Practical Method for log[erfc(a)] Approximation in the Extreme Tail

Accurate evaluation of extremely small Gaussian tail probabilities is essential in statistical meta-analyses, in which large z-scores (often exceeding 8 or 9) must be converted into p-values. Meanwhile, direct numerical integration of complementary error function erfc(a) suffers from severe underflow in floating-point arithmetic. In this paper, a simple and robust approximation scheme for log[erfc(a)] is proposed based on a geometric tangent construction. This approach yields explicit lower and upper bounds, closed-form asymptotic expansions up to order a^-8, and numerically stable formulas suitable for implementation in statistical software. Numerical comparisons demonstrate that the lower and upper bounds become extremely tight for a>=6, making the proposed method practical for large-scale meta-analytic computations.