What's Your Approach

Define

$f(x) = \dfrac{1}{2} \tan\left(\dfrac{x}{2}\right) + \dfrac{1}{4} \tan\left(\dfrac{x}{4}\right) + \dfrac{1}{8} \tan\left(\frac{x}{8}\right) +\cdots.$

Then $f\left(\dfrac{3\pi}{8}\right)$ can be written as $\dfrac{a}{b\pi^{c}} + d - \sqrt{e}$

If $\dfrac{a}{b}$ is in simplest form i.e. $\gcd(a,b)=1$, and $a,b,c,d,e\in\mathbb N$, find the value of $a + b + c + d + e$.