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Merged
OlivierHnt merged 1 commit into from
Sep 5, 2026
+63 −3
Merged

ForwardDiff with Interval partials #769

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35 changes: 32 additions & 3 deletions ext/IntervalArithmeticForwardDiffExt.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -28,6 +28,20 @@ Base.:(==)(x::Dual, y::Interval) = value(x) == y
Base.:<(x::Interval, y::Dual) = x < value(y)
Base.:<(x::Dual, y::Interval) = value(x) < y

# ForwardDiff's `^` methods use `iszero(partials(x))`, which is undecidable for
# non-thin intervals. Use a recursive thin-zero test instead. `NestedInterval`
# supports intervals nested in up to four `Dual` layers.
const NestedInterval = let
U = Interval
for _ in 1:4
U = Union{U, Dual{T,<:U} where {T}}
end
U
end
_isthinzero(x::Interval) = isthinzero(x)
_isthinzero(x::Real) = iszero(x)
_isthinzero(d::Dual) = _isthinzero(value(d)) && all(_isthinzero, partials(d))

function Base.:(^)(x::Dual{Txy,<:Interval}, y::Dual{Txy,<:Interval}) where {Txy}
vx, vy = value(x), value(y)
expv = vx^vy
Expand All @@ -49,16 +63,31 @@ function Base.:(^)(x::Dual{Tx,<:Interval}, y::Dual{Ty,<:Interval}) where {Tx,Ty}
end
end

function Base.:(^)(x::Dual{Tx,<:Interval}, y::Interval) where {Tx}
function Base.:(^)(x::Dual{Tx,<:NestedInterval}, y::Interval) where {Tx}
v = value(x)
expv = v^y
if isthinzero(y) || all(isthinzero, values(partials(x)))
if isthinzero(y) || all(_isthinzero, values(partials(x)))
return Dual{Tx}(expv, zero(partials(x)))
else
return Dual{Tx}(expv, partials(x) * y * v^(y - interval(1)))
end
end

# A `y::Real` method would be ambiguous with ForwardDiff's generated `^`
# methods and its `Dual^Dual` tag methods, so define the relevant concrete
# exponent types instead.
for R in (:Integer, :Rational, :AbstractFloat, :Irrational)
@eval function Base.:(^)(x::Dual{Tx,<:NestedInterval}, y::$R) where {Tx}
v = value(x)
expv = v^y
if iszero(y) || all(_isthinzero, values(partials(x)))
return Dual{Tx}(expv, zero(partials(x)))
else
return Dual{Tx}(expv, partials(x) * y * v^(y - 1))
end
end
end

function Base.:(^)(x::Interval, y::Dual{Ty,<:Interval}) where {Ty}
v = value(y)
expv = x^v
Expand All @@ -76,7 +105,7 @@ Base.:(^)(x::ExactReal, y::Dual{<:Any,I}) where {I<:Interval} = convert(I, x)^y
function Base.:(^)(x::Dual{Tx}, y::ExactReal) where {Tx}
v = value(x)
expv = v^y
if iszero(y.value) || all(iszero, values(partials(x)))
if iszero(y.value) || all(_isthinzero, values(partials(x)))
return Dual{Tx}(expv, zero(partials(x)))
else
return Dual{Tx}(expv, partials(x) * y * v^(y - 1))
Expand Down
31 changes: 31 additions & 0 deletions test/interval_tests/forwarddiff.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -105,4 +105,35 @@ end
@exact g(x) = 2^x + 6sin(x^3) - 33
@test isguaranteed(ForwardDiff.derivative(f, interval(1)))
end

@testset "thick partials" begin
# Differentiating by a real leaves interval-valued constants in the partials.
x, w = 2.0, interval(-0.5, 0.5)

# Test real, interval, and exact exponents.
for n ∈ (4, 4.0, interval(4), exact(4))
@test isequal_interval(ForwardDiff.derivative(t -> (x + t*w)^n, 0), interval(4x^3) * w)
@test isequal_interval(ForwardDiff.derivative(t -> ForwardDiff.derivative(s -> (x + s*w)^n, t), 0),
interval(12x^2) * w * w)
end

# Each derivative order adds a `Dual` layer.
ϕ(t) = (x + t*w)^4
dϕ(t) = ForwardDiff.derivative(ϕ, t)
ddϕ(t) = ForwardDiff.derivative(dϕ, t)
dddϕ(t) = ForwardDiff.derivative(ddϕ, t)
ddddϕ(t) = ForwardDiff.derivative(dddϕ, t)

@test isequal_interval(dϕ(0) , interval(4x^3) * w)
@test isequal_interval(ddϕ(0) , interval(12x^2) * w * w)
@test isequal_interval(dddϕ(0) , interval(24x) * w * w * w)
@test isequal_interval(ddddϕ(0), interval(24) * w * w * w * w)

# Test multivariate first- and second-order derivatives.
ψ(v) = (v[1]*w + v[2])^5
@test all(isequal_interval.(ForwardDiff.gradient(ψ, [0, 1]), [interval(5) * w, interval(5)]))
@test all(isequal_interval.(ForwardDiff.hessian(ψ, [0, 1]),
[interval(20) * w * w interval(20) * w
interval(20) * w interval(20) ]))
end
end
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