To compute the logarithm of an integer efficiently using pen and paper, one can first write the number as a product of numbers close to $1$. For example, $$10 = \left(\frac{64}{63}\right)^{63}\left(\frac{245}{243}\right)^{40}\left(\frac{126}{125}\right)^{73}\left(\frac{225}{224}\right)^{90}.$$ Then one can use $$\ln\left(\frac{p}{q}\right)=2 \operatorname{arctanh}\left(\frac{p-q}{p+q}\right)=2\sum_{k=0}^{\infty} \frac{1}{2k+1}\left(\frac{p-q}{p+q}\right)^{2k+1} $$ to reach the desired accuracy.
In the exact same way, one can compute $\pi$ by breaking up $i$ as a product of numbers close to $1$ on the unit circle.
For example, $$ i = [18+i]^7[21+i]^{5}[47+i]^{5}[57+i]^3, $$ with $\forall z \in \mathbb{C^*}, [z]=\frac{z}{\overline{z}}=e^{2i \arg(z)}$.
Therefore $$ e^{i\frac{\pi}{2}} = e^{i \ 14\operatorname{arctan}\left(\frac{1}{18}\right)} e^{i \ 10\operatorname{arctan}\left(\frac{1}{21}\right)} e^{i \ 10\operatorname{arctan}\left(\frac{1}{47}\right)} e^{i \ 6\operatorname{arctan}\left(\frac{1}{57}\right)},$$
Please note $i \operatorname{arctan}(x) = \operatorname{arctanh}(i x) $ to see the similarity between the two results.
However the breakup of the positive integer is much easier than the breakup of $i$ in my opinion. It is easy enough to divide fractions one by the other until you reach a satisfactory result by using their "prime coordinates"; whereas the breakup of $i$ involves trial and error or trying to solve diophantine equations and it seems impossible to go as far with only four factors. That is to say, you get much smaller exponents before having to add a factor to get your factors closer to $1$.
Why is that? Why is it so much harder to decompose $i$ along the unit circle than it is to decompose an integer? Is there an equivalent to the "prime coordinates" used to decompose an integer to make your calculations easier when decomposing $i$? Can the roots of one help you?