Error Function Calculator (erf)

Enter any real number into the x value field and this Error Function Calculator (erf) instantly computes erf(x), the complementary error function erfc(x), the inverse error function erf⁻¹(y), and inverse complementary error function erfc⁻¹(y). Switch between forward and inverse calculation modes using the function type selector. You can also set the desired decimal precision for all outputs. Also try the Gamma Function Calculator.

Function Type *

Choose whether to compute the error function or its inverse.

Input value x for erf(x) and erfc(x). Values beyond ±4 approach ±1 asymptotically.

Input value y for erf⁻¹(y). Must be strictly between -1 and 1.

Number of decimal places shown in results.

Results

erf(x)

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erfc(x) = 1 − erf(x)

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erf⁻¹(y)

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erfc⁻¹(y)

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Results Table

Frequently Asked Questions

What is the error function (erf)?

The error function, often abbreviated erf and also called the Gaussian error function, is a special mathematical function defined as erf(x) = (2/√π) ∫₀ˣ exp(−t²) dt. It arises naturally in probability, statistics, and physics — particularly in solutions to the heat equation and in the cumulative distribution function of the normal distribution. Its values range from −1 to 1. See also our use the Euler's Number Calculator.

What is the complementary error function erfc(x)?

The complementary error function is simply erfc(x) = 1 − erf(x). It represents the probability that a normally distributed random variable falls outside a given range. Because erf(x) approaches 1 as x → ∞, erfc(x) approaches 0 for large positive x.

What is the inverse error function erf⁻¹(y)?

The inverse error function erf⁻¹(y) returns the value x such that erf(x) = y, where y must be in the open interval (−1, 1). It is widely used in statistics to convert probabilities back into standard deviation units and in signal processing applications.

How do I calculate erf by hand?

Exact closed-form calculation is not possible, but erf(x) can be approximated using its Taylor series: erf(x) ≈ (2/√π)(x − x³/3 + x⁵/10 − x⁷/42 + ...). For small x this converges quickly, but for large x numerical methods or lookup tables are more practical. This calculator uses a high-accuracy polynomial approximation (Abramowitz & Stegun). You might also find our calculate Shortest Path Cost, Optimal Path & Number of Hops — Dijkstra Shortest Path useful.

What values can x take in erf(x)?

The error function is defined for all real numbers x, from −∞ to +∞. However, for |x| > 4, erf(x) is essentially ±1 (within about 10⁻⁸ of the limit), so values beyond that range have diminishing practical difference.

How is the error function related to the normal distribution?

The cumulative distribution function (CDF) of the standard normal distribution Φ(x) is directly related to erf: Φ(x) = (1 + erf(x/√2)) / 2. This means you can convert between normal distribution probabilities and erf values using a simple formula.

What precision does this calculator use?

The calculator uses a well-known rational polynomial approximation (from Abramowitz & Stegun, formula 7.1.26) which provides accuracy to approximately 1.5 × 10⁻⁷. You can display results with 2 to 15 decimal places using the Decimal Precision field.

Where is the error function used in real life?

The error function appears in heat transfer (diffusion equations), signal processing (Gaussian noise analysis), statistics (normal distribution probabilities), finance (option pricing models), and physics (quantum mechanics and electromagnetic diffusion). Any process modeled by a Gaussian distribution will involve erf somewhere in its solution.