Weird Calculator

Lynn has a calculator with only two buttons that perform $\boxed{+1}$ and $\boxed{\div 2}$.

She also has a screen with $9$ significant figures, and it displays $0$ when she gets the calculator.

If she wants to display $\pi$ up to the eighth decimal $(3.14159265)$, what is the fewest number of taps she needs to do?

Hint: The first 64 bits of $\pi$ in binary are given below:

$\pi \approx$11.00100100001111110110101010001000100001011010001100001000110100$_2$

Note: You may want to use the code environment below.