diff --git a/src/scalar.jl b/src/scalar.jl
index 13384cc..db3503e 100644
--- a/src/scalar.jl
+++ b/src/scalar.jl
@@ -6,8 +6,11 @@ $(TYPEDEF)
 
 Transform a scalar (real number) to another scalar.
 
-Subtypes mustdefine `transform`, `transform_and_logjac`, and `inverse`; other
-methods of of the interface should have the right defaults.
+Subtypes must define `transform`, `transform_and_logjac`, and `inverse`.
+Other methods of of the interface should have the right defaults.
+
+!!! NOTE
+    This type is for code organization within the package, and is not part of the public API.
 """
 abstract type ScalarTransform <: AbstractTransform end
 
@@ -26,7 +29,11 @@ function inverse_at!(x::AbstractVector, index::Int, t::ScalarTransform, y::Real)
     index + 1
 end
 
-inverse_eltype(t::ScalarTransform, y::T) where {T <: Real} = float(T)
+function inverse_eltype(t::ScalarTransform, y::Real)
+    # NOTE this is a shortcut to get sensible types for all subtypes of ScalarTransform, which
+    # we test for. If it breaks it should be extended accordingly.
+    return Base.promote_typejoin_union(Base.promote_op(inverse, typeof(t), typeof(y)))
+end
 
 _domain_label(::ScalarTransform, index::Int) = ()
 
diff --git a/test/runtests.jl b/test/runtests.jl
index f972660..73b3e22 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -699,3 +699,34 @@ end
     U = transform(t, x)
     @test isfinite(logabsdet(U)[1])
 end
+
+@testset "inverse_eltype of scalar transforms with parameters" begin
+    # `Float64` parameters and `Float32` input
+    for t in (as(Real, 0.5, ∞), as(Real, -∞, 2.1), as(Real, 0.5, 2.1))
+        @test @inferred(inverse_eltype(t, 1.1f0)) === Float64
+        @test @inferred(inverse(t, 1.1f0)) isa Float64
+    end
+
+    # Derivatives wrt parameters of the transforms
+    d1 = ForwardDiff.derivative(5.3) do x
+        return @inferred only(inverse(as(Vector, as(Real, x, ∞), 1), [10]))
+    end
+    d2 = ForwardDiff.derivative(5.3) do x
+        return @inferred inverse(as(Real, x, ∞), 10)
+    end
+    @test d1 == d2
+    d1 = ForwardDiff.derivative(-3) do x
+        return @inferred only(inverse(as(Vector, as(Real, -∞, x), 1), [-6.1]))
+    end
+    d2 = ForwardDiff.derivative(-3) do x
+        return @inferred inverse(as(Real, -∞, x), -6.1)
+    end
+    @test d1 == d2
+    d1 = ForwardDiff.gradient([-0.3, 4.7]) do x
+        return @inferred only(inverse(as(Vector, as(Real, x[1], x[2]), 1), [2.3]))
+    end
+    d2 = ForwardDiff.gradient([-0.3, 4.7]) do x
+        return @inferred inverse(as(Real, x[1], x[2]), 2.3)
+    end
+    @test d1 == d2
+end