s13ad returns the value of the sine integral
 $Six=∫0xsin uudu,$
.

# Syntax

C#
```public static double s13ad(
double x
)```
Visual Basic
```Public Shared Function s13ad ( _
x As Double _
) As Double```
Visual C++
```public:
double x
)```
F#
```static member s13ad :
x : float -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.

#### Return Value

s13ad returns the value of the sine integral
 $Six=∫0xsin uudu,$
.

# Description

s13ad calculates an approximate value for $\mathrm{Si}\left(x\right)$.
For $\left|x\right|\le 16.0$ it is based on the Chebyshev expansion
 $Six=x∑r=0′arTrt,t=2x162-1.$
For $16<\left|x\right|<{x}_{\mathrm{hi}}$, where ${x}_{\mathrm{hi}}$ is an implementation-dependent number,
 $Six=signxπ2-fxcos xx-gxsin xx2$
where $f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{f}_{r}{T}_{r}\left(t\right)$ and $g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{g}_{r}{T}_{r}\left(t\right)$, $t=2{\left(\frac{16}{x}\right)}^{2}-1$.
For $\left|x\right|\ge {x}_{\mathrm{hi}}$, $\mathrm{Si}\left(x\right)=\frac{1}{2}\pi \mathrm{sign} x$ to within machine precision.

# References

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

# Error Indicators and Warnings

There are no failure exits from s13ad. The parameter _ifail has been included for consistency with other methods in this chapter.

# Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃δsin xSix.$
The equality may 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, then since the factor relating $\delta$ to $\epsilon$ is always less than one, the accuracy will be limited by machine precision.

None.

None.

# Example

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