This work provides a new exact expression for the pairwise error probability under perfect channel state information. Using this expression, the average union upper bounds for (1,7/5,7/5) and (1,5/7,5/7) turbo coding schemes are presented based on transfer function approach assuming fully interleaved Rayleigh fading channels. Derivation for the pairwise error probability leads to a compact approximation which overlaps with the exact expression. This new approximation is computationally more efficient than the existing ones, and leads to a much quicker general ion of union bounds for performance analysis.