diff --git a/Project.toml b/Project.toml
index 5772a980..8dacd88c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ForwardDiff"
 uuid = "f6369f11-7733-5829-9624-2563aa707210"
-version = "1.3.0"
+version = "1.3.1"
 
 [deps]
 CommonSubexpressions = "bbf7d656-a473-5ed7-a52c-81e309532950"
diff --git a/src/partials.jl b/src/partials.jl
index be55c1b3..49f8364b 100644
--- a/src/partials.jl
+++ b/src/partials.jl
@@ -49,6 +49,8 @@ Base.mightalias(x::AbstractArray, y::Partials) = false
 
 Base.isequal(a::Partials{N}, b::Partials{N}) where {N} = isequal(a.values, b.values)
 Base.:(==)(a::Partials{N}, b::Partials{N}) where {N} = a.values == b.values
+Base.:(<)(a::Partials{N}, b::Partials{N}) where {N} = a.values < b.values
+Base.isless(a::Partials{N}, b::Partials{N}) where {N} = isless(a.values, b.values)
 
 const PARTIALS_HASH = hash(Partials)
 
diff --git a/test/DualTest.jl b/test/DualTest.jl
index 35d3116b..df9f9dcb 100644
--- a/test/DualTest.jl
+++ b/test/DualTest.jl
@@ -259,6 +259,8 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
     # M is the length of M_PARTIALS, which affects:
     # NESTED_FDNUM = Dual{TestTag}(Dual{TestTag}(PRIMAL, M_PARTIALS), NESTED_PARTIALS)
 
+    NAN_PARTIALS = Partials{N,float(V)}(map(x -> oftype(float(x), NaN), PARTIALS.values))
+
     @test (FDNUM == Dual{TestTag}(PRIMAL, PARTIALS2)) == (PARTIALS == PARTIALS2)
     @test isequal(FDNUM, Dual{TestTag}(PRIMAL, PARTIALS2)) == (PARTIALS == PARTIALS2)
     @test isequal(NESTED_FDNUM, Dual{TestTag}(Dual{TestTag}(PRIMAL, M_PARTIALS2), NESTED_PARTIALS2)) == ((M_PARTIALS == M_PARTIALS2) && (NESTED_PARTIALS == NESTED_PARTIALS2))
@@ -276,10 +278,17 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
         @test NESTED_FDNUM != NESTED_FDNUM2
     end
 
+    @test (Dual{TestTag}(PRIMAL, NAN_PARTIALS) == Dual{TestTag}(PRIMAL, NAN_PARTIALS)) === (N == 0)
+    @test isequal(Dual{TestTag}(PRIMAL, NAN_PARTIALS), Dual{TestTag}(PRIMAL, NAN_PARTIALS))
+
     @test isless(Dual{TestTag}(1, PARTIALS), Dual{TestTag}(2, PARTIALS2))
     @test isless(Dual{TestTag}(1, PARTIALS), Dual{TestTag}(1, PARTIALS2)) === isless(PARTIALS, PARTIALS2)
     @test !(isless(Dual{TestTag}(1, PARTIALS), Dual{TestTag}(1, PARTIALS)))
     @test !(isless(Dual{TestTag}(2, PARTIALS), Dual{TestTag}(1, PARTIALS2)))
+    @test isless(Dual{TestTag}(1, PARTIALS), Dual{TestTag}(2, NAN_PARTIALS))
+    @test isless(Dual{TestTag}(1, PARTIALS), Dual{TestTag}(1, NAN_PARTIALS)) === (N > 0)
+    @test !isless(Dual{TestTag}(1, NAN_PARTIALS), Dual{TestTag}(1, PARTIALS))
+    @test !(isless(Dual{TestTag}(2, PARTIALS), Dual{TestTag}(1, NAN_PARTIALS)))
 
     @test isless(Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS), Dual{TestTag}(Dual{TestTag}(2, M_PARTIALS2), NESTED_PARTIALS2))
     @test isless(Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS), Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS2)) === isless(NESTED_PARTIALS, NESTED_PARTIALS2)
@@ -290,6 +299,10 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
     @test (Dual{TestTag}(1, PARTIALS) < Dual{TestTag}(1, PARTIALS2)) === (PARTIALS < PARTIALS2)
     @test !(Dual{TestTag}(1, PARTIALS) < Dual{TestTag}(1, PARTIALS))
     @test !(Dual{TestTag}(2, PARTIALS) < Dual{TestTag}(1, PARTIALS2))
+    @test Dual{TestTag}(1, PARTIALS) < Dual{TestTag}(2, NAN_PARTIALS)
+    @test !(Dual{TestTag}(1, PARTIALS) < Dual{TestTag}(1, NAN_PARTIALS))
+    @test !(Dual{TestTag}(1, NAN_PARTIALS) < Dual{TestTag}(1, PARTIALS))
+    @test !(Dual{TestTag}(2, PARTIALS) < Dual{TestTag}(1, NAN_PARTIALS))
 
