Merge pull request #1068 from mcabbott/bcast_second

Don't use dual numbers in broadcasting within 2nd derivatives

src/lib/broadcast.jl

@@ -175,7 +175,7 @@ _dual_safearg(x) = false
   # Avoid generic broadcasting in two easy cases:
   if T == Bool
     return (f.(args...), _ -> nothing) 
-  elseif T <: Real && isconcretetype(T) && _dual_purefun(F) && all(_dual_safearg, args)
+  elseif T <: Real && isconcretetype(T) && _dual_purefun(F) && all(_dual_safearg, args) && !isderiving()
     return broadcast_forward(f, args...)
   end
   len = inclen(args)

test/features.jl

@@ -558,4 +558,9 @@ end
   @test gradient((x,p) -> sum(x .^ p), [1.0,2.0,4.0], -1)[1] ≈ [-1.0, -0.25, -0.0625]
   @test gradient((x,p) -> sum(z -> z^p, x), [1.0,2.0,4.0], -1)[1] ≈ [-1.0, -0.25, -0.0625]
   @test gradient((x,p) -> mapreduce(z -> z^p, +, x), [1.0,2.0,4.0], -1)[1] ≈ [-1.0, -0.25, -0.0625]
+
+  # second order
+  @test gradient(x -> sum(gradient(y -> sum(y.^2), x)[1]), [1, 2])[1] ≈ [2, 2]
+  @test gradient(x -> sum(gradient(y -> sum(sin.(y)), x)[1]), [1, 2])[1] ≈ [-0.8414709848078965, -0.9092974268256817]
+  @test gradient(x -> sum(abs, gradient(y -> sum(log.(2 .* exp.(y)) .^ 2), x)[1]), [1, 2])[1] ≈ [2,2]
 end