# NAG FL Interfaces10abf (sinh)

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## 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 <nag.h>
 double s10abf_ (const double *x, Integer *ifail)
The routine may be called by the names s10abf or nagf_specfun_sinh.

## 3Description

s10abf calculates an approximate value for the hyperbolic sine of its argument, $\mathrm{sinh}x$.
For $|x|\le 1$ it uses the Chebyshev expansion
 $sinh⁡x=x×y(t)=x∑′r=0arTr(t)$
where $t=2{x}^{2}-1$.
For $1<|x|\le {E}_{1}\text{, }\mathrm{sinh}x=\frac{1}{2}\left({e}^{x}-{e}^{-x}\right)$
where ${E}_{1}$ is a machine-dependent constant , details of which are given in the Users' Note for your implementation.
For $|x|>{E}_{1}$, 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.
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).

## 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: $|{\mathbf{x}}|\le {E}_{1}$.
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.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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
 $|ε|≃ |xcoth⁡x×δ|.$
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 $|x|\ge 2$
 $ε∼xδ=Δ$
where $\Delta$ is the absolute error in the argument.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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)
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