Error Function Calculator

The error function, denoted as erf(x), is a special, non-elementary sigmoid function that appears frequently in probability, statistics, and partial differential equations. Mathematically, it is defined as the integral: erf(x) = (2 / √π) ∫ exp(-t²) dt from 0 to x. Depending on the calculation mode selected, this tool evaluates direct error functions (like res1 = erf(x1)), complementary error functions (like res2 = 1 - erf(x2)), or inverse error functions to map probability values back to their original domain thresholds.

Suppose you want to compute the standard error function for an input variable x1 = 1.5. Using the governing equation res1 = erf(x1), the calculator evaluates the integral of the Gaussian distribution from 0 to 1.5. The resulting value is approximately 0.9661, meaning that about 96.61% of the cumulative area under the standard normal distribution curve falls within 1.5 standard deviations of the mean.

Double-check your input values before calculating, especially when dealing with decimal inputs in inverse modes where values must strictly fall between 0 and 1. Ensure your chosen operational mode matches your specific mathematical requirement, such as differentiating between standard error functions and complementary counterparts to avoid sign errors.