Fix rrule for broadcasted over empty Tuple{} (#834)

* Fix rrule for broadcasted over empty Tuple{}

When broadcasting over an empty tuple, `Broadcast.combine_eltypes` returns
`Union{}`. Since `Union{} <: Number` is true, the code entered
`may_bc_derivatives` which tried to construct `Tuple{Union{}, ...}` and
errored. Fix by treating `Union{}` as a trivial non-differentiable case
alongside `Bool`.

Fixes #830

Co-Authored-By: Chris Rackauckas <[email protected]>
Co-Authored-By: Claude Opus 4.6 <[email protected]>

* Update test/rulesets/Base/broadcast.jl

Co-authored-by: David Müller-Widmann <[email protected]>

* Update test/rulesets/Base/broadcast.jl

Co-authored-by: David Müller-Widmann <[email protected]>

* Add multi-argument test for broadcast over empty tuples

Test atan.((), ()) to cover the multi-arg path through the trivial
broadcast rrule when eltype is Union{}.

Co-Authored-By: Chris Rackauckas <[email protected]>
Co-Authored-By: Claude Opus 4.6 <[email protected]>

---------

Co-authored-by: ChrisRackauckas-Claude <[email protected]>
Co-authored-by: Claude Opus 4.6 <[email protected]>
Co-authored-by: David Müller-Widmann <[email protected]>

Author: 4 people

src/rulesets/Base/broadcast.jl

@@ -28,7 +28,7 @@ end
 
 function rrule(cfg::RCR, ::typeof(broadcasted), ::BroadcastStyle, f::F, args::Vararg{Any,N}) where {F,N}
     T = Broadcast.combine_eltypes(f, args)
-    if T === Bool  # TODO use nondifftype here
+    if T === Bool || T === Union{}  # TODO use nondifftype here
         # 1: Trivial case: non-differentiable output, e.g. `x .> 0`
         @debug("split broadcasting trivial", f, T)
         bc_trivial_back(_) = (TRI_NO..., ntuple(Returns(ZeroTangent()), length(args))...)

test/rulesets/Base/broadcast.jl

@@ -177,6 +177,20 @@ BT1 = Broadcast.BroadcastStyle(Tuple)
     end
 
     @testset "bugs" begin
+        @testset "broadcast over empty tuple" begin  # https://github.com/JuliaDiff/ChainRules.jl/issues/830
+            y, bk = rrule(CFG, copy∘broadcasted, BT1, isone, ())
+            @test y == ()
+            @test bk(Tangent{Tuple{}}()) == (NoTangent(), NoTangent(), NoTangent(), ZeroTangent())
+
+            y2, bk2 = rrule(CFG, copy∘broadcasted, BT1, sin, ())
+            @test y2 == ()
+            @test bk2(Tangent{Tuple{}}()) == (NoTangent(), NoTangent(), NoTangent(), ZeroTangent())
+
+            # Multi-argument case
+            y3, bk3 = rrule(CFG, copy∘broadcasted, BT1, atan, (), ())
+            @test y3 == ()
+            @test bk3(Tangent{Tuple{}}()) == (NoTangent(), NoTangent(), NoTangent(), ZeroTangent(), ZeroTangent())
+        end
         @testset "unbroadcast with NTuple" begin  # https://github.com/JuliaDiff/ChainRules.jl/pull/661
             @test ChainRules.unbroadcast((1, 2, [3]), [4, 5, [6]]) isa Tangent   # earlier, NTuple demanded same type
             @test ChainRules.unbroadcast(broadcasted(-, (1, 2), 3), (4, 5)) == (4, 5)  # earlier, called ndims(::Tuple)