Define interval arithmetics using broadcasting

src/IntervalSets.jl

@@ -3,6 +3,7 @@ module IntervalSets
 using Base: @pure
 import Base: eltype, convert, show, in, length, isempty, isequal, issubset, ==, hash,
              union, intersect, minimum, maximum, extrema, range, ⊇
+import Base.Broadcast: broadcasted
 
 using Statistics
 import Statistics: mean

src/interval.jl

@@ -225,6 +225,63 @@ end
 ClosedInterval{T}(i::AbstractUnitRange{I}) where {T,I<:Integer} = ClosedInterval{T}(minimum(i), maximum(i))
 ClosedInterval(i::AbstractUnitRange{I}) where {I<:Integer} = ClosedInterval{I}(minimum(i), maximum(i))
 
+for op in (+, -, *)
+    @eval begin
+        broadcasted(::typeof($op), i::AbstractInterval, x) = $op.(i, (x..x))
+        broadcasted(::typeof($op), x, i::AbstractInterval) = $op.((x..x), i)
+    end
+end
+
+broadcasted(::typeof(+), d1::AbstractInterval, d2::AbstractInterval) =
+    Interval{
+        isleftclosed(d1) && isleftclosed(d2) ? :closed : :open,
+        isrightclosed(d1) && isrightclosed(d2) ? :closed : :open,
+    }((endpoints(d1) .+ endpoints(d2))...)
+
+broadcasted(::typeof(-), d1::AbstractInterval, d2::AbstractInterval) =
+    Interval{
+        isleftclosed(d1) && isrightclosed(d2) ? :closed : :open,
+        isrightclosed(d1) && isleftclosed(d2) ? :closed : :open,
+    }((endpoints(d1) .- reverse(endpoints(d2)))...)
+
+@inline foldlargs(op, x) = x
+@inline foldlargs(op, x1, x2, xs...) = foldlargs(op, op(x1, x2), xs...)
+@inline extremaby(f, x, xs...) =
+    foldlargs((x, x), xs...) do (min, max), x
+        if f(min) > f(x)
+            (x, max)
+        elseif f(max) < f(x)
+            (min, x)
+        else
+            (min, max)
+        end
+    end
+
+_value(::Val{x}) where x = x
+
+function broadcasted(::typeof(*), d1::AbstractInterval, d2::AbstractInterval)
+    l1, r1 = endpoints(d1)
+    l2, r2 = endpoints(d2)
+    candidates = (
+        (l1 * l2, Val(isleftclosed(d1) && isleftclosed(d2) ? :closed : :open)),
+        (l1 * r2, Val(isleftclosed(d1) && isrightclosed(d2) ? :closed : :open)),
+        (r1 * l2, Val(isrightclosed(d1) && isleftclosed(d2) ? :closed : :open)),
+        (r1 * r2, Val(isrightclosed(d1) && isrightclosed(d2) ? :closed : :open)),
+    )
+    (left, L), (right, R) = extremaby(first, candidates...)
+    return Interval{_value(L), _value(R)}(left, right)
+end
+
+broadcasted(::typeof(/), d1::AbstractInterval, d2::AbstractInterval) =
+    MethodError(broadcasted, (/, d1, d2))
+# Defining this to be a method error so that the `x` below is not of
+# type `AbstractInterval`.
+
+broadcasted(::typeof(/), i::AbstractInterval, x) =
+    Interval{
+        isleftclosed(i) ? :closed : :open,
+        isrightclosed(i) ? :closed : :open,
+    }((endpoints(i) ./ x)...)
 
 
 Base.promote_rule(::Type{Interval{L,R,T1}}, ::Type{Interval{L,R,T2}}) where {L,R,T1,T2} = Interval{L,R,promote_type(T1, T2)}

test/runtests.jl

@@ -693,4 +693,33 @@ struct IncompleteInterval <: AbstractInterval{Int} end
         @test_throws ErrorException endpoints(I)
         @test_throws ErrorException closedendpoints(I)
     end
+
+    @testset "Interval arithmetic" begin
+        @test (1..2) .+ 1 == 2..3
+        @test 1 .+ (1..2) == 2..3
+        @test (1..2) .+ (3..4) == 4..6
+        @test (1..2) .+ (3..5) == 4..7
+        @test Interval{:open,:closed}(1, 2) .+ Interval{:closed,:open}(3, 4) ==
+            Interval{:open,:open}(4, 6)
+
+        @test (1..2) .- 1 == 0..1
+        @test 1 .- (1..2) == -1 .. 0
+        @test (1..2) .- (3..4) == -3 .. -1
+        @test (1..2) .- (3..5) == -4 .. -1
+        @test Interval{:open,:closed}(1, 2) .- Interval{:closed,:open}(3, 4) ==
+            Interval{:open,:closed}(-3, -1)
+
+        @test 3 .* (1..2) == 3..6
+        @test (1..2) .* 3 == 3..6
+        @test (1 .. 2) .* (3 .. 4) == 3 .. 8
+        @test (-1 .. 2) .* (3 .. 4) == -4 .. 8
+        @test (1 .. -2) .* (3 .. 4) == -8 .. 4
+        @test (1 .. 2) .* (-3 .. 4) == -6 .. 8
+        @test (1 .. 2) .* (3 .. -4) == -8 .. 6
+        @test Interval{:open,:closed}(1, 2) .* Interval{:closed,:open}(3, 4) ==
+            Interval{:open,:open}(3, 8)
+
+        @test (2..4) ./ 2 == 1..2
+        @test Interval{:open,:closed}(2, 4) ./ 2 == Interval{:open,:closed}(1, 2)
+    end
 end