Relative Humidity approximation from Dew Point and Temperature

I would like to find a formula to approximate RH based on known air temperature and dew point. I need the approximation to be valid at least between -20...+40 degrees Celsius. I read the accepted answer on this question, but unfortunately

• the link for the document where the tabulated values were retrieved from is no longer working
• using those values yield weird results. I tried it (the RH formula with the constants in the first row of the table) with T = 16.15 and TD (dew point) = -4.45, and it gives 215% RH (could it be $$100^{m*...}$$ instead of $$100 *10^{m*...}$$?...)

Any help would be appreciated. Thanks.

UPDATE

Thanks @BarocliniCplusplus for his answer. If anyone reading this needs an implementation in Python to approximate the RH, here is one:

RH = 100*(math.exp((17.625*TD)/(243.04+TD))/math.exp((17.625*T)/(243.04+T)))

where T is the temperature, and TD the dew point. This approximation is taken from this article (see "Conclusions" in the article for more details).

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 "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.
• $L_v$ is the latent heat of vaporization (the amount of energy it takes to turn water vapor into liquid water). Technically, [t does vary with temperature] [en.wikipedia.org/wiki/…, but it is assumed to be a small amount. I usually use 2.5E6 J/kg, Obviously, if the temperature is below freezing, then it is more likely ice would form rather than liquid water (in which case, follow the hyperlinks for deposition). – BarocliniCplusplus Dec 15 '20 at 16:39