Closed Form for $\sum_{n=0}^{\infty}x^{2^n}$

Question

I am looking to find a closed form for $$f(x)=\sum_{n=0}^{\infty}x^{2^n}$$ $f$ exists on $|x|<1$ and we can immediately see that $f$ satisfies the functional equation $$f(x^2)=f(x)-x$$ Thus we only need to know $f$ on either $(-1, 0]$ or $[0,1)$. I really don't know how to proceed from here though. I would have thought this closed form was already known since it's so close to a geometric series, but I haven't been able to find it.