The Dedekind eta function \begin{equation} \eta(\tau)=e^{\frac{\pi i \tau}{12}} \prod_{n=1}^\infty (1-e^{2n\pi i \tau}) =q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n) \end{equation} where the Eulear function \begin{equation} \phi(q) = q^{-\frac{1}{24}} \eta(\tau) = q^{-\frac{1}{24}} \sum_{n=-\infty}^\infty (-1)^n q^{\frac{3n^2-n}{2}} \end{equation} The derivation of the Euler function is connected to the Pentagonal number theorem
In order to compute the Dedekind eta function numerically, one needed to known the error. For the first expression, it might be useful to simply compute until the first significant digits appear at $N$th place, However, it's slow and it's hard to actually obtain the error estimate. The second expression is fast, but it's an infinite sum that might contribute to a certain number at fixed $n$.
What is the error function for the numerical Dedekind eta function?