Do not use widen for 128-bit ints in MultiplicativeInverses

[sumiya11]

Hello! This PR improves MultiplicativeInverses for Int128 and UInt128. PR consists of 3 tiny points:

• Previously, division with multiplicative inverses used widen. It widened 128-bit ints to BigInt, which hindered performance. The solution in this PR is to replace widen with higher half multiplication for Int128 and UInt128.

# 1.8.2
using BenchmarkTools
a, b = rand(UInt128), rand(UInt128)
d = Base.MultiplicativeInverses.UnsignedMultiplicativeInverse(b)

@btime div($a, $d)
  645.455 ns (12 allocations: 216 bytes)

# this pr
using BenchmarkTools
a, b = rand(UInt128), rand(UInt128)
d = Base.MultiplicativeInverses.UnsignedMultiplicativeInverse(b)

@btime div($a, $d)
  6.800 ns (0 allocations: 0 bytes)

• Added tests for MultiplicativeInverses for all bit integer types;
• Draw attention to the fact that signed division with MultiplicativeInverses does not throw when the result overflows (see tests)

As for the latter, it's alright with me, but I think it would be good to mention this fact somewhere

Also, I've coded the higher half mult. from scratch, so let me know if there were functions for this in Base already 😊

[oscardssmith]

This is a really nice improvement! I think we should consider making mulhi an exported function (e.g. I wouldn't mind using it in Primes.jl).

[oscardssmith]

Also, do you happen to know if there's an efficient way to compute a*b%n for Int128 without widening to BigInt?

[sumiya11]

Also, do you happen to know if there's an efficient way to compute a*b%n for Int128 without widening to BigInt?

Nope :(

I think we should consider making mulhi an exported function (e.g. I wouldn't mind using it in Primes.jl).

Sounds like a good idea. I would also use it somewhere in fast modular arithmetic

[sumiya11]

ping !

[oscardssmith]

Needs a rebase but other than that, I think it's ready.

[oscardssmith]

I just did the rebase and will merge when tests pass.