diff --git a/DifferentiationInterface/Project.toml b/DifferentiationInterface/Project.toml
index d50bdcf11..b2fa8cf7f 100644
--- a/DifferentiationInterface/Project.toml
+++ b/DifferentiationInterface/Project.toml
@@ -69,7 +69,7 @@ ForwardDiff = "0.10.36,1"
 GPUArraysCore = "0.2"
 GTPSA = "1.4.0"
 LinearAlgebra = "1"
-Mooncake = "0.4.175"
+Mooncake = "0.5.0"
 PolyesterForwardDiff = "0.1.2"
 ReverseDiff = "1.15.1"
 SparseArrays = "1"
diff --git a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/DifferentiationInterfaceMooncakeExt.jl b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/DifferentiationInterfaceMooncakeExt.jl
index 561c06cc8..b27dc0472 100644
--- a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/DifferentiationInterfaceMooncakeExt.jl
+++ b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/DifferentiationInterfaceMooncakeExt.jl
@@ -29,7 +29,9 @@ using Mooncake:
     NoRData,
     primal,
     _copy_output,
-    _copy_to_output!!
+    _copy_to_output!!,
+    primal_to_tangent!!,
+    tangent_to_primal!!
 
 const AnyAutoMooncake{C} = Union{AutoMooncake{C}, AutoMooncakeForward{C}}
 
diff --git a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_onearg.jl b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_onearg.jl
index 61946a0d3..298fbc546 100644
--- a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_onearg.jl
+++ b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_onearg.jl
@@ -19,10 +19,14 @@ function DI.prepare_pushforward_nokwarg(
     _sig = DI.signature(f, backend, x, tx, contexts...; strict)
     config = get_config(backend)
     cache = prepare_derivative_cache(
-        f, x, map(DI.unwrap, contexts)...; config.debug_mode, config.silence_debug_messages
+        f, x, map(DI.unwrap, contexts)...; config
     )
-    dx_righttype = zero_tangent(x)
     df = zero_tangent(f)
+    if config.friendly_tangents
+        dx_righttype = zero_tangent(x)
+    else
+        dx_righttype = nothing
+    end
     context_tangents = map(zero_tangent_unwrap, contexts)
     prep = MooncakeOneArgPushforwardPrep(_sig, cache, dx_righttype, df, context_tangents)
     return prep
@@ -38,8 +42,7 @@ function DI.value_and_pushforward(
     ) where {F, C, X}
     DI.check_prep(f, prep, backend, x, tx, contexts...)
     ys_and_ty = map(tx) do dx
-        dx_righttype =
-            dx isa tangent_type(X) ? dx : _copy_to_output!!(prep.dx_righttype, dx)
+        dx_righttype = isnothing(prep.dx_righttype) ? dx : primal_to_tangent!!(prep.dx_righttype, dx)
         y_dual = value_and_derivative!!(
             prep.cache,
             Dual(f, prep.df),
@@ -47,7 +50,11 @@ function DI.value_and_pushforward(
             map(Dual_unwrap, contexts, prep.context_tangents)...,
         )
         y = primal(y_dual)
-        dy = _copy_output(tangent(y_dual))
+        if isnothing(prep.dx_righttype)
+            dy = _copy_output(tangent(y_dual))
+        else
+            dy = tangent_to_primal!!(_copy_output(y), tangent(y_dual))
+        end
         return y, dy
     end
     y = first(ys_and_ty[1])
diff --git a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_twoarg.jl b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_twoarg.jl
index dc8f8c1f0..0930c90dd 100644
--- a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_twoarg.jl
+++ b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/forward_twoarg.jl
@@ -25,11 +25,15 @@ function DI.prepare_pushforward_nokwarg(
         y,
         x,
         map(DI.unwrap, contexts)...;
-        config.debug_mode,
-        config.silence_debug_messages,
+        config,
     )
-    dx_righttype = zero_tangent(x)
