xref: /aosp_15_r20/external/eigen/doc/InsideEigenExample.dox (revision bf2c37156dfe67e5dfebd6d394bad8b2ab5804d4)

namespace Eigen {

/** \page TopicInsideEigenExample What happens inside Eigen, on a simple example

\eigenAutoToc

<hr>


Consider the following example program:

\code
#include<Eigen/Core>

int main()
{
  int size = 50;
  // VectorXf is a vector of floats, with dynamic size.
  Eigen::VectorXf u(size), v(size), w(size);
  u = v + w;
}
\endcode

The goal of this page is to understand how Eigen compiles it, assuming that SSE2 vectorization is enabled (GCC option -msse2).

\section WhyInteresting Why it's interesting

Maybe you think, that the above example program is so simple, that compiling it shouldn't involve anything interesting. So before starting, let us explain what is nontrivial in compiling it correctly -- that is, producing optimized code -- so that the complexity of Eigen, that we'll explain here, is really useful.

Look at the line of code
\code
  u = v + w;   //   (*)
\endcode

The first important thing about compiling it, is that the arrays should be traversed only once, like
\code
  for(int i = 0; i < size; i++) u[i] = v[i] + w[i];
\endcode
The problem is that if we make a naive C++ library where the VectorXf class has an operator+ returning a VectorXf, then the line of code (*) will amount to:
\code
  VectorXf tmp = v + w;
  VectorXf u = tmp;
\endcode
Obviously, the introduction of the temporary \a tmp here is useless. It has a very bad effect on performance, first because the creation of \a tmp requires a dynamic memory allocation in this context, and second as there are now two for loops:
\code
  for(int i = 0; i < size; i++) tmp[i] = v[i] + w[i];
  for(int i = 0; i < size; i++) u[i] = tmp[i];
\endcode
Traversing the arrays twice instead of once is terrible for performance, as it means that we do many redundant memory accesses.

The second important thing about compiling the above program, is to make correct use of SSE2 instructions. Notice that Eigen also supports AltiVec and that all the discussion that we make here applies also to AltiVec.

SSE2, like AltiVec, is a set of instructions allowing to perform computations on packets of 128 bits at once. Since a float is 32 bits, this means that SSE2 instructions can handle 4 floats at once. This means that, if correctly used, they can make our computation go up to 4x faster.

However, in the above program, we have chosen size=50, so our vectors consist of 50 float's, and 50 is not a multiple of 4. This means that we cannot hope to do all of that computation using SSE2 instructions. The second best thing, to which we should aim, is to handle the 48 first coefficients with SSE2 instructions, since 48 is the biggest multiple of 4 below 50, and then handle separately, without SSE2, the 49th and 50th coefficients. Something like this:

\code
  for(int i = 0; i < 4*(size/4); i+=4) u.packet(i)  = v.packet(i) + w.packet(i);
  for(int i = 4*(size/4); i < size; i++) u[i] = v[i] + w[i];
\endcode

So let us look line by line at our example program, and let's follow Eigen as it compiles it.

\section ConstructingVectors Constructing vectors

Let's analyze the first line:

\code
  Eigen::VectorXf u(size), v(size), w(size);
\endcode

First of all, VectorXf is the following typedef:
\code
  typedef Matrix<float, Dynamic, 1> VectorXf;
\endcode

The class template Matrix is declared in src/Core/util/ForwardDeclarations.h with 6 template parameters, but the last 3 are automatically determined by the first 3. So you don't need to worry about them for now. Here, Matrix\<float, Dynamic, 1\> means a matrix of floats, with a dynamic number of rows and 1 column.

The Matrix class inherits a base class, MatrixBase. Don't worry about it, for now it suffices to say that MatrixBase is what unifies matrices/vectors and all the expressions types -- more on that below.

When we do
\code
  Eigen::VectorXf u(size);
\endcode
the constructor that is called is Matrix::Matrix(int), in src/Core/Matrix.h. Besides some assertions, all it does is to construct the \a m_storage member, which is of type DenseStorage\<float, Dynamic, Dynamic, 1\>.

