# NAG Library Routine Document

## 1Purpose

s01baf returns a value of the shifted logarithmic function, $\mathrm{ln}\left(1+x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s01baf ( x,
 Real (Kind=nag_wp) :: s01baf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s01baf_ (const double *x, Integer *ifail)

## 3Description

s01baf computes values of $\mathrm{ln}\left(1+x\right)$, retaining full relative precision even when $\left|x\right|$ is small. The routine is based on the Chebyshev expansion
 $ln⁡1+p2+2px- 1+p2-2px- =4∑k=0∞p2k+1 2k+1 T2k+1x-.$
Setting $\stackrel{-}{x}=\frac{x\left(1+{p}^{2}\right)}{2p\left(x+2\right)}$, and choosing $p=\frac{q-1}{q+1}$, $q=\sqrt[4]{2}$ the expansion is valid in the domain $x\in \left[\frac{1}{\sqrt{2}}-1,\sqrt{2}-1\right]$.
Outside this domain, $\mathrm{ln}\left(1+x\right)$ is computed by the standard logarithmic function.

## 4References

Lyusternik L A, Chervonenkis O A and Yanpolskii A R (1965) Handbook for Computing Elementary Functions p. 57 Pergamon Press

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>-1.0$.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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, ${\mathbf{x}}\le -1.0$.
The result is returned as zero.
${\mathbf{ifail}}=-99$
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

The returned result should be accurate almost to machine precision, with a limit of about $20$ significant figures due to the precision of internal constants. Note however that if $x$ lies very close to $-1.0$ and is not exact (for example if $x$ is the result of some previous computation and has been rounded), then precision will be lost in the computation of $1+x$, and hence $\mathrm{ln}\left(1+x\right)$, in s01baf.

## 8Parallelism and Performance

s01baf is not threaded in any implementation.

Empirical tests show that the time taken for a call of s01baf usually lies between about $1.25$ and $2.5$ times the time for a call to the standard logarithm function.

## 10Example

The example program 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 (s01bafe.f90)

### 10.2Program Data

Program Data (s01bafe.d)

### 10.3Program Results

Program Results (s01bafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017