public static double s01ba(
	double x,
	out int ifail
)

Public Shared Function s01ba ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s01ba(
	double x, 
	[OutAttribute] int% ifail
)

static member s01ba : 
        x : float * 
        ifail : int byref -> float 

s01ba returns a value of the shifted logarithmic function, $\ln \left(1+x\right)$.

s01ba computes values of $\ln \left(1+x\right)$, retaining full relative precision even when $\left|x\right|$ is small. The method is based on the Chebyshev expansion 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, $\ln \left(1+x\right)$ is computed by the standard logarithmic function.

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

Errors or warnings detected by the method:

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 $\ln \left(1+x\right)$, in s01ba.

None.

Empirical tests show that the time taken for a call of s01ba usually lies between about $1.25$ and $2.5$ times the time for a call to the standard logarithm function.

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