diff --git a/Project.toml b/Project.toml
index 494b4b779..9e5605a3c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "0.5.0"
+version = "0.5.1"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
@@ -11,7 +11,7 @@ Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 
 [compat]
 ChainRulesCore = "0.7"
-ChainRulesTestUtils = "0.2.1"
+ChainRulesTestUtils = "0.2.2"
 Compat = "3"
 FiniteDifferences = "0.9"
 Reexport = "0.2"
diff --git a/src/rulesets/Base/base.jl b/src/rulesets/Base/base.jl
index db7672c32..562498342 100644
--- a/src/rulesets/Base/base.jl
+++ b/src/rulesets/Base/base.jl
@@ -4,7 +4,7 @@
 
 @scalar_rule(abs(x::Real), sign(x))
 @scalar_rule(abs2(x), 2x)
-@scalar_rule(exp(x), Ω)
+@scalar_rule(exp(x::Real), Ω)
 @scalar_rule(exp10(x), Ω * log(oftype(x, 10)))
 @scalar_rule(exp2(x), Ω * log(oftype(x, 2)))
 @scalar_rule(expm1(x), exp(x))
@@ -45,7 +45,8 @@
 @scalar_rule(sinh(x), cosh(x))
 @scalar_rule(tanh(x), 1-Ω^2)
 
-@scalar_rule(acosh(x), inv(sqrt(x^2 - 1)))
+# Can't multiply though sqrt in acosh because of negative complex case for x
+@scalar_rule(acosh(x), inv(sqrt(x - 1) * sqrt(x + 1)))
 @scalar_rule(acoth(x), inv(1 - x^2))
 @scalar_rule(acsch(x), -inv(x^2 * sqrt(1 + x^-2)))
 @scalar_rule(acsch(x::Real), -inv(abs(x) * sqrt(1 + x^2)))
@@ -66,7 +67,9 @@
 @scalar_rule(-(x, y), (One(), -1))
 @scalar_rule(/(x, y), (inv(y), -(x / y / y)))
 @scalar_rule(\(x, y), (-(y / x / x), inv(x)))
-@scalar_rule(^(x, y), (ifelse(iszero(y), zero(Ω), y * x^(y - 1)), Ω * log(x)))
+
+#log(complex(x)) is require so it give correct complex answer for x<0 
+@scalar_rule(^(x, y), (ifelse(iszero(y), zero(Ω), y * x^(y - 1)), Ω * log(complex(x))))
 
 @scalar_rule(cbrt(x), inv(3 * Ω^2))
 @scalar_rule(inv(x), -Ω^2)
@@ -117,7 +120,7 @@ end
 
 function rrule(::typeof(*), x::Number, y::Number)
     function times_pullback(ΔΩ)
-        return (NO_FIELDS,  @thunk(ΔΩ * y'), @thunk(x' * ΔΩ))
+        return (NO_FIELDS,  @thunk(ΔΩ * y), @thunk(x * ΔΩ))
     end
     return x * y, times_pullback
 end
@@ -132,3 +135,14 @@ function rrule(::typeof(identity), x)
     end
     return x, identity_pullback
 end
+
+function rrule(::typeof(identity), x::Tuple)
+    # `identity(::Tuple)` returns multiple outputs;because that is how we think of
+    # returning a tuple, so its pullback needs to accept multiple inputs.
+    # `identity(::Tuple)` has one input, so its pullback should return 1 matching output
+    # see https://github.com/JuliaDiff/ChainRulesCore.jl/issues/152
+    function identity_pullback(ȳs...)
+        return (NO_FIELDS, Composite{typeof(x)}(ȳs...))
+    end
+    return x, identity_pullback
+end
diff --git a/src/rulesets/LinearAlgebra/dense.jl b/src/rulesets/LinearAlgebra/dense.jl
index 322de69e6..cd4a37be8 100644
--- a/src/rulesets/LinearAlgebra/dense.jl
+++ b/src/rulesets/LinearAlgebra/dense.jl
@@ -40,17 +40,17 @@ end
 ##### `det`
 #####
 
