nag_tanh (s10aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_tanh (s10aac)


    1  Purpose
    7  Accuracy

1  Purpose

nag_tanh (s10aac) returns a value for the hyperbolic tangent, tanhx.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_tanh (double x)

3  Description

nag_tanh (s10aac) calculates an approximate value for the hyperbolic tangent of its argument, tanhx.
For x1 it is based on the Chebyshev expansion
where -1x1,  -1t1,   and  t=2x2-1.
For 1<x<E1 (see the Users' Note for your implementation for value of E1)
tanhx=e2x-1 e2x+1 .  
For xE1, tanhx=signx 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:     x doubleInput
On entry: the argument x of the function.

6  Error Indicators and Warnings


7  Accuracy

If δ and ε are the relative errors in the argument and the result respectively, then in principle,
ε 2x sinh2x δ .  
That is, a relative error in the argument, x, is amplified by a factor approximately 2x sinh2x , in the result.
The equality should hold if δ is greater than the machine precision (δ due to data errors etc.) but if δ 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
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.

9  Further Comments


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)

nag_tanh (s10aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016