Support derivs of abs; add eps and clamp

ext/ThickNumbersForwardDiffExt.jl

@@ -1,7 +1,14 @@
 module ThickNumbersForwardDiffExt
 
 using ThickNumbers
-using ForwardDiff: ForwardDiff, Dual, Partials, Tag
+using ForwardDiff: ForwardDiff, Dual, Partials, Tag, value, partials, npartials, tagtype
+using ForwardDiff.DiffRules: DiffRules, @define_diffrule
+# Supports up to third-order derivatives
+const ThickDual = Union{Dual{T,<:ThickNumber} where T,
+                        Dual{T1,<:Dual{T2,<:ThickNumber}} where {T1,T2},
+                        Dual{T1,<:Dual{T2,<:Dual{T3,<:ThickNumber}}} where {T1,T2,T3},
+}
+const ThickLike = Union{ThickNumber, ThickDual}
 
 function ForwardDiff.derivative(f::F, x::TN) where {F,TN<:ThickNumber}
     T = typeof(Tag(f, TN))
@@ -27,4 +34,24 @@ Base.promote_rule(::Type{TN}, ::Type{Dual{T,V,N}}) where {TN<:ThickNumber,T,V<:N
 promote_dual(::Type{TN}, ::Type{V}) where {TN<:ThickNumber,V} = promote_type(TN, V)
 promote_dual(::Type{TN}, ::Type{Dual{T,V,N}}) where {TN<:ThickNumber,T,V,N} = Dual{T, promote_dual(TN, V), N}
 
+### Special functions
+
+## First and higher-order derivatives of `abs`
+function DiffRules._abs_deriv(x::ThickNumber)
+    sb = signbit(x)
+    lv, hv = loval(x), hival(x)
+    return lohi(basetype(typeof(x)), true ∈ sb ? -one(lv) : one(lv), false ∈ sb ? one(hv) : -one(hv))
+end
+# Second derivative of abs spans from 0 to either 0 or Inf (if 0 is included in the range)
+_abs_deriv2(x::ThickNumber) = iszero(mig(x)) ? lohi(typeof(x), 0, typemax(valuetype(x))) : zero(x)
+@define_diffrule DiffRules._abs_deriv(x) = :($(_abs_deriv2)($x))
+eval(ForwardDiff.unary_dual_definition(:DiffRules, :_abs_deriv))
+# Third and higher derivatives of abs span from -Inf to Inf, or is zero if 0 is not included in the range
+_abs_deriv3(x::ThickNumber{T}) where T = iszero(mig(x)) ? lohi(typeof(x), typemin(T), typemax(T)) : zero(x)
+@define_diffrule ThickNumbersForwardDiffExt._abs_deriv2(x) = :($(_abs_deriv3)($x))
+@define_diffrule ThickNumbersForwardDiffExt._abs_deriv3(x) = :($(_abs_deriv3)($x))
+eval(ForwardDiff.unary_dual_definition(:ThickNumbersForwardDiffExt, :_abs_deriv2))
+eval(ForwardDiff.unary_dual_definition(:ThickNumbersForwardDiffExt, :_abs_deriv3))
+
+
 end

src/ThickNumbers.jl

@@ -157,6 +157,7 @@ Interval{$Int}(1, 2)
 ```
 """
 basetype(::Type{TN}) where TN<:ThickNumber = error("basetype not defined for $TN")
+basetype(x::ThickNumber) = basetype(typeof(x))
 
 # Optional specializations
 
@@ -526,5 +527,16 @@ Base.:(-)(a::TN) where TN<:ThickNumber{T} where T<:Integer = lohi(TN,
     Base.Checked.checked_neg(loval(a))
 )
 
+# Functions
+
+Base.eps(::Type{TN}) where TN<:ThickNumber{T} where T = (e = eps(T); lohi(TN, e, e))
+Base.eps(x::ThickNumber) = lohi(typeof(x), eps(mig(x)), eps(mag(x)))
+
+Base.signbit(x::ThickNumber) = lohi(basetype(typeof(x)), signbit(hival(x)), signbit(loval(x)))
+
+Base.abs(x::ThickNumber) = lohi(typeof(x), mig(x), mag(x))
+
+Base.clamp(x::ThickNumber, lo::Real, hi::Real) = lohi(basetype(x), max(lo, loval(x)), min(hi, hival(x)))
+
 
 end # module

test/extensions/forwarddiff.jl

@@ -20,4 +20,16 @@ using IntervalArith
     @test df2(b) ⩪ b
     ddf2(x) = ForwardDiff.derivative(df2, x)
     @test ddf2(b) ≐ 1
+
+    # abs
+    dabs(x) = ForwardDiff.derivative(abs, x)
+    ddabs(x) = ForwardDiff.derivative(dabs, x)
+    dddabs(x) = ForwardDiff.derivative(ddabs, x)
+    @test dabs(Interval(1.0, 2.0)) === Interval(1.0, 1.0)
+    @test ddabs(Interval(1.0, 2.0)) === Interval(0.0, 0.0)
+    @test dddabs(Interval(1.0, 2.0)) === Interval(0.0, 0.0)
+    @test dabs(Interval(-1.0, 2.0)) === Interval(-1.0, 1.0)
+    @test ddabs(Interval(-1.0, 2.0)) === Interval(0.0, Inf)
+    abs3 = dddabs(Interval(-1.0, 2.0))
+    @test abs3 === Interval(-Inf, Inf) || isnan_tn(abs3)
 end

test/runtests.jl

@@ -121,6 +121,13 @@ using IntervalArith
     @test [Interval(1, 2), Interval(0, 1+eps())] ⩪ [Interval(1, 2*(1+eps())), Interval(0, 1)]
 end
 
+@testset "Specializations" begin
+    @test loval(signbit(Interval(-1, 1))) === false
+    @test hival(signbit(Interval(-1, 1))) === true
+    @test eps(Interval(1.0, 1000.0)) === Interval(eps(1.0), eps(1000.0))
+    @test clamp(Interval(-1, 4), 2, 3) === Interval(2, 3)
+end
+
 include(joinpath("extensions", "runtests.jl"))
 
 cleanup()