Expression Templates

Expression templates are “structures representing a computation at compile-time, which 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 at the center of lazy evaluation and the center of this post.

What problem do expression templates solve? Thanks to expression templates, you can eliminate superfluous temporary objects in expressions. What do I mean by 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 (lines 12 and 15), knows its length (lines 18 – 20), and supports the reading (lines 23 – 25) and writing (lines 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 (lines 34 – 41), the overloaded * operator (lines 44 – 51), and the overloaded output operator (lines 54 – 62), the objects x, y, and result feel like numbers.

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

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

Expression templates

There are no temporaries needed for the expression result[i]= x[i]+x[i] + y[i]*y[i]. The assignment triggers the evaluation. Sadly, even in this simple usage, the code is 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 (lines 94 and 100). The expression tree is only created but not evaluated. Lazy, of course. The assignment operator (lines 22 – 27) triggers the evaluation of the expression tree that needs no temporaries.

The result is the same.

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

Under the hood

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

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.

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.

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Expression Templates

A first naive approach

Modernes C++ Mentoring