diff --git a/Project.toml b/Project.toml
index cd04fd278..210d77792 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "0.7.7"
+version = "0.7.8"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
diff --git a/src/rulesets/Base/base.jl b/src/rulesets/Base/base.jl
index 82609236a..cc2481f09 100644
--- a/src/rulesets/Base/base.jl
+++ b/src/rulesets/Base/base.jl
@@ -138,7 +138,7 @@ end
     ),
     (!(islow | ishigh), islow, ishigh),
 )
-@scalar_rule x \ y (-((y / x) / x), inv(x))
+@scalar_rule x \ y (-(Ω / x), one(y) / x)
 
 function frule((_, ẏ), ::typeof(identity), x)
     return (x, ẏ)
diff --git a/src/rulesets/Base/fastmath_able.jl b/src/rulesets/Base/fastmath_able.jl
index 04bb97bb2..0089ccbc0 100644
--- a/src/rulesets/Base/fastmath_able.jl
+++ b/src/rulesets/Base/fastmath_able.jl
@@ -133,7 +133,7 @@ let
 
         @scalar_rule x + y (One(), One())
         @scalar_rule x - y (One(), -1)
-        @scalar_rule x / y (inv(y), -((x / y) / y))
+        @scalar_rule x / y (one(x) / y, -(Ω / y))
         #log(complex(x)) is required so it gives correct complex answer for x<0
         @scalar_rule(x ^ y,
             (ifelse(iszero(x), zero(Ω), y * Ω / x), Ω * log(complex(x))),
@@ -141,7 +141,7 @@ let
         # x^y for x < 0 errors when y is not an integer, but then derivative wrt y
         # is undefined, so we adopt subgradient convention and set derivative to 0.
         @scalar_rule(x::Real ^ y::Real,
-            (ifelse(iszero(x), zero(Ω), y * Ω / x), Ω * log(ifelse(x ≤ 0, one(x), x))),
+            (ifelse(iszero(x), zero(Ω), y * Ω / x), Ω * log(oftype(Ω, ifelse(x ≤ 0, one(x), x)))),
         )
         @scalar_rule(
             rem(x, y),
diff --git a/test/rulesets/Base/fastmath_able.jl b/test/rulesets/Base/fastmath_able.jl
index 8ccdbe67d..49e30d55f 100644
--- a/test/rulesets/Base/fastmath_able.jl
+++ b/test/rulesets/Base/fastmath_able.jl
@@ -137,7 +137,29 @@ const FASTABLE_AST = quote
             Δz = randn(typeof(f(x, y)))
 
             frule_test(f, (x, Δx), (y, Δy))
-            rrule_test(f, Δz, (x, x̄), (y, ȳ))
+            rrule_test(f, Δz, (x, x̄), (y, ȳ))    
+        end
+
+        @testset "$f(x::$T, y::$T) type check" for f in (/, +, -,\, hypot, ^), T in (Float32, Float64)
+            x, Δx, x̄ = 10rand(T, 3)
+            y, Δy, ȳ = rand(T, 3)
+            @assert T == typeof(f(x, y))
+            Δz = randn(typeof(f(x, y)))
+
+            @test frule((Zero(), Δx, Δy), f, x, y) isa Tuple{T, T}
+            _, ∂x, ∂y = rrule(f, x, y)[2](Δz)
+            @test extern.((∂x, ∂y)) isa Tuple{T, T}
+
+            if f != hypot
+                # Issue #233                
+                @test frule((Zero(), Δx, Δy), f, x, 2) isa Tuple{T, T}
+                _, ∂x, ∂y = rrule(f, x, 2)[2](Δz)
+                @test extern.((∂x, ∂y)) isa Tuple{T, T}
+                
+                @test frule((Zero(), Δx, Δy), f, 2, y) isa Tuple{T, T}
+                _, ∂x, ∂y = rrule(f, 2, y)[2](Δz)
+                @test extern.((∂x, ∂y)) isa Tuple{T, T}
+            end
         end
 
         @testset "^(x::$T, n::$T)" for T in (Float64, ComplexF64)