How to round floats to integers while preserving their sum?

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One option you could try is “cascade rounding”.

For this algorithm you keep track of two running totals: one of floating point numbers so far, and one of the integers so far.
To get the next integer you add the next fp number to your running total, round the running total, then subtract the integer running total from the rounded running total:-

number  running total   integer integer running total
   1.3       1.3          1           1
   1.7       3.0          2           3
   1.9       4.9          2           5
   2.2       8.1          3           8
   2.8      10.9          3          11
   3.1      14.0          3          14

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