My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <bitset>
       
#include <cmath>
       
 
       
int main()
       
{
       
  // in my tests I found that all tested squares are well below one million
       
  const unsigned int Limit = 1000000;
       
 
       
  // record all square numbers
       
  // [n] => [true, if n is a perfect square]
       
  std::bitset<Limit> isSquare; // default is false / zero
       
  for (unsigned int i = 1; i*i < isSquare.size(); i++)
       
    isSquare[i*i] = true;
       
 
       
  // substitute in (3):
       
  // (9) z = c^2 - x
       
  for (unsigned int a = 3; ; a++)
       
  {
       
    // a and b must be both odd or both even
       
    unsigned int minB = (a % 2 == 0) ? 2 : 1;
       
    for (unsigned int b = minB; b < a; b += 2) // keep parity, skip every second number
       
    {
       
      // from (8): compute x
       
      auto x = (a*a + b*b) / 2;
       
      // from (9): compute y
       
      auto y = a*a - x;
       
      // ensure x > y
       
      if (x <= y)
       
        break;
       
 
       
      for (unsigned int c = (unsigned int)sqrt(x) + 1; ; c++)
       
      {
       
        // from (10): compute z
       
        auto z = c*c - x;
       
        // ensure y > z
       
        if (y <= z)
       
          break;
       
 
       
        // check whether d^2, e^2 and f^2 are perfect squares
       
        if (isSquare[x - z] &&
       
            isSquare[y + z] &&
       
            isSquare[y - z])
       
        {
       
          // found a solution, stop program
       
          std::cout << x + y + z << std::endl;
       
          return 0;
       
        }
       
      }
       
    }
       
  }
       
 
       
  // never reached
       
}

This solution contains 6 empty lines, 14 comments and 3 preprocessor commands.