I am unable to open either of the xlsx files, (at least with Bytescout XLS viewer). Can someone check that they are valid?
To put the conversion into a single equation:
curvedvalue = floor(0.5 + pow(value/255.0, 0.5) * 255);
As value goes from 0 to 255, curvedvalue also goes from 0 to 255, but in an exponentially curved way.
For example, if value is 64 ( 1/4) , curvedvalue should come out 128 (1/2).
This might help debug.
64 / 255 = 0.251
pow(0.251, 0.5) = 0.5001
0.5001 * 255 = 127.75
floor (0.5 + 127.75) = 128
Note that though your input and output values are integers, the intermediate calculations must be floating point.
Important only for your Arduino code, not the spreadsheet.