David Abrahams <[EMAIL PROTECTED]> writes:
> I've attached the ReST/LitRE source to our sample chapter "A Deeper
> Look at Metafunctions" for your amusement.
Whoops; forgot of course!
-----------------------
|metafunctions-title|
-----------------------
.. include:: header.rst
.. raw:: latex
\renewcommand{\thetable}{3.\arabic{table}}
\renewcommand{\thepage}{3-\arabic{page}}
.. sectnum::
:prefix: 3.
:depth: 3
With the foundation laid so far, we're ready to explore one of the
most basic uses for 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.
Dimensional Analysis
====================
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
error-prone. 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*\ :sup:`2`
And indeed, acceleration is commonly measured in "meters per second
squared." It follows that the dimensions of force must be:
*ml*\ /*t*\ :sup:`2`
.. include:: nopara.rst
and force is commonly measured in kg(m/s\ :sup:`2`), or
"kilogram-meters 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++* [BN94]_. We will recast their approach here in
metaprogramming terms.
.. [BN94] John J. Barton and Lee R. Nackman. *Scientific and
Engineering C++: an Introduction with Advanced Techniques and
Examples.* Reading, MA: Addison Wesley. ISBN
0-201-53393-6. 1994.
Representing Dimensions
-----------------------
An international standard called *Syst�me
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 that, 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. [#divisor]_ 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:
.. parsed-literal::
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};
...
.. [#divisor] Divisors just contribute negative exponents, since
1/*x* = *x*\ :sup:`-1`.
In that representation, force would be::
dimension const force = {1, 1, -2, 0, 0, 0, 0};
.. @compile(2)
.. include:: nopara.rst
that is, *mlt*\ :sup:`-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 built-in
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;
.. sidebar:: Namespace Aliases
.. line-block::
``namespace`` *alias* ``=`` *namespace-name*\ ``;``
declares *alias* to be a synonym for *namespace-name*. Many
examples in this book will use ``mpl::`` to indicate
``boost::mpl::``, but will omit the alias that makes it legal
C++.
.. @ignore() # nonsense isn't worth testing
prefix +=['''
#include <boost/mpl/int.hpp>
#include <boost/mpl/vector.hpp>
''']
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;
...
.. @ # We explained about the implicit namespace alias above
prefix.append("""
namespace boost{namespace mpl {}}
namespace mpl = boost::mpl;
""")
compile('all')
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:
.. parsed-literal::
// 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\ :sup:`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\ :sup:`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;
.. @stack[0].replace('hpp>', 'hpp>\nnamespace {')
stack[0].append('}')
compile('all', pop = None)
Representing Quantities
-----------------------
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 number type ``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;
};
.. @ quantity_declaration = len(stack) - 1 # Remember position for later
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; // compile-time type error
.. @ example.wrap('void f() {', '}')
compile('all', pop = 1, expect_error = True)
Implementing Addition and Subtraction
-------------------------------------
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());
}
.. @ example.append('void test1() {') # function prologue
These operators enable us to write code like:
.. parsed-literal::
quantity<float,\ **length**> len1( 1.0f );
quantity<float,\ **length**> len2( 2.0f );
len1 = len1 + len2; // OK
.. include:: nopara.rst
but prevent us from trying to add incompatible dimensions:
.. parsed-literal::
len1 = len2 + quantity<float,\ **mass**>( 3.7f ); // **error**
.. @ stack[-1].append('}') # last 2 examples together
compile('all', pop = 1, expect_error = True)
stack[-1].append('}') # just the prior example
compile('all', pop = None)
Implementing Multiplication
---------------------------
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*\ :sup:`a`)(*x*\ :sup:`b`) == *x* :sup:`(a + b)`
.. include:: nopara.rst
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);
.. @ example.wrap('namespace shield{','}')
compile()
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 |reference|, 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
));
.. @ compile(pop = None)
.. sidebar:: ``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 |diagnostics| for an in-depth discussion.
