Cost scaling and optimal strategy for synthesis

Solution 1:

With a bit of work I figured that the formula for the cost of synthesising a level L object with probability 0 <= x < 1 is

cost = floor(L^2 * tan(x * pi/2) + 8 * L)

In case you're wondering, to find it I first wrote down the costs of synthesising a few items of different levels with various success rates, sampled every 5%. Then — using Sage, an open-source computer algebra system — I plotted the data points for the highest level object (which had the highest "resolution") and then tried to fit them with a couple of likely functions. Finally, I checked my solution against my other data sets.