/// <summary>
/// Returns the Inverse Hyperbolic Sine
/// <para>Asinh(x) = ln(x + Sqrt(x^2 + 1))</para>
/// </summary>
/// <param name="x">Asinh argument</param>
public static double Asinh(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Asinh(x: {0}): NaNs not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
return(x);
}
// odd function
if (x < 0)
{
return(-Math2.Asinh(-x));
}
// For |x| < eps^(1/4), use a taylor series
// Asinh(x) = z - z^3/6 + 3 * z^5/40 ...
// http://functions.wolfram.com/ElementaryFunctions/ArcSinh/06/01/03/01/
//
// otherwise use varying forms of
// Ln(x + Sqrt(x * x + 1))
// http://functions.wolfram.com/ElementaryFunctions/ArcSinh/02/
if (x < 0.75)
{
if (x < DoubleLimits.RootMachineEpsilon._4)
{
return(x * (1.0 - x * x / 6));
}
// rearrangment of log(x + sqrt(x^2 + 1)) to preserve digits:
return(Math2.Log1p(x + Math2.Sqrt1pm1(x * x)));
}
// for large x, result = Math.Log(2*x), but watch for overflows
if (x > 1 / DoubleLimits.RootMachineEpsilon._2)
{
return(Constants.Ln2 + Math.Log(x));
}
return(Math.Log(x + Math.Sqrt(x * x + 1)));
}