# NAG FL Interfaces07aaf (tan)

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## 1Purpose

s07aaf returns the value of the circular tangent, $\mathrm{tan}x$, via the function name.

## 2Specification

Fortran Interface
 Function s07aaf ( x,
 Real (Kind=nag_wp) :: s07aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s07aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s07aaf or nagf_specfun_tan.

## 3Description

s07aaf calculates an approximate value for the circular tangent of its argument, $\mathrm{tan}x$. It is based on the Chebyshev expansion
 $tan⁡θ=θy(t)=θ∑′r=0crTr(t)$
where $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$ and $-1.
The reduction to the standard range is accomplished by taking
 $x=Nπ/2+θ$
where $N$ is an integer and $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$,
i.e., $\theta =x-\left(\frac{2x}{\pi }\right)\frac{\pi }{2}$ where $N=\left[\frac{2x}{\pi }\right]=\text{​ the nearest integer to ​}\frac{2x}{\pi }$.
From the properties of $\mathrm{tan}x$ it follows that
 $tan⁡x={ tan⁡θ, Neven -1/tan⁡θ, Nodd }$

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
2: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$ and the constant ${F}_{1}=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{x}}|\le {F}_{1}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$ and the constant ${F}_{2}=⟨\mathit{\text{value}}⟩$.
The routine has been called with an argument that is too close to an odd multiple of $\pi /2$, at which the function is infinite; the routine has returned a value with the correct sign but a more or less arbitrary but large magnitude.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

If $\delta$ and $\epsilon$ 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 $2x/\mathrm{sin}2x$ 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 $\delta$ is greater than the machine precision ($\delta$ is a result of data errors etc.) but if $\delta$ 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 2
In the principal range it is possible to preserve relative accuracy even near the zero of $\mathrm{tan}x$ at $x=0$ but at the other zeros only absolute accuracy is possible. Near the infinities of $\mathrm{tan}x$ both the relative and absolute errors become infinite and the routine must fail (indicated by ${\mathbf{ifail}}={\mathbf{2}}$).
If $N$ is odd and $|\theta |\le x{F}_{2}$ the routine 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 routine fails and it returns the value
 $-sign⁡θ (1|xF2|) ≃-sign⁡θtan(π2-|xF2|)$
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 routine is called with a large argument. If $|x|>{F}_{1}$ the routine fails (indicated by ${\mathbf{ifail}}={\mathbf{1}}$) and returns zero. (See the Users' Note for your implementation for specific values of ${F}_{1}$ and ${F}_{2}$.)

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s07aaf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s07aafe.f90)

### 10.2Program Data

Program Data (s07aafe.d)

### 10.3Program Results

Program Results (s07aafe.r)