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

template<class T>
calculated-result-type acosh(const T x);

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

Computes the reciprocal of (the restriction to the range of [0;+∞[) the hyperbolic cosine function, at x. Values returned are positive.

If x is in the range ]-∞;+1[ then returns the result of domain_error.

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.

Accuracy

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

Testing

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.

Implementation

For sufficiently large x, we can use the approximation:

For x sufficiently close to 1 we can use the approximation:

Otherwise for x close to 1 we can use the following rearrangement of the primary definition to preserve accuracy:

Otherwise the primary definition is used: