Avoiding Temporaries with Expression Templates

8 March 2022

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Expression templates are typically used in linear algebra and 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). In other words, expression templates are only evaluated when needed.

ExpressionTemplates

I provide you with this post only the critical ideas of expression templates. To use them, you should study further content, such as

C++ Templates: The Complete Guide by David Vandervoorde, Nicolai M. Josuttis, and Douglas Gregor (http://www.tmplbook.com/)
Boost Basic Linear Algebra Library (https://www.boost.org/doc/libs/1_59_0/libs/numeric/ublas/doc/index.html)
Expression Templates Revisited by Klaus Iglberger (https://www.youtube.com/watch?v=hfn0BVOegac). Klaus's talk demystifies many performance-related myths about expression templates.

What problem do expression templates solve? Thanks to expression templates, you can get rid of 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 1 and 2), knows its length (line 3), and supports the reading (line 4) and writing (line 4) by index.

// 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){}                                           // (1)

  // MyVector with initial size and value
  MyVector(const std::size_t n, const double initialValue) : cont(n, initialValue){}  // (2)
  
  // size of underlying container
  std::size_t size() const{                                                           // (3)
    return cont.size(); 
  }

  // index operators
  T operator[](const std::size_t i) const{                                            // (4)
    return cont[i]; 
  }

  T& operator[](const std::size_t i){                                                 // (5)
    return cont[i]; 
  }

};

// function template for the + operator
template<typename T> 
MyVector<T> operator+ (const MyVector<T>& a, const MyVector<T>& b){                   // (6)
  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){                  // (7)
   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){                 // (8)
  std::cout << '\n';
  for (int i = 0; i < cont.size(); ++i) {
    os << cont[i] << ' ';
  }
  os << '\n';
  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 << '\n';
  
}

Thanks to the overloaded + operator (line 6), the overloaded * operator (line 7), and the overloaded output operator (line 8) the objects x, y, and result behave like numbers.

vectorArithmeticOperatorOverloading

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

Temporaries

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. Lazy evaluation means that an expression is only evaluated when needed.

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Expression templates

ExpressionTree

There are no temporaries needed for the expression result[i] = x[i] + x[i] + y[i] * y[i]. The assignment triggers the evaluation. Sadly, 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>                                      // (3)
  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()));   // (1)
}

// 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()));  // (2)
}

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

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 trees (lines 1 and 2). The expression trees are only created but not evaluated. Lazy, of course. The assignment operator (line 3) triggers the evaluation of the expression tree that needs no temporaries.

The result is the same.

vectorArithmeticExpressionTemplates

Thanks to the compiler explorer, I can visualize the magic of the program vectorArithmeticExpressionTemplates.cpp.

Under the hood

Here are the essential assembler instructions for the final assignment in the main function: result= x + x + y * y.

godbolt

The expression tree in the assembler snippet looks scary, but you can see the structure with a sharp eye. For simplicity reasons, I ignored std::allocator in my graphic.

Exression

What's next?

A policy is a generic function or class whose behavior can be configured. Let me introduce them in my next post.

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