You may wonder, isn't it overengineering to have the storage in a separate class? The reason is that the Matrix class template covers all kinds of matrices and vector: both fixed-size and dynamic-size. The storage method is not the same in these two cases. For fixed-size, the matrix coefficients are stored as a plain member array. For dynamic-size, the coefficients will be stored as a pointer to a dynamically-allocated array. Because of this, we need to abstract storage away from the Matrix class. That's DenseStorage.

Let's look at this constructor, in src/Core/DenseStorage.h. You can see that there are many partial template specializations of DenseStorages here, treating separately the cases where dimensions are Dynamic or fixed at compile-time. The partial specialization that we are looking at is:
\code
template<typename T, int _Cols> class DenseStorage<T, Dynamic, Dynamic, _Cols>
\endcode

Here, the constructor called is DenseStorage::DenseStorage(int size, int rows, int columns)
with size=50, rows=50, columns=1.

Here is this constructor:
\code
inline DenseStorage(int size, int rows, int) : m_data(internal::aligned_new<T>(size)), m_rows(rows) {}
\endcode

Here, the \a m_data member is the actual array of coefficients of the matrix. As you see, it is dynamically allocated. Rather than calling new[] or malloc(), as you can see, we have our own internal::aligned_new defined in src/Core/util/Memory.h. What it does is that if vectorization is enabled, then it uses a platform-specific call to allocate a 128-bit-aligned array, as that is very useful for vectorization with both SSE2 and AltiVec. If vectorization is disabled, it amounts to the standard new[].

As you can see, the constructor also sets the \a m_rows member to \a size. Notice that there is no \a m_columns member: indeed, in this partial specialization of DenseStorage, we know the number of columns at compile-time, since the _Cols template parameter is different from Dynamic. Namely, in our case, _Cols is 1, which is to say that our vector is just a matrix with 1 column. Hence, there is no need to store the number of columns as a runtime variable.

When you call VectorXf::data() to get the pointer to the array of coefficients, it returns DenseStorage::data() which returns the \a m_data member.

When you call VectorXf::size() to get the size of the vector, this is actually a method in the base class MatrixBase. It determines that the vector is a column-vector, since ColsAtCompileTime==1 (this comes from the template parameters in the typedef VectorXf). It deduces that the size is the number of rows, so it returns VectorXf::rows(), which returns DenseStorage::rows(), which returns the \a m_rows member, which was set to \a size by the constructor.

\section ConstructionOfSumXpr Construction of the sum expression

Now that our vectors are constructed, let's move on to the next line:

\code
u = v + w;
\endcode

The executive summary is that operator+ returns a "sum of vectors" expression, but doesn't actually perform the computation. It is the operator=, whose call occurs thereafter, that does the computation.

Let us now see what Eigen does when it sees this:

\code
v + w
\endcode

Here, v and w are of type VectorXf, which is a typedef for a specialization of Matrix (as we explained above), which is a subclass of MatrixBase. So what is being called is

\code
MatrixBase::operator+(const MatrixBase&)
\endcode

The return type of this operator is
\code
CwiseBinaryOp<internal::scalar_sum_op<float>, VectorXf, VectorXf>
\endcode
The CwiseBinaryOp class is our first encounter with an expression template. As we said, the operator+ doesn't by itself perform any computation, it just returns an abstract "sum of vectors" expression. Since there are also "difference of vectors" and "coefficient-wise product of vectors" expressions, we unify them all as "coefficient-wise binary operations", which we abbreviate as "CwiseBinaryOp". "Coefficient-wise" means that the operations is performed coefficient by coefficient. "binary" means that there are two operands -- we are adding two vectors with one another.

Now you might ask, what if we did something like

\code
v + w + u;
\endcode

The first v + w would return a CwiseBinaryOp as above, so in order for this to compile, we'd need to define an operator+ also in the class CwiseBinaryOp... at this point it starts looking like a nightmare: are we going to have to define all operators in each of the expression classes (as you guessed, CwiseBinaryOp is only one of many) ? This looks like a dead end!

The solution is that CwiseBinaryOp itself, as well as Matrix and all the other expression types, is a subclass of MatrixBase. So it is enough to define once and for all the operators in class MatrixBase.

