Error Function Calculator

What does this calculator do?

Enter a value y and a choice of function, and this tool solves for x in either erf(x) = y (the inverse error function, erf⁻¹) or erfc(x) = y (the inverse complementary error function, erfc⁻¹). It's the reverse direction of the more familiar error function: instead of plugging in x to get erf(x)/erfc(x), you supply the output and solve for the input.

How to use it

1. Choose whether you're solving erf(x) = y or erfc(x) = y.
2. Enter y — strictly between -1 and 1 for erf⁻¹, or strictly between 0 and 2 for erfc⁻¹ (those are the only values erf/erfc ever actually produce).
3. Read x from the result table, along with a round-trip check row that plugs x back into the forward function so you can confirm it lands back on y.

The math

erf(x) and erfc(x) have no closed-form inverse, so both are computed via the exact relationship between the error function and the standard normal distribution's quantile function Φ⁻¹ (the "probit" function):

Φ(z) = ½ · (1 + erf(z / √2))

Solving for x when erf(x) = y gives:

erf⁻¹(y) = Φ⁻¹((y + 1) / 2) / √2

and since erfc(x) = 1 − erf(x):

erfc⁻¹(y) = erf⁻¹(1 − y)

Φ⁻¹ itself is evaluated with Peter Acklam's rational approximation, accurate to a relative error under 1.15×10⁻⁹ across the full (0, 1) domain — including the extreme tails, where a lower-precision approach would lose digits.

Relation to the forward functions

If you need erf(x)/erfc(x) directly — plugging in x to get the function's value, rather than solving for x — see the Complementary Error Function Calculator, which covers that forward direction.

Example use cases

• Probability and statistics — finding the z-score (or an erf-based equivalent) that corresponds to a given tail probability, e.g. working backward from a confidence level to a critical value.
• Diffusion and heat transfer — solving for the depth or time in a semi-infinite diffusion/conduction problem when the concentration or temperature ratio (expressed via erfc) is already known.
• Signal processing — recovering the argument of a Q-function or bit-error-rate formula when the error rate itself is the known quantity.

Privacy

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

Related tools

For the forward direction — computing erf(x) and erfc(x) directly from x — see the Complementary Error Function Calculator. For general descriptive statistics, see the Statistics Calculator.