Can't you use a bit shift, nested in a couple of for loops

Don't use the code it's crap

ie

for (i = 0; i++; i<16){       //outer loop increments the bits set
   for (j= 0; j++; j<16){     //inner loop shifts the set bits along
      variable_to_shift >> j; 
   }
    
   number_of_bits_set = i;
}

Obviously there is a bit more than this, but not a lot I would suspect.

Don;t judge the code, it's not thought out

Anyway just a thought.