Question

Whats the reverse function of x XOR (x/2)?

Is there a system of rules for equation solving, similar to algebra, but with logic operators?

Answer 1

Suppose we have a number x of N bits. You could write this as:

b(N-1) b(N-2) b(N-3) ... b(0)

where b(i) is bit number i in the number (where 0 is the least significant bit).

x / 2 is the same as x shifted left 1 bit. Let’s assume unsigned numbers. So:

x / 2 = 0 b(N-1) b(N-2) ... b(1)

Now we XOR x with x / 2:

x ^ (x / 2) = b(N-1)^0 b(N-2)^b(N-1) b(N-3)^b(N-2) ... b(0)^b(1)

Note that the rightmost bit (the most significant bit) of this is b(N-1)^0 which is b(N-1). In other words, you can get bit b(N-1) from the result immediately. When you have this bit, you can calculate b(N-2) because the second bit of the result is b(N-2)^b(N-1) and you already know b(N-1). And so on, you can compute all bits b(N-1) to b(0) of the original number x.