# NAG FL Interfaces11abf (arcsinh)

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## 1Purpose

s11abf returns the value of the inverse hyperbolic sine, $\mathrm{arcsinh}x$, via the function name.

## 2Specification

Fortran Interface
 Function s11abf ( x,
 Real (Kind=nag_wp) :: s11abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s11abf_ (const double *x, Integer *ifail)
The routine may be called by the names s11abf or nagf_specfun_arcsinh.

## 3Description

s11abf calculates an approximate value for the inverse hyperbolic sine of its argument, $\mathrm{arcsinh}x$.
For $|x|\le 1$ it is based on the Chebyshev expansion
 $arcsinh⁡x=x×y(t)=x∑′r=0crTr(t), where ​t=2x2-1.$
For $|x|>1$ it uses the fact that
 $arcsinh⁡x=sign⁡x×ln(|x|+x2+1) .$
This form is used directly for $1<|x|<{10}^{k}$, 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 $|x|$ to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate
 $arcsinh⁡x=sign⁡x×(ln⁡2+ln|x|).$
NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

None.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and the result, respectively, then in principle
 $|ε|≃ | x 1+x2 arcsinh⁡x δ| .$
That is, the relative error in the argument, $x$, is amplified by a factor at least $\frac{x}{\sqrt{1+{x}^{2}}\mathrm{arcsinh}x}$, in the result.
The equality should hold if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is simply due to round-off in the machine representation it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the amplification factor is shown in the following graph: Figure 1
It should be noted that this factor is always less than or equal to one. For large $x$ we have the absolute error in the result, $E$, in principle, given by
 $E∼δ.$
This means that eventually accuracy is limited by machine precision.

## 8Parallelism and Performance

s11abf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s11abfe.f90)

### 10.2Program Data

Program Data (s11abfe.d)

### 10.3Program Results

Program Results (s11abfe.r)