  We derive a general formula of the minimum achievable rate for
fixed-to-variable length coding with a regular cost function by allowing the
error probability up to a constant $\varepsilon$. For a fixed-to-variable
length code, we call the set of source sequences that can be decoded without
error the dominant set of source sequences. For any two regular cost functions,
it is revealed that the dominant set of source sequences for a code attaining
the minimum achievable rate with a cost function is also the dominant set for a
code attaining the minimum achievable rate with the other cost function. We
also give a general formula of the second-order minimum achievable rate.
