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NAG Library Manual

# NAG Library Function Documentnag_tanh (s10aac)

## 1  Purpose

nag_tanh (s10aac) returns a value for the hyperbolic tangent, $\mathrm{tanh}x$.

## 2  Specification

 #include #include
 double nag_tanh (double x)

## 3  Description

nag_tanh (s10aac) calculates an approximate value for the hyperbolic tangent of its argument, $\mathrm{tanh}x$.
For $\left|x\right|\le 1$ it is based on the Chebyshev expansion
 $tanh⁡x=x×yt=x∑′r=0arTrt$
where $-1\le x\le 1\text{, }-1\le t\le 1\text{, and }t=2{x}^{2}-1$.
For $1<\left|x\right|<{E}_{1}$ (see the Users' Note for your implementation for value of ${E}_{1}$)
 $tanh⁡x=e2x-1 e2x+1 .$
For $\left|x\right|\ge {E}_{1}$, $\mathrm{tanh}x=\mathrm{sign}x$ to within the representation accuracy of the machine and so this approximation is used.

## 4  References

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

## 5  Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.

None.

## 7  Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and the result respectively, then in principle,
 $ε≃ 2x sinh⁡2x δ .$
That is, a relative error in the argument, $x$, is amplified by a factor approximately $\frac{2x}{\mathrm{sinh}2x}$, in the result.
The equality should hold if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is due simply to the round-off in the machine representation it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the amplification factor is shown in the following graph:
Figure 1
It should be noted that this factor is always less than or equal to $1.0$ and away from $x=0$ the accuracy will eventually be limited entirely by the precision of machine representation.

## 8  Parallelism and Performance

nag_tanh (s10aac) is not threaded in any implementation.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (s10aace.c)

### 10.2  Program Data

Program Data (s10aace.d)

### 10.3  Program Results

Program Results (s10aace.r)