How can I speed up the binary GCD algorithm using __builtin_ctz?

Question

clang and GCC have a int __builtin_ctz(unsigned) function. This counts the trailing zeros in an integer. The Wikipedia article on this family of functions mentions that the binary GCD algorithm can be sped up using __builtin_ctz, but I don't understand how.

The sample implementation of the binary GCD looks like this:

unsigned int gcd(unsigned int u, unsigned int v)
{
    // simple cases (termination)
    if (u == v)
        return u;

    if (u == 0)
        return v;

    if (v == 0)
        return u;

    // look for factors of 2
    if (~u & 1) // u is even
        if (v & 1) // v is odd
            return gcd(u >> 1, v);
        else // both u and v are even
            return gcd(u >> 1, v >> 1) << 1;

    if (~v & 1) // u is odd, v is even
        return gcd(u, v >> 1);

    // reduce larger argument
    if (u > v)
        return gcd(u - v, v);

    return gcd(v - u, u);
}

My suspicion is that I could use __builtin_ctz as follows:

constexpr unsigned int gcd(unsigned int u, unsigned int v)
{
    // simplified first three ifs
    if (u == v || u == 0 || v == 0)
        return u | v;

    unsigned ushift = __builtin_ctz(u);
    u >>= ushift;

    unsigned vshift = __builtin_ctz(v);
    v >>= vshift;

    // Note sure if max is the right approach here.
    // In the if-else block you can see both arguments being rshifted
    // and the result being leftshifted only once.
    // I expected to recreate this behavior using max.
    unsigned maxshift = std::max(ushift, vshift);

    // The only case which was not handled in the if-else block before was
    // the odd/odd case.
    // We can detect this case using the maximum shift.
    if (maxshift != 0) {
        return gcd(u, v) << maxshift;
    }

    return (u > v) ? gcd(u - v, v) : gcd(v - u, u);
}

int main() {
    constexpr unsigned result = gcd(5, 3);
    return result;
}

Unfortunately this does not work yet. The program results in 4, when it should be 1. So what am I doing wrong? How can I use __builtin_ctz correctly here? See my code so far on GodBolt.

Answer 1

Here's my iterative implementation from the comments:

While tail-recursive algorithms are often elegant, iterative implementations are almost always faster in practice. (Modern compilers can actually perform this transform in very simple cases.)

unsigned ugcd (unsigned u, unsigned v)
{
    unsigned t = u | v;

    if (u == 0 || v == 0)
        return t; /* return (v) or (u), resp. */

    int g = __builtin_ctz(t);

    while (u != 0)
    {
        u >>= __builtin_ctz(u);
        v >>= __builtin_ctz(v);

        if (u >= v)
            u = (u - v) / 2;
        else
            v = (v - u) / 2;
    }

    return (v << g); /* scale by common factor. */
}

As mentioned, the |u - v| / 2 step is typically implemented as a very efficient, unconditional right shift, e.g., shr r32, to divide by (2) - as both (u), (v) are odd, and therefore |u - v| must be even.

It's not strictly necessary, as the 'oddifying' step: u >>= __builtin_clz(u); will effectively perform this operation in the next iteration.

Supposing that (u) or (v) have a 'random' bit distribution, the probability of (n) trailing zeroes, via tzcnt, is ~ (1/(2^n)). This instruction is an improvement over bsf, the implementation for __builtin_clz prior to Haswell, IIRC.