 Coupled binary exponentiation

To compute two nonnegative integer powers in one while loop, maintain a separate accumulator, current base and remaining exponent for each exponentiation by squaring. The paired loop invariant is $p{x}^k=A^n$ and $q{y}^l=B^m$. For each positive remaining exponent, transfer its odd factor to its accumulator and halve the exponent while squaring its base. A zero remaining exponent is left unchanged. The mathematical integer $k+l$ is a loop variant: at least one positive term strictly decreases in every iteration. Both powers are therefore complete when the loop stops. The iteration count is $O(\log (\max (n,m)+1))$, with constant space and a constant number of multiplications per iteration. This is an exact-arithmetic identity; floating-point rounding introduces a separate error in a numerical implementation.