Relative Humidity approximation from Dew Point and Temperature

The equation for relative humidity is $$RH=100times frac{e}{e_s(T)}=100times frac{e_s(T_d)}{e_s(T)} tag{1}$$ 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:

$$e_s(T)= e_s(273 mathrm{K})expleft[frac{L_v}{R_v}left(273.15^{-1}-T^{-2}right)right] tag{2}$$ 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 "pure" 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.