Suggestions for interesting PI calculation

This is a personal interest project that has come to a dead-end. I'm looking for comments and suggestions.

Looking for interesting ways to calculate PI, I seem to have take an approach similar to the one for Viète's formula. I've come up with this extensible expression:

$${pi} \approx {2}^{3}\cdot\sqrt{2-\sqrt{2+\sqrt{2}}}$$ $${pi} \approx {2}^{4}\cdot\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$$ $${pi} \approx {2}^{5}\cdot\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$$ $${pi} \approx {2}^{6}\cdot\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}$$

As a programmer, this form has the very interesting term ${2}^{n}$, which may point to a possible efficient bitwise implementation, but the recursive roots are obviously not good. I would like to convert this form into a series but this is where my mathematical knowledge has hit its limits.

So, some questions:

• Does this form exist already?
• Is it worth exploring more?
• If so, can you suggest methods I can research?

I see a similar form appear in this question, but it doesn't help me move forward, I believe.