﻿ s10ab Method
s10ab returns the value of the hyperbolic sine, $\mathrm{sinh} x$.

# Syntax

C#
```public static double s10ab(
double x,
out int ifail
)```
Visual Basic
```Public Shared Function s10ab ( _
x As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s10ab(
double x,
[OutAttribute] int% ifail
)```
F#
```static member s10ab :
x : float *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s10ab returns the value of the hyperbolic sine, $\mathrm{sinh} x$.

# Description

s10ab 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 method 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.

# References

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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
The method has been called with an argument too large in absolute magnitude. There is a danger of setting overflow. The result is the value of $\mathrm{sinh} x$ at the closest argument for which a valid call could be made. (See [Description] and the Users' Note for your implementation.)
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

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.

None.

None.

# Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

Example program (C#): s10abe.cs

Example program data: s10abe.d

Example program results: s10abe.r