# Error Function Calculator

URL: https://speedycalculator.com/calculators/math/error-function/
Last updated: 2026-10-30

> The error function calculator evaluates erf(x), the complementary error function erfc(x) and their inverses, which appear throughout probability, statistics and diffusion problems.

## Inputs

- Choose the function
- Input x

## Results

- Result
- erf(x)
- erfc(x)

## Formula

```
erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt
erfc(x) = 1 − erf(x)
erfc⁻¹(x) = erf⁻¹(1 − x)
```

## Explanation

The error function measures the probability that a normally distributed error lies within a given range – hence its name. It also solves the heat equation for a sudden temperature change and describes diffusion of particles.

### How to use the Error Function Calculator

- **Choose the function**
- **Input x**

Results update as you type, and everything is calculated in your browser.

### How it works

erf is computed from the standard normal distribution, erf(x) = 2Φ(x√2) − 1, using a double-precision algorithm. The inverses use an accurate inverse normal function.

### Inputs explained

FieldWhat to enterChoose the functionChoose one: Error function erf(x), Complementary error function erfc(x), Inverse error function erf⁻¹(x), Inverse complementary erfc⁻¹(x)Input xA number

### What each result means

- **erf(x)** = erf([Input x])
- **erfc(x)** = erfc([Input x])

### Example values

How the main result changes when **Input x** changes and everything else stays at the example values above:

Input xResult0.250.27630.3750.40410.50.52050.6250.62320.750.711210.8427

## Worked example

Inputs — Choose the function: Error function erf(x); Input x: 0.5.

Results — **Result = 0.5204998778**; erf(x) = 0.5204998778; erfc(x) = 0.4795001222.

## Key terms

- **Error function**: erf(x) = (2/√π)∫₀ˣ e^(−t²) dt.
- **Complementary error function**: erfc(x) = 1 − erf(x).

## FAQ

**What is erf(1)?**
About 0.8427: 84.27% of a normal distribution with σ = 1/√2 lies within ±1.

**What is the domain of erf⁻¹?**
−1 < x < 1. For erfc⁻¹ it is 0 < x < 2.

**How is erf related to the normal distribution?**
Φ(z) = ½(1 + erf(z/√2)).

**Is erf odd?**
Yes: erf(−x) = −erf(x).


## Sources

- [NIST DLMF – Chapter 7: Error functions](https://dlmf.nist.gov/7)
- [Wolfram MathWorld – Erf](https://mathworld.wolfram.com/Erf.html)