Consider a $BPL$ machine (namely, a probabilistic algorithm that uses logspace and polynomially many random bits). It is known (Saks-Zhou) that $BPL \subseteq DSPACE(log^{1.5}(n))$.
My question is about a $BPL$ machine that uses only polylog many bits of randomness. In one of Goldreich's papers, it is mentioned in passing that a language decided by such a $BPL$ machine is actually in deterministic logspace $L$. But I am unable to find any explanations of this remark anywhere.
Why can it be derandomized completely in logspace?