Specialize inv

[jishnub]

Given that many of the constants defined here are reciprocals of each other, we may specialize inv and literal_pow to improve floating-point accuracy.

On main

ulia> sqrt2π*invsqrt2π
1.0000000000000002

julia> sqrt2*invsqrt2
1.0000000000000002

This PR

julia> sqrt2π*invsqrt2π == 1
true

julia> sqrt2*invsqrt2 == 1
true

[JeffreySarnoff]

before going into your PR more deeply, the else clause below is suspect

if VERSION > v"1.6.0"
    _one(x) = one(x)
else
    _one(x) = true
end

as is this result in v1.6.1

julia> sqrt2π * invsqrt2π
true

[devmotion]

one and zero for AbstractIrrationals was added in JuliaLang/julia#34773, so this is the expected result in recent Julia versions.

[JeffreySarnoff]

Good; still shouldn't sqrt2π * invsqrt2π === true be avoided?

[devmotion]

The advantage of true is that it is the least invasive identity. I assume this was also the motivation behind the definition in base.

[jishnub]

Yes indeed, that's the idea, true is equal to 1 and sits the lowest in the promotion hierarchy for Real numbers. If this is not desired, perhaps one may define an analogous real identity type that isn't a Bool but gets promoted identically.

[JeffreySarnoff]

Where would this approach leave e.g. sqrt4π * invsqrt2π? Is there a more general approach that covers that and similar cases? Would handling those relations be worth the overhead [i.e. what would be a parsimonious and effective way to handle exact integer ratios and their reciprocal results]?