# NAG Library Routine Document

## 1Purpose

s10abf returns the value of the hyperbolic sine, $\mathrm{sinh}x$, via the function name.

## 2Specification

Fortran Interface
 Function s10abf ( x,
 Real (Kind=nag_wp) :: s10abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s10abf_ (const double *x, Integer *ifail)

## 3Description

s10abf calculates an approximate value for the hyperbolic sine of its argument, $\mathrm{sinh}x$.
For $\left|x\right|\le 1$ it uses the Chebyshev expansion
 $sinh⁡x=x×yt=x∑′r=0arTrt$
where $t=2{x}^{2}-1$.
For $1<\left|x\right|\le {E}_{1}\text{, }\mathrm{sinh}x=\frac{1}{2}\left({e}^{x}-{e}^{-x}\right)$
where ${E}_{1}$ is a machine-dependent constant, details of which are given in the Users' Note for your implementation.
For $\left|x\right|>{E}_{1}$, the routine fails owing to the danger of setting overflow in calculating ${e}^{x}$. The result returned for such calls is $\mathrm{sinh}\left(\mathrm{sign}x{E}_{1}\right)$, i.e., it returns the result for the nearest valid argument.

## 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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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: $\left|{\mathbf{x}}\right|\le {E}_{1}$.
The routine has been called with an argument too large in absolute magnitude. There is a danger of overflow. The result returned is the value of $\mathrm{sinh}x$ at the closest argument for which a valid call could be made.
${\mathbf{ifail}}=-99$
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
 $ε≃ xcoth⁡x×δ.$
That is the relative error in the argument, $x$, is amplified by a factor, approximately $x\mathrm{coth}x$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is a result of data errors etc.) but, if $\delta$ is simply a result of round-off in the machine representation of $x$, then it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the error amplification factor can be seen in the following graph:
Figure 1
It should be noted that for $\left|x\right|\ge 2$
 $ε∼xδ=Δ$
where $\Delta$ is the absolute error in the argument.

## 8Parallelism and Performance

s10abf 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 (s10abfe.f90)

### 10.2Program Data

Program Data (s10abfe.d)

### 10.3Program Results

Program Results (s10abfe.r)