# Documentation

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# `inverf`

Inverse of the error function

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

## Syntax

```inverf(`x`)
```

## Description

`inverf(x)` computes the inverse of the error function. This function is defined for all complex arguments `x`.

The inverse function `inverf` is singular at the points `x = -1` and `x = 1`.

The inverses of the error functions return floating-point values only for floating-point arguments that belong to a particular interval. Thus, the inverse of the error function `inverf(x)` returns floating-point values for real values `x` from the interval `[-1, 1]`. The implemented exact values are: `inverf(-1) = -∞`, ```inverf(0) = 0```, `inverf(1) = ∞`. For all other arguments, the error functions return symbolic function calls.

MuPAD® can simplify expressions that contain error functions and their inverses. For real values `x`, the system applies the following simplification rules:

• ```inverf(erf(x)) = inverf(1 - erfc(x)) = inverfc(1 - erf(x)) = inverfc(erfc(x)) = x```

• ```inverf(-erf(x)) = inverf(erfc(x) - 1) = inverfc(1 + erf(x)) = inverfc(2 - erfc(x)) = -x```

For any value `x`, the system applies the following simplification rules:

• `inverf(-x) = -inverf(x)`

• `inverfc(2 - x) = -inverfc(x)`

• `erf(inverf(x)) = erfc(inverfc(x)) = x`

• `erf(inverfc(x)) = erfc(inverf(x)) = 1 - x`

## Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable `DIGITS`, which determines the numerical working precision.

## Examples

### Example 1

You can call the inverse of the error function with exact and symbolic arguments:

`inverf(-1), inverf(0), inverf(1), inverf(x + 1), inverf(1/5), inverf(1/sqrt(2))`

To approximate exact results with floating-point numbers, use `float`:

`float(inverf(1/5)), float(inverf(1/sqrt(2)))`

Alternatively, use floating-points value as arguments:

`inverf(0.2), inverf(1/sqrt(2.0))`

For floating-point arguments `x` from the interval ```[-1, 1]```, `inverf` returns floating-point values:

`inverf(-0.5), inverf(0.85)`

For floating-point arguments outside of this interval, `inverf` returns symbolic function calls:

`inverf(-5.3), inverf(10.0)`

### Example 2

`diff`, `float`, `limit`, `rewrite`, `series`, and other functions handle expressions involving the inverse of the error function:

`diff(inverf(x), x)`

`float(ln(3 + inverf(1/sqrt(PI))))`

```limit(1/inverf(x), x = 0, Right); limit(1/inverf(x), x = 0, Left)```

`rewrite(inverfc(x), inverf)`

`series(inverf(x), x = 0)`

## Parameters

 `x` Arithmetical expression

## Return Values

Arithmetical expression