Calculation results
| erf(x) | ||
| erfc(x) | ||
| erf-1(y) | ||
| erfc-1(y) |
Created by CalculatorOK Team · Last updated: Jun 18, 2025
Error Function Calculator
What the error function does
Special functions rarely get attention outside engineering, yet the error function quietly underpins half of statistics and much of physics. The Error Function Calculator evaluates erf(x) and its inverse for any value in range, with adjustable Precision. Diffusion modeling, heat flow, and normal probability work all call on it, which is why both directions share one panel.
The definition behind the number
By definition, erf(x) = 2 / sqrt(pi) times the integral from 0 to x of e^(-t^2) dt. The bridge to the bell curve is Phi(z) = 0.5 * (1 + erf(z / sqrt(2))). Plug in z = 1: erf(0.7071) is about 0.6827, so Phi(1) = 0.5 * 1.6827 = 0.8413, the familiar 84.13 percent at one standard deviation.
How to use the calculator
Select the Error function mode, enter your Value, adjust Precision, and press Calculate. For the reverse job, switch to the Inverse error function mode and enter a value strictly between -1 and 1, because erf only approaches those limits at infinity. The inverse returns the x whose erf equals your input, handy for recovering thresholds from known probabilities.
Two spot checks make the panel less abstract. In Error function mode, a Value of 1.414214 with Precision 4 returns 0.9545, and 0.5 times that plus 0.5 lands on 0.9772, the share of a normal distribution within two standard deviations of the mean. In Inverse error function mode, a Value of 0.5 returns 0.4769, and multiplying by sqrt(2) gives 0.6745, the deviate marking the 75th percentile. Diffusion modeling leans on the same pair of moves: given a concentration profile you evaluate erf to predict penetration depth, then use the inverse to solve for the time or distance hitting a threshold.
Frequently Asked Questions
What values can erf output?
The error function stays bounded between -1 and 1, approaching those limits asymptotically. It equals 0 at x = 0 and climbs smoothly toward 1 for large positive inputs.
Where does the formula come from?
It is defined as 2 over sqrt(pi) times the integral from 0 to x of e^(-t^2) dt. The constant is tuned so erf of infinity equals exactly 1.
Why does statistics care about erf?
Because the normal CDF is Phi(z) = 0.5 * (1 + erf(z / sqrt(2))). Every normal probability and critical value ultimately passes through the error function during computation.
Is erf an odd function?
Yes, perfectly symmetric: erf(-x) = -erf(x). That cuts verification work in half, since checking any positive value fixes its matching negative one with just the sign flipped.
How wide is the inverse domain?
The inverse accepts inputs strictly between -1 and 1 only, because erf reaches exactly 1 or -1 only at infinity. Endpoint inputs have no finite answer to return.