-    dy_righttype = zero_tangent(y)
+    if config.friendly_tangents
+        dx_righttype = zero_tangent(x)
+        dy_righttype = zero_tangent(y)
+    else
+        dx_righttype = nothing
+        dy_righttype = nothing
+    end
     df! = zero_tangent(f!)
     context_tangents = map(zero_tangent_unwrap, contexts)
     prep = MooncakeTwoArgPushforwardPrep(_sig, cache, dx_righttype, dy_righttype, df!, context_tangents)
@@ -48,7 +52,7 @@ function DI.value_and_pushforward(
     DI.check_prep(f!, y, prep, backend, x, tx, contexts...)
     ty = map(tx) do dx
         dx_righttype =
-            dx isa tangent_type(X) ? dx : _copy_to_output!!(prep.dx_righttype, dx)
+            isnothing(prep.dx_righttype) ? dx : primal_to_tangent!!(prep.dx_righttype, dx)
         y_dual = zero_dual(y)
         value_and_derivative!!(
             prep.cache,
@@ -57,7 +61,11 @@ function DI.value_and_pushforward(
             Dual(x, dx_righttype),
             map(Dual_unwrap, contexts, prep.context_tangents)...,
         )
-        dy = _copy_output(tangent(y_dual))
+        if isnothing(prep.dx_righttype)
+            dy = _copy_output(tangent(y_dual))
+        else
+            dy = tangent_to_primal!!(_copy_output(y), tangent(y_dual))
+        end
         return dy
     end
     return y, ty
@@ -89,9 +97,9 @@ function DI.value_and_pushforward!(
     DI.check_prep(f!, y, prep, backend, x, tx, contexts...)
     foreach(tx, ty) do dx, dy
         dx_righttype =
-            dx isa tangent_type(X) ? dx : _copy_to_output!!(prep.dx_righttype, dx)
+            isnothing(prep.dx_righttype) ? dx : primal_to_tangent!!(prep.dx_righttype, dx)
         dy_righttype =
-            dy isa tangent_type(Y) ? dy : _copy_to_output!!(prep.dy_righttype, dy)
+            isnothing(prep.dy_righttype) ? dy : primal_to_tangent!!(prep.dy_righttype, dy)
         value_and_derivative!!(
             prep.cache,
             Dual(f!, prep.df!),
@@ -99,7 +107,7 @@ function DI.value_and_pushforward!(
             Dual(x, dx_righttype),
             map(Dual_unwrap, contexts, prep.context_tangents)...,
         )
-        dy === dy_righttype || copyto!(dy, dy_righttype)
+        isnothing(prep.dy_righttype) || tangent_to_primal!!(dy, dy_righttype)
     end
     return y, ty
 end
diff --git a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/onearg.jl b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/onearg.jl
index ab9818735..49ac0b656 100644
--- a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/onearg.jl
+++ b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/onearg.jl
@@ -1,9 +1,8 @@
 ## Pullback
 
-struct MooncakeOneArgPullbackPrep{SIG, Tcache, DY, N} <: DI.PullbackPrep{SIG}
+struct MooncakeOneArgPullbackPrep{SIG, Tcache, N} <: DI.PullbackPrep{SIG}
     _sig::Val{SIG}
     cache::Tcache
-    dy_righttype::DY
     args_to_zero::NTuple{N, Bool}
 end
 
@@ -12,18 +11,14 @@ function DI.prepare_pullback_nokwarg(
     ) where {F, C}
     _sig = DI.signature(f, backend, x, ty, contexts...; strict)
     config = get_config(backend)
-    cache = prepare_pullback_cache(
-        f, x, map(DI.unwrap, contexts)...; config.debug_mode, config.silence_debug_messages
-    )
-    y = f(x, map(DI.unwrap, contexts)...)
-    dy_righttype = zero_tangent(y)
+    cache = prepare_pullback_cache(f, x, map(DI.unwrap, contexts)...; config)
     contexts_tup_false = map(_ -> false, contexts)
     args_to_zero = (
         false,  # f
         true,  # x
         contexts_tup_false...,
     )
-    prep = MooncakeOneArgPullbackPrep(_sig, cache, dy_righttype, args_to_zero)
+    prep = MooncakeOneArgPullbackPrep(_sig, cache, args_to_zero)
     return prep
 end
 
