To combine corresponding elements from two sequences, we'll use
MPL's transform
algorithm. transform
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.
template <class Sequence1, class Sequence2, class BinaryOperation> struct transform; // returns a Sequence
The signature above should look familiar if you're acquainted with the
STL transform
algorithm that accepts two runtime sequences
as inputs:
template < class InputIterator1, class InputIterator2 , class OutputIterator, class BinaryOperation > void transform( InputIterator1 start1, InputIterator2 finish1 , InputIterator2 start2 , OutputIterator result, BinaryOperation func);
Now we just need to pass a BinaryOperation
that adds or
subtracts in order to multiply or divide dimensions with
mpl::transform
. If you look through the MPL reference manual, you'll
come across plus
and minus
metafunctions that do just what
you'd expect:
#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 ));
BOOST_STATIC_ASSERT
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 assert(...)
,
BOOST_STATIC_ASSERT
can also be used at class scope,
allowing us to put assertions in our metafunctions. See
Chapter 8 for an indepth discussion.
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 transform
algorithm in the implementation of operator*
yields a compiler
error:
#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) { ... }
It fails because the protocol says that metafunction arguments
must be types, and plus
is not a type, but a class template.
Somehow we need to make metafunctions like plus
fit the
metadata mold.
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 metafunction class:
struct plus_f { template <class T1, class T2> struct apply { typedef typename mpl::plus<T1,T2>::type type; }; };
A Metafunction Class is a class with a publicly accessible
nested metafunction called apply
.
Whereas a metafunction is a template but not a type, a metafunction class wraps that template within an ordinary nontemplated class, which is a type. Since metafunctions operate on and return types, a metafunction class can be passed as an argument to, or returned from, another metafunction.
Finally, we have a BinaryOperation
type that we can pass to
transform
without causing a compilation error:
template <class T, class D1, class D2> quantity< T , typename mpl::transform<D1,D2,plus_f>::type // new dimensions > operator*(quantity<T,D1> x, quantity<T,D2> y) { typedef typename mpl::transform<D1,D2,plus_f>::type dim; return quantity<T,dim>( x.value() * y.value() ); }
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^{2}) times the mass of the laptop:
quantity<float,mass> m(5.0f); quantity<float,acceleration> a(9.8f); std::cout << "force = " << (m * a).value();
Our operator*
multiplies the runtime values (resulting in
6.0f), and our metaprogram code uses transform
to sum the
metasequences of fundamental dimension exponents, so that the
result type contains a representation of a new list of exponents,
something like:
vector_c<int,1,1,2,0,0,0,0>
However, if we try to write:
quantity<float,force> f = m * a;
we'll run into a little problem. Although the result of
m * a
does indeed represent a force with exponents of mass,
length, and time 1, 1, and 2 respectively, the type returned by
transform
isn't a specialization of vector_c
. Instead,
transform
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
vector_c<int,1,1,2,0,0,0,0>
, 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
force
names a different type, so the assignment above will fail.
In order to resolve the problem, we can add an implicit conversion
from the multiplication's result type to quantity<float,force>
.
Since we can't predict the exact types of the dimensions involved
in any computation, this conversion will have to be templated,
something like:
template <class T, class Dimensions> struct quantity { // converting constructor template <class OtherDimensions> quantity(quantity<T,OtherDimensions> const& rhs) : m_value(rhs.value()) { } ...
Unfortunately, such a general conversion undermines our whole purpose, allowing nonsense such as:
// Should yield a force, not a mass! quantity<float,mass> bogus = m * a;
Luckily, we can correct that problem using another MPL algorithm,
equal
, which tests that two sequences have the same elements:
template <class OtherDimensions> quantity(quantity<T,OtherDimensions> const& rhs) : m_value(rhs.value()) { BOOST_STATIC_ASSERT(( mpl::equal<Dimensions,OtherDimensions>::type::value )); }
Now, if the dimensions of the two quantities fail to match, the assertion will cause a compilation error.
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