# NAG FL Interfaces11aaf (arctanh)

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## 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 <nag.h>
 double s11aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s11aaf or nagf_specfun_arctanh.

## 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
 $arctanh⁡x=x×y(t)=x∑′r=0arTr(t)$
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
 $arctanh⁡x=12ln(1+x 1-x ) .$
For $|x|\ge 1$, the routine fails as $\mathrm{arctanh}x$ is undefined.
NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: $|{\mathbf{x}}|<1.0$.
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).

## 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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{x}}|<1$.
The routine has been called with an argument greater than or equal to $1.0$ in magnitude, for which $\mathrm{arctanh}$ is not defined.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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 $|x|=1$. However, in the region where $|x|$ is close to one, $1-|x|\sim \delta$, the above analysis is inapplicable since $x$ is bounded by definition, $|x|<1$. 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 $|x|\sim 1$, 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)