This paper deals with the random number generation problem under the framework of a separate coding system for correlated memoryless sources posed and investigated by Slepian and Wolf. Two correlated data sequences with length n are separately encoded to nR1, nR2 bit messages at each location and those are sent to the information processing center where the encoder wish to generate an approximation of the sequence of independent uniformly distributed random variables with length nR3 from two received random messages. The admissible rate region is defined by the set of all the triples (R1,R2,R3) for which the approximation error goes to zero as n tends to infinity. In this paper we examine the asymptotic behavior of the approximation error inside and outside the admissible rate region. We derive an explicit lower bound of the optimal exponent for the approximation error to vanish and show that it can be attained by the universal codes. Furthermore, we derive an explicit lower bound of the optimal exponent for the approximation error to tend to 2 as n goes to infinity outside the admissible rate region.