diff --git a/benchmark/pinn.jl b/benchmark/pinn.jl
index 5ebc910..c2ba229 100644
--- a/benchmark/pinn.jl
+++ b/benchmark/pinn.jl
@@ -6,11 +6,14 @@ using Plots
 const input = 2
 const hidden = 16
 
-model = Chain(Dense(input => hidden, sin),
-              Dense(hidden => hidden, sin),
-              Dense(hidden => 1),
-              first)
-trial(model, x) = x[1] * (1 - x[1]) * x[2] * (1 - x[2]) * model(x)
+# model = Chain(Dense(input => hidden, exp),
+#               Dense(hidden => hidden, exp),
+#               Dense(hidden => 1),
+#               first)
+# trial(model, x) = x[1] * (1 - x[1]) * x[2] * (1 - x[2]) * model(x)
+
+model = Chain(Dense(input => 1, exp), first)
+trial(model, x) = model(x)
 
 M = 100
 data = [rand(Float32, input) for _ in 1:M]
@@ -25,7 +28,7 @@ function loss_by_finitediff(model, x)
 end
 function loss_by_taylordiff(model, x)
     f(x) = trial(model, x)
-    error = derivative(f, x, Float32[1, 0], 2) + derivative(f, x, Float32[0, 1], 2) +
+    error = derivative(f, x, Float32[1, 0], Val(3)) + derivative(f, x, Float32[0, 1], Val(3)) +
             sin(π * x[1]) * sin(π * x[2])
     abs2(error)
 end
diff --git a/src/chainrules.jl b/src/chainrules.jl
index 2b10231..2ea9fca 100644
--- a/src/chainrules.jl
+++ b/src/chainrules.jl
@@ -1,4 +1,4 @@
-import ChainRulesCore: rrule, RuleConfig, ProjectTo
+import ChainRulesCore: rrule, RuleConfig, ProjectTo, backing
 using ZygoteRules: @adjoint
 
 function contract(a::TaylorScalar{T, N}, b::TaylorScalar{S, N}) where {T, S, N}
@@ -39,7 +39,8 @@ end
 function rrule(::typeof(value), t::TaylorScalar{T, N}) where {N, T}
     value_pullback(v̄::NTuple{N, T}) = NoTangent(), TaylorScalar(v̄)
     # for structural tangent, convert to tuple
-    value_pullback(v̄) = NoTangent(), TaylorScalar(map(x -> convert(T, x), Tuple(v̄)))
+    value_pullback(v̄::Tangent{P, NTuple{N, T}}) where P = NoTangent(), TaylorScalar{T, N}(backing(v̄))
+    value_pullback(v̄) = NoTangent(), TaylorScalar{T, N}(map(x -> convert(T, x), Tuple(v̄)))
     return value(t), value_pullback
 end
 
diff --git a/src/derivative.jl b/src/derivative.jl
index 33ce200..b035ee8 100644
--- a/src/derivative.jl
+++ b/src/derivative.jl
@@ -28,8 +28,11 @@ end
     return extract_derivative(f(t), N)
 end
 
+# Need to rewrite like this to help Zygote infer types
+make_taylor(t0::T, t1::T, ::Val{N}) where {T, N} = TaylorScalar{T, N}(t0, t1)
+
 @inline function derivative(f, x::Vector{T}, l::Vector{T},
-                            ::Val{N}) where {T <: Number, N}
-    t = map(TaylorScalar{T, N}, x, l)
+                            vN::Val{N}) where {T <: Number, N}
+    t = map((t0, t1) -> make_taylor(t0, t1, vN), x, l) # i.e. map(TaylorScalar{T, N}, x, l)
     return extract_derivative(f(t), N)
 end
diff --git a/src/primitive.jl b/src/primitive.jl
index d31de26..31b828a 100644
--- a/src/primitive.jl
+++ b/src/primitive.jl
@@ -15,6 +15,14 @@ import Base: hypot, max, min
 @inline cbrt(t::TaylorScalar) = ^(t, 1 / 3)
 @inline inv(t::TaylorScalar) = one(t) / t
 
+exp(t::TaylorScalar{T, 2}) where T = let v = value(t), e1 = exp(v[1])
+    TaylorScalar{T, 2}((e1, e1 * v[2]))
+end
+
+exp(t::TaylorScalar{T, 3}) where T = let v = value(t), e1 = exp(v[1])
+    TaylorScalar{T, 3}((e1, e1 * v[2], e1 * v[3] + e1 * v[2] * v[2]))
+end
+
 for func in (:exp, :expm1, :exp2, :exp10)
     @eval @generated function $func(t::TaylorScalar{T, N}) where {T, N}
         ex = quote
@@ -37,7 +45,7 @@ for func in (:exp, :expm1, :exp2, :exp10)
         if $(QuoteNode(func)) == :expm1
             ex = :($ex; v1 = expm1(v[1]))
         end
-        ex = :($ex; TaylorScalar($([Symbol('v', i) for i in 1:N]...)))
+        ex = :($ex; TaylorScalar{T, N}(tuple($([Symbol('v', i) for i in 1:N]...))))
         return :(@inbounds $ex)
     end
 end