Error Function Calculator

Using the error function calculator

Pick one of the two modes with the Calculate choice.

• Error function: type any real number x. The calculator prints erf(x) and the complementary erfc(x).
• Inverse error function: type a number y from -1 to 1. It prints erf⁻¹(y), the x for which erf(x) = y, and erfc⁻¹(y).

The Precision field sets how many decimals are printed, from 2 to 25 (14 by default). Whole numbers are printed without decimals, and a value that needs only one or two decimals is shown with exactly two.

Everything is computed in ordinary double-precision arithmetic. erf and erfc come from rational Chebyshev approximations whose relative error is about 1e-16, and erfc is evaluated directly instead of as 1 − erf, so it keeps its leading digits far out in the tail. A computer stores only about 15 to 16 significant digits, so the calculator never prints more than 15, even when the Precision box asks for 25 decimals; the last printed digit can still be one unit off. Use an arbitrary-precision library if you need more.

What is the error function?

The error function, also called the Gauss error function, measures how much of a bell-shaped curve lies within a symmetric window around its centre. It first appeared in the theory of measurement errors, hence the name. It is S-shaped, odd (erf(−x) = −erf(x)), equals 0 at x = 0 and tends to 1 as x grows. By x = 3 it is within 0.00003 of 1.

Its most common use is as a rescaled normal distribution. Take a normally distributed variable centred on 0 with spread σ (its standard deviation): the chance that it lands between −a and a is erf(a ÷ (σ√2)). Because of that link the error function underlies p-values, confidence intervals and the normal distribution calculator, and it shows up in diffusion and heat-conduction problems as well.

Error function formula

erf(x) = (2 ÷ √π) × ∫ from 0 to x of e^(−t²) dtThe area under the Gaussian curve from 0 to x, scaled so that the total area to infinity is 1.

The integral has no closed form, but it can be written as an alternating power series that converges for every x:

erf(x) = (2 ÷ √π) × ( x − x³ ÷ 3 + x⁵ ÷ 10 − x⁷ ÷ 42 + x⁹ ÷ 216 − … )The terms are (−1)ⁿ x^(2n+1) ÷ (n! (2n+1)).

Example with x = 0.5. The first five terms are 0.5, −0.041667, +0.003125, −0.000186 and +0.000009, which sum to 0.461281. Multiplying by 2 ÷ √π = 1.128379 gives 0.52050, in agreement with the calculator's 0.52049988.

Inverse error function

The inverse error function undoes erf: if y = erf(x) then x = erf⁻¹(y). It is defined for −1 < y < 1 and grows without bound as y approaches ±1. There is no simple formula, so numerical root-finding is used: start from a rough guess and refine it with Newton or Halley steps until erf(x) matches y. The inverse complementary function satisfies erfc⁻¹(y) = erf⁻¹(1 − y) for y between 0 and 2.

In the calculator the second inverse value is shown for the entered y as erf⁻¹(1 − |y|), so erfc⁻¹(0.95) equals erf⁻¹(0.05). At the ends of the range the equation has no finite solution, and the calculator prints ∞ (or −∞) for erf⁻¹(1) and erf⁻¹(−1), and ∞ for erfc⁻¹(0).

Error function table

Values of erf and erfc to six decimals, computed from the exact function:

Worked examples

From erf to a normal probability. What is the chance that a standard normal variable is below 1.5? Compute z ÷ √2 = 1.5 ÷ 1.41421 = 1.06066, look up erf(1.06066) = 0.866386, and apply Φ(z) = ½ (1 + erf(z ÷ √2)): Φ(1.5) = ½ × 1.866386 = 0.933193.

From erf⁻¹ to a quantile. The 97.5th percentile of the standard normal is z = √2 × erf⁻¹(2p − 1) with p = 0.975, so z = 1.41421 × erf⁻¹(0.95). Entering y = 0.95 gives erf⁻¹(0.95) ≈ 1.38590, hence z ≈ 1.41421 × 1.38590 = 1.95996, the familiar 1.96 behind 95% confidence intervals. Quantiles for other confidence levels are in the z-score calculator and the p-value calculator.

Accuracy and common mistakes

• Asking for more digits than exist. Setting the precision to 25 does not make the answer 25 digits correct. A double-precision number holds about 15 significant digits, so that is where the calculator stops printing.
• Mixing up erf and the normal CDF. erf(1) is 0.8427, but the probability of a standard normal below 1 is 0.8413. They differ by the √2 rescaling.
• Large x. erf(x) rounds to exactly 1 for x above about 5.9 in double precision, which is why erfc exists. erfc(6) is 2.15 × 10⁻¹⁷ and the calculator shows it as soon as the Precision box has room for 17 decimals; a value below 10⁻²⁵ would print as 0 whatever the accuracy.
• Inputs outside the domain. The inverse needs y between -1 and 1; anything else is rejected.