# NAG Library Function Documentnag_cosh (s10acc)

## 1  Purpose

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

## 2  Specification

#include <nag.h>
#include <nags.h>

double  nag_cosh (double x,  NagError *fail)

## 3  Description

nag_cosh (s10acc) calculates an approximate value for the hyperbolic cosine, $\mathrm{cosh}x$.
For $\left|x\right|\le {E}_{1}$, (where ${E}_{1}$ is a machine-dependent constant) $\mathrm{cosh}x=\frac{1}{2}\left({e}^{x}+{e}^{-x}\right)$.
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.

## 4  References

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

## 5  Arguments

1:     x – doubleInput
On entry: the argument $x$ of the function.
2:     fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le 〈\mathit{\text{value}}〉$.
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.

## 7  Accuracy

If $\delta $ and $\epsilon $ are the relative errors in the argument and result, respectively, then in principle

$ε ≃ x tanh⁡x δ .$
That is, the relative error in the argument, $x$, is amplified by a factor at least $x\mathrm{tanh}x$ in the result. 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.
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 $\left|x\right|\gtrsim 2$

$ε ∼ x δ = Δ$
where $\Delta $ is the absolute error in the argument $x$.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s10acce.c)

### 9.2  Program Data

Program Data (s10acce.d)

### 9.3  Program Results

Program Results (s10acce.r)
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