Error Function Calculator
The error function erf(x) is a special function that arises whenever a Gaussian, or bell curve, is integrated. It equals two over the square root of pi times the integral of e to the minus t squared from zero to x, and it is the backbone of normal-distribution probability. This calculator returns erf(x), the complementary error function erfc(x) = 1 minus erf(x), and the matching cumulative normal probability, for any real input. The approximation used is accurate to roughly seven significant figures.
Error function definition
erf(x) = (2 / sqrt(pi)) * integral from 0 to x of e^(-t^2) dt
erfc(x) = 1 - erf(x)
erf(-x) = -erf(x) (odd function)
Normal CDF(x) = 0.5 * (1 + erf(x / sqrt(2)))
The integral has no elementary antiderivative, so the calculator uses a rational approximation with bounded error to evaluate erf, then derives erfc and the cumulative normal value.
About the error function
- erf(0) is exactly 0 and the function is odd about the origin.
- erf grows toward 1 for large positive x and toward minus 1 for large negative x.
- It defines normal-distribution probabilities and z-score lookups.
- It appears in heat conduction and diffusion solutions across physics.
- erfc is preferred in the far tail where erf rounds to 1 and loses precision.
Error function: frequently asked questions
What is the error function?
The error function erf(x) is two over the square root of pi times the integral from 0 to x of e to the minus t squared. It measures the probability that a normally distributed value falls within a scaled range and ranges from -1 to 1.
What is the complementary error function?
The complementary error function is erfc(x) = 1 - erf(x). It is the tail probability and is often more numerically stable for large positive x, where erf(x) is very close to 1.
How accurate is this calculator?
It uses a standard rational approximation with a maximum absolute error around 1.5 times 10 to the minus 7, which is accurate to about seven significant figures across the whole real line.
How does erf relate to the normal distribution?
The cumulative normal distribution is one half times (1 + erf(x divided by the square root of 2)). The error function is therefore the engine behind normal-distribution probabilities and z-score tables.
What are the limiting values of erf?
erf(0) is 0, erf tends to 1 as x grows large and positive, and erf tends to -1 as x grows large and negative. The function is odd, so erf(-x) equals minus erf(x).
Official sources
- NIST Digital Library of Mathematical Functions: Error Functions: Definitions.
- NIST Digital Library of Mathematical Functions: Error Functions: Approximations.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.