Improve upon: $\sqrt[4]{3^4+2^4+\frac{1}{2+(\frac{2}{3})^2}} \approx \pi$

Question

So here we have an approximate value of $\pi$. $$\sqrt[4]{3^4+2^4+\frac{1}{2+(\frac{2}{3})^2}} \approx \pi$$

$$3.14159265262 \ldots \approx 3.14159265358\ldots$$

How could one get a better approximation?

Highly appreciated,

Cro-Magnon

Answer 1

$\pi^4$ happens to have a very good rational approximation $$ \pi^4 \approx 97 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{4}}} = \dfrac{2143}{22}$$ with error approximately $1.25 \times 10^{-7}$, which comes from the fact that the continued fraction $$ \pi^4 = [97;2,2,3,1,16539,1,6,7,\ldots]$$ has a large element $16539$. You're writing $97 = 3^4 + 2^4$ and $ 2 + 1/4 = (3/2)^2$.

Another example is

$$ (\pi + 7)^{4/9} = [2,1,4,23571, \ldots]$$ so that $$ \pi \approx \left(3 - \dfrac{1}{5}\right)^{9/4} - 7 $$ with error approximately $1.4 \times 10^{-5}$. And another: $$ \pi \approx 801^{1/5} - \dfrac{2}{3}$$ with error approximately $9.1 \times 10^{-8}$.

Answer 2

"Improve upon" is vague. Here is one way to improve upon the OP. The expression given yields 9 digits of accuracy, and uses eighteen symbols (surd, two fractions, three plusses, two parentheses, and ten digits).

Meanwhile, the more mundane expression $$\frac{312689}{99532}$$ achieves 11 digits of accuracy using only twelve symbols.

Answer 3

You have the following interesting formula $$\pi \approx\frac{\ln(640320^3+744)}{\sqrt{163}}$$ which gives $30$ "digits of accuracy".

(Thanks for the English expression in quotes to @vadim123. The best rational approximation I knew was $\pi \approx \frac{22}{17}+\frac{37}{47}+\frac{88}{83}$ which gives $9$ exacts digits).