# NAG Library Routine Document

## 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 <nagmk26.h>
 double s07aaf_ (const double *x, Integer *ifail)

## 3Description

s07aaf calculates an approximate value for the circular tangent of its argument, $\mathrm{tan}x$. It is based on the Chebyshev expansion
 $tan⁡θ=θyt=θ∑′r=0crTrt$
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
 $tan⁡x= tan⁡θ, Neven -1/tan⁡θ, Nodd$
NIST Digital Library of Mathematical Functions

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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:
${\mathbf{ifail}}=1$
The routine has been called with an argument that is too large; the default result returned is zero.
${\mathbf{ifail}}=2$
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.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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 (error $2$).
If $N$ is odd and $\left|\theta \right|\le x{F}_{2}$ 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
 $-sign⁡θ 1xF2 ≃-sign⁡θtanπ2-xF2$
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 routine is called with a large argument. If $\left|x\right|>{F}_{1}$ the routine fails (error $1$) and returns zero. (See the Users' Note for your implementation for specific values of ${F}_{1}$ and ${F}_{2}$.)

## 8Parallelism and Performance

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)