# NAG FL Interfaces10aaf (tanh)

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## 1Purpose

s10aaf returns a value for the hyperbolic tangent, $\mathrm{tanh}x$, via the function name.

## 2Specification

Fortran Interface
 Function s10aaf ( x,
 Real (Kind=nag_wp) :: s10aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s10aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s10aaf or nagf_specfun_tanh.

## 3Description

s10aaf calculates an approximate value for the hyperbolic tangent of its argument, $\mathrm{tanh}x$.
For $|x|\le 1$ it is based on the Chebyshev expansion
 $tanh⁡x=x×y(t)=x∑′r=0arTr(t)$
where $-1\le x\le 1\text{, }-1\le t\le 1\text{, and }t=2{x}^{2}-1$.
For $1<|x|<{E}_{1}$ (see the Users' Note for your implementation for value of ${E}_{1}$)
 $tanh⁡x=e2x-1 e2x+1 .$
For $|x|\ge {E}_{1}$, $\mathrm{tanh}x=\mathrm{sign}x$ to within the representation accuracy of the machine and so this approximation is used.

## 4References

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,
 $|ε|≃ | 2x sinh⁡2x δ| .$
That is, a relative error in the argument, $x$, is amplified by a factor approximately $\frac{2x}{\mathrm{sinh}2x}$, 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 due simply to the 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:
It should be noted that this factor is always less than or equal to $1.0$ and away from $x=0$ the accuracy will eventually be limited entirely by the precision of machine representation.

## 8Parallelism and Performance

s10aaf 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 (s10aafe.f90)

### 10.2Program Data

Program Data (s10aafe.d)

### 10.3Program Results

Program Results (s10aafe.r)