.. @prefix.append('#include <boost/static_assert.hpp>')
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) { ... }
.. @ example.replace('{ ... }',';')
compile('all', pop = 1, expect_error = True)
prefix +=['#include <boost/mpl/transform.hpp>']
.. include:: nopara.rst
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;
};
};
.. admonition:: Definition
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
non-templated 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:
.. parsed-literal::
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\ :sup:`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();
.. @example.wrap('#include <iostream>\nvoid ff() {', '}')
compile('all', pop = 1)
Our ``operator*`` multiplies the runtime values (resulting in
6.0f), and our metaprogram code uses ``transform`` 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::
mpl::vector_c<int,1,1,-2,0,0,0,0>
.. @example.wrap('''
#include <boost/mpl/vector_c.hpp>
typedef''', 'xxxx;')
compile()
.. include:: nopara.rst
However, if we try to write::
quantity<float,force> f = m * a;
.. @ 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)
.. include:: nopara.rst
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
``mpl::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())
{
}
...
.. @ example.append("""
explicit quantity(T x)
: m_value(x)
{}
T value() const { return m_value; }
private:
T m_value;
};""")
stack[quantity_declaration] = example
ignore()
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;
.. @ example.wrap('void bogus2%s {' % ma_function_args, '}')
bogus_example = example
compile('all', pop = 1)
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
));
}
.. @ 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)
Now, if the dimensions of the two quantities fail to match, the
assertion will cause a compilation error.
Implementing Division
---------------------
Division is similar to multiplication, but instead of adding
exponents, we must subtract them. Rather than writing out a near
duplicate of ``plus_f``, we can use the following trick to make
``minus_f`` much simpler::
struct minus_f
{
template <class T1, class T2>
struct apply
: mpl::minus<T1,T2> {};
};
.. @ # The following is OK because we showed how to get at mpl_plus
prefix.append('#include <boost/mpl/minus.hpp>')
compile(1)
Here ``minus_f::apply`` uses inheritance to expose the nested
``type`` of its base class, ``mpl::minus``, so we don't have to
write::
typedef typename ...::type type
.. @ignore()
We don't have to write
``typename`` here (in fact, it would be illegal), because the
compiler knows that dependent names in ``apply``\ 's
*base-specifier-list* must be base classes. [#plus_too]_ This powerful
simplification is known as **metafunction forwarding**; we'll apply
it often as the book goes on. [#edg]_
.. [#plus_too] In case you're wondering, the same approach could
have been applied to ``plus_f``, but since it's a little subtle,
we introduced the straightforward but verbose formulation
first.
.. [#edg] Users of EDG-based compilers should consult |performance|
for a caveat about metafunction forwarding. You can tell whether
you have an EDG compiler by checking the preprocessor symbol
``__EDG_VERSION__``, which is defined by all EDG-based compilers.
Syntactic tricks notwithstanding, writing trivial classes to wrap
existing metafunctions is going to get boring pretty quickly. Even
though the definition of ``minus_f`` was far less verbose than that
of ``plus_f``, it's still an awful lot to type. Fortunately, MPL gives
us a *much* simpler way to pass metafunctions around. Instead of
building a whole metafunction class, we can invoke ``transform``
this way:
.. parsed-literal::
typename mpl::transform<D1,D2, **mpl::minus<_1,_2>** >::type
.. @# 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
Those funny looking arguments (``_1`` and ``_2``) are known as
**placeholders**, and they signify that when the ``transform``\ 's
``BinaryOperation`` is invoked, its first and second arguments will
be passed on to ``minus`` in the positions indicated by ``_1`` and
``_2``, respectively. The whole type ``mpl::minus<_1,_2>`` is
known as a **placeholder expression**.
.. Note:: MPL's placeholders are in the ``mpl::placeholders``
namespace and defined in ``boost/mpl/placeholders.hpp``. In
this book we will usually assume that you have written::
#include<boost/mpl/placeholders.hpp>
using namespace mpl::placeholders;
so that they can be accessed without qualification.