Since MatrixBase is the common base class of different subclasses, the aspects that depend on the subclass must be abstracted from MatrixBase. This is called polymorphism.

The classical approach to polymorphism in C++ is by means of virtual functions. This is dynamic polymorphism. Here we don't want dynamic polymorphism because the whole design of Eigen is based around the assumption that all the complexity, all the abstraction, gets resolved at compile-time. This is crucial: if the abstraction can't get resolved at compile-time, Eigen's compile-time optimization mechanisms become useless, not to mention that if that abstraction has to be resolved at runtime it'll incur an overhead by itself.

Here, what we want is to have a single class MatrixBase as the base of many subclasses, in such a way that each MatrixBase object (be it a matrix, or vector, or any kind of expression) knows at compile-time (as opposed to run-time) of which particular subclass it is an object (i.e. whether it is a matrix, or an expression, and what kind of expression).

The solution is the <a href="http://en.wikipedia.org/wiki/Curiously_Recurring_Template_Pattern">Curiously Recurring Template Pattern</a>. Let's do the break now. Hopefully you can read this wikipedia page during the break if needed, but it won't be allowed during the exam.

In short, MatrixBase takes a template parameter \a Derived. Whenever we define a subclass Subclass, we actually make Subclass inherit MatrixBase\<Subclass\>. The point is that different subclasses inherit different MatrixBase types. Thanks to this, whenever we have an object of a subclass, and we call on it some MatrixBase method, we still remember even from inside the MatrixBase method which particular subclass we're talking about.

This means that we can put almost all the methods and operators in the base class MatrixBase, and have only the bare minimum in the subclasses. If you look at the subclasses in Eigen, like for instance the CwiseBinaryOp class, they have very few methods. There are coeff() and sometimes coeffRef() methods for access to the coefficients, there are rows() and cols() methods returning the number of rows and columns, but there isn't much more than that. All the meat is in MatrixBase, so it only needs to be coded once for all kinds of expressions, matrices, and vectors.

So let's end this digression and come back to the piece of code from our example program that we were currently analyzing,

\code
v + w
\endcode

Now that MatrixBase is a good friend, let's write fully the prototype of the operator+ that gets called here (this code is from src/Core/MatrixBase.h):

\code
template<typename Derived>
class MatrixBase
{
  // ...

  template<typename OtherDerived>
  const CwiseBinaryOp<internal::scalar_sum_op<typename internal::traits<Derived>::Scalar>, Derived, OtherDerived>
  operator+(const MatrixBase<OtherDerived> &other) const;

  // ...
};
\endcode

Here of course, \a Derived and \a OtherDerived are VectorXf.

As we said, CwiseBinaryOp is also used for other operations such as substration, so it takes another template parameter determining the operation that will be applied to coefficients. This template parameter is a functor, that is, a class in which we have an operator() so it behaves like a function. Here, the functor used is internal::scalar_sum_op. It is defined in src/Core/Functors.h.

Let us now explain the internal::traits here. The internal::scalar_sum_op class takes one template parameter: the type of the numbers to handle. Here of course we want to pass the scalar type (a.k.a. numeric type) of VectorXf, which is \c float. How do we determine which is the scalar type of \a Derived ? Throughout Eigen, all matrix and expression types define a typedef \a Scalar which gives its scalar type. For example, VectorXf::Scalar is a typedef for \c float. So here, if life was easy, we could find the numeric type of \a Derived as just
\code
typename Derived::Scalar
\endcode
Unfortunately, we can't do that here, as the compiler would complain that the type Derived hasn't yet been defined. So we use a workaround: in src/Core/util/ForwardDeclarations.h, we declared (not defined!) all our subclasses, like Matrix, and we also declared the following class template:
\code
template<typename T> struct internal::traits;
\endcode
In src/Core/Matrix.h, right \em before the definition of class Matrix, we define a partial specialization of internal::traits for T=Matrix\<any template parameters\>. In this specialization of internal::traits, we define the Scalar typedef. So when we actually define Matrix, it is legal to refer to "typename internal::traits\<Matrix\>::Scalar".

