How does Uniswap's LowGasSafeMath work?

Question

I'm having trouble understanding LowGasSafeMath. For example, how does this function work?

function mul(uint256 x, uint256 y) internal pure returns (uint256 z) {
    require(x == 0 || (z = x * y) / x == y);
}

The first thing that puzzled me is that this returns a value without a return statement. Then, how does the logic inside the require statement work? As a programmer, I'm familiar with boolean logic, but haven't seen it used like this before.

Answer 1

Return is done with returns (uint256 z). This means that when the function ends the variable z is returned. The function checks if there was an overflow in the multiplication, because then the result would be wrong. To make sure it is correct, the result of z = x* y is decided by x, if this equals y then the calculation is correct (one could also decide by y). To avoid division by 0, it is checked if x ==0. If this is the case, the whole statement is true and therefore not evaluated further. Because multiplication by 0 is 0, no calculation is needed, because z = 0 by default.

It is the same as

if (x==0) {
    return 0;
}
z = x * y;
require(z / x == y);
return z;