# NAG Library Routine Document

## 1Purpose

s11aaf returns the value of the inverse hyperbolic tangent, $\mathrm{arctanh}x$, via the function name.

## 2Specification

Fortran Interface
 Function s11aaf ( x,
 Real (Kind=nag_wp) :: s11aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s11aaf_ (const double *x, Integer *ifail)

## 3Description

s11aaf calculates an approximate value for the inverse hyperbolic tangent of its argument, $\mathrm{arctanh}x$.
For ${x}^{2}\le \frac{1}{2}$ it is based on the Chebyshev expansion
 $arctanh⁡x=x×yt=x∑′r=0arTrt$
where $-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$, $\text{ }-1\le t\le 1$, $\text{ and }t=4{x}^{2}-1$.
For $\frac{1}{2}<{x}^{2}<1$, it uses
 $arctanh⁡x=12ln1+x 1-x .$
For $\left|x\right|\ge 1$, the routine fails as $\mathrm{arctanh}x$ is undefined.

## 4References

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

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: $\left|{\mathbf{x}}\right|<1.0$.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The routine has been called with an argument greater than or equal to $1.0$ in magnitude, for which $\mathrm{arctanh}$ is not defined. On soft failure, the result is returned as zero.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃ x 1-x2 arctanh⁡x ×δ .$
That is, the relative error in the argument, $x$, is amplified by at least a factor $\frac{x}{\left(1-{x}^{2}\right)\mathrm{arctanh}x}$ 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 then 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
The factor is not significantly greater than one except for arguments close to $\left|x\right|=1$. However in the region where $\left|x\right|$ is close to one, $1-\left|x\right|\sim \delta$, the above analysis is inapplicable since $x$ is bounded by definition, $\left|x\right|<1$. In this region where arctanh is tending to infinity we have
 $ε∼1/ln⁡δ$
which implies an obvious, unavoidable serious loss of accuracy near $\left|x\right|\sim 1$, e.g., if $x$ and $1$ agree to $6$ significant figures, the result for $\mathrm{arctanh}x$ would be correct to at most about one figure.

## 8Parallelism and Performance

s11aaf 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 (s11aafe.f90)

### 10.2Program Data

Program Data (s11aafe.d)

### 10.3Program Results

Program Results (s11aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017