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NAG C Library Manual

# NAG Library Function Documentnag_arcsinh (s11abc)

## 1  Purpose

nag_arcsinh (s11abc) returns the value of the inverse hyperbolic sine, $\mathrm{arcsinh}x$.

## 2  Specification

 #include #include
 double nag_arcsinh (double x)

## 3  Description

nag_arcsinh (s11abc) calculates an approximate value for the inverse hyperbolic sine of its argument, $\mathrm{arcsinh}x$.
For $\left|x\right|\le 1$ the function is based on a Chebyshev expansion.
For $\left|x\right|>1$
 $arcsinh⁡x = sign⁡x × ln x + x 2 + 1 .$
This form is used directly for $1<\left|x\right|<{10}^{k}$, where $k=n/2+1$, and the machine uses approximately $n$ decimal place arithmetic.
For $\left|x\right|\ge {10}^{k},\sqrt{{x}^{2}+1}$ is equal to $\left|x\right|$ to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate
 $arcsinh⁡x = sign⁡x × ln⁡2 + ln x$

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.

None.

## 7  Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and the result, respectively, then in principle
 $ε ≃ x 1 + x 2 arcsinh⁡x δ .$
That is, the relative error in the argument, $x$, is amplified by a factor at least
 $x 1 + x 2 arcsinh⁡x$
in the result.
The equality should hold if $\delta$ is greater than the machine precision($\delta$ due to data errors etc.), but if $\delta$ is simply due to round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation round-off.
It should be noted that this factor is always less than or equal to one. For large $x$ we have the absolute error in the result, $E$, in principle, given by
 $E ∼ δ .$
This means that eventually accuracy is limited by machine precision.

None.

## 9  Example

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

### 9.1  Program Text

Program Text (s11abce.c)

### 9.2  Program Data

Program Data (s11abce.d)

### 9.3  Program Results

Program Results (s11abce.r)