The first one is already explained, a^b + 2*(a&b) is computing the sum and carry in parallel, and then adding them up. The second formula is more elaborate, an

The first one is already explained, a^b + 2*(a&b) is computing the sum and carry in parallel, and then adding them up. The second formula is more elaborate, and makes heavy use of the above identity: 2*(a|b)-(a^b) = 2*((a&b)^a^b)-(a^b) (express boolean or in terms of and and xor) = 2*(a&b) ^ 2*(a^b) - (a^b) = 2*(a&b) + 2*(a^b) - (a^b) (there can't be carries, so xor equals addition in this case) = 2*(a^b) + 4*(a&b) - 2*(a&b) - (a^b) = 2*((a^b) + 2*(a&b)) - 2*(a&b) - (a^b) = 2*(a+b) - (a+b) = a+b.