diff --git a/src/Ops.jl b/src/Ops.jl
index 11c75820a..a873de4ca 100644
--- a/src/Ops.jl
+++ b/src/Ops.jl
@@ -548,6 +548,18 @@ end
 #     )
 #     return TracedRArray{T,N}((), res, size(x))
 # end
+@noinline function bitcast_convert(
+    ::Type{U},
+    x::TracedRNumber{T};
+    location=mlir_stacktrace("bitcast_convert", @__FILE__, @__LINE__),
+) where {T,U}
+    res = MLIR.IR.result(
+        stablehlo.bitcast_convert(
+            x.mlir_data; result_0=mlir_type(TracedRArray{U,0}, ()), location
+        ),
+    )
+    return TracedRNumber{U}((), res)
+end
 
 @noinline function fft(
     x::TracedRArray{T,N};
diff --git a/src/TracedRNumber.jl b/src/TracedRNumber.jl
index 75bfc708c..e7410a0b4 100644
--- a/src/TracedRNumber.jl
+++ b/src/TracedRNumber.jl
@@ -94,7 +94,6 @@ for (jlop, hloop) in (
     (:(Base.:*), :multiply),
     (:(Base.:/), :divide),
     (:(Base.:^), :power),
-    (:(Base.mod), :remainder),
     (:(Base.rem), :remainder),
 )
     @eval function $(jlop)(
@@ -109,13 +108,30 @@ function Base.rem(
 ) where {T}
     return Ops.remainder(lhs, TracedUtils.promote_to(TracedRNumber{T}, rhs))
 end
-
 function Base.rem(
     @nospecialize(lhs::Number), @nospecialize(rhs::TracedRNumber{T})
 ) where {T}
     return Ops.remainder(TracedUtils.promote_to(TracedRNumber{T}, lhs), rhs)
 end
 
+# Based on https://github.com/JuliaLang/julia/blob/39255d47db7657950ff1c82137ecec5a70bae622/base/float.jl#L608-L617
+function Base.mod(
+    @nospecialize(x::Reactant.TracedRNumber{T}), @nospecialize(y::Reactant.TracedRNumber{T})
+) where {T}
+    r = rem(x, y)
+    return ifelse(r == 0, copysign(r, y), ifelse((r > 0) ⊻ (y > 0), r + y, r))
+end
+function Base.mod(
+    @nospecialize(lhs::TracedRNumber{T}), @nospecialize(rhs::Number)
+) where {T}
+    return mod(lhs, TracedUtils.promote_to(TracedRNumber{T}, rhs))
+end
+function Base.mod(
+    @nospecialize(lhs::Number), @nospecialize(rhs::TracedRNumber{T})
+) where {T}
+    return mod(TracedUtils.promote_to(TracedRNumber{T}, lhs), rhs)
+end
+
 function Base.div(@nospecialize(lhs::TracedRNumber{T}), rhs) where {T<:Integer}
     return Ops.divide(lhs, TracedUtils.promote_to(TracedRNumber{T}, rhs))
 end
@@ -224,6 +240,12 @@ for (T1, T2) in zip((Bool, Integer), (Bool, Integer))
                 TracedUtils.promote_to(TracedRNumber{$(T)}, y),
             )
         end
+        function Base.xor(x::TracedRNumber{<:$(T1)}, y::TracedRNumber{<:$(T2)})
+            return Ops.xor(
+                TracedUtils.promote_to(TracedRNumber{$(T)}, x),
+                TracedUtils.promote_to(TracedRNumber{$(T)}, y),
+            )
+        end
         Base.:!(x::TracedRNumber{<:$(T1)}) = Ops.not(x)
     end
 end
@@ -391,4 +413,20 @@ function Base.typed_hvncat(
     return Base.typed_hvncat(T, dims, row_first, xs...)
 end
 
+for (Ti, Tf) in ((Int16, Float16), (Int32, Float32), (Int64, Float64))
+    @eval begin
+        Base.signbit(x::TracedRNumber{$(Ti)}) = x < 0
+        Base.signbit(x::TracedRNumber{$(Tf)}) = signbit(Ops.bitcast_convert($(Ti), x))
+    end
+end
+Base.signbit(::TracedRNumber{<:Unsigned}) = ConcreteRNumber(false)
+
+Base.copysign(x::TracedRNumber, y::TracedRNumber) = ifelse(signbit(y), -1, 1) * abs(x)
+function Base.copysign(x::TracedRNumber{T}, y::S) where {T,S<:Number}
+    return copysign(x, TracedUtils.promote_to(TracedRNumber{S}, y))
 end
+function Base.copysign(x::S, y::TracedRNumber{T}) where {S<:Number,T}
+    return copysign(TracedUtils.promote_to(TracedRNumber{S}, x), y)
+end
+
+end # module TracedRNumberOverrides
diff --git a/test/basic.jl b/test/basic.jl
index 08c6bff81..99a19ceb0 100644
--- a/test/basic.jl
+++ b/test/basic.jl
@@ -829,13 +829,10 @@ end
     a = [-1.1, 7.7, -3.3, 9.9, -5.5]
     b = [6.6, -2.2, -8.8, 4.4, -10.1]
 
-    # Currently broken because `mod` is JIT-ed to an HLO operator with same semantic as
-    # Julia's `rem`, rather than `mod`.
     expected_mod = mod.(a, b)
-    @test_broken Reactant.@jit(mod.(Reactant.to_rarray(a), Reactant.to_rarray(b))) ≈
-        expected_mod
-    @test_broken Reactant.@jit(mod.(a, Reactant.to_rarray(b))) ≈ expected_mod
-    @test_broken Reactant.@jit(mod.(Reactant.to_rarray(a), b)) ≈ expected_mod
+    @test Reactant.@jit(mod.(Reactant.to_rarray(a), Reactant.to_rarray(b))) ≈ expected_mod
+    @test Reactant.@jit(mod.(a, Reactant.to_rarray(b))) ≈ expected_mod
+    @test Reactant.@jit(mod.(Reactant.to_rarray(a), b)) ≈ expected_mod
 
     expected_rem = rem.(a, b)
     @test Reactant.@jit(rem.(Reactant.to_rarray(a), Reactant.to_rarray(b))) ≈ expected_rem
@@ -843,6 +840,26 @@ end
     @test Reactant.@jit(rem.(Reactant.to_rarray(a), b)) ≈ expected_rem
 end
 
+@testset "xor" begin
+    for a in (true, false), b in (true, false)
+        @test @jit(xor(ConcreteRNumber(a), ConcreteRNumber(b))) == xor(a, b)
+    end
+end
+
+@testset "signbit" begin
+    for x in (-4, -3.14, -0.0f0, 0.0, 0, 5, 6.28f0)
+        @test @jit(signbit(ConcreteRNumber(x))) == signbit(x)
+    end
+end
+
+@testset "copysign" begin
+    for a in (-3.14, -2, 0.0, 2.71, 42), b in (-7, -0.57, -0.0, 1, 3.14)
+        # Make sure also the return type is correct
+        @test Reactant.to_number(@jit(copysign(ConcreteRNumber(a), ConcreteRNumber(b)))) ===
+            copysign(a, b)
+    end
+end
+
 @testset "reduce integers" begin
     x = rand(Bool, 100)
     x_ra = Reactant.to_rarray(x)