NewtonNonlinearSolver returns nan when initialized with root value

It seems like the Newton solver fails when initialized with the root value at which also $f'(x) = 0$. A minimal working example:

from diffrax import NewtonNonlinearSolver

def fun(x, args): return x**3
solver = NewtonNonlinearSolver(rtol=1e-3, atol=1e-6)
sol = solver(fun, 0., None)
print(sol.root, fun(sol.root, None))  # nan nan

It looks like the conditions here would catch it, where it not for at_least_two iterations. Of course, Newtons method can't do much in case of $f'(x) = 0$, but I would expect the solver to stop when it figures out it is already at the root.

Start with diffrax/nonlinear_solver/newton.py at the linked condition around line 128, then reproduce the provided x**3 example with an initial value of 0. Check how the solver handles an initial point that is already a root when the derivative is zero. Done means the example returns root 0.0 and a non-NaN function value.