About the Error Function

The error function (erf) is a special function in mathematics that describes the probability of a random variable falling within a range in a normal distribution. It's widely used in statistics, physics, and engineering for problems involving diffusion and normal distributions.

How to Use This Calculator

Enter a real number in the input field
Results will show erf(x) and complementary erfc(x) = 1 - erf(x)
The graph shows erf(x) from x = -3 to 3

Examples

erf(0) = 0
erf(0.5) ≈ 0.5205
erf(1) ≈ 0.8427
erf(2) ≈ 0.9953

Reference Table

x	erf(x)	erfc(x)
0	0	1
0.5	0.5205	0.4795
1	0.8427	0.1573
2	0.9953	0.0047
3	0.99998	0.00002

Notes on Approximation

This calculator uses a polynomial approximation (Abramowitz and Stegun, 7.1.26) accurate to within 1.5×10⁻⁷. For extreme values (|x| > 4), consider specialized approximations.

Frequently Asked Questions

What is the error function used for?

The error function is used in probability, statistics, and partial differential equations. It's essential for calculating probabilities in normal distributions and solving heat/diffusion equations.

How accurate is this approximation?

The approximation has a maximum error of 1.5×10⁻⁷ across all real numbers. For most practical purposes, this is sufficient.

What is the relationship between erf and erfc?

erfc(x) = 1 - erf(x). The complementary error function is useful for large x values where erf(x) approaches 1.

Error Function (erf) Calculator

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Error Function erf Calculator - Fast Approximation

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