Cache divgcd(::Int8, ::Int8)

[MilesCranmer]

This creates cached_divgcd with a specialized method for Int8 which caches all results at compilation time. Can get a 4x speedup on vectorized computations with normal Rational{Int8}.

This defines a CRational struct that uses this cached_divgcd for relevant rational calculations

@oscardssmith

[github-actions]

Benchmark Results

	main	a86d431...	t[main]/t[a86d431...]
creation/Quantity(x)	4 ± 0.1 ns	3.2 ± 0.1 ns	1.25
creation/Quantity(x, length=y)	3.3 ± 0.1 ns	3.3 ± 0.1 ns	1
time_to_load	0.138 ± 0.0032 s	0.137 ± 0.0024 s	1.01
with_numbers/*real	3.3 ± 0.1 ns	3.2 ± 0.1 ns	1.03
with_numbers/^int	28.4 ± 2 ns	0.0595 ± 0.0028 μs	0.477
with_numbers/^int * real	28.3 ± 2.2 ns	0.0594 ± 0.0026 μs	0.476
with_quantity/*y	17.8 ± 0.4 ns	0.0341 ± 1e-06 μs	0.522
with_quantity//y	17.8 ± 0.4 ns	0.039 ± 0.0002 μs	0.456
with_quantity/^y	27.4 ± 2.1 ns	0.0589 ± 0.0012 μs	0.465
with_self/dimension	1.6 ± 0.1 ns	1.6 ± 0.1 ns	1
with_self/inv	18.5 ± 0 ns	16.5 ± 0 ns	1.12
with_self/ustrip	1.6 ± 0.1 ns	1.6 ± 0.1 ns	1

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[oscardssmith]

I've convinced myself that the right type here is actually a Int8/6, since the math then becomes trivial and doing a search has not lead me to any use-cases that need a power that isn't a multiple of 1/6th.

[MilesCranmer]

Wow, check out the speeds here! So caching the gcd is enough to make normal Rational{Int8} fast. Then it is much safer from overflow too, because it always reduces to lowest coprimes.

[MilesCranmer]

I do think the best option would be to parametrize the Dimensions type by this choice. And just have the default be Rational{Int} I can probably get away with Int8/6 as well in my stuff, though I’m sure some people will have some weird system where they need large rational powers… For example, constants associated with a polytropic index in astrophysics can pretty much have any rational power on the units you can write down.

[oscardssmith]

The type piracy here isn't great but the performance is. One improvement here is that you can make the table 2x smaller by making x be smaller than y (since gcd commutes).

[MilesCranmer]

Surprisingly it is slower. I guess the <(::Int8,: :Int8) incurs more of a penalty than just the lookup?

[MilesCranmer]

@oscardssmith two new changes:

1. I made a custom CRational type to avoid type piracy. It's an internal type, so I only overload the methods we actually need.
2. I switched to using SafeInt8 again.

The speed goes down a bit, but it's so much safer than normal Ratios.jl that I think this is worth it.

When you have a chance could you maybe try to get Dimensions{R} working for a user-specified R?

[oscardssmith]

will do

[MilesCranmer]

Superseded by the faster & cleaner #21