According to the survey paper on autodiff (linked) Autodiff works on inputs that cannot be specified in closed form but can be described by a sequence of code, each component of which is differentiable. Autodiff also works on code that can be symbolically differentiated but in this latter case the benefits are less obvious and more subtle. However, I haven't been able to come up with an example where autodiff works but symbolic diff will not work.
My question is: does there exist a simple example of autodiff with a code input that will not work with symbolic diff?
Note: I realize simple is somewhat arbitrarily defined so let's say in less than 20 lines of code so that the answer isn't too long to read. If 20 lines isn't enough then something like the minimal number of lines in code would work.