-function frule((_, ẋ), ::typeof(det), x)
+function frule((_, ẋ), ::typeof(det), x::Union{Number, AbstractMatrix})
     Ω = det(x)
     # TODO Performance optimization: probably there is an efficent
     # way to compute this trace without during the full compution within
     return Ω, Ω * tr(inv(x) * ẋ)
 end
 
-function rrule(::typeof(det), x)
+function rrule(::typeof(det), x::Union{Number, AbstractMatrix})
     Ω = det(x)
     function det_pullback(ΔΩ)
-        return NO_FIELDS, @thunk(Ω * ΔΩ * inv(x)')
+        return NO_FIELDS, Ω * ΔΩ * transpose(inv(x))
     end
     return Ω, det_pullback
 end
@@ -59,15 +59,15 @@ end
 ##### `logdet`
 #####
 
-function frule((_, Δx), ::typeof(logdet), x)
+function frule((_, Δx), ::typeof(logdet), x::Union{Number, AbstractMatrix})
     Ω = logdet(x)
     return Ω, tr(inv(x) * Δx)
 end
 
-function rrule(::typeof(logdet), x)
+function rrule(::typeof(logdet), x::Union{Number, AbstractMatrix})
     Ω = logdet(x)
     function logdet_pullback(ΔΩ)
-        return (NO_FIELDS, @thunk(ΔΩ * inv(x)'))
+        return (NO_FIELDS, ΔΩ * transpose(inv(x)))
     end
     return Ω, logdet_pullback
 end
@@ -81,6 +81,8 @@ function frule((_, Δx), ::typeof(tr), x)
 end
 
 function rrule(::typeof(tr), x)
+    # This should really be a FillArray
+    # see https://github.com/JuliaDiff/ChainRules.jl/issues/46
     function tr_pullback(ΔΩ)
         return (NO_FIELDS, @thunk Diagonal(fill(ΔΩ, size(x, 1))))
     end
@@ -121,14 +123,11 @@ function rrule(::typeof(/), A::AbstractVecOrMat{<:Real}, B::AbstractVecOrMat{<:R
     C, dC_pb = rrule(adjoint, Cᵀ)
     function slash_pullback(Ȳ)
         # Optimization note: dAᵀ, dBᵀ, dC are calculated no matter which partial you want
-        # this is not a problem if you want the 2nd or 3rd, but if you want the first, it
-        # is fairly wasteful
         _, dC = dC_pb(Ȳ)
-        _, dBᵀ, dAᵀ = dS_pb(extern(dC))
+        _, dBᵀ, dAᵀ = dS_pb(unthunk(dC))
 
-        # need to extern as  dAᵀ, dBᵀ  are generally `Thunk`s, which don't support adjoint
-        ∂A = @thunk last(dA_pb(extern(dAᵀ)))
-        ∂B = @thunk last(dA_pb(extern(dBᵀ)))
+        ∂A = last(dA_pb(unthunk(dAᵀ)))
+        ∂B = last(dA_pb(unthunk(dBᵀ)))
 
         (NO_FIELDS, ∂A, ∂B)
     end
diff --git a/src/rulesets/LinearAlgebra/structured.jl b/src/rulesets/LinearAlgebra/structured.jl
index 45f026edb..666ffe7dc 100644
--- a/src/rulesets/LinearAlgebra/structured.jl
+++ b/src/rulesets/LinearAlgebra/structured.jl
@@ -6,8 +6,15 @@
 #####
 
