# NAG FL Interfaces09aaf (arcsin)

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## 1Purpose

s09aaf returns the value of the inverse circular sine, $\mathrm{arcsin}x$, via the function name. The value is in the principal range $\left(-\pi /2,\pi /2\right)$.

## 2Specification

Fortran Interface
 Function s09aaf ( x,
 Real (Kind=nag_wp) :: s09aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s09aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s09aaf or nagf_specfun_arcsin.

## 3Description

s09aaf calculates an approximate value for the inverse circular sine, $\mathrm{arcsin}x$. It is based on the Chebyshev expansion
 $arcsin⁡x=x×y(x)=x∑′r=0arTr(t)$
where $-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$ and $t=4{x}^{2}-1$.
For ${x}^{2}\le \frac{1}{2}\text{, }\mathrm{arcsin}x=x×y\left(x\right)$.
For $\frac{1}{2}<{x}^{2}\le 1\text{, }\mathrm{arcsin}x=\mathrm{sign}x\left\{\frac{\pi }{2}-\mathrm{arcsin}\sqrt{1-{x}^{2}}\right\}$.
For ${x}^{2}>1\text{, }\mathrm{arcsin}x$ is undefined and the routine fails.
NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: $|{\mathbf{x}}|\le 1.0$.
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, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: $|{\mathbf{x}}|\le 1$.
${\mathbf{ifail}}=-99$
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
 $|ε|≃ | x arcsin⁡x 1-x2 ×δ| .$
That is, a relative error in the argument $x$ is amplified by at least a factor $\frac{x}{\mathrm{arcsin}x\sqrt{1-{x}^{2}}}$ in the result.
The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is a result of data errors etc.) but if $\delta$ is produced simply by round-off error in the machine it is possible that rounding in internal calculations may lose an extra figure in the result.
This factor stays close to one except near $|x|=1$ where its behaviour is shown in the following graph. Figure 1
For $|x|$ close to unity, $1-|x|\sim \delta$, the above analysis is no longer applicable owing to the fact that both argument and result are subject to finite bounds, ($|x|\le 1$ and $|\mathrm{arcsin}x|\le \frac{1}{2}\pi$). In this region $\epsilon \sim \sqrt{\delta }$; that is the result will have approximately half as many correct significant figures as the argument.
For $|x|=1$ the result will be correct to full machine precision.

## 8Parallelism and Performance

s09aaf 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 (s09aafe.f90)

### 10.2Program Data

Program Data (s09aafe.d)

### 10.3Program Results

Program Results (s09aafe.r)