@@ -37,9 +32,8 @@ function DI.value_and_pullback(
     ) where {F, Y, C}
     DI.check_prep(f, prep, backend, x, ty, contexts...)
     dy = only(ty)
-    dy_righttype = dy isa tangent_type(Y) ? dy : _copy_to_output!!(prep.dy_righttype, dy)
     new_y, (_, new_dx) = value_and_pullback!!(
-        prep.cache, dy_righttype, f, x, map(DI.unwrap, contexts)...;
+        prep.cache, dy, f, x, map(DI.unwrap, contexts)...;
         prep.args_to_zero
     )
     return new_y, (_copy_output(new_dx),)
@@ -55,10 +49,8 @@ function DI.value_and_pullback(
     ) where {F, Y, C}
     DI.check_prep(f, prep, backend, x, ty, contexts...)
     ys_and_tx = map(ty) do dy
-        dy_righttype =
-            dy isa tangent_type(Y) ? dy : _copy_to_output!!(prep.dy_righttype, dy)
         y, (_, new_dx) = value_and_pullback!!(
-            prep.cache, dy_righttype, f, x, map(DI.unwrap, contexts)...;
+            prep.cache, dy, f, x, map(DI.unwrap, contexts)...;
             prep.args_to_zero
         )
         y, _copy_output(new_dx)
@@ -121,9 +113,7 @@ function DI.prepare_gradient_nokwarg(
     ) where {F, C}
     _sig = DI.signature(f, backend, x, contexts...; strict)
     config = get_config(backend)
-    cache = prepare_gradient_cache(
-        f, x, map(DI.unwrap, contexts)...; config.debug_mode, config.silence_debug_messages
-    )
+    cache = prepare_gradient_cache(f, x, map(DI.unwrap, contexts)...; config)
     contexts_tup_false = map(_ -> false, contexts)
     args_to_zero = (
         false,  # f
diff --git a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/twoarg.jl b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/twoarg.jl
index 2ee11b5ae..ff41be77a 100644
--- a/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/twoarg.jl
+++ b/DifferentiationInterface/ext/DifferentiationInterfaceMooncakeExt/twoarg.jl
@@ -1,7 +1,6 @@
-struct MooncakeTwoArgPullbackPrep{SIG, Tcache, DY, F, N} <: DI.PullbackPrep{SIG}
+struct MooncakeTwoArgPullbackPrep{SIG, Tcache, F, N} <: DI.PullbackPrep{SIG}
     _sig::Val{SIG}
     cache::Tcache
-    dy_righttype::DY
     target_function::F
     args_to_zero::NTuple{N, Bool}
 end
@@ -27,10 +26,8 @@ function DI.prepare_pullback_nokwarg(
         y,
         x,
         map(DI.unwrap, contexts)...;
-        debug_mode = config.debug_mode,
-        silence_debug_messages = config.silence_debug_messages,
+        config,
     )
-    dy_righttype_after = zero_tangent(y)
     contexts_tup_false = map(_ -> false, contexts)
     args_to_zero = (
         false,  # target_function
@@ -39,9 +36,7 @@ function DI.prepare_pullback_nokwarg(
         true,  # x
         contexts_tup_false...,
     )
-    prep = MooncakeTwoArgPullbackPrep(
-        _sig, cache, dy_righttype_after, target_function, args_to_zero
-    )
+    prep = MooncakeTwoArgPullbackPrep(_sig, cache, target_function, args_to_zero)
     return prep
 end
 
@@ -56,12 +51,10 @@ function DI.value_and_pullback(
     ) where {F, C}
     DI.check_prep(f!, y, prep, backend, x, ty, contexts...)
     dy = only(ty)
-    # Prepare cotangent to add after the forward pass.
-    dy_righttype_after = copyto!(prep.dy_righttype, dy)
     # Run the reverse-pass and return the results.
     y_after, (_, _, _, dx) = value_and_pullback!!(
         prep.cache,
-        dy_righttype_after,
+        dy,
         prep.target_function,
         f!,
         y,