complementary error function
/ the symbol erfc is read 'erf-see' /
When you care about a rare event — the chance a measurement is far out in the tail of a bell curve — you want the small remaining area, not the bulk. Subtracting two numbers that are both near 1 to get a tiny answer is numerically dangerous. The complementary error function gives that tail area directly and accurately. It is simply what is left over: erfc(x) = 1 minus erf(x).
Equivalently, erfc(x) = (2 over square root of pi) times integral from x to infinity of e^{-t^2} dt — the area under the Gaussian bell beyond x, rather than up to x. So erfc(0) = 1, erfc(infinity) = 0, and the two functions always sum to erf(x) plus erfc(x) = 1. For large x the tail decays extremely fast, with the asymptotic estimate erfc(x) is approximately e^{-x^2} over (x times square root of pi); this tells you instantly how rare a far-out value is. The reason erfc earns its own name is numerical: computing 1 minus erf(x) for large x loses nearly all significant digits, whereas a dedicated erfc routine keeps full precision.
erfc is the standard way to express tail probabilities of the normal distribution, so it underlies the Q-function in communications (probability of a bit error), reliability and survival analysis, and any worst-case noise budget. It also describes how a diffusion front has not yet arrived deep in a material. Whenever the quantity of interest is 'how much is beyond this threshold', erfc, not erf, is the right tool.
erfc(2) is approximately 0.00468, the small Gaussian tail beyond x = 2. The fast asymptotic estimate e^{-4}/(2 times square root of pi) is approximately 0.00517 — the right order of magnitude, improving for larger x.
For tail values, computing erfc directly beats forming 1 minus erf, which would cancel away the significant digits.
erfc and erf carry the same information (they sum to 1); erfc exists mainly for accuracy in the tail, where 1 minus erf(x) suffers catastrophic cancellation.