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.
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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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.
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.
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.
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).
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.
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.
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.
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.