Rounding cost my insurance company quite a bit of money, until they figured out how to avoid the expense.
Say my insurance bill is $100 for the year, to be paid monthly. The monthly amount will need to be rounded to a cent, so is $8.33. At the end of 12 months, the total amount collected is only 99.96, which is 0.04 short of the intended amount.
My insurance company used to send a mailing in the 11th month telling me that my final month's payment would be $8.37, to make the correct total for the year. Of course, the cost of the mailing far exceeded the extra amount collected.
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This is an example of a situation where rounding error can be corrected relatively easily. My insurance company now details each month's bill amount in the initial mailing, saving the cost of an 11th-month mailing. Here, an entire company's way of doing business was affected by rounding error.
Consider space flight. It involves predicting the future positions of numerous celestial bodies and the resulting gravitational influence on the free flight of a spacecraft. The equations involved cannot be solved in closed form, so numerical solution is used.
Generally, future positions are computed from present positions and some estimate of the applied forces. When the intermediate results of these computations are rounded, which they must be due to the finite precision of numbers in the computer, the errors accumulate and predicted positions can vary wildly from actual positions. (This is one reason why mid-course corrections are part of the flight plan.)
While the equations for predicting space flight may be "well-behaved", the equations for predicting weather are not. Small changes in the initial conditions can cause large changes in predicted weather. (This is sometimes referred to as "the butterfly effect".) Thus, the nature of the rounding of numbers used in computation can have a significant influence on the computation results.