.. @ prefix.append(str(example)) # move to common prefix
ignore()
Here's our division operator written using placeholder
expressions:
.. parsed-literal::
template <class T, class D1, class D2>
quantity<
T
, typename mpl::transform<D1,D2,\ **mpl::minus<_1,_2>** >::type
>
operator/(quantity<T,D1> x, quantity<T,D2> y)
{
typedef typename
mpl::transform<D1,D2,\ **mpl::minus<_1,_2>** >::type dim;
return quantity<T,dim>( x.value() / y.value() );
}
.. @compile('all', pop = 1)
This code is considerably simpler. We can simplify it even further
by factoring the code that calculates the new dimensions into its
own metafunction:
.. parsed-literal::
template <class D1, class D2>
struct **divide_dimensions**
: mpl::transform<D1,D2,mpl::minus<_1,_2> > // forwarding again
{};
template <class T, class D1, class D2>
quantity<T, typename **divide_dimensions<D1,D2>**::type>
operator/(quantity<T,D1> x, quantity<T,D2> y)
{
return quantity<T, typename **divide_dimensions<D1,D2>**::type>(
x.value() / y.value());
}
.. @compile('all', pop = None)
Now we can verify our "force-on-a-laptop" computation by reversing
it, as follows::
quantity<float,mass> m2 = f/a;
float rounding_error = std::abs((m2 - m).value());
.. @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')
If we got everything right, ``rounding_error`` 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 ``a/f`` instead of ``f/a``, there would have
been a compilation error, preventing a mistake from propagating
throughout our program.
Higher-Order Metafunctions
==========================
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 "first class metadata" allows
``transform`` 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 [Hudak89]_ has
earned it a fancy title: **higher-order functional programming**.
A function that operates on another function is known as a
**higher-order function**. It follows that ``transform`` is a
higher-order
metafunction: a metafunction that operates on another metafunction.
"Conception, Evolution, and Application of Functional Programming
Languages."
.. [Hudak89] Paul Hudak. "Conception, Evolution, and Application of
Functional Programming Languages," 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.
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 ``twice``, which |-| given a unary
metafunction *f* and arbitrary metadata *x* |-| computes:
*twice*\ (*f*, *x*) := *f*\ (*f*\ (*x*))
This might seem like a trivial example, and in fact it is. You
won't find much use for ``twice`` in real code. We hope you'll
bear with us anyway: Because it doesn't do much more than accept
and invoke a metafunction, ``twice`` captures all the essential
elements of "higher-orderness" without any distracting details.
If *f* is a metafunction class, the definition of ``twice`` is
straightforward::
template <class F, class X>
struct twice
{
typedef typename F::template apply<X>::type once; // f(x)
typedef typename F::template apply<once>::type type; // f(f(x))
};
.. @ 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()
.. include:: nopara.rst
Or, applying metafunction forwarding::
template <class F, class X>
struct twice
: F::template apply<
typename F::template apply<X>::type
>
{};
.. @ example.append(twice_test)
compile()
.. admonition:: C++ Language Note
The C++ standard requires the ``template`` keyword when we use a
**dependent name** that refers to a member template.
``F::apply`` may or may not name a template, *depending* on the
particular ``F`` that is passed. See |typename| for more
information about ``template``.
Given the need to sprinkle our code with the ``template`` 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::
template <class UnaryMetaFunctionClass, class Arg>
struct apply1
: UnaryMetaFunctionClass::template apply<Arg>
{};
Now ``twice`` is just::
template <class F, class X>
struct twice
: apply1<F, typename apply1<F,X>::type>
{};
To see ``twice`` at work, we can apply it to a little metafunction
class built around the ``add_pointer`` metafunction::
struct add_pointer_f
{
template <class T>
struct apply : boost::add_pointer<T> {};
};
.. include:: nopara.rst
Now we can use ``twice`` with ``add_pointer_f`` to build
pointers-to-pointers::
BOOST_STATIC_ASSERT((
boost::is_same<
twice<add_pointer_f, int>::type
, int**
>::value
));
.. @ apply1 = stack[-4]
add_pointer_f = stack[-2]
compile('all', pop = 0)
Handling Placeholders
=====================
Our implementation of ``twice`` already works with metafunction
classes. Ideally, we would
like it to work with placeholder expressions too, much the same as
``mpl::transform`` allows us to pass either form. For example, we
would like to be able to write:
.. parsed-literal::
template <class X>
struct two_pointers
: twice<**boost::add_pointer<_1>**, X>
{};
.. @ example.append('typedef two_pointers<int>::type intstar2;')
compile('all', pop = 1, expect_error = True)
But when we look at the implementation of ``boost::add_pointer``,
it becomes clear that the current definition of ``twice`` can't
work that way. ::
template <class T>
struct add_pointer
{
typedef T* type;
};
.. @ compile()
To be invokable by ``twice``, ``boost::add_pointer<_1>`` would have
to be a metafunction class, along the lines of ``add_pointer_f``.
Instead, it's just a nullary metafunction returning the almost
senseless type ``_1*``. Any attempt to use ``two_pointers`` will
fail when ``apply1`` reaches for a nested ``::apply``
metafunction in ``boost::add_pointer<_1>`` and finds that it
doesn't exist.
We've determined that we don't get the behavior we want
automatically, so what next? Since ``mpl::transform`` can do this
sort of thing, there ought to be a way for us to do it too |-| and
so there is.
The ``lambda`` Metafunction
---------------------------
We can *generate* a metafunction class from
``boost::add_pointer<_1>``, using MPL's ``lambda`` metafunction:
.. parsed-literal::
template <class X>
struct two_pointers
: twice<**typename mpl::lambda<boost::add_pointer<_1> >::type**, X>
{};
BOOST_STATIC_ASSERT((
boost::is_same<
two_pointers<int>::type
, int**
>::value
));
.. @ prefix.append('#include <boost/mpl/lambda.hpp>')
compile('all')
We'll refer to metafunction classes like ``add_pointer_f`` and
placeholder expressions like ``boost::add_pointer<_1>``
as **lambda expressions**. The term, meaning "unnamed function
object," was introduced in the 1930s by the logician Alonzo Church
as part of a fundamental theory of computation he called the
*lambda-calculus*. [#lambda]_ MPL uses the somewhat obscure word
``lambda`` because of its well-established precedent in functional
programming languages.
.. [#lambda] See http://en.wikipedia.org/wiki/Lambda_calculus for
an in-depth treatment, including a reference to Church's paper
proving that the equivalence of lambda expressions is in general
not decidable.
Although its primary purpose is to turn placeholder expressions
into metafunction classes, ``mpl::lambda`` can accept any lambda
expression, even if it's already a metafunction class. In that
case, ``lambda`` returns its argument unchanged. MPL algorithms
like ``transform`` call ``lambda`` internally, before invoking the
resulting metafunction class, so that they work equally well with
either kind of lambda expression. We can apply the same strategy
to ``twice``::
template <class F, class X>
struct twice
: apply1<
typename mpl::lambda<F>::type
, typename apply1<
typename mpl::lambda<F>::type
, X
>::type
>
{};
Now we can use ``twice`` with metafunction classes *and*
placeholder expressions:
.. parsed-literal::
int* x;
twice<**add_pointer_f**, int>::type p = &x;
twice<**boost::add_pointer<_1>**, int>::type q = &x;
.. @ stack[-2:] = [ apply1, stack[-2], add_pointer_f, stack[-1]]
compile('all')
The ``apply`` Metafunction
--------------------------
Invoking the result of ``lambda`` is such a common pattern
that MPL provides an ``apply`` metafunction to do just
that. Using ``mpl::apply``, our flexible version of ``twice``
becomes::
#include <boost/mpl/apply.hpp>
template <class F, class X>
struct twice
: mpl::apply<F, typename mpl::apply<F,X>::type>
{};
.. @ example.append(twice_test + '''
BOOST_MPL_ASSERT((boost::is_same<twice<boost::add_pointer<_1>,int>::type,int**>));
''')
compile()
prefix.append('#include <boost/mpl/apply.hpp>')
You can think of ``mpl::apply`` as being just like the ``apply1``
template that we wrote, with two additional features:
1. While ``apply1`` operates only on metafunction classes, the first
argument to ``mpl::apply`` can be any lambda expression
(including those built with placeholders).
2. While ``apply1`` accepts only one additional argument to which
the metafunction class will be applied, ``mpl::apply`` can
invoke its first argument on any number from zero to five
additional arguments. [#arity]_ For example:
.. parsed-literal::
// binary lambda expression applied to 2 additional arguments
mpl::apply<
mpl::plus<_1,_2>
, **mpl::int_<6>**
, **mpl::int_<7>**
>::type::value // == 13
.. [#arity] See the Configuration Macros section of the |reference|
for a description of how to change the maximum number of
arguments handled by ``mpl::apply``.
