Before you try to invent another simple formula try this. It is simple but that does not mean easy to do longhand because the subject is quite involved.
It takes the bold step that if we go flying we quickly get into the realms of dry air very quickly and can neglect dew point etc
google richard shelquist + density altitude + calculator
I sometimes find a range of altitudes that work well with richard's approximation then use something like the "polyfit.exe" program for 10 points of interest and the quadratic or cubic function. Google polyfit.exe. On a calculator use the standard deviation test and chose which gives the lower value. Its usually but always the quadratic version. If your home brew formula works it will show up bonzer. If your formula is cr*p stdev will tell you. Bin it!!
My experience of home grown rules is woeful. I make incorrect assumptions! My "rules" fall flat and it is not easy for someone else to spot where I go wrong.
No, let Shelquist do it for you. Then download any interpolation formula, excel will do; test how good the formula is using standard deviation test as I have said; choose the formula that gives the lowest value of stdev. Program your calculator and save.
Look for Richard's section that starts as follows:
Simpler Methods of Calculation... If you really want to know the actual density altitude, it will need to be calculated in the general manner that has been described above. However, there are simple approximations which have been developed over the years. For example, a particularly convenient form of density altitude approximation is obtained by simply ignoring the actual moisture content in the air. Here is such an equation which has been used by the National Weather Service (see
ref 13) to calculate the approximate density altitude without any need to know the humidity, dew point pressure: