listing 1
The simplest square root.

unsigned long sqrt(unsigned long a){
  unsigned long x = 1;
  while(x*x <= a)
    ++x;
  return ‹x;
}



listing 2
Improving efficiency.

unsigned long sqrt(unsigned long a){
  unsigned long square = 1;
  unsigned long delta = 3;
  while(square <= a){
    square += delta;
    delta += 2;
  }
    return (delta/2 - 1);
} 



listing 3
The binary square root.

unsigned short sqrt(unsigned long a){
  unsigned long rem = 0;
  unsigned long root = 0;
  unsigned long divisor = 0;
  for(int i=0; i<16; i++){
    root <<= 1;
    rem = ((rem << 2) + (a >> 30));
    a <<= 2;
    divisor = (root<<1) + 1;
    if(divisor <= rem){
      rem -= divisor;
      root++;
    }
  }
  return (unsigned short)(root);
}



listing 4
An improved version.

unsigned short sqrt(unsigned long a){
  unsigned long rem = 0;
  unsigned long root = 0;
  for(int i=0; i<16; i++){
    root <<= 1;
    rem = ((rem << 2) + (a >> 30));
    a <<= 2;
    root ++;
    if(root <= rem){
      rem -= root;
      root++;
    }
    else
      root‹;
  }
  return (unsigned short)(root >> 1);
}