Error Function
Compute the error function erf(x) for any real number using Horner-form rational approximation. Free online erf calculator for statistics, probability, thermodynamics, and signal processing applications.
About This Calculator
The Error Function Calculator computes erf(x) using the Horner-form rational approximation by Abramowitz and Stegun, accurate to approximately 1.5 x 10⁻^7. The error function is defined as erf(x) = (2/sqrtpi) integral0ˣ e^(-t^2) dt and is a non-elementary sigmoid-shaped special function.
erf(x) is used in: probability theory (Gaussian error distribution), statistics (cumulative distribution function of the normal distribution), thermodynamics (diffusion equations), quantum mechanics (Fermi-Dirac integrals), and signal processing (error probability calculations).
Frequently Asked Questions
What is the error function erf(x)?
The error function erf(x) is a special function defined as the integral of the Gaussian distribution: erf(x) = (2/sqrtpi) integral0ˣ e^(-t^2) dt. It appears in probability, statistics (cumulative normal distribution), physics (diffusion), and signal processing.
What is the range of erf(x)?
erf(x) ranges from -1 to 1. As x -> +∞, erf(x) -> 1. As x -> -∞, erf(x) -> -1. At x = 0, erf(0) = 0. The function is odd: erf(-x) = -erf(x).
What is the complementary error function?
The complementary error function erfc(x) = 1 - erf(x). It is used for computing the tail probability of the normal distribution. This calculator also displays erfc(x) alongside erf(x).
Is this calculator free?
Yes, all calculators on Calculy are completely free to use.