.. @ prefix+=['#include <boost/mpl/plus.hpp>']
example.wrap('enum { is13 = ','''};
BOOST_STATIC_ASSERT(is13 == 13);''')
compile()
.. admonition:: Guideline
When writing a metafunction that invokes one of its arguments,
use ``mpl::apply`` so that it works with lambda expressions.
More Lambda Capabilities
========================
Lambda expressions provide much more than just the ability to pass a
metafunction as an argument. The two capabilities described next
combine to make lambda expressions an invaluable part of almost every
metaprogramming task.
Partial Metafunction Application
--------------------------------
Consider the lambda expression ``mpl::plus<_1,_1>``. A single
argument is directed to both of ``plus``\ 's parameters, thereby
adding a number to itself. Thus, a *binary* metafunction,
``plus``, is used to build a *unary* lambda expression. In other
words, we've created a whole new computation! We're not done yet,
though: By supplying a non-placeholder as one of the arguments, we
can build a unary lambda expression that adds a fixed value, say
42, to its argument::
mpl::plus<_1, mpl::int_<42> >
.. @ apply_test = 'enum { value = mpl::apply<', ''' >::type::value };
BOOST_STATIC_ASSERT(value == %d);'''
example.wrap(apply_test[0],
', mpl::int_<3>' + apply_test[1] % 45)
compile()
The process of binding argument values to a subset of a function's
parameters is known in the world of functional programming as
**partial function application**.
Metafunction Composition
------------------------
Lambda expressions can also be used to assemble more interesting
computations from simple metafunctions. For example, the following
expression, which multiplies the sum of two numbers by their
difference, is a **composition** of the three metafunctions ``multiplies``,
``plus``, and ``minus``::
mpl::multiplies<mpl::plus<_1,_2>, mpl::minus<_1,_2> >
.. @ example.wrap(apply_test[0],
', mpl::int_<5>,mpl::int_<3>' + apply_test[1] % 16)
# Can't exactly justify this yet, but there's no way to get
# it into the text
prefix += ['#include <boost/mpl/multiplies.hpp>']
compile()
When evaluating a lambda expression, MPL checks to see if any of its
arguments are themselves lambda expressions, and evaluates each one
that it finds. The results of these inner evaluations are substituted
into the outer expression before it is evaluated.
Lambda Details
==============
Now that you have an idea of the semantics of MPL's ``lambda``
facility, let's formalize that understanding and look at things a
little more deeply.
Placeholders
------------
The definition of "placeholder" may surprise you:
.. admonition:: Definition
A **placeholder** is a metafunction class of the form ``mpl::arg<X>``.
Implementation
..............
The convenient names ``_1``, ``_2``,... ``_5`` are actually
``typedef``\ s for specializations of ``mpl::arg`` that simply
select the *N*\ th argument for any *N*. [#config]_ The
implementation of placeholders looks something like this:
.. [#config] MPL provides five placeholders by default. See
the Configuration Macros section of |reference| for a
description of how to change the number of placeholders
provided.
.. parsed-literal::
namespace boost { namespace mpl { namespace placeholders {
template <int N> struct arg; // forward declarations
struct void\_;
template <>
struct arg<**1**>
{
template <
class **A1**, class A2 = void\_, ... class A\ *m* = void\_>
struct apply
{
typedef **A1** type; // return the first argument
};
};
typedef **arg<1> _1**;
template <>
struct arg<**2**>
{
template <
class A1, class **A2**, class A3 = void\_, ...class A\ *m* = void\_
>
struct apply
{
typedef **A2** type; // return the second argument
};
};
typedef **arg<2> _2**;
*more specializations and typedefs...*
}}}
.. @example.replace('...','')
Remember that invoking a metafunction class is the same as invoking
its nested ``apply`` metafunction. When a placeholder in a lambda
expression is evaluated, it is invoked on the expression's actual
arguments, returning just one of them. The results are then
substituted back into the lambda expression and the evaluation
process continues.
The Unnamed Placeholder
.......................
