Error Propagation (complicated case)
Logarithmic differentiation is your friend $$G=\frac{\left(\frac{4 \pi ^2}{T^2}+\frac{1}{\tau^2}\right)\, r^2 \, d\, S}{4M\, L}=\frac{\left(\frac{4 \pi ^2 \tau ^2}{T^2}+1\right)\, d\, S}{4M\, L}$$ $$\log(G)=\log\left(\frac{4 \pi ^2 \tau ^2}{T^2}+1\right)+\log(d)+\log(S)-\log(M)-\log(L)-\log(4)$$ Differentiate to obtain, for any variable $X$,$$\frac1 G\frac{\partial G}{\partial X}=\text{something}$$ and, when done,use $$\frac{\partial G}{\partial X}=G \times\text{something}$$