Binary GCD Algorithm - Efficiency

Efficiency

Akhavi and Vallée proved that binary GCD can be about 60% more efficient (in terms of the number of bit operations) on average than the Euclidean algorithm. However, although this algorithm outperforms the traditional Euclidean algorithm, its asymptotic performance is the same, and it is considerably more complex thanks to the availability of division instruction in all modern microprocessors.

In addition, real computers operate on more than one bit at a time, and even assembly language binary GCD implementations have to compete against carefully designed hardware circuits for integer division. Overall, Knuth (1998) reports a 15% gain over Euclidean GCD, and according to one comparison, the greatest gain was about 60%, while on some popular architectures even good implementations of binary GCD were marginally slower than the Euclidean algorithm.

In general, with implementations of binary GCD similar to the above C code, the gain in speed over the Euclidean algorithm is always less in practice than in theory. The reason is that the code uses many data-dependent branches. Many branches may be removed by computing min(a,b) and |a-b| using mixtures of Boolean algebra and arithmetic.

The only data-dependent branch that these techniques do not remove is the loop condition marked Loop X, which can be replaced by a single count trailing zeros (CTZ) operation and shift. Depending on platform, CTZ may be performed either by a single hardware instruction, by an equivalent instruction sequence, or with the aid of a lookup table.

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