# atanh() Function & Examples

Use the atanh() trigonometry function in Calcul.io. Review its syntax, edit working examples, understand the result, and explore related math functions.

## atanh

#### atanh(x)

Try it yourself:

```calculio
atanh(0.5)
```

[acosh](https://calcul.io/function/acosh/index.md)

[asinh](https://calcul.io/function/asinh/index.md)

---

## atanh(): the inverse hyperbolic tangent

**atanh(x)** returns the inverse hyperbolic tangent of *x*. It answers the question: which value *y* has `tanh(y) = x`? It is useful when rearranging models involving hyperbolic curves, logarithms, or transformed correlation-like values.

## Formula and valid inputs

For real numbers, `atanh(x) = 0.5 · ln((1 + x)/(1 − x))`. Its real domain is strictly between −1 and 1. For example, `atanh(0)` is 0 and `atanh(0.5)` is about 0.5493. As x approaches 1 or −1, the result grows without bound; inputs at or beyond those endpoints need complex-number handling or are not real-valued.

## Working with hyperbolic functions

The natural companion is [tanh()](https://calcul.io/function/tanh/index.md): applying it to a valid atanh result recovers the original input. The related [asinh()](https://calcul.io/function/asinh/index.md) and [acosh()](https://calcul.io/function/acosh/index.md) functions invert the other hyperbolic functions. When you need the logarithmic form explicitly, use [log()](https://calcul.io/function/log/index.md); do not confuse atanh with the ordinary inverse tangent [atan()](https://calcul.io/function/atan/index.md).

## Practical care

Keep full precision until the end of a calculation, particularly near ±1. Check that a measurement or normalized value is in range before calling atanh, and remember that rounding can accidentally turn 0.999999999 into 1. Use [abs()](https://calcul.io/function/abs/index.md) to test distance from the boundary when validating data.

[All functions](https://calcul.io/functions/index.md)