Complementary Error Function Calculator

What does this calculator do?

Enter a real number x and this tool computes two closely related values: erf(x), the error function, and erfc(x), the complementary error function, where erfc(x) = 1 − erf(x). Both are computed with an Abramowitz & Stegun rational approximation accurate to about 7-8 significant digits.

How to use it

1. Enter a value for x (between -6 and 6 — beyond that range, erfc(x) is numerically indistinguishable from 0 or 2 at double precision, so the result stops being meaningful).
2. Read erf(x) and erfc(x) from the result table.

The math

The error function is defined by the integral:

erf(x) = (2/√π) · ∫₀ˣ e^(−t²) dt

It has no closed form in terms of elementary functions, so it's evaluated with a numerical approximation — here, the Abramowitz & Stegun 7.1.26 rational approximation, with a maximum error of about 1.5×10⁻⁷ across the whole real line. The complementary error function is simply:

erfc(x) = 1 − erf(x)

Both functions are odd/symmetric around erf: erf(−x) = −erf(x) and erfc(−x) = 2 − erfc(x). At x = 0, erf(0) = 0 and erfc(0) = 1. As x grows, erf(x) approaches 1 (and erfc(x) approaches 0) very quickly — by x = 3, erfc(x) is already under 0.0001.

Relation to the normal distribution

erf and erfc are directly related to the cumulative distribution function (CDF) of the standard normal distribution, Φ(z):

Φ(z) = ½ · erfc(−z / √2) = ½ · (1 + erf(z / √2))

So if you already have a value from this calculator, you can recover a standard-normal probability from it without a separate z-table — useful when working through statistics problems that are stated in terms of erf rather than Φ.

Example use cases

• Probability and statistics — computing tail probabilities of a normally distributed variable, working through problems from textbooks that state results in terms of erf/erfc rather than Φ.
• Heat transfer and diffusion — erfc(x) appears directly in the classic solution for transient heat conduction into a semi-infinite solid, and in Fick's second law solutions for diffusion into a semi-infinite medium (e.g. carburizing, doping profiles).
• Signal processing — bit-error-rate formulas for digital communication channels (e.g. BPSK over an AWGN channel) are commonly expressed using Q(x), which is itself ½ · erfc(x/√2).

Privacy

All calculations run locally in your browser — your input is never sent to a server.

Related tools

For general descriptive statistics — mean, standard deviation, variance — see the Statistics Calculator. For hypothesis testing built on the same normal-CDF math, see the A/B Test Significance Calculator.