Anyway, we have declared our operator+. In our case, where \a Derived and \a OtherDerived are VectorXf, the above declaration amounts to:
\code
class MatrixBase<VectorXf>
{
  // ...

  const CwiseBinaryOp<internal::scalar_sum_op<float>, VectorXf, VectorXf>
  operator+(const MatrixBase<VectorXf> &other) const;

  // ...
};
\endcode

Let's now jump to src/Core/CwiseBinaryOp.h to see how it is defined. As you can see there, all it does is to return a CwiseBinaryOp object, and this object is just storing references to the left-hand-side and right-hand-side expressions -- here, these are the vectors \a v and \a w. Well, the CwiseBinaryOp object is also storing an instance of the (empty) functor class, but you shouldn't worry about it as that is a minor implementation detail.

Thus, the operator+ hasn't performed any actual computation. To summarize, the operation \a v + \a w just returned an object of type CwiseBinaryOp which did nothing else than just storing references to \a v and \a w.

\section Assignment The assignment

<div class="warningbox">
<strong>PLEASE HELP US IMPROVING THIS SECTION.</strong>
This page reflects how %Eigen worked until 3.2, but since %Eigen 3.3 the assignment is more sophisticated as it involves an Assignment expression, and the creation of so called evaluator which are responsible for the evaluation of each kind of expressions.
</div>

At this point, the expression \a v + \a w has finished evaluating, so, in the process of compiling the line of code
\code
u = v + w;
\endcode
we now enter the operator=.

What operator= is being called here? The vector u is an object of class VectorXf, i.e. Matrix. In src/Core/Matrix.h, inside the definition of class Matrix, we see this:
\code
    template<typename OtherDerived>
    inline Matrix& operator=(const MatrixBase<OtherDerived>& other)
    {
      eigen_assert(m_storage.data()!=0 && "you cannot use operator= with a non initialized matrix (instead use set()");
      return Base::operator=(other.derived());
    }
\endcode
Here, Base is a typedef for MatrixBase\<Matrix\>. So, what is being called is the operator= of MatrixBase. Let's see its prototype in src/Core/MatrixBase.h:
\code
    template<typename OtherDerived>
    Derived& operator=(const MatrixBase<OtherDerived>& other);
\endcode
Here, \a Derived is VectorXf (since u is a VectorXf) and \a OtherDerived is CwiseBinaryOp. More specifically, as explained in the previous section, \a OtherDerived is:
\code
CwiseBinaryOp<internal::scalar_sum_op<float>, VectorXf, VectorXf>
\endcode
So the full prototype of the operator= being called is:
\code
VectorXf& MatrixBase<VectorXf>::operator=(const MatrixBase<CwiseBinaryOp<internal::scalar_sum_op<float>, VectorXf, VectorXf> > & other);
\endcode
This operator= literally reads "copying a sum of two VectorXf's into another VectorXf".

Let's now look at the implementation of this operator=. It resides in the file src/Core/Assign.h.

What we can see there is:
\code
template<typename Derived>
template<typename OtherDerived>
inline Derived& MatrixBase<Derived>
  ::operator=(const MatrixBase<OtherDerived>& other)
{
  return internal::assign_selector<Derived,OtherDerived>::run(derived(), other.derived());
}
\endcode

OK so our next task is to understand internal::assign_selector :)

Here is its declaration (all that is still in the same file src/Core/Assign.h)
\code
template<typename Derived, typename OtherDerived,
         bool EvalBeforeAssigning = int(OtherDerived::Flags) & EvalBeforeAssigningBit,
         bool NeedToTranspose = Derived::IsVectorAtCompileTime
                && OtherDerived::IsVectorAtCompileTime
                && int(Derived::RowsAtCompileTime) == int(OtherDerived::ColsAtCompileTime)
                && int(Derived::ColsAtCompileTime) == int(OtherDerived::RowsAtCompileTime)
                && int(Derived::SizeAtCompileTime) != 1>
struct internal::assign_selector;
\endcode

So internal::assign_selector takes 4 template parameters, but the 2 last ones are automatically determined by the 2 first ones.

