NAG FL Interface
s10aaf (tanh)

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1 Purpose

s10aaf returns a value for the hyperbolic tangent, tanhx, via the function name.

2 Specification

Fortran Interface
Function s10aaf ( x, ifail)
Real (Kind=nag_wp) :: s10aaf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s10aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s10aaf or nagf_specfun_tanh.

3 Description

s10aaf calculates an approximate value for the hyperbolic tangent of its argument, tanhx.
For |x|1 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 |x|E1, tanhx=signx to within the representation accuracy of the machine and so this approximation is used.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

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:
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 10−1 100 0 0.5 1 1.5 2 2.5 3 ε/δ x 2*x/sinh(2 * x)
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

Background information to multithreading can be found in the Multithreading documentation.
s10aaf 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 (s10aafe.f90)

10.2 Program Data

Program Data (s10aafe.d)

10.3 Program Results

Program Results (s10aafe.r)