s10aa returns a value for the hyperbolic tangent, tanhx.

Syntax

C#
public static double s10aa(
	double x
)
Visual Basic
Public Shared Function s10aa ( _
	x As Double _
) As Double
Visual C++
public:
static double s10aa(
	double x
)
F#
static member s10aa : 
        x : float -> float 

Parameters

x
Type: System..::..Double
On entry: the argument x of the function.

Return Value

s10aa returns a value for the hyperbolic tangent, tanhx.

Description

s10aa calculates an approximate value for the hyperbolic tangent of its argument, tanhx.
For x1 it is based on the Chebyshev expansion
tanhx=x×yt=xr=0arTrt
where -1x1,  -1t1,   and  t=2x2-1.
For 1<x<E1 (see the Users' Note for your implementation for value of E1)
tanhx=e2x-1e2x+1.
For xE1, tanhx=signx to within the representation accuracy of the machine and so this approximation is used.

References

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

Error Indicators and Warnings

None.

Accuracy

If δ and ε are the relative errors in the argument and the result respectively, then in principle,
ε2xsinh2xδ.
That is, a relative error in the argument, x, is amplified by a factor approximately 2xsinh2x, 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.

Parallelism and Performance

None.

Further Comments

None.

Example

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

Example program (C#): s10aae.cs

Example program data: s10aae.d

Example program results: s10aae.r

See Also