...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/asinh.hpp>

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

Computes the reciprocal of the hyperbolic sine function.

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.

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.

Generally accuracy is to within 1 or 2 machine 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 large x we can use the approximation:

While for very small x we can use the approximation:

For 0.5 > x > ε the following rearrangement of the primary definition is used:

Otherwise evaluation is via the primary definition: