#1638 - Efficient way to obtain precise Interval bounds in 1D

[schillic]

Closes #1638.

The first commit is only related but it makes the convert faster.

julia> @btime convert(Interval, rand(Hyperrectangle, dim=1));  # master
  265.718 ns (7 allocations: 608 bytes)

julia> @btime convert(Interval, rand(Hyperrectangle, dim=1));  # this branch
  225.956 ns (6 allocations: 512 bytes)

[mforets]

I haven't looked at the PR yet but here is a comment with respect to the benchmark: If the set is hyperrectangular, convert / overapproximate should be very cheap:

julia> using LazySets, BenchmarkTools

julia> x = rand(Hyperrectangle, dim=1);

julia> @btime convert(Interval, $x)
  81.583 ns (3 allocations: 288 bytes)
Interval{Float64,IntervalArithmetic.Interval{Float64}}([0.0447304, 1.20319])

julia> f(x) = Interval(center(x)[1] - radius_hyperrectangle(x, 1), center(x)[1] + radius_hyperrectangle(x)[1])
f (generic function with 1 method)

julia> @btime f($x)
  1.759 ns (0 allocations: 0 bytes)
Interval{Float64,IntervalArithmetic.Interval{Float64}}([0.0447304, 1.20319])

[schillic]

If the set is hyperrectangular, convert / overapproximate should be very cheap

Yes, we already had a fast method for AbstractHyperrectangles.

[mforets]

True, we have it but it is suboptimal:

function convert(::Type{Interval}, H::AbstractHyperrectangle) # master
    @assert dim(H) == 1 "can only convert a one-dimensional $(typeof(H)) to `Interval`"
    return Interval([low(H); high(H)])
end

function convert3(::Type{Interval}, H::AbstractHyperrectangle)
    @assert dim(H) == 1 "can only convert a one-dimensional $(typeof(H)) to `Interval`"
    r = radius_hyperrectangle(H, 1)
    c = center(H)[1] # TODO: change to center(H, 1) when available (see #.....)
    return Interval(c-r, c+r)
end

julia> using BenchmarkTools; x= rand(Hyperrectangle, dim=1);

julia> @btime convert(Interval, $x) # master
  81.593 ns (3 allocations: 288 bytes)
Interval{Float64,IntervalArithmetic.Interval{Float64}}([-0.222182, 0.716047])

julia> @btime LazySets.convert3(Interval, $x) # new function
  7.583 ns (0 allocations: 0 bytes)
Interval{Float64,IntervalArithmetic.Interval{Float64}}([-0.222182, 0.716047])

julia> @btime LazySets.convert3(Interval, $x) # new function without assertion
  1.780 ns (0 allocations: 0 bytes)
Interval{Float64,IntervalArithmetic.Interval{Float64}}([-0.222182, 0.716047])

I know this discussion is unrelated to the PR, i will make a separate issue..

[mforets]

lgtm 🎉