 function rrule(::Type{<:Diagonal}, d::AbstractVector)
-    function Diagonal_pullback(ȳ)
-        return (NO_FIELDS, @thunk(diag(ȳ)))
+    function Diagonal_pullback(ȳ::AbstractMatrix)
+        return (NO_FIELDS, diag(ȳ))
+    end
+    function Diagonal_pullback(ȳ::Composite)
+        # TODO: Assert about the primal type in the Composite, It should be Diagonal
+        # infact it should be exactly the type of `Diagonal(d)`
+        # but right now Zygote loses primal type information so we can't use it.
+        # See https://github.com/FluxML/Zygote.jl/issues/603
+        return (NO_FIELDS, ȳ.diag)
     end
     return Diagonal(d), Diagonal_pullback
 end
diff --git a/test/rulesets/Base/base.jl b/test/rulesets/Base/base.jl
index a6d56debb..9a3142ef4 100644
--- a/test/rulesets/Base/base.jl
+++ b/test/rulesets/Base/base.jl
@@ -35,14 +35,15 @@
             test_scalar(acsc, 1/x)
             test_scalar(acot, 1/x)
         end
-        @testset "Inverse hyperbolic" for x = (0.5, Complex(0.5, 0.25))
+        @testset "Inverse hyperbolic" for x = (0.5, Complex(0.5, 0.25),  Complex(-2.1 -3.1im))
             test_scalar(asinh, x)
-            test_scalar(acosh, x + 1)  # +1 accounts for domain
+            test_scalar(acosh, x + 1)  # +1 accounts for domain for real
             test_scalar(atanh, x)
             test_scalar(asech, x)
             test_scalar(acsch, x)
             test_scalar(acoth, x + 1)
         end
+
         @testset "Inverse degrees" for x = (0.5, Complex(0.5, 0.25))
             test_scalar(asind, x)
             test_scalar(acosd, x)
@@ -100,7 +101,23 @@
         end
     end
 
-    @testset "*(x, y)" begin
+    @testset "*(x, y) (scalar)" begin
+        # This is pretty important so testing it fairly heavily
+        test_points = (0.0, -2.1, 3.2, 3.7+2.12im, 14.2-7.1im)
+        @testset "$x * $y; (perturbed by: $perturb)" for
+            x in test_points, y in test_points, perturb in test_points
+
+            # give small off-set so as can't slip in symmetry
+            x̄ = ẋ = 0.5 + perturb
+            ȳ = ẏ = 0.6 + perturb
+            Δz = perturb
+
+            frule_test(*, (x, ẋ), (y, ẏ))
+            rrule_test(*, Δz, (x, x̄), (y, ȳ))
+        end
+    end
+
+    @testset "matmul *(x, y)" begin
         x, y = rand(3, 2), rand(2, 5)
         z, pullback = rrule(*, x, y)
 
@@ -125,10 +142,26 @@
         rrule_test(f, Δz, (x, x̄), (y, ȳ))
     end
 
+    @testset "x^n for x<0" begin
+        rng = MersenneTwister(123456)
+        x = -15*rand(rng)
+        Δx, x̄ = 10rand(rng, 2)
+        y, Δy, ȳ = rand(rng, 3)
+        Δz = rand(rng)
+
+        frule_test(^, (-x, Δx), (y, Δy))
+        rrule_test(^, Δz, (-x, x̄), (y, ȳ))
+    end
+
     @testset "identity" begin
         rng = MersenneTwister(1)
         rrule_test(identity, randn(rng), (randn(rng), randn(rng)))
         rrule_test(identity, randn(rng, 4), (randn(rng, 4), randn(rng, 4)))
+
+        rrule_test(
+            identity, Tuple(randn(rng, 3)),
+            (Composite{Tuple}(randn(rng, 3)...), Composite{Tuple}(randn(rng, 3)...))
+        )
     end
 
     @testset "Constants" for x in (-0.1, 6.4, 1.0+0.5im, -10.0+0im, 0+200im)
diff --git a/test/rulesets/LinearAlgebra/structured.jl b/test/rulesets/LinearAlgebra/structured.jl
index 1731aad32..07b8ef986 100644
--- a/test/rulesets/LinearAlgebra/structured.jl
+++ b/test/rulesets/LinearAlgebra/structured.jl
@@ -6,7 +6,15 @@
         rrule_test(Diagonal, D, (randn(rng, N), randn(rng, N)))
         # Concrete type instead of UnionAll
         rrule_test(typeof(D), D, (randn(rng, N), randn(rng, N)))
+
+        # TODO: replace this with a `rrule_test` once we have that working
+        # see https://github.com/JuliaDiff/ChainRulesTestUtils.jl/issues/24
+        res, pb = rrule(Diagonal, [1, 4])
+        @test pb(10*res) == (NO_FIELDS, [10, 40])
+        comp = Composite{typeof(res)}(; diag=10*res.diag)  # this is the structure of Diagonal
+        @test pb(comp) == (NO_FIELDS, [10, 40])
     end
+    
     @testset "::Diagonal * ::AbstractVector" begin
         rng, N = MersenneTwister(123456), 3
         rrule_test(