SPECIAL FUNCTIONS / GAUSSIAN COMPLEMENT
Evaluate erfc(x) alongside erf(x) and their defining sum-to-one identity.
METHOD / WORKED EXAMPLE
erfc packages the remaining Gaussian-integral share after erf. It is common in upper-tail and diffusion formulas, but the argument scaling still belongs to the model using it.
WORKED DEFAULT
For x = 1, erfc(1) is approximately 0.15729921 and erf(1) is approximately 0.84270079. Their sum returns 1 within the displayed numerical precision.
READ THE RESULT
A small erfc value represents the complement of erf at the same argument. It is not automatically a p-value, failure rate, or standard-normal tail without the correct transformation.
ASSUMPTIONS AND LIMITS
Very small tails can require a dedicated erfc algorithm with higher relative accuracy; do not treat rounded zero as proof of an impossible event.
COMMON QUESTIONS
The identity is exactly one minus erf, but dedicated erfc implementations can preserve relative accuracy when the complement is extremely small and subtraction would cancel leading digits. This bounded page exposes both values for clarity. For stringent tail calculations, use a library that evaluates erfc directly and documents error across the required domain.
For a standard normal variable, the upper-tail probability above z equals one half times erfc(z/sqrt(2)). The square-root-of-two scaling and one-half factor are both required. This page shows that transformed expression as context, but users must still define whether their event is one-sided, two-sided, central, or conditioned in another way.
No. A very small positive value can round to zero or underflow in finite arithmetic even though the mathematical function remains positive for finite positive x. Increase numerical precision or use a log-tail or scaled complementary-error function when rare-event magnitude matters. Never infer impossibility solely from a rounded floating-point display.
RELATED TOOLS
Use boundary
Evaluate erf(x) on the bounded real domain, form erfc(x) = 1 - erf(x), and expose the identity plus a correctly scaled standard-normal upper-tail relation.
erfc(x) = 1 - erf(x).