# erfcinv

Inverse complementary error function

## Description

example

````erfcinv(x)` returns the value of the Inverse Complementary Error Function for each element of `x`. For inputs outside the interval `[0 2]`, `erfcinv` returns `NaN`. Use the `erfcinv` function to replace expressions containing `erfinv(1-x)` for greater accuracy when `x` is close to `1`.```

## Examples

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### Find Inverse Complementary Error Function

```erfcinv(0.3) ```
```ans = 0.7329 ```

Find the inverse complementary error function of the elements of a vector.

```V = [-10 0 0.5 1.3 2 Inf]; erfcinv(V) ```
```ans = NaN Inf 0.4769 -0.2725 -Inf NaN ```

Find the inverse complementary error function of the elements of a matrix.

```M = [0.1 1.2; 1 0.9]; erfcinv(M) ```
```ans = 1.1631 -0.1791 0 0.0889 ```

### Avoid Roundoff Errors Using Inverse Complementary Error Function

You can use the inverse complementary error function `erfcinv` in place of `erfinv(1-x)` to avoid roundoff errors when `x` is close to `0`.

Show how to avoid roundoff by calculating `erfinv(1-x)` using `erfcinv(x)` for `x = 1e-100`. The original calculation returns `Inf` while `erfcinv(x)` returns the correct result.

```x = 1e-100; erfinv(1-x) erfcinv(x) ```
```ans = Inf ans = 15.0656 ```

## Input Arguments

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### `x` — Inputreal number | vector of real numbers | matrix of real numbers | multidimensional array of real numbers

Input, specified as a real number, or a vector, matrix, or multidimensional array of real numbers. `x` cannot be sparse.

Data Types: `single` | `double`

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### Inverse Complementary Error Function

The inverse complementary error function `erfcinv(x)` is defined as $\text{erfcinv}\left(\text{erfc}\left(x\right)\right)=x.$

### Tips

• You can also find the inverse standard normal probability distribution using the Statistics and Machine Learning Toolbox™ function `norminv`. The relationship between the inverse complementary error function `erfcinv` and `norminv` is

$\text{norminv}\left(p\right)=\left(-\sqrt{2}\right)×\text{erfcinv}\left(2p\right).$

• For expressions of the form `erfcinv(1-x)`, use the inverse error function `erfinv` instead. This substitution maintains accuracy. When `x` is close to `1`, then `1 - x` is a small number and might be rounded down to `0`. Instead, replace `erfcinv(1-x)` with `erfinv(x)`.