How to find degree of polynomial represented as a circuit?

I know very little about arithmetic circuits, so maybe it is something well-known.

Given a small circuit consisted of $\{1,x,-,+,*\}$ defining one variable polynomial. Let be additionally known that degree of this polynomial does not exceed $d$ and all the coefficients are small. I wonder if exists a fast way to find actual degree of this polynomial?
Using FFT and some small field, one can do it in $O(d)$ time (regardless $polylog$ factors), but this time is sufficiently to reconstruct the the entire polynomial, so I hope computing degree only can be done more efficient.

If the coefficients are small, then for large $x$ the polynomial is roughly equal to $Cx^d$. If you plug in both $x$ and, say, $2x$, then the ratio should be roughly $2^d$. Given bounds on the coefficients and the degree, this method can be made rigorous.

Another idea uses difference sequences. For a polynomial $P(x) $, the polynomial $P(x+1) - P(x)$ has degree smaller by one, the polynomial $P(x+2)-2P(x+1)+P(x) $ has degree smaller by two, and so on (in general, the coefficients are binomial coefficients with alternating signs). Using polynomial identity testing you can find the degree by binary search. This seems less efficient than your approach.

StackExchange Computer Science Q#27724, answer score: 2