# NAG Library Function Document

## 1Purpose

nag_cosh (s10acc) returns the value of the hyperbolic cosine, $\mathrm{cosh}x$.

## 2Specification

 #include #include
 double nag_cosh (double x, NagError *fail)

## 3Description

nag_cosh (s10acc) calculates an approximate value for the hyperbolic cosine, $\mathrm{cosh}x$.
For $\left|x\right|\le {E}_{1}\text{, }\mathrm{cosh}x=\frac{1}{2}\left({e}^{x}+{e}^{-x}\right)$.
For $\left|x\right|>{E}_{1}$, the function fails owing to danger of setting overflow in calculating ${e}^{x}$. The result returned for such calls is $\mathrm{cosh}{E}_{1}$, i.e., it returns the result for the nearest valid argument. The value of machine-dependent constant ${E}_{1}$ may be given in the Users' Note for your implementation.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
2:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le {E}_{1}$.
The function has been called with an argument too large in absolute magnitude. There is a danger of overflow. The result returned is the value of $\mathrm{cosh}x$ at the nearest valid argument.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃xtanh⁡x×δ.$
That is, the relative error in the argument, $x$, is amplified by a factor, at least $x\mathrm{tanh}x$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is due to 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 is shown by the following graph:
Figure 1
It should be noted that near $x=0$ where this amplification factor tends to zero the accuracy will be limited eventually by the machine precision. Also for $\left|x\right|\ge 2$
 $ε∼xδ=Δ$
where $\Delta$ is the absolute error in the argument $x$.

## 8Parallelism and Performance

nag_cosh (s10acc) 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 (s10acce.c)

### 10.2Program Data

Program Data (s10acce.d)

### 10.3Program Results

Program Results (s10acce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017