We prove the following very accurate approximation formula for the factorial function:n!p ???-???(? + 1 + 72(3(¾¾¾!+!)2332800 - (^ +1? This gives better results than the following approximation formula, at- n -n I 1 1 31 139 9871?! Pá V27rnne n\ n +---1--------H---,V 6 72n 6480n2 155520?3 6531840?4'which is established by the author [5] and C. Mortici [16] independently, and gives similar results with32 32 ? n 176 128, r- (?\n 8/???? 32176 ~? ?! Pá ?/? — \ 16?4 + — ?3 + — ?2 + —— ? Ve/ V 3 9 4053 9 405 1215which is established by C. Mortici in his very new paper [8].