Entry 16

Ramanujan gave an unusual approximation to \(\pi\)
$$\pi \approx \frac{4}{\sqrt{522}}\ln\left[\Big(\frac{5+\sqrt{29}}{\sqrt{2}}\Big)^3(5\sqrt{29}+11\sqrt{6})\left(\sqrt{\frac{9+3\sqrt{6}}{4}}+\sqrt{\frac{5+3\sqrt{6}}{4}} \right)^6\right]$$ which is good to \(31\) digits. This in fact belongs to a family. First, define the fundamental units $$\begin{aligned}
U_{2} &= 1+\sqrt{2}\\
U_{29} &= \frac{5+\sqrt{29}}{2}\\
U_{58} &= 99+13\sqrt{58}\\
U_{174} &= 1451+110\sqrt{174}\end{aligned}$$ Note that \(174 = 6\times29\). These are involved in fundamental solutions to Pell equations.  For example, for \(x^2-58y^2 = -1\), it is \((x, y) = (99, 13)\), (see the values for \(U_{58}\)). Then with increasing precision $$\begin{aligned}
\pi &\approx \frac{1}{\sqrt{58}} \ln \Big[ 2^6 (U_{29})^{12} \Big]\\
\pi &\approx \frac{1}{2\sqrt{58}} \ln \left[ 2^9 \left((U_2)^3\, U_{29} \sqrt{U_{58}} \,\right)^6 \right]\\
\pi &\approx \frac{1}{3\sqrt{58}} \ln \left[ 2^6 (U_{29})^{12}\, (U_{174})^2 \left( \sqrt{\frac{9+3\sqrt{6}}{4} } + \sqrt{\frac{5+3\sqrt{6}}{4}}\right)^{24}\right]\\
\pi &\approx \frac{1}{4\sqrt{58}} \ln \Big[ 2^9 \left((U_2)^3\, U_{29} \sqrt{2\,U_{58}} \,\right)^3 \big(\sqrt{v+1} +\sqrt{v}\big)^{12}\Big]\end{aligned}$$ where \(v = 2^{-1/2}(U_2)^6\,(U_{29})^3\).

Beautifully consistent, aren't they?  The last is by this author and is accurate to \(42\) digits. The expression inside the log function is the exact value of \(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24}\) where  \(\eta(\tau)\) is the Dedekind eta function, and  \(\tau = \frac{\sqrt{-58}}{2},  \tau = \frac{2\sqrt{-58}}{2},  \tau = \frac{3\sqrt{-58}}{2},  \tau = \frac{4\sqrt{-58}}{2}\), respectively.

Anyone can find a nice expression for the next step?