Russian Peasants Multiplication Algorithm

Here is a cool algorithm that calculates multiplication by addition and bit-shifting only. This method is exponentially faster than the literal implantation of multiplication in which x is added y times.

int multiply (unsigned x, unsigned y) {
  unsigned a,b;
  unsigned c = 0;
  // make sure a < b so that we add b a times
  if (x<y) {
    a = x;
    b = y;
  } else {
    a = y;
    b = x;
  }

  while (a > 0) {
    if (a & 1)
      c += b;
    a >>= 1;
    b <<= 1;
  }

  return c;
}

To prove that it works, consider the function
f(a,b,c) = ab + c
before the loop, during the loop, and after the loop.

Before the loop, we have
f = xy + 0 = xy

During the loop, we have for even a,
a' = a/2
b' = 2b
c' = c
f' = ab + c = xy

and for odd a,
a' = (a-1)/2
b' = 2b
c' = c + b
f' = ab + c = xy

Hence, it must be the case that f(a,b,c) = xy for any number of iterations in the loop by induction. Now, the ending condition of the loop yields a = 0, so it must be the case that
c = f - ab = f = xy

In other words, the return value is guaranteed to be the product of the input x and y!