x & (x-1) will clear the lowest set bit of x

x & ~(x-1) extracts the lowest set bit of x (all others are clear). Pretty patterns when applied to a linear sequences.

x & (x + (1 << n)) = x, with the run of set bits (possibly length 0) starting at bit n cleared.

x & ~(x + (1 << n)) = the run of set bits (possibly length 0) in x, starting at bit n.

x | (x + 1) = x with the lowest cleared bit set.

x | ~(x + 1) = extracts the lowest cleared bit of x (all others are set).

x | (x - (1 << n)) = x, with the run of cleared bits (possibly length 0) starting at bit n set.

x | ~(x - (1 << n)) = the lowest run of cleared bits (possibly length 0) in x, starting at bit n are the only clear bits.

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x ^ 0s = x

x & 0s = 0

x | 0s = x

x ^ 1s = ~x

x & 1s = x

x | 1s = 1s

x ^ x = 0

x & x = x

x | x = x