...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

#include <boost/math/special_functions/atanh.hpp>

template<class T>calculated-result-typeatanh(const T x); template<class T, class Policy>calculated-result-typeatanh(const T x, const Policy&);

Computes the reciprocal of the hyperbolic tangent function, at x.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

If x is in the range `]-∞;-1[`

or in the range `]+1;+∞[`

then returns the result of domain_error.

If x is in the range `[-1;-1+ε[`

, then the result of -overflow_error is returned, with ε
denoting
numeric_limits<T>::epsilon().

If x is in the range `]+1-ε;+1]`

, then the result of overflow_error is returned, with ε
denoting
numeric_limits<T>::epsilon().

The return type of this function is computed using the *result
type calculation rules*: the return type is `double`

when T is an integer type, and T
otherwise.

Generally accuracy is to within 1 or 2 epsilon across all supported platforms.

This function is tested using a combination of random test values designed to give full function coverage computed at high precision using the "naive" formula:

along with a selection of sanity check values computed using functions.wolfram.com to at least 50 decimal digits.

For sufficiently small x we can use the approximation:

Otherwise the primary definition:

or it's equivalent form:

is used.