Expression Templates

15 February 2017

Contents[Show]

Expression templates are "structures representing a computation at compile-time, which structures are evaluated only as needed to produce efficient code for the entire computation" (https://en.wikipedia.org/wiki/Expression_templates). As needed, now we are in the centre of lazy evaluation and in the centre of this post.

What problem do expression templates solve? Thanks to expression templates, you can get rid of superfluous temporary objects in expressions. What do I mean with superfluous temporary objects? My implementation of the class MyVector.

A first naive approach

MyVector is a simple wrapper for a std::vector<T>. The wrapper has two constructors (line 12 and 15), knows its length (line 18 - 20) and supports the reading (line 23 - 25) and writing (line 27 - 29) index access.

// vectorArithmeticOperatorOverloading.cpp

#include <iostream>
#include <vector>

template<typename T>
class MyVector{
  std::vector<T> cont;   

public:
  // MyVector with initial size
  MyVector(const std::size_t n) : cont(n){}  

  // MyVector with initial size and value
  MyVector(const std::size_t n, const double initialValue) : cont(n, initialValue){}
  
  // size of underlying container
  std::size_t size() const{ 
    return cont.size(); 
  }

  // index operators
  T operator[](const std::size_t i) const{ 
    return cont[i]; 
  }

  T& operator[](const std::size_t i){ 
    return cont[i]; 
  }

};

// function template for the + operator
template<typename T> 
MyVector<T> operator+ (const MyVector<T>& a, const MyVector<T>& b){
  MyVector<T> result(a.size());
  for (std::size_t s= 0; s <= a.size(); ++s){
    result[s]= a[s]+b[s];
  }
  return result;
}

// function template for the * operator
template<typename T>
MyVector<T> operator* (const MyVector<T>& a, const MyVector<T>& b){
   MyVector<T> result(a.size());
  for (std::size_t s= 0; s <= a.size(); ++s){
    result[s]= a[s]*b[s]; 
  }
  return result;
}

// function template for << operator
template<typename T>
std::ostream& operator<<(std::ostream& os, const MyVector<T>& cont){  
  std::cout << std::endl;
  for (int i=0; i<cont.size(); ++i) {
    os << cont[i] << ' ';
  }
  os << std::endl;
  return os;
} 

int main(){

  MyVector<double> x(10,5.4);
  MyVector<double> y(10,10.3);

  MyVector<double> result(10);
  
  result= x+x + y*y;
  
  std::cout << result << std::endl;
  
}

Thanks to the overloaded + operator (line 34 - 41), the overloaded * operator (line 44 - 51) and the overloaded output operator (line 54 - 62) the objects x, y and result feel like numbers.

vectorArithmeticOperatorOverloading

Why is this implementation naive? The answer is in the expression result= x+x + y*y. In order to evaluate the expression, three temporary objects are needed to hold the result of each arithmetic sub expression.

Temporaries

How can I get rid of the temporaries? The idea is simple. Instead of performing the vector operations greedy, I create the expression tree for result[i] at compile time in a lazy manner.

Expression templates

ExpressionTree

There are no temporaries need for the expression result[i]= x[i]+x[i] + y[i]*y[i]. The assignment triggers the evaluation. Sad to say, but the code is even in this simple usage not so easy to digest.

// vectorArithmeticExpressionTemplates.cpp

#include <cassert>
#include <iostream>
#include <vector>

template<typename T, typename Cont= std::vector<T> >
class MyVector{
  Cont cont;   

public:
  // MyVector with initial size
  MyVector(const std::size_t n) : cont(n){}

  // MyVector with initial size and value
  MyVector(const std::size_t n, const double initialValue) : cont(n, initialValue){}

  // Constructor for underlying container
  MyVector(const Cont& other) : cont(other){}

  // assignment operator for MyVector of different type
  template<typename T2, typename R2>
  MyVector& operator=(const MyVector<T2, R2>& other){
    assert(size() == other.size());
    for (std::size_t i = 0; i < cont.size(); ++i) cont[i] = other[i];
    return *this;
  }

