diff --git a/Project.toml b/Project.toml
index 919cb0c0..40b29d60 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "Mooncake"
 uuid = "da2b9cff-9c12-43a0-ae48-6db2b0edb7d6"
 authors = ["Will Tebbutt, Hong Ge, and contributors"]
-version = "0.4.3"
+version = "0.4.4"
 
 [deps]
 ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
diff --git a/src/rrules/low_level_maths.jl b/src/rrules/low_level_maths.jl
index 33b53c13..9569ccfe 100644
--- a/src/rrules/low_level_maths.jl
+++ b/src/rrules/low_level_maths.jl
@@ -5,7 +5,7 @@ for (M, f, arity) in DiffRules.diffrules(; filter_modules=nothing)
         continue  # Skip rules for methods not defined in the current scope
     end
     (f == :rem2pi || f == :ldexp) && continue # not designed for Float64s
-    (f in [:+, :*, :sin, :cos, :exp]) && continue # use other functionality to implement these
+    (f in [:+, :*, :sin, :cos, :exp, :-, :abs2, :inv, :abs, :/, :\]) && continue # use other functionality to implement these
     if arity == 1
         dx = DiffRules.diffrule(M, f, :x)
         pb_name = Symbol("$(M).$(f)_pb!!")
@@ -84,11 +84,19 @@ function generate_hand_written_rrule!!_test_cases(rng_ctor, ::Val{:low_level_mat
         end
         arity > 2 && return
         (f == :rem2pi || f == :ldexp || f == :(^)) && return
-        (f == :+ || f == :*) && return # use intrinsics instead
+        (f in [:+, :*, :sin, :cos, :exp, :-, :abs2, :inv, :abs, :/, :\]) && return # use other functionality to implement these
         f = @eval $M.$f
         push!(test_cases, (false, :stability, nothing, f, rand_inputs(rng, Float64, f, arity)...))
         push!(test_cases, (true, :stability, nothing, f, rand_inputs(rng, Float32, f, arity)...))
     end
+
+    # test cases for additional rules written in this file.
+    push!(test_cases, (false, :stability_and_allocs, nothing, sin, 1.1))
+    push!(test_cases, (true, :stability_and_allocs, nothing, sin, Float32(1.1)))
+    push!(test_cases, (false, :stability_and_allocs, nothing, cos, 1.1))
+    push!(test_cases, (true, :stability_and_allocs, nothing, cos, Float32(1.1)))
+    push!(test_cases, (false, :stability_and_allocs, nothing, exp, 1.1))
+    push!(test_cases, (true, :stability_and_allocs, nothing, exp, Float32(1.1)))
     memory = Any[]
     return test_cases, memory
 end
diff --git a/test/front_matter.jl b/test/front_matter.jl
index d738d2c5..9b32e431 100644
--- a/test/front_matter.jl
+++ b/test/front_matter.jl
@@ -61,7 +61,9 @@ using Mooncake:
     InvalidFDataException,
     InvalidRDataException,
     verify_fdata_value,
-    verify_rdata_value
+    verify_rdata_value,
+    is_primitive,
+    MinimalCtx
 
 using .TestUtils:
     test_rule,
diff --git a/test/rrules/low_level_maths.jl b/test/rrules/low_level_maths.jl
index 32c696d4..ce6834a4 100644
--- a/test/rrules/low_level_maths.jl
+++ b/test/rrules/low_level_maths.jl
@@ -1,3 +1,20 @@
 @testset "low_level_maths" begin
     TestUtils.run_rrule!!_test_cases(StableRNG, Val(:low_level_maths))
+
+    # These are all examples of signatures which we do _not_ want to make primitives,
+    # because they are very shallow wrappers around lower-level primitives for which we
+    # already have rules.
+    @testset "$T, $C" for T in [Float16, Float32, Float64], C in [DefaultCtx, MinimalCtx]
+        @test !is_primitive(C, Tuple{typeof(+), T})
+        @test !is_primitive(C, Tuple{typeof(-), T})
+        @test !is_primitive(C, Tuple{typeof(abs2), T})
+        @test !is_primitive(C, Tuple{typeof(inv), T})
+        @test !is_primitive(C, Tuple{typeof(abs), T})
+
+        @test !is_primitive(C, Tuple{typeof(+), T, T})
+        @test !is_primitive(C, Tuple{typeof(-), T, T})
+        @test !is_primitive(C, Tuple{typeof(*), T, T})
+        @test !is_primitive(C, Tuple{typeof(/), T, T})
+        @test !is_primitive(C, Tuple{typeof(\), T, T})
+    end
 end