Error Function Calculator: erf(x), erfc(x) and the Inverse Error Function

How it works

erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt · erfc(x) = 1 − erf(x) = (2/√π) ∫ₓ^∞ e^(−t²) dt · erf(−x) = −erf(x) · erfcx(x) = e^(x²)·erfc(x) · Φ(z) = ½[1 + erf(z/√2)] · erf⁻¹(y) = x such that erf(x) = y · erfc⁻¹(y) = erf⁻¹(1 − y)

The error function erf(x) is the area under the bell curve e^(−t²) from 0 to x, scaled so that erf(∞) = 1. It rises from −1 to 1, is odd (erf(−x) = −erf(x)), and erf(x) is the chance that a normal variable with mean 0 and variance ½ lands within ±x. The complementary error function erfc(x) = 1 − erf(x) is the tail beyond x; it appears in heat conduction and diffusion (the temperature or concentration in a semi-infinite solid follows erfc(x / 2√(αt))), in bit-error rates and in the normal distribution, where Φ(z) = ½[1 + erf(z/√2)]. This calculator sums the power series 2/√π·e^(−x²)·Σ 2ⁿx^(2n+1)/(1·3·…·(2n+1)) for |x| ≤ 2.5, where every term is positive so no digits cancel, and switches to the Laplace continued fraction for erfc above 2.5. Working with ln erfc means tiny tails such as erfc(10) = 2.088 × 10⁻⁴⁵ keep all their digits instead of rounding to 0. The inverse functions start from Winitzki’s closed-form approximation and polish it with Newton’s method, and the result is checked by plugging it back in. Use the normal distribution calculator when you have a mean, standard deviation and a value; use this page when a formula is written with erf, erfc or erf⁻¹ directly, as in physics, engineering and signal processing. Each result also shows the derivative, the scaled function erfcx and the matching normal z and Φ(z).

Worked example

With the default x = 0.5: x² = 0.25 and e^(−0.25) = 0.77880078. The series gives erf(0.5) = 1.1283792 × 0.77880078 × (0.5 + 0.08333333 + 0.00833333 + …) = 0.5204998778, so erfc(0.5) = 1 − 0.5204998778 = 0.4795001222. The derivative is (2/√π)·e^(−0.25) = 0.8787825789, erfcx(0.5) = e^(0.25) × 0.4795001222 = 0.6156903442, and the matching normal value is z = 0.5 × √2 = 0.7071067812 with Φ(z) = (1 + 0.5204998778)/2 = 0.7602499389.

Sources: NIST DLMF §7.2: Definitions of erf, erfc and their limits · Omni Calculator: Error function (modes and table of erf values) · Analyze Math: Error function calculator (erf and the normal CDF)

Frequently asked questions