There's one special placeholder, known as the **unnamed
placeholder**, that we haven't yet defined:
.. parsed-literal::
namespace boost { namespace mpl { namespace placeholders {
**typedef arg<-1> _;** // the unnamed placeholder
}}}
.. @ stack[-2].prepend('namespace shield {')
example.append('}') # so we don't conflict with the prefix
compile('all')
The details of its implementation aren't important; all you really
need to know about the unnamed placeholder is that it gets special
treatment. When a lambda expression is being transformed into a
metafunction class by ``mpl::lambda``,
the *n*\ th appearance of the unnamed placeholder *in a given
template specialization* is replaced with ``_``\ *n*.
So, for example, every row of Table |metafunctions|.1
below contains two equivalent lambda expressions.
.. table:: Unnamed Placeholder Semantics
+----------------------------+------------------------------+
|:: |:: |
| | |
| mpl::plus<_,_> | mpl::plus<_1,_2> |
+----------------------------+------------------------------+
|:: |:: |
| | |
| boost::is_same< | boost::is_same< |
| _ | _1 |
| , boost::add_pointer<_> | , boost::add_pointer<_1> |
| > | > |
+----------------------------+------------------------------+
|:: |:: |
| | |
| mpl::multiplies< | mpl::multiplies< |
| mpl::plus<_,_> | mpl::plus<_1,_2> |
| , mpl::minus<_,_> | , mpl::minus<_1,_2> |
| > | > |
+----------------------------+------------------------------+
.. @ for n in range(len(stack)):
stack[n].wrap('typedef ', 'type%d;' % n)
compile('all')
Especially when used in simple lambda expressions, the unnamed
placeholder often eliminates just enough syntactic "noise" to
significantly improve readability.
Placeholder Expression Definition
---------------------------------
Now that you know just what *placeholder* means, we can define
*placeholder expression*:
.. admonition:: Definition
A placeholder expression is either:
- a placeholder
*or*
- a template specialization with at least one argument that
is a placeholder expression.
In other words, a placeholder expression always involves a
placeholder.
.. DWA: I'm still not sure we shouldn't be at least mentioning the
pitfall, but for now it's commented out.
Lambda and Nullary Metafunctions
--------------------------------
The definition of *placeholder expression* above has an interesting
implication: an ordinary nullary metafunction is never a placeholder
expression. In other words, even though ``add_pointer<int>`` is a
nullary metafunction, it won't be invoked in the expression below;
the assertion will always fail:
.. parsed-literal::
BOOST_STATIC_ASSERT((
mpl::apply<
boost::is_same<**boost::add_pointer<int>**,_1>
, int\*
>::type::value
));
In order to allow a nullary metafunction to be used as a lambda
expression, MPL provides this definition of ``arg``:
.. parsed-literal::
// primary template definition (not a specialization)
template <class F>
struct arg
{
template <class A1 = void\_, class A2 = void\_, ... class *Am* = void\_>
struct apply : F
{
};
};
When applied to a lambda expression's actual arguments, ``arg<F>``
ignores them and simply returns ``F::type``. In other words, if
``F`` is a nullary metafunction, ``arg<F>`` is a metafunction class
that invokes ``F`` and returns the result. So we can transform
add_pointer<int> into a placeholder and get the desired result
with:
.. parsed-literal::
BOOST_STATIC_ASSERT((
mpl::apply<
boost::is_same<
**mpl::arg<boost::add_pointer<int> >**
, _1
>
, int\*
>::type::value
));
Lambda and Non-Metafunction Templates
-------------------------------------
There is just one detail of placeholder expressions that we haven't
discussed yet. MPL uses a special rule to make it easier to
integrate ordinary templates into metaprograms: After all of the
placeholders have been replaced with actual
arguments, if the resulting template specialization *X* doesn't
have a nested ``::type``, the result is just
*X* itself.
