faster gcd

[nolta]

Average speedup for random inputs ranges from 20% 8% (Int8) to 70% (Int64).

[vtjnash]

is this actually a faster method, or just a way to avoid rem

[nolta]

If i'm reading this correctly, the binary algorithm should be about 1.8x faster for large N.

[carlobaldassi]

This implementation requires trailing_zeros and bit shift operations, so it depends on the internal representation - I wouldn't expect all possible user-defined integer types to require implementing those functions, or conforming to a specific representation pattern. In other words, the bit representation shouldn't be part of the interface an Integer type is expected to have.

I think that in the interest of generality it would be useful to keep the old version (for user-defined integers), while this more optimized one could be specialized on the concrete Integer types of Base.

[nolta]

That's fair. I guess i'm a little squeamish about including code in Base which isn't used by any of the types in Base.

[carlobaldassi]

Yes, I understand. It would also need to be tested explicitly via invoke or by constructing an Integer for that sole purpose, which is also somehow a possible hint of a problem.

One more issue is that of extending the specialized version: using a Union type would not be extensible, so a traits-based version would be better in that respect. But then we'd need to introduce a "standard binary representation" trait, or something along those lines.

On the other hand, gcd is a general function, which other functions depend upon (e.g. Rational constructor), and with a very clear mathematical meaning in terms of elementary operations, so I believe that the generic function with the abstract Integer signature should follow that.

Maybe having a concrete example in mind would be useful to further the discussion, e.g. ModInts.

[nolta]

Ok, i've restricted this to Union(64,128). The code is type unstable for ints < Int, so the speedup is only ~1.1x for random Int8/16/32, but then jumps to 1.7x for Int64 and 2.1x for Int128.

[eschnett]

Type-unstable? Couldn't that be fixed rather easily by promoting the arguments or converting the result of trailing_zeros etc?

[nolta]

Sure, but i'm lazy.

[ivarne]

Should we have a comment in the source to inform source code readers why we have two implementations of gcd for Integers? I don't think it is totally obvious.

[nolta]

Ok, comment added.

[StefanKarpinski]

Sure, but i'm lazy.

You have a strange definition of lazy.