transcendental functions type-unstable for Interval{Rational}

[lucaferranti]

e.g.

julia> @code_warntype sqrt(1//2..3//4)
Variables
  #self#::Core.Const(sqrt)
  a@_2::Interval{Rational{Int64}}
  domain::Interval{Rational{Int64}}
  a@_4::Interval{Rational{Int64}}

Body::Union{Interval{Rational{Int64}}, Interval{Float64}}
1 ─       (a@_4 = a@_2)
│   %2  = Core.apply_type(IntervalArithmetic.Interval, $(Expr(:static_parameter, 1)))::Core.Const(Interval{Rational{Int64}})
│         (domain = (%2)(0, IntervalArithmetic.Inf))
│         (a@_4 = a@_4 ∩ domain::Core.Const([0//1, 1//0]))
│   %5  = IntervalArithmetic.isempty(a@_4)::Bool
└──       goto #3 if not %5
2 ─       return a@_4
3 ─ %8  = Base.getproperty(a@_4, :lo)::Rational{Int64}
│   %9  = IntervalArithmetic.sqrt(%8, RoundingMode{:Down}())::Float64
│   %10 = Base.getproperty(a@_4, :hi)::Rational{Int64}
│   %11 = IntervalArithmetic.sqrt(%10, RoundingMode{:Up}())::Float64
│   %12 = IntervalArithmetic.Interval(%9, %11)::Interval{Float64}
└──       return %12

[lucaferranti]

what should e.g. sqrt(1//2..3//4) return anyway? empty interval? change to Interval{Float64}? to Interval{BigFloat}? I think the 1788 way would probably be to print a warning and return an empty interval?

[dpsanders]

I think it should call sqrt(float(x))

[lbenet]

I fully agree with David: If the user is working with Interval{Rational{Int}}, it is because it makes sense, so I guess the user deals at most with rational functions with integer or rational coefficients. If by some reason this is not the case, promotion has to set in, which in this case means return Interval{Float64} (or if the original interval is Interval{Rational{BigInt}}, then returning Interval{BigFloat}), and then it is the responsibility of the user to stick to this type or convert later to Interval{Rational{Int}}.

The reason to have Interval{Rational{T}} is, if I recall correctly, that some arithmetic operations are exact for rationals (e.g. rational functions with integer or rational coefficients), so no rounding needs to be done.

[Kolaru]

I don't remember the standard ever mentioning bounds that are not floats, so I think that we are free to promote here.

[OlivierHnt]

This is fixed on master, note that the return type is Interval{Rational{Int64}}. While returning Interval{Float64} gives a sharper answer, we should avoid type promotion when a valid Interval{Rational{Int64}} can be returned. After all,Rational{Int64} is a supported type for the interval bounds. Also, there is no loss here, the sharpest result can always be retrieved via sqrt(float(x)) which makes the bound type promotion explicit.