diff --git a/NEWS.md b/NEWS.md
index ba95817..4229e69 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,3 +1,9 @@
+# 0.7.1
+
+time_exp has been removed in favour of assuming that whichever AD library is being used can
+successfully AD through the matrix exponential. Guard rails to prevent mis-use of previous
+rule have been remved.
+
 # 0.7
 
 Mooncake.jl (and probably Enzyme.jl) is now able to differentiate everything in
diff --git a/Project.toml b/Project.toml
index cc4fd34..5ceb77e 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "TemporalGPs"
 uuid = "e155a3c4-0841-43e1-8b83-a0e4f03cc18f"
 authors = ["Will Tebbutt and contributors"]
-version = "0.7.1"
+version = "0.7.2"
 
 [deps]
 AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
@@ -14,12 +14,6 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
 
-[weakdeps]
-Mooncake = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6"
-
-[extensions]
-TemporalGPsMooncakeExt = "Mooncake"
-
 [compat]
 AbstractGPs = "0.5.17"
 BenchmarkTools = "1"
@@ -28,7 +22,7 @@ BlockDiagonals = "0.1.7"
 FillArrays = "0.13.0 - 0.13.7, 1"
 JET = "0.9"
 KernelFunctions = "0.9, 0.10.1"
-Mooncake = "0.4.3"
+Mooncake = "0.4.41"
 StaticArrays = "1"
 StructArrays = "0.5, 0.6"
 julia = "1.6"
diff --git a/ext/TemporalGPsMooncakeExt.jl b/ext/TemporalGPsMooncakeExt.jl
deleted file mode 100644
index e87c56e..0000000
--- a/ext/TemporalGPsMooncakeExt.jl
+++ /dev/null
@@ -1,22 +0,0 @@
-module TemporalGPsMooncakeExt
-
-using Mooncake, TemporalGPs
-import Mooncake:
-    rrule!!,
-    CoDual,
-    primal,
-    @is_primitive,
-    zero_fcodual,
-    MinimalCtx
-
-@is_primitive MinimalCtx Tuple{typeof(TemporalGPs.time_exp), Matrix{<:Real}, Real}
-function rrule!!(::CoDual{typeof(TemporalGPs.time_exp)}, A::CoDual, t::CoDual{Float64})
-    _A = primal(A)
-    B_dB = zero_fcodual(TemporalGPs.time_exp(_A, primal(t)))
-    B = primal(B_dB)
-    dB = tangent(B_dB)
-    time_exp_pb(::NoRData) = NoRData(), NoRData(), sum(dB .* (_A * B))
-    return B_dB, time_exp_pb
-end
-
-end
diff --git a/src/gp/lti_sde.jl b/src/gp/lti_sde.jl
index fba5d96..b37b3f2 100644
--- a/src/gp/lti_sde.jl
+++ b/src/gp/lti_sde.jl
@@ -130,9 +130,6 @@ function add_proj_mean(hs::AbstractVector, m)
     return map((h, m) -> h + vcat(m, Zeros(length(h) - 1)), hs, m)
 end
 
-# Really just a hook for AD.
-time_exp(A, t) = exp(A * t)
-
 # Generic constructors for base kernels.
 
 function broadcast_components(
@@ -140,7 +137,7 @@ function broadcast_components(
 ) where {T}
     P = Symmetric(x0.P)
     t = vcat([first(t) - 1], t)
-    As = map(Δt -> time_exp(F, T(Δt)), diff(t))
+    As = map(Δt -> exp(F * T(Δt)), diff(t))
     as = Fill(Zeros{T}(size(first(As), 1)), length(As))
     Qs = map(A -> P - A * P * A', As)
     Hs = Fill(H, length(As))
@@ -152,7 +149,7 @@ function broadcast_components(
     (F, q, H)::Tuple, x0::Gaussian, t::Union{StepRangeLen, RegularSpacing}, ::StorageType{T}
 ) where {T}
     P = Symmetric(x0.P)
-    A = time_exp(F, T(step(t)))
+    A = exp(F * T(step(t)))
     As = Fill(A, length(t))
     as = Fill(Zeros{T}(size(F, 1)), length(t))
     Q = Symmetric(P) - A * Symmetric(P) * A'
@@ -187,8 +184,6 @@ function stationary_distribution(k::SimpleKernel, ::ArrayStorage{T}) where {T<:R
     return Gaussian(collect(x.m), collect(x.P))
 end
 
-safe_to_product(::Kernel) = false
-
 # Matern-1/2
 
 function to_sde(::Matern12Kernel, ::SArrayStorage{T}) where {T<:Real}
@@ -205,8 +200,6 @@ function stationary_distribution(::Matern12Kernel, ::SArrayStorage{T}) where {T<
     )
 end
 
-safe_to_product(::Matern12Kernel) = true
-
 # Matern - 3/2
 
 function to_sde(::Matern32Kernel, ::SArrayStorage{T}) where {T<:Real}
@@ -224,8 +217,6 @@ function stationary_distribution(::Matern32Kernel, ::SArrayStorage{T}) where {T<
     )
 end
 
-safe_to_product(::Matern32Kernel) = true
-
 # Matern - 5/2
 
 function to_sde(::Matern52Kernel, ::SArrayStorage{T}) where {T<:Real}
@@ -243,8 +234,6 @@ function stationary_distribution(::Matern52Kernel, ::SArrayStorage{T}) where {T<
     return Gaussian(m, P)
 end
 