EvalBeforeAssigning is here to enforce the EvalBeforeAssigningBit. As explained <a href="TopicLazyEvaluation.html">here</a>, certain expressions have this flag which makes them automatically evaluate into temporaries before assigning them to another expression. This is the case of the Product expression, in order to avoid strange aliasing effects when doing "m = m * m;" However, of course here our CwiseBinaryOp expression doesn't have the EvalBeforeAssigningBit: we said since the beginning that we didn't want a temporary to be introduced here. So if you go to src/Core/CwiseBinaryOp.h, you'll see that the Flags in internal::traits\<CwiseBinaryOp\> don't include the EvalBeforeAssigningBit. The Flags member of CwiseBinaryOp is then imported from the internal::traits by the EIGEN_GENERIC_PUBLIC_INTERFACE macro. Anyway, here the template parameter EvalBeforeAssigning has the value \c false.

NeedToTranspose is here for the case where the user wants to copy a row-vector into a column-vector. We allow this as a special exception to the general rule that in assignments we require the dimesions to match. Anyway, here both the left-hand and right-hand sides are column vectors, in the sense that ColsAtCompileTime is equal to 1. So NeedToTranspose is \c false too.

So, here we are in the partial specialization:
\code
internal::assign_selector<Derived, OtherDerived, false, false>
\endcode

Here's how it is defined:
\code
template<typename Derived, typename OtherDerived>
struct internal::assign_selector<Derived,OtherDerived,false,false> {
  static Derived& run(Derived& dst, const OtherDerived& other) { return dst.lazyAssign(other.derived()); }
};
\endcode

OK so now our next job is to understand how lazyAssign works :)

\code
template<typename Derived>
template<typename OtherDerived>
inline Derived& MatrixBase<Derived>
  ::lazyAssign(const MatrixBase<OtherDerived>& other)
{
  EIGEN_STATIC_ASSERT_SAME_MATRIX_SIZE(Derived,OtherDerived)
  eigen_assert(rows() == other.rows() && cols() == other.cols());
  internal::assign_impl<Derived, OtherDerived>::run(derived(),other.derived());
  return derived();
}
\endcode

What do we see here? Some assertions, and then the only interesting line is:
\code
  internal::assign_impl<Derived, OtherDerived>::run(derived(),other.derived());
\endcode

OK so now we want to know what is inside internal::assign_impl.

Here is its declaration:
\code
template<typename Derived1, typename Derived2,
         int Vectorization = internal::assign_traits<Derived1, Derived2>::Vectorization,
         int Unrolling = internal::assign_traits<Derived1, Derived2>::Unrolling>
struct internal::assign_impl;
\endcode
Again, internal::assign_selector takes 4 template parameters, but the 2 last ones are automatically determined by the 2 first ones.

These two parameters \a Vectorization and \a Unrolling are determined by a helper class internal::assign_traits. Its job is to determine which vectorization strategy to use (that is \a Vectorization) and which unrolling strategy to use (that is \a Unrolling).

We'll not enter into the details of how these strategies are chosen (this is in the implementation of internal::assign_traits at the top of the same file). Let's just say that here \a Vectorization has the value \a LinearVectorization, and \a Unrolling has the value \a NoUnrolling (the latter is obvious since our vectors have dynamic size so there's no way to unroll the loop at compile-time).

So the partial specialization of internal::assign_impl that we're looking at is:
\code
internal::assign_impl<Derived1, Derived2, LinearVectorization, NoUnrolling>
\endcode

Here is how it's defined:
\code
template<typename Derived1, typename Derived2>
struct internal::assign_impl<Derived1, Derived2, LinearVectorization, NoUnrolling>
{
  static void run(Derived1 &dst, const Derived2 &src)
  {
    const int size = dst.size();
    const int packetSize = internal::packet_traits<typename Derived1::Scalar>::size;
    const int alignedStart = internal::assign_traits<Derived1,Derived2>::DstIsAligned ? 0
                           : internal::first_aligned(&dst.coeffRef(0), size);
    const int alignedEnd = alignedStart + ((size-alignedStart)/packetSize)*packetSize;

    for(int index = 0; index < alignedStart; index++)
      dst.copyCoeff(index, src);

    for(int index = alignedStart; index < alignedEnd; index += packetSize)
    {
      dst.template copyPacket<Derived2, Aligned, internal::assign_traits<Derived1,Derived2>::SrcAlignment>(index, src);
    }

    for(int index = alignedEnd; index < size; index++)
      dst.copyCoeff(index, src);
  }
};
\endcode

Here's how it works. \a LinearVectorization means that the left-hand and right-hand side expression can be accessed linearly i.e. you can refer to their coefficients by one integer \a index, as opposed to having to refer to its coefficients by two integers \a row, \a column.

