Error Function Calculator

Background.

The bell curve has no tidy antiderivative. Integrate e^(−t²) from zero up to some point x and no combination of polynomials, exponentials or logarithms gives the answer, so the result was given a name of its own. Scaled by 2/√π so that it runs from −1 to 1, it is the error function, erf(x).

Its home ground is probability. A normal distribution's cumulative probability is Φ(z) = ½[1 + erf(z/√2)], and the chance of a value landing within k standard deviations of the mean is simply erf(k/√2). Enter 1/√2 (0.7071) and the page returns 0.6826894723, the familiar 68.27%; enter √2 and it returns 0.9544998742, the two-sigma 95.45%. The same function in its complementary form, erfc = 1 − erf, describes how heat or a dissolved substance spreads into a thick body from a surface held at a fixed level, so it turns up in heat-transfer and diffusion work as often as in statistics.

The curve is easy to picture. It is odd, so erf(−x) = −erf(x). It climbs through the origin with slope 2/√π ≈ 1.1284 and then flattens quickly toward ±1: by x = 2 it has reached 0.9953, by x = 3 0.99998, and from about 5 onward the page simply shows 1.

The value comes from Abramowitz and Stegun's formula 7.1.26, a five-term rational approximation whose error never exceeds 1.5 × 10⁻⁷. That is ample for probabilities and engineering, but the display carries ten decimal places, so treat everything from the seventh decimal place on as approximate. Inputs are real numbers from −8 to 8; the complex error function and the inverse erf are not computed.

What is error function calculator?

The error function is erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt: the area under the curve e^(−t²) from 0 to x, scaled so that erf(x) approaches 1 as x grows without bound. It is an odd function with values between −1 and 1 and erf(0) = 0. Its complement is erfc(x) = 1 − erf(x). In probability terms, erf(x) is the chance that a normally distributed variable with mean 0 and variance ½ falls between −x and x.

How to use this calculator.

1. 01Enter x, any real number from −8 to 8. Negative inputs are fine and return the negative of erf(|x|).
2. 02For a normal-distribution question, convert first. The probability within k standard deviations of the mean is erf(k/√2), so enter k × 0.70711. For the cumulative probability Φ(z), enter z/√2 and then take ½ × (1 + result).
3. 03For erfc(x), the form used in diffusion and heat-conduction solutions, subtract the result from 1. Beyond about x = 3 that subtraction keeps only a few correct digits, because erfc is tiny there.
4. 04For |x| below about 0.001, the approximation's small fixed error becomes a large share of the answer; 2x/√π, about 1.12838 × x, is the more accurate figure there.

The formula.

The defining integral has no closed form, so the page uses approximation 7.1.26 from Abramowitz and Stegun's Handbook of Mathematical Functions. For x ≥ 0 it sets t = 1 ÷ (1 + 0.3275911x) and computes erf(x) ≈ 1 − (a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵) × e^(−x²), with a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027 and a₅ = 1.061405429. Negative x is handled through the symmetry erf(−x) = −erf(x). The Gaussian factor e^(−x²) carries most of the shape and the polynomial in t corrects it. The published error bound, 1.5 × 10⁻⁷ for all x ≥ 0, is absolute, so it matters most where erf itself is small, close to the origin. The five coefficients add up to 0.999999999 rather than 1, which is why x = 0 is treated separately and returned as exactly 0.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.