﻿ s11ac Method
s11ac returns the value of the inverse hyperbolic cosine, $\mathrm{arccosh} x$. The result is in the principal positive branch.

# Syntax

C#
```public static double s11ac(
double x,
out int ifail
)```
Visual Basic
```Public Shared Function s11ac ( _
x As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s11ac(
double x,
[OutAttribute] int% ifail
)```
F#
```static member s11ac :
x : float *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}\ge 1.0$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s11ac returns the value of the inverse hyperbolic cosine, $\mathrm{arccosh} x$. The result is in the principal positive branch.

# Description

s11ac calculates an approximate value for the inverse hyperbolic cosine, $\mathrm{arccosh} x$. It is based on the relation
 $arccosh x=lnx+x2-1.$
This form is used directly for $1, where $k=n/2+1$, and the machine uses approximately $n$ decimal place arithmetic.
For $x\ge {10}^{k}$, $\sqrt{{x}^{2}-1}$ is equal to $\sqrt{x}$ to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate
 $arccosh x=ln 2+ln x.$

# References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
The method has been called with an argument less than $1.0$, for which $\mathrm{arccosh} x$ is not defined. The result returned is zero.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result respectively, then in principle
 $ε≃xx2-1arccosh x×δ.$
That is the relative error in the argument is amplified by a factor at least $\frac{x}{\sqrt{{x}^{2}-1}\mathrm{arccosh} x}$ in the result. The equality should apply if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is simply a result of round-off in the machine representation it is possible that an extra figure may be lost in internal calculation and round-off. The behaviour of the amplification factor is shown in the following graph:
Figure 1
It should be noted that for $x>2$ the factor is always less than $1.0$. For large $x$ we have the absolute error $E$ in the result, in principle, given by
 $E∼δ.$
This means that eventually accuracy is limited by machine precision. More significantly for $x$ close to $1$, $x-1\sim \delta$, the above analysis becomes inapplicable due to the fact that both function and argument are bounded, $x\ge 1$, $\mathrm{arccosh} x\ge 0$. In this region we have
 $E∼δ.$
That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

None.

None.

# Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

Example program (C#): s11ace.cs

Example program data: s11ace.d

Example program results: s11ace.r