For example, ``mpl::apply<std::vector<_>, T>`` is always just
``std::vector<T>``. If it weren't for this behavior, we would
have to build trivial metafunctions to create ordinary template
specializations in lambda expressions:
.. parsed-literal::
// trivial std::vector generator
template<class U>
struct make_vector { typedef std::vector<U> type; };
typedef mpl::apply<**make_vector<_>**, T>::type vector_of_t;
Instead, we can simply write:
.. parsed-literal::
typedef mpl::apply<**std::vector<_>**, T>::type vector_of_t;
.. @ # ensure indentation works. The ReST parser will push the 2nd line left
while stack:
stack[-1].prepend('#include <vector>\ntypedef int T;')
compile()
The Importance of Being Lazy
----------------------------
Recall the definition of ``always_int`` from the previous chapter::
struct always_int
{
typedef int type;
};
Nullary metafunctions might not seem very important at first, since
something like ``add_pointer<int>`` could be replaced by ``int*`` in
any lambda expression where it appears. Not all nullary
metafunctions are that simple, though:
.. parsed-literal::
typedef mpl::vector<int, char*, double&> seq;
typedef **mpl::transform<seq, boost::add_pointer<_> >** calc_ptr_seq;
.. @ example.prepend('''
#include <boost/mpl/vector.hpp>
#include <boost/mpl/transform.hpp>
''')
compile('all')
Note that ``calc_ptr_seq`` is a nullary metafunction, since it has
``transform``\ 's nested ``::type``. A C++ template is not
instantiated until we actually "look inside it," though. Just
naming ``calc_ptr_seq`` does not cause it to be evaluated, since we
haven't accessed its ``::type`` yet.
Metafunctions can be invoked *lazily*, rather than immediately upon
supplying all of their arguments. We can use **lazy evaluation** to
improve compilation time when a metafunction result is only going
to be used conditionally. We can sometimes also avoid contorting
program structure by *naming* an invalid computation without
actually performing it. That's what we've done with
``calc_ptr_seq`` above, since you can't legally form ``double&*``.
Laziness and all of its virtues will be a recurring theme
throughout this book.
Details
=======
By now you should have a fairly complete view of the fundamental
concepts and language of both template metaprogramming in general
and of the Boost Metaprogramming Library. This section
reviews the highlights.
Metafunction forwarding.
The technique of using public derivation to
supply the nested ``type`` of a metafunction by accessing the one
provided by its base class.
Metafunction class.
The most basic way to formulate a compile-time
function so that it can be treated as polymorphic metadata; that
is, as a type. A metafunction class is a class with a nested
metafunction called ``apply``.
MPL.
Most of this book's examples will use the Boost
Metaprogramming Library. Like the Boost type traits headers,
MPL
headers follow a simple convention:
.. parsed-literal::
#include <boost/mpl/*component-name*.hpp>
If the component's name ends in an underscore, however, the
corresponding MPL header name does not include the trailing
underscore. For example, ``mpl::bool_`` can be found in
``<boost/mpl/bool.hpp>``. Where the library deviates from this
convention, we'll be sure to point it out to you.
.. @ignore()
Higher-order function.
A function that operates on or returns a function. Making
metafunctions polymorphic with other metadata is a key
ingredient in higher-order metaprogramming.
Lambda expression.
Simply put, a lambda expression is callable metadata. Without
some form of callable metadata, higher-order metafunctions
would be impossible. Lambda expressions have two basic forms:
*metafunction classes* and *placeholder expressions*.
Placeholder expression.
A kind of lambda expression that, through the use of
placeholders, enables in-place *partial metafunction
application* and *metafunction composition*. As you will see
throughout this book, these features give us the truly amazing
ability to build up almost any kind of complex type computation
from more primitive metafunctions, right at its point of use::
// find the position of a type x in some_sequence such that:
// x is convertible to 'int'
// && x is not 'char'
// && x is not a floating type
typedef mpl::find_if<
some_sequence
, mpl::and_<
boost::is_convertible<_1,int>
, mpl::not_<boost::is_same<_1,char> >
, mpl::not_<boost::is_float<_1> >
>
>::type iter;
Placeholder expressions make good on the promise of algorithm reuse
without forcing us to write new metafunction classes. The
corresponding capability is often sorely missed in the runtime
world of the STL, since it is often much easier to write a loop
by hand than it is to use standard algorithms, despite their
correctness and efficiency advantages.
.. @ example.prepend('''
#include <boost/mpl/and.hpp>
#include <boost/mpl/not.hpp>
#include <boost/mpl/find_if.hpp>
#include <boost/type_traits/is_convertible.hpp>
#include <boost/type_traits/is_float.hpp>
typedef mpl::vector<char, double, short, long> some_sequence;
''')
compile()
The ``lambda`` metafunction.
A metafunction that transforms a lambda expression into a
corresponding metafunction class. For detailed information on
``lambda`` and the lambda evaluation process,
please see the |reference|.
The ``apply`` metafunction.
A metafunction that invokes its first argument, which must be a
lambda expression, on its remaining arguments. In general, to
invoke a lambda expression, you should always pass it to
``mpl::apply`` along with the arguments you want to apply it
to in lieu of using ``lambda`` and invoking the result "manually."
Lazy evaluation.
A strategy of delaying evaluation until a result is
required, thereby avoiding any unneccessary computation and any
associated unneccessary errors. Metafunctions are only invoked
when we access their nested ``::type``\ s, so we can supply all
of their arguments without performing any computation and
delay evaluation to the last possible moment.
Exercises
=========
|metafunctions|-0.
Use ``BOOST_STATIC_ASSERT`` to add error checking to the ``binary``
template presented in section |intro|.4.1 so
that ``binary<N>::value`` causes a compilation error if ``N``
contains digits other than ``0`` or ``1``.
|metafunctions|-1.
Turn ``vector_c<int,1,2,3>`` into a type sequence with elements
(2,3,4) using ``transform``.
|metafunctions|-2.
Turn ``vector_c<int,1,2,3>`` into a type sequence with elements
(1,4,9) using ``transform``.
|metafunctions|-3.
Turn ``T`` into ``T****`` by using ``twice`` twice.
|metafunctions|-4.
Turn ``T`` into ``T****`` using ``twice`` on itself.
|metafunctions|-5.
There's still a problem with the dimensional analysis code in
section |metafunctions|.1.
Hint: What happens when you do::
f = f + m * a;
Repair this example using techniques shown in this
chapter.
.. @ example.wrap('''void will_fail%s
{ quantity<float,force> f(m*a);
''' % ma_function_args, '}')
stack[:0] = dimensional_analysis # stick support code in
compile('all', expect_error = True)
|metafunctions|-6.
Build a lambda expression that has functionality equivalent to
``twice``. Hint: ``mpl::apply`` is a metafunction!
|metafunctions|-7*.
What do you think would be the semantics of the following
constructs::
typedef mpl::lambda<mpl::lambda<_1> >::type t1;
typedef mpl::apply<_1,mpl::plus<_1,_2> >::type t2;
typedef mpl::apply<_1,std::vector<int> >::type t3;
typedef mpl::apply<_1,std::vector<_1> >::type t4;
typedef mpl::apply<mpl::lambda<_1>,std::vector<int> >::type t5;
typedef mpl::apply<mpl::lambda<_1>,std::vector<_1> >::type t6;
typedef mpl::apply<mpl::lambda<_1>,mpl::plus<_1,_2> >::type t7;
typedef mpl::apply<_1,mpl::lambda< mpl::plus<_1,_2> > >::type t8;
.. @example.prepend('#include <vector>')
compile()
Show the steps used to
arrive at your answers and write tests verifying your assumptions.
Did the library behavior match your reasoning? If not, analyze the
failed tests to discover the actual expression semantics.
Explain why your assumptions were different, what
behavior you find more coherent, and why.
|metafunctions|-8*.
Our dimensional analysis framework dealt with dimensions, but it
entirely ignored the issue of *units*. A length can be
represented in inches, feet, or meters. A force can be
represented in newtons or in kg m/sec\ :sup:`2`. Add the
ability to specify units and test your code. Try to make your
interface as syntactically friendly as possible for the user.
.. Along with Alan Turing, Church was one of the founders of computer
science.
.. yes, this is easier to read, but the real beauty of ``apply`` is that
it not only works on metafunction classes, but on metafunctions passed
with placeholder arguments.
.. Naturally, MPL defines a series of these numbered placeholder
types, so that we can handle metafunctions with more arguments.
.. : in order to pass a template in the ``BinaryOperation`` position,
``transform`` would have to be declared::
template <
class Sequence1, class Sequence2
, template <class,class> class BinaryOperation
>
struct transform; // returning a sequence
.. "Programming with types"
.. Once again, a nested ``value`` member is optional for
metafunctions, and required for integral type wrappers.
.. include:: chapter-numbers.rst
--
Dave Abrahams
Boost Consulting
www.boost-consulting.com
