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This sample mod provides a file, savable_infinity.lua, which when modimport'ed allows storing correctly the following numerical values as savedata:
Plus and minus infinity (i.e., math.huge and -math.huge);
NaN (Not a Number, i.e., the result of invalid arithmetical operations, such as 0/0).
The game's save system is patched transparently, so other than modimport'ing savable_infinity.lua no other steps need to be taken: simply feel free to store the above values in savedata.
The patching of the game's DataDumper function uses the fact that Don't Starve's savedata is simply Lua code to store these values as arithmetical expressions: plus infinity is saved as 1/0, negative infinity as -1/0 and NaN as 0/0. The method used for the patching is more hackish than I would like (making heavy use of Lua's debug library), but it is the cleanest method of doing so I could think of, other than simply overriding vanilla's dumper.lua.
The rest of the mode code, beyond savable_infinity.lua, is just a test suite to see the added functionality in action.
The file savable_infinity.lua is modimport'ed in modworldgenmain.lua, so that all savedata is stored correctly (but it can safely be modimport'ed in modmain.lua instead, if worldgen savedata is not a concern).
In modmain.lua, the global variable ENCODE_SAVES is set to false, causing save files to be stored as plain Lua instead of zip compressed Lua, allowing visual inspection of the saves' contents. Furthermore, modmain.lua adds the included infinitysavetest component to the world entity: this component simply returns the mentioned values in its OnSave method and checks the validity of the loaded savedata in its OnLoad method, which prints the following to log.txt: