How to prove that the C statement -x, ~x+1, and ~(x-1) yield the same results?

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Consider what you get when you add a number to its bitwise complement. The bitwise complement of an n-bit integer x has a 1 everywhere x has a 0, and vice versa. So it’s clear to see:

x + ~x = 0b11…11 (n-bit value of all ones)

Regardless of the number of bits in x. Further, note that adding one to an n-bit number filled with all ones will make it wrap to zero. Thus we see:

x + ~x + 1 = 0b11…11 + 1 = 0
and ~x + 1 = -x.

Similarly, note (x – 1) + ~(x – 1) = 0b11…11. Then (x – 1) + ~(x – 1) + 1 = 0, and ~(x – 1) = -x.

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