Relative Humidity approximation from Dew Point and Temperature
The equation for relative humidity is $$RH=100\times \frac{e}{e_s(T)}=100\times \frac{e_s(T_d)}{e_s(T)} \tag{1}$$
Where $T_d$ is the dewpoint temperature and $T$ is the temperature, $e$ is the water vapor pressure, and $e_s$ is the saturation vapor pressure, which is also known as the Clausius Clapeyeron equation. While the previous link has a decent definition and equation, my preferred equation (particularly because it is derivable) is the low temperature approximation:
$$e_s(T)= e_s(273 \mathrm{K})\exp\left[\frac{L_v}{R_v}\left(273.15^{-1}-T^{-2}\right)\right] \tag{2}$$
where $T$ is the temperature or dewpoint temperature in Kelvin, $L_v$ is the latent heat of vaporization,$e_s(273 \mathrm{K})=6.11 hPa$, and $R_v$ is the specific gas constant for water vapor. Note that $(2)$ is for liquid water. You may be able to replace $L_v$ for $L_s$ for sublimation/deposition. Also note, that this is the &quot;pure&quot; form of the relative humidity equation. The presence of solutes (Cloud condensation nuclei) may lower the saturation vapor pressure, increasing the actual relative humidity. I'll let combining $(1)$ and $(2)$ be an exercise left for the reader.