As we said at the beginning, vectorization works with blocks of 4 floats. Here, \a PacketSize is 4.

There are two potential problems that we need to deal with:
\li first, vectorization works much better if the packets are 128-bit-aligned. This is especially important for write access. So when writing to the coefficients of \a dst, we want to group these coefficients by packets of 4 such that each of these packets is 128-bit-aligned. In general, this requires to skip a few coefficients at the beginning of \a dst. This is the purpose of \a alignedStart. We then copy these first few coefficients one by one, not by packets. However, in our case, the \a dst expression is a VectorXf and remember that in the construction of the vectors we allocated aligned arrays. Thanks to \a DstIsAligned, Eigen remembers that without having to do any runtime check, so \a alignedStart is zero and this part is avoided altogether.
\li second, the number of coefficients to copy is not in general a multiple of \a packetSize. Here, there are 50 coefficients to copy and \a packetSize is 4. So we'll have to copy the last 2 coefficients one by one, not by packets. Here, \a alignedEnd is 48.

Now come the actual loops.

First, the vectorized part: the 48 first coefficients out of 50 will be copied by packets of 4:
\code
  for(int index = alignedStart; index < alignedEnd; index += packetSize)
  {
    dst.template copyPacket<Derived2, Aligned, internal::assign_traits<Derived1,Derived2>::SrcAlignment>(index, src);
  }
\endcode

What is copyPacket? It is defined in src/Core/Coeffs.h:
\code
template<typename Derived>
template<typename OtherDerived, int StoreMode, int LoadMode>
inline void MatrixBase<Derived>::copyPacket(int index, const MatrixBase<OtherDerived>& other)
{
  eigen_internal_assert(index >= 0 && index < size());
  derived().template writePacket<StoreMode>(index,
    other.derived().template packet<LoadMode>(index));
}
\endcode

OK, what are writePacket() and packet() here?

First, writePacket() here is a method on the left-hand side VectorXf. So we go to src/Core/Matrix.h to look at its definition:
\code
template<int StoreMode>
inline void writePacket(int index, const PacketScalar& x)
{
  internal::pstoret<Scalar, PacketScalar, StoreMode>(m_storage.data() + index, x);
}
\endcode
Here, \a StoreMode is \a #Aligned, indicating that we are doing a 128-bit-aligned write access, \a PacketScalar is a type representing a "SSE packet of 4 floats" and internal::pstoret is a function writing such a packet in memory. Their definitions are architecture-specific, we find them in src/Core/arch/SSE/PacketMath.h:

The line in src/Core/arch/SSE/PacketMath.h that determines the PacketScalar type (via a typedef in Matrix.h) is:
\code
template<> struct internal::packet_traits<float>  { typedef __m128  type; enum {size=4}; };
\endcode
Here, __m128 is a SSE-specific type. Notice that the enum \a size here is what was used to define \a packetSize above.

And here is the implementation of internal::pstoret:
\code
template<> inline void internal::pstore(float*  to, const __m128&  from) { _mm_store_ps(to, from); }
\endcode
Here, __mm_store_ps is a SSE-specific intrinsic function, representing a single SSE instruction. The difference between internal::pstore and internal::pstoret is that internal::pstoret is a dispatcher handling both the aligned and unaligned cases, you find its definition in src/Core/GenericPacketMath.h:
\code
template<typename Scalar, typename Packet, int LoadMode>
inline void internal::pstoret(Scalar* to, const Packet& from)
{
  if(LoadMode == Aligned)
    internal::pstore(to, from);
  else
    internal::pstoreu(to, from);
}
\endcode

OK, that explains how writePacket() works. Now let's look into the packet() call. Remember that we are analyzing this line of code inside copyPacket():
\code
derived().template writePacket<StoreMode>(index,
    other.derived().template packet<LoadMode>(index));
\endcode