     @test Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) < Dual{TestTag}(Dual{TestTag}(2, M_PARTIALS2), NESTED_PARTIALS2)
     @test (Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) < Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS2)) === (NESTED_PARTIALS < NESTED_PARTIALS2)
@@ -300,6 +313,10 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
     @test (Dual{TestTag}(1, PARTIALS) <= Dual{TestTag}(1, PARTIALS2)) === (PARTIALS <= PARTIALS2)
     @test Dual{TestTag}(1, PARTIALS) <= Dual{TestTag}(1, PARTIALS)
     @test !(Dual{TestTag}(2, PARTIALS) <= Dual{TestTag}(1, PARTIALS2))
+    @test Dual{TestTag}(1, PARTIALS) <= Dual{TestTag}(2, NAN_PARTIALS)
+    @test (Dual{TestTag}(1, PARTIALS) <= Dual{TestTag}(1, NAN_PARTIALS)) === (N == 0)
+    @test (Dual{TestTag}(1, NAN_PARTIALS) <= Dual{TestTag}(1, PARTIALS)) === (N == 0)
+    @test !(Dual{TestTag}(2, PARTIALS) <= Dual{TestTag}(1, NAN_PARTIALS))
 
     @test Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) <= Dual{TestTag}(Dual{TestTag}(2, M_PARTIALS2), NESTED_PARTIALS2)
     @test (Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) <= Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS2)) === (NESTED_PARTIALS <= NESTED_PARTIALS2)
@@ -310,6 +327,10 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
     @test (Dual{TestTag}(1, PARTIALS) > Dual{TestTag}(1, PARTIALS2)) === (PARTIALS > PARTIALS2)
     @test !(Dual{TestTag}(1, PARTIALS) > Dual{TestTag}(1, PARTIALS))
     @test !(Dual{TestTag}(1, PARTIALS) > Dual{TestTag}(2, PARTIALS2))
+    @test !(Dual{TestTag}(1, PARTIALS) > Dual{TestTag}(2, NAN_PARTIALS))
+    @test !(Dual{TestTag}(1, PARTIALS) > Dual{TestTag}(1, NAN_PARTIALS))
+    @test !(Dual{TestTag}(1, NAN_PARTIALS) > Dual{TestTag}(1, PARTIALS))
+    @test Dual{TestTag}(2, PARTIALS) > Dual{TestTag}(1, NAN_PARTIALS)
 
     @test Dual{TestTag}(Dual{TestTag}(2, M_PARTIALS), NESTED_PARTIALS) > Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS2), NESTED_PARTIALS2)
     @test (Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) > Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS2)) === (NESTED_PARTIALS > NESTED_PARTIALS2)
@@ -320,6 +341,10 @@ ForwardDiff.:≺(::Type{OuterTestTag}, ::Type{TestTag}) = false
     @test (Dual{TestTag}(1, PARTIALS) >= Dual{TestTag}(1, PARTIALS2)) === (PARTIALS >= PARTIALS2)
     @test Dual{TestTag}(1, PARTIALS) >= Dual{TestTag}(1, PARTIALS)
     @test !(Dual{TestTag}(1, PARTIALS) >= Dual{TestTag}(2, PARTIALS2))
+    @test !(Dual{TestTag}(1, PARTIALS) >= Dual{TestTag}(2, NAN_PARTIALS))
+    @test ((Dual{TestTag}(1, PARTIALS) >= Dual{TestTag}(1, NAN_PARTIALS))) === (N == 0)
+    @test ((Dual{TestTag}(1, NAN_PARTIALS) >= Dual{TestTag}(1, PARTIALS))) === (N == 0)
+    @test Dual{TestTag}(2, PARTIALS) >= Dual{TestTag}(1, NAN_PARTIALS)
 
     @test Dual{TestTag}(Dual{TestTag}(2, M_PARTIALS), NESTED_PARTIALS) >= Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS2), NESTED_PARTIALS2)
     @test (Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS) >= Dual{TestTag}(Dual{TestTag}(1, M_PARTIALS), NESTED_PARTIALS2)) === (NESTED_PARTIALS >= NESTED_PARTIALS2)
diff --git a/test/PartialsTest.jl b/test/PartialsTest.jl
index 23ebff3b..0e73160b 100644
--- a/test/PartialsTest.jl
+++ b/test/PartialsTest.jl
@@ -60,6 +60,19 @@ samerng() = MersenneTwister(1)
     @test (PARTIALS == PARTIALS2) == (N == 0)
     @test isequal(PARTIALS, copy(PARTIALS))
     @test isequal(PARTIALS, PARTIALS2) == (N == 0)
+    @test !(PARTIALS < copy(PARTIALS))
+    @test PARTIALS <= copy(PARTIALS)
+    @test !(PARTIALS > copy(PARTIALS))
+    @test !isless(PARTIALS, copy(PARTIALS))
+    for f in (<, <=, >, isless)
+        @test f(PARTIALS, PARTIALS2) === f(VALUES, VALUES2)
+    end
+    NAN_PARTIALS = Partials{N,float(T)}(map(x -> oftype(float(x), NaN), VALUES))
+    @test !(PARTIALS < NAN_PARTIALS)
+    @test (PARTIALS <= NAN_PARTIALS) === (N == 0)
+    @test !(PARTIALS > NAN_PARTIALS)
+    @test isless(PARTIALS, NAN_PARTIALS) === (N > 0)
+    @test !isless(NAN_PARTIALS, PARTIALS)
 
     @test hash(PARTIALS) == hash(copy(PARTIALS))
     @test hash(PARTIALS, hash(1)) == hash(copy(PARTIALS), hash(1))