-safe_to_product(::Matern52Kernel) = true
-
 # Cosine
 
 function to_sde(::CosineKernel, ::SArrayStorage{T}) where {T}
@@ -260,8 +249,6 @@ function stationary_distribution(::CosineKernel, ::SArrayStorage{T}) where {T<:R
     return Gaussian(m, P)
 end
 
-safe_to_product(::CosineKernel) = true
-
 # ApproxPeriodicKernel
 
 # The periodic kernel is approximated by a sum of cosine kernels with different frequencies.
@@ -319,8 +306,6 @@ function stationary_distribution(kernel::ApproxPeriodicKernel{N}, storage::Array
     return Gaussian(m, P)
 end
 
-safe_to_product(::ApproxPeriodicKernel) = true
-
 # Constant
 
 function TemporalGPs.to_sde(::ConstantKernel, ::SArrayStorage{T}) where {T<:Real}
@@ -334,9 +319,6 @@ function TemporalGPs.stationary_distribution(k::ConstantKernel, ::SArrayStorage{
     return TemporalGPs.Gaussian(SVector{1, T}(0), SMatrix{1, 1, T}(T(only(k.c))))
 end
 
-safe_to_product(::ConstantKernel) = true
-
-
 # Scaled
 
 function to_sde(k::ScaledKernel, storage::StorageType{T}) where {T<:Real}
@@ -349,8 +331,6 @@ function stationary_distribution(k::ScaledKernel, storage::StorageType)
     return stationary_distribution(k.kernel, storage)
 end
 
-safe_to_product(k::ScaledKernel) = safe_to_product(k.kernel)
-
 function lgssm_components(k::ScaledKernel, ts::AbstractVector, storage_type::StorageType)
     As, as, Qs, emission_proj, x0 = lgssm_components(k.kernel, ts, storage_type)
     σ = sqrt(convert(eltype(storage_type), only(k.σ²)))
@@ -378,8 +358,6 @@ function stationary_distribution(
     return stationary_distribution(k.kernel, storage)
 end
 
-safe_to_product(::TransformedKernel{<:Kernel, <:ScaleTransform}) = false
-
 function lgssm_components(
     k::TransformedKernel{<:Kernel, <:ScaleTransform},
     ts::AbstractVector,
@@ -396,12 +374,8 @@ apply_stretch(a, ts::RegularSpacing) = RegularSpacing(a * ts.t0, a * ts.Δt, ts.
 
 # Product
 
-safe_to_product(k::KernelProduct) = all(safe_to_product, k.kernels)
-
 function lgssm_components(k::KernelProduct, ts::AbstractVector, storage::StorageType)
 
-    safe_to_product(k) || throw(ArgumentError("Not all kernels in k are safe to product."))
-
     sde_kernels = to_sde.(k.kernels, Ref(storage))
     F_kernels = getindex.(sde_kernels, 1)
     F = foldl(_kron_add, F_kernels)
diff --git a/test/gp/lti_sde.jl b/test/gp/lti_sde.jl
index ad9cdfb..6fdf7f9 100644
--- a/test/gp/lti_sde.jl
+++ b/test/gp/lti_sde.jl
@@ -13,7 +13,6 @@ end
 
 println("lti_sde:")
 @testset "lti_sde" begin
-
     @testset "ApproxPeriodicKernel" begin
         k = ApproxPeriodicKernel()
         @test k isa ApproxPeriodicKernel{7}
@@ -207,10 +206,4 @@ println("lti_sde:")
             )
         end
     end
-    @testset "time_exp AD" begin
-        test_rule(
-            Xoshiro(123), t -> TemporalGPs.time_exp([1.0 2.0; 3.0 4.0], t), rand();
-            is_primitive=false,
-        )
-    end
 end
diff --git a/test/models/linear_gaussian_conditionals.jl b/test/models/linear_gaussian_conditionals.jl
index d6fb055..306bf06 100644
--- a/test/models/linear_gaussian_conditionals.jl
+++ b/test/models/linear_gaussian_conditionals.jl
@@ -178,7 +178,7 @@ println("linear_gaussian_conditionals:")
                 x_post_large, lml_large = posterior_and_lml(x, model, y_missing)
 
                 # Check that they give roughly the same answer.
-                @test x_post_vanilla ≈ x_post_large rtol=1e-8 atol=1e-8
+                @test x_post_vanilla ≈ x_post_large rtol=1e-5 atol=1e-5
                 @test lml_vanilla ≈ lml_large rtol=1e-8 atol=1e-8
 
                 # Check that everything infers and AD gives the right answer.
diff --git a/test/space_time/to_gauss_markov.jl b/test/space_time/to_gauss_markov.jl
index 18de0ee..b8c6bfc 100644
--- a/test/space_time/to_gauss_markov.jl
+++ b/test/space_time/to_gauss_markov.jl
@@ -81,7 +81,10 @@
             @test std.(marginals(fx_post_naive)) ≈ std.(marginals(fx_post_sde))
 
             y_post = rand(rng, fx_post_naive)
-            @test logpdf(fx_post_naive, y_post) ≈ logpdf(fx_post_sde, y_post)
+            @test isapprox(
+                logpdf(fx_post_naive, y_post), logpdf(fx_post_sde, y_post);
+                atol=1e-6, rtol=1e-6,
+            )
 
             # No statistical tests run on `rand`, which seems somewhat dangerous, but there's
             # not a lot to be done about it unfortunately.