Here, \a other is our sum expression \a v + \a w. The .derived() is just casting from MatrixBase to the subclass which here is CwiseBinaryOp. So let's go to src/Core/CwiseBinaryOp.h:
\code
class CwiseBinaryOp
{
  // ...
    template<int LoadMode>
    inline PacketScalar packet(int index) const
    {
      return m_functor.packetOp(m_lhs.template packet<LoadMode>(index), m_rhs.template packet<LoadMode>(index));
    }
};
\endcode
Here, \a m_lhs is the vector \a v, and \a m_rhs is the vector \a w. So the packet() function here is Matrix::packet(). The template parameter \a LoadMode is \a #Aligned. So we're looking at
\code
class Matrix
{
  // ...
    template<int LoadMode>
    inline PacketScalar packet(int index) const
    {
      return internal::ploadt<Scalar, LoadMode>(m_storage.data() + index);
    }
};
\endcode
We let you look up the definition of internal::ploadt in GenericPacketMath.h and the internal::pload in src/Core/arch/SSE/PacketMath.h. It is very similar to the above for internal::pstore.

Let's go back to CwiseBinaryOp::packet(). Once the packets from the vectors \a v and \a w have been returned, what does this function do? It calls m_functor.packetOp() on them. What is m_functor? Here we must remember what particular template specialization of CwiseBinaryOp we're dealing with:
\code
CwiseBinaryOp<internal::scalar_sum_op<float>, VectorXf, VectorXf>
\endcode
So m_functor is an object of the empty class internal::scalar_sum_op<float>. As we mentioned above, don't worry about why we constructed an object of this empty class at all -- it's an implementation detail, the point is that some other functors need to store member data.

Anyway, internal::scalar_sum_op is defined in src/Core/Functors.h:
\code
template<typename Scalar> struct internal::scalar_sum_op EIGEN_EMPTY_STRUCT {
  inline const Scalar operator() (const Scalar& a, const Scalar& b) const { return a + b; }
  template<typename PacketScalar>
  inline const PacketScalar packetOp(const PacketScalar& a, const PacketScalar& b) const
  { return internal::padd(a,b); }
};
\endcode
As you can see, all what packetOp() does is to call internal::padd on the two packets. Here is the definition of internal::padd from src/Core/arch/SSE/PacketMath.h:
\code
template<> inline __m128  internal::padd(const __m128&  a, const __m128&  b) { return _mm_add_ps(a,b); }
\endcode
Here, _mm_add_ps is a SSE-specific intrinsic function, representing a single SSE instruction.

To summarize, the loop
\code
  for(int index = alignedStart; index < alignedEnd; index += packetSize)
  {
    dst.template copyPacket<Derived2, Aligned, internal::assign_traits<Derived1,Derived2>::SrcAlignment>(index, src);
  }
\endcode
has been compiled to the following code: for \a index going from 0 to the 11 ( = 48/4 - 1), read the i-th packet (of 4 floats) from the vector v and the i-th packet from the vector w using two __mm_load_ps SSE instructions, then add them together using a __mm_add_ps instruction, then store the result using a __mm_store_ps instruction.

There remains the second loop handling the last few (here, the last 2) coefficients:
\code
  for(int index = alignedEnd; index < size; index++)
    dst.copyCoeff(index, src);
\endcode
However, it works just like the one we just explained, it is just simpler because there is no SSE vectorization involved here. copyPacket() becomes copyCoeff(), packet() becomes coeff(), writePacket() becomes coeffRef(). If you followed us this far, you can probably understand this part by yourself.

We see that all the C++ abstraction of Eigen goes away during compilation and that we indeed are precisely controlling which assembly instructions we emit. Such is the beauty of C++! Since we have such precise control over the emitted assembly instructions, but such complex logic to choose the right instructions, we can say that Eigen really behaves like an optimizing compiler. If you prefer, you could say that Eigen behaves like a script for the compiler. In a sense, C++ template metaprogramming is scripting the compiler -- and it's been shown that this scripting language is Turing-complete. See <a href="http://en.wikipedia.org/wiki/Template_metaprogramming"> Wikipedia</a>.

*/

}