Expression templates

Expression templates are a C++ template metaprogramming technique that builds structures representing a computation at compile time, which structures are evaluated only as needed to produce efficient code for the entire computation. Expression templates thus allow programmers to bypass the normal order of evaluation of the C++ language and achieve optimizations such as loop fusion.

Expression templates were invented independently by Todd Veldhuizen and David Vandevoorde; it was Veldhuizen who gave them their name. They are a popular technique for the implementation of linear algebra software.

Consider a library representing vectors and operations on them. One common mathematical operation is to add two vectors u and v, element-wise, to produce a new vector. The obvious C++ implementation of this operation would be an overloaded operator+ that returns a new vector object:

Users of this class can now write Vec x = a + b; where a and b are both instances of Vec.

A problem with this approach is that more complicated expressions such as Vec x = a + b + c are implemented inefficiently. The implementation first produces a temporary vector to hold a + b, then produces another vector with the elements of c added in. Even with return value optimization this will allocate memory at least twice and require two loops.

Delayed evaluation solves this problem, and can be implemented in C++ by letting operator+ return an object of a custom type, say VecSum, that represents the unevaluated sum of two vectors, or a vector with a VecSum, etc. Larger expressions then effectively build expression trees that are evaluated only when assigned to an actual Vec variable. But this requires traversing such trees to do the evaluation, which is in itself costly.

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