  // size of underlying container
  std::size_t size() const{ 
    return cont.size(); 
  }

  // index operators
  T operator[](const std::size_t i) const{ 
    return cont[i]; 
  }

  T& operator[](const std::size_t i){ 
    return cont[i]; 
  }

  // returns the underlying data
  const Cont& data() const{ 
    return cont; 
  }

  Cont& data(){ 
    return cont; 
  }
};

// MyVector + MyVector
template<typename T, typename Op1 , typename Op2>
class MyVectorAdd{
  const Op1& op1;
  const Op2& op2;

public:
  MyVectorAdd(const Op1& a, const Op2& b): op1(a), op2(b){}

  T operator[](const std::size_t i) const{ 
    return op1[i] + op2[i]; 
  }

  std::size_t size() const{ 
    return op1.size(); 
  }
};

// elementwise MyVector * MyVector
template< typename T, typename Op1 , typename Op2 >
class MyVectorMul {
  const Op1& op1;
  const Op2& op2;

public:
  MyVectorMul(const Op1& a, const Op2& b ): op1(a), op2(b){}

  T operator[](const std::size_t i) const{ 
    return op1[i] * op2[i]; 
  }

  std::size_t size() const{ 
    return op1.size(); 
  }
};

// function template for the + operator
template<typename T, typename R1, typename R2>
MyVector<T, MyVectorAdd<T, R1, R2> >
operator+ (const MyVector<T, R1>& a, const MyVector<T, R2>& b){
  return MyVector<T, MyVectorAdd<T, R1, R2> >(MyVectorAdd<T, R1, R2 >(a.data(), b.data()));
}

// function template for the * operator
template<typename T, typename R1, typename R2>
MyVector<T, MyVectorMul< T, R1, R2> >
operator* (const MyVector<T, R1>& a, const MyVector<T, R2>& b){
   return MyVector<T, MyVectorMul<T, R1, R2> >(MyVectorMul<T, R1, R2 >(a.data(), b.data()));
}

// function template for < operator
template<typename T>
std::ostream& operator<<(std::ostream& os, const MyVector<T>& cont){  
  std::cout << std::endl;
  for (int i=0; i<cont.size(); ++i) {
    os << cont[i] << ' ';
  }
  os << std::endl;
  return os;
} 

int main(){

  MyVector<double> x(10,5.4);
  MyVector<double> y(10,10.3);

  MyVector<double> result(10);
  
  result= x+x + y*y;
  
  std::cout << result << std::endl;
  
}

The key difference between the first naive implementation and this implementation with expression templates is that the overloaded + and + operators return in the case of the expression tree proxy objects. These proxies represent the expression tree (line 94 and 100). The expression tree is only created but not evaluated. Lazy, of course. The assignment operator (line 22 - 27) triggers the evaluation of the expression tree that needs no temporaries.

The result is the same.

vectorArithmeticExpressionTemplates

If you were not able to follow my explanation, no problem. The assembler code of the program vectorArithmeticExpressionTemplates.cpp shows the magic.

Modernes C++ Mentoring

Stay informed about my mentoring programs.

- - - - "Fundamentals for C++ Professionals" (open)
- - - - "Design Patterns and Architectural Patterns with C++" (reopens May 2023)

Subscribe via E-Mail.

Under the hood

Thanks to the compiler explorer on godbolt.org, it's quite easy to have the assembler instructions.

godbolt

The expression tree in line 60 is not so beautiful. But with a sharp eye, you can see the structure. For simplicity reasons, I ignored the std::allocator in my graphic.

Exression

What's next?

With the next post, I will start the rework of my blog. That means I will rework old posts and write new posts to complete my stories. I will start in the next post with the multithreading features of C++17 and C++20. Here is an overview of all my posts.

{tooltip} title page small {end-texte} Go to Leanpub/cpplibrary "What every professional C++ programmer should know about the C++ standard library". {end-tooltip} Get your e-book. Support my blog.