How to calculate relative humidity from temperature, dew point, and pressure?
You can refer to this question for more detail on the origin of this formula (based on the Magnus approximation), but if you do some algebra to the expression there for dew point ($TD$) as function of temperature ($T$) and relative humidity ($RH$), you get
$RH=100 \, e^{\Large \left(\frac{c\, b (TD-T)}{(c+T) (c+TD)}\right)}$
With $b=17.625$ and $c=243.04$.
In this case, where $TD$ is one of your input variables, there is no need to consider the pressure, pressure have no effect in $RH$, or more accurately, the pressure dependence is already considered through $TD$.
The pressure would come into play if you are computing $TD$ from water vapour pressure, because water vapour pressure is what have a small dependence in atmospheric pressure.
The Magnus approximation above is considered valid for:
$0^oC &lt; T &lt; 60^oC$
$1\% &lt; RH &lt; 100\%$
$0^oC &lt; TD &lt; 50^oC$
There are also other equivalent formulas that extends their validity range by changing the constants, like this one
$RH=100\cdot10^{\Large m\left( \frac{TD}{TD+T_n}-\frac{T}{T+T_n}\right)}$
Where values for the constants $m$ and $T_n$ depend on temperature and are tabulated:

See this document for more details.
There are also very simple approximations to these formulas, like
$RH \approx 100 - 5 (T-TD)$
You can find a discussion on the accuracy of this approximation here.