In this paper, we develop the density evolution of bit-flipping algorithm (BFA) for regular low-density parity-check (LDPC) codes and analyze the evolution of probability density for the bit-decision error. In contrast to the sum-product algorithm (SPA) and min-sum algorithm (MSA), the density evolution for BFA has remained untouched, which motivates this work. In the developed density evolution for BFA, we first introduce a state variable which is a function of bit-decision result and the flipping function value at each iteration. Then, the state probability is derived, with which the density of bit-decision error is obtained. To confirm the derivations, we design regular LDPC codes and compare the empirical evaluations of bit-error rates and the derived probability of bit-decision errors.