Error Function Calculator — erf(x) and erfc(x)
Enter any real number x to compute the Gaussian error function erf(x) and the complementary error function erfc(x). The curve plot shows how erf varies from −1 to +1 across its full range.
Gaussian error function — ranges from −1 to +1
How does this calculator work?
erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt, ranging from −1 to +1. erfc(x) = 1 − erf(x). Normal probability P(−z ≤ Z ≤ z) = erf(z/√2). Computed here via the Abramowitz & Stegun A&S 7.1.26 polynomial approximation (error < 1.5 × 10⁻⁷). The curve plot shows erf across [−3, 3].
Formula
How this is calculated
The Gaussian error function erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt appears throughout probability theory, statistics, and the physical sciences. Its most direct connection is to the normal distribution: the probability that a standard normal variable falls in (−z, z) equals erf(z/√2). It is also fundamental to the solutions of the diffusion equation (heat conduction, Brownian motion) and appears in digital communications and signal processing.
The function is odd (erf(−x) = −erf(x)), equals 0 at x = 0, and approaches ±1 asymptotically as x → ±∞. For practical purposes erf(x) = ±1 to 8 decimal places beyond |x| ≈ 3.3. The complementary error function erfc(x) = 1 − erf(x) is numerically important for large positive x because computing 1 − erf(x) directly loses precision; the scaled form erfc(x) × e^(x²) stays well-behaved even as erfc itself approaches zero.
This calculator uses the Abramowitz & Stegun polynomial approximation (7.1.26), which gives a maximum absolute error of 1.5 × 10⁻⁷. For higher precision (more than 7 significant digits), arbitrary-precision libraries such as MPFR or Boost.Math implement the Lanczos approximation or Chebyshev series.
Frequently asked questions
For a standard normal random variable Z, P(−z ≤ Z ≤ z) = erf(z / √2). The cumulative distribution function Φ(x) = P(Z ≤ x) = (1 + erf(x / √2)) / 2. So you can convert any erf value to a normal-distribution probability with a simple rescaling.
erfinv(p) is the value x such that erf(x) = p. It is used to find z-scores: erfinv(0.95) ≈ 1.386, meaning erf(1.386) ≈ 0.95. Many maths libraries expose erfinv directly; it can also be approximated by Newton iteration starting from a rational approximation. This calculator currently shows erf and erfc but not the inverse — use a statistics package for erfinv.
For large x, erfc(x) becomes very small very fast while e^(x²) grows very fast; their product erfcx(x) = erfc(x) × e^(x²) remains a well-conditioned quantity of order 1/x√π. This is the form used in numerical computations when evaluating erfc for large x to avoid catastrophic cancellation. It is useful in statistics (extreme tails) and in physics (plasma dispersion functions).
Also known as
TG we-Calculate Editorial Team. (2026). Error Function Calculator — erf(x) and erfc(x) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/error-function-calculator
TG we-Calculate Editorial Team. "Error Function Calculator — erf(x) and erfc(x)." TG we-Calculate. 2026. https://we-calculate.com/calculator/error-function-calculator.
TG we-Calculate Editorial Team, "Error Function Calculator — erf(x) and erfc(x)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/error-function-calculator
@misc{wecalculate_error_function_calculator, title = {Error Function Calculator — erf(x) and erfc(x)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/error-function-calculator}}, year = {2026}, note = {TG we-Calculate} }
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