# NAG Library Routine Document

## 1Purpose

s09abf returns the value of the inverse circular cosine, $\mathrm{arccos}x$, via the function name; the result is in the principal range $\left(0,\pi \right)$.

## 2Specification

Fortran Interface
 Function s09abf ( x,
 Real (Kind=nag_wp) :: s09abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s09abf_ (const double *x, Integer *ifail)

## 3Description

s09abf calculates an approximate value for the inverse circular cosine, $\mathrm{arccos}x$. It is based on the Chebyshev expansion
 $arcsin⁡x=x×yt=x∑′r=0arTrt$
where $\frac{-1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}\text{, and }t=4{x}^{2}-1$.
For ${x}^{2}\le \frac{1}{2}\text{, }\mathrm{arccos}x=\frac{\pi }{2}-\mathrm{arcsin}x$.
For $-1\le x<\frac{-1}{\sqrt{2}}\text{, }\mathrm{arccos}x=\pi -\mathrm{arcsin}\sqrt{1-{x}^{2}}$.
For $\frac{1}{\sqrt{2}}.
For $\left|x\right|>1\text{, }\mathrm{arccos}x$ is undefined and the routine fails.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: $\left|{\mathbf{x}}\right|\le 1.0$.
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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le 1$.
${\mathbf{ifail}}=-99$
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 the result, respectively, then in principle
 $ε≃ x arccos⁡x 1-x2 ×δ .$
The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is due to data errors etc.), but if $\delta$ is due simply to round-off in the machine it is possible that rounding etc. in internal calculations may lose one extra figure.
The behaviour of the amplification factor $\frac{x}{\mathrm{arccos}x\sqrt{1-{x}^{2}}}$ is shown in the graph below.
In the region of $x=0$ this factor tends to zero and the accuracy will be limited by the machine precision. For $\left|x\right|$ close to one, $1-\left|x\right|\sim \delta$, the above analysis is not applicable owing to the fact that both the argument and the result are bounded $\left|x\right|\le 1$, $0\le \mathrm{arccos}x\le \pi$.
In the region of $x\sim -1$ we have $\epsilon \sim \sqrt{\delta }$, that is the result will have approximately half as many correct significant figures as the argument.
In the region $x\sim +1$, we have that the absolute error in the result, $E$, is given by $E\sim \sqrt{\delta }$, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.
Figure 1

## 8Parallelism and Performance

s09abf 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 (s09abfe.f90)

### 10.2Program Data

Program Data (s09abfe.d)

### 10.3Program Results

Program Results (s09abfe.r)