Implement rand(::Interval)

[lbenet]

Is it worth to have a function returning a "random" point inside agiven interval? I use something like this for testing in TaylorModels, but implementing it there sounds like type pyracy. (I think we actually had something alike in v0.20)

[OlivierHnt]

Oh I'm sorry, I did not read this issue properly... From the title I was mislead into thinking you needed to create a random interval, i.e. rand(::Type{<:Interval}).
Maybe to avoid ambiguities we should call the function you want randpick (or something like this) instead of rand.

What you want is a feature in Distributions.jl.
I am not an expert on random numbers, and they probably do something smart to have a good distribution.

What is your usage? If it's for testing I would suggest to use Distributions directly (e.g. randpick(x::Interval) = rand(Uniform(bounds(x)...))).
If there is a more fundamental reason for having this feature, we could provide a somewhat naive version

julia> randpick(x::Interval{T}) where {T<:AbstractFloat} = max(inf(x), floatmin(T)) + rand(T) * (min(sup(x), floatmax(T)) - max(inf(x), floatmin(T)))

julia> randpick(interval(1, 2))
1.412138491886071

julia> randpick(interval(1, Inf))
4.194213273535214e307

julia> randpick(emptyinterval())
NaN

julia> randpick(nai())
NaN

[OlivierHnt]

Ah in fact, isn't what you want just equivalent to calling mid(x, rand())?

[lbenet]

What you want is a feature in Distributions.jl. I am not an expert on random numbers, and they probably do something smart to have a good distribution.

What is your usage? If it's for testing I would suggest to use Distributions directly (e.g. randpick(x::Interval) = rand(Uniform(bounds(x)...))). If there is a more fundamental reason for having this feature, we could provide a somewhat naive version

julia> randpick(x::Interval{T}) where {T<:AbstractFloat} = max(inf(x), floatmin(T)) + rand(T) * (min(sup(x), floatmax(T)) - max(inf(x), floatmin(T)))

julia> randpick(interval(1, 2))
1.412138491886071

julia> randpick(interval(1, Inf))
4.194213273535214e307

julia> randpick(emptyinterval())
NaN

julia> randpick(nai())
NaN

My usage is testing, assuming a uniform distribution: simply pick-up a pseudorandom number between inf(a) and sup(a). What we had at some point was even more naïve that randpick as you wrote it above (without the important subtleties of floatmin and floatmax).

Ah in fact, isn't what you want just equivalent to calling mid(x, rand())?

Didn't think about that! That a good enough solution!

Thanks again!

[dpsanders]

That is a cute solution, but I still think we should have rand(x) for an interval x defined in that way, since this is a general method in Julia for random sampling from a set.

[OlivierHnt]

Ah yes (cf. https://docs.julialang.org/en/v1/stdlib/Random/#A-simple-sampler-without-pre-computed-data).
So following the Random API we get

Random.rand(rng::Random.AbstractRNG, x::Random.SamplerTrivial{Interval{T}}) where {T} = mid(x[], rand(T))

[OlivierHnt]

This has revealed a bug 😅

julia> mid(interval(2), 0.4969816845401611)
1.9999999999999998

[dpsanders]

Ouch! How did that happen?

[dpsanders]

I guess we just need to clamp the result to the interval.

[OlivierHnt]

We compute the midpoint via β * (hi + lo * (1 / β - 1)) and this gives $\pm 1$ ulp error (maybe that can be more).
I'll make a quick PR.