erf(): The Gaussian Error Function

erf() calculates the error function, a smooth function closely connected to the normal distribution and diffusion processes. It integrates a Gaussian curve, so it appears in probability, heat flow, signal analysis, and statistics.

Interpreting values

For real input, erf(x) lies between −1 and 1, is odd, and approaches those limits as magnitude grows. erf(0) is zero. It relates a standard normal cumulative probability to an input scaled by √2, which makes it useful when translating between Gaussian models and probability statements.

Use exp() in explicit Gaussian formulas, sqrt() for standard-deviation scaling, and abs() for symmetric error magnitudes. For a tail probability, avoid subtracting nearly equal values when precision matters; choose a numerically suitable formulation.