s07aa Method
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s07aa returns the value of the circular tangent, tan⁡x.

Syntax

C#
public static double s07aa(
	double x,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Function s07aa ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double s07aa(
	double x, 
	[OutAttribute] int% ifail
)
F#
static member s07aa : 
        x:float * 
        ifail:int byref -> float

Parameters

x
Type: System..::.Double
On entry: the argument x of the function.
ifail
Type: System..::.Int32 %
On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Return Value

s07aa returns the value of the circular tangent, tan⁡x.

Description

s07aa calculates an approximate value for the circular tangent of its argument, tan⁡x. It is based on the Chebyshev expansion
tan⁡θ=θyt=θ∑'r=0crTrt
where - π4<θ< π4  and -1<t<+1,  t=2 4θπ 2-1.
The reduction to the standard range is accomplished by taking
x=Nπ/2+θ
where N is an integer and -π4<θ<π4 ,
i.e., θ=x- 2xπ π2  where N = 2x π = ​ the nearest integer to ​ 2x π .
From the properties of tan⁡x it follows that
tan⁡x= tan⁡θ, Neven -1/tan⁡θ, Nodd

References

Error Indicators and Warnings

Accuracy

If δ and ε are the relative errors in the argument and result respectively, then in principle
ε≥2x sin⁡2x δ.
That is a relative error in the argument, x, is amplified by at least a factor 2x/sin⁡2x in the result.
Similarly if E is the absolute error in the result this is given by
E≥xcos2x δ.
The equalities should hold if δ is greater than the machine precision (δ is a result of data errors etc.) but if δ is simply the round-off error in the machine it is possible that internal calculation rounding will lose an extra figure.
The graphs below show the behaviour of these amplification factors.
Figure 1
Figure 1
Figure 2
Figure 2
In the principal range it is possible to preserve relative accuracy even near the zero of tan⁡x at x=0 but at the other zeros only absolute accuracy is possible. Near the infinities of tan⁡x both the relative and absolute errors become infinite and the method must fail (error 2).
If N is odd and θ≤xF2 the method could not return better than two figures and in all probability would produce a result that was in error in its most significant figure. Therefore the method fails and it returns the value
-sign⁡θ 1xF2 ≃-sign⁡θtanπ2-xF2
which is the value of the tangent at the nearest argument for which a valid call could be made.
Accuracy is also unavoidably lost if the method is called with a large argument. If x>F1 the method fails (error 1) and returns zero. (See the Library Constants section of the Introduction for specific values of F1 and F2.)

Further Comments

Example

See Also