Forced periodicity (modulo) breaks ForwardDiff

It seems that modulo periodicity that seems to break ForwardDiff:

If I give it a very simple function that should be differentiable, but use a modulo for time, then this happens:

#+begin_src julia
function f(t)
    sin(pi * 2 * (t % 1))^2
end
#+end_src

#+RESULTS:
: f (generic function with 1 method)

#+begin_src julia
ForwardDiff.derivative(f, 1)
#+end_src

#+RESULTS:
: NaN

#+begin_src julia
derivative(f, 1)
#+end_src

#+RESULTS:
: -5.736299254432828e-15

So the result from the CD is close to zero, as expected. But the ForwardDiff gives NaN. If I take the modulo out, then AutoDiff works.

So while I took care that my functions are continuous, smooth and differentiable at the period boundaries, AutoDiff will not work for them. It does work for cosine series, which are naturally periodic, but not with piecewise continuous functions with modulo periodicity.

Start by reproducing the issue with the Julia function f(t) and the ForwardDiff.derivative(f, 1) call shown in the report, then compare it with derivative(f, 1). Investigate how the modulo expression is handled at the period boundary; done means the ForwardDiff result is finite and agrees with the expected near-zero derivative.