Inverse Error Function Calculator

Inverse Error Function:

\[ \text{erfinv}(y) \text{ such that } \text{erf}(\text{erfinv}(y)) = y \]

Inverse Error Function (erfinv):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Is the Inverse Error Function?

The inverse error function (erfinv) is the inverse of the error function (erf). For a given y between -1 and 1, erfinv(y) returns the value x such that erf(x) = y. It's commonly used in probability, statistics, and partial differential equations.

2. How Does the Calculator Work?

The calculator uses an approximation formula for the inverse error function:

\[ \text{erfinv}(y) \approx \text{sign}(y) \sqrt{ \sqrt{ \left( \frac{2}{\pi a} + \frac{\ln(1 - y^2)}{2} \right)^2 - \frac{\ln(1 - y^2)}{a} } - \left( \frac{2}{\pi a} + \frac{\ln(1 - y^2)}{2} \right) } \]

Where:

\( y \) — Input value between -1 and 1
\( a \) — Constant (0.147)
\( \text{sign}(y) \) — Sign of y (1 or -1)

Explanation: This approximation provides good accuracy (within about 0.0001) for most practical purposes.

3. Applications of Inverse Error Function

Details: The inverse error function is used in statistics for transforming probabilities, in communications theory, and in solving certain heat conduction problems.

4. Using the Calculator

Tips: Enter a value between -1 and 1 (exclusive). The function is undefined at y = ±1. The calculator will return NaN for invalid inputs.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between erf and erfinv?
A: They are inverse functions: erf(erfinv(y)) = y and erfinv(erf(x)) = x.

Q2: What's the domain and range of erfinv?
A: Domain is (-1, 1), range is all real numbers.

Q3: How accurate is this approximation?
A: The approximation used here is accurate to about 4 decimal places for most inputs.

Q4: Are there exact formulas for erfinv?
A: No simple closed-form exact formula exists, which is why approximations are used.

Q5: What's the derivative of erfinv?
A: The derivative is \( \frac{d}{dy} \text{erfinv}(y) = \frac{\sqrt{\pi}}{2} e^{[\text{erfinv}(y)]^2} \).