...one of the most highly
regarded and expertly designed C++ library projects in the
world. — Herb Sutter and Andrei
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
the hyperbolic cosine
function, at x. Values returned are positive.
If x is in the range
]-∞;+1[ then returns the result of
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:
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: