Whether $BPL=L$ (which is conjectured to be equal), or even whether $BPL\subseteq NL$, is a big open problem in theoretical computer science. It is well known that $L-NC^1\subseteq L\subseteq NL\subseteq L-AC^1$. In this work we will show that $BPL\subseteq L-AC^1$, which was not known before. Our proof is based on modifying the Richardson Iteration method for boosting precision in approximating matrix powering, which was developed in a line of works [AKM+20][PV21][CDR+21][CDST22][PP22][CHT+23]. We also improve the algorithm for approximating counting in low-depth $L$-uniform $AC$ circuit from additive error setting to multiplicative error setting.