Pathological floating point problems
This page is a solution to the task Pathological floating point problems in the Rosetta Code, written in Fōrmulæ.
Contents
Description (from Rosetta Code)
Most programmers are familiar with the inexactness of floating point calculations in a binary processor.
The classic example being: 0.1 + 0.2 = 0.30000000000000004 In many situations the amount of error in such calculations is very small and can be overlooked or eliminated with rounding. There are pathological problems however, where seemingly simple, straightforward calculations are extremely sensitive to even tiny amounts of imprecision. This task's purpose is to show how your language deals with such classes of problems. Task 1. A sequence that seems to converge to a wrong limit. Consider the sequence:
As n grows larger, the series should converge to 6 but small amounts of error will cause it to approach 100. Display the values of the sequence where n = 3, 4, 5, 6, 7, 8, 20, 30, 50 & 100 to at least 16 decimal places. n = 3 18.5 n = 4 9.378378 n = 5 7.801153 n = 6 7.154414 n = 7 6.806785 n = 8 6.5926328 n = 20 6.0435521101892689 n = 30 6.006786093031205758530554 n = 50 6.0001758466271871889456140207471954695237 n = 100 6.000000019319477929104086803403585715024350675436952458072592750856521767230266 Task 2. The Chaotic Bank Society. The Chaotic Bank Society is offering a new investment account to their customers. You first deposit $e  1 where e is 2.7182818... the base of natural logarithms. After each year, your account balance will be multiplied by the number of years that have passed, and $1 in service charges will be removed. So ...
What will your balance be after 25 years? Starting balance: $e1 Balance = (Balance * year)  1 for 25 years Balance after 25 years: $0.0399387296732302 Task 3. Siegfried Rump's example. (extra credit) Consider the following function, designed by Siegfried Rump in 1988.
compute f(a,b) where a=77617.0 and b=33096.0 f(77617.0, 33096.0) = 0.827396059946821 Demonstrate how to solve at least one of the first two problems, or both, and the third if you're feeling particularly jaunty. See also

Task 0. Classic example
Task 1. A sequence that seems to converge to a wrong limit
The correct value converges to 6.
Task 2. The Chaotic Bank Society
If it works correctly, it must not diverge, the balance after 25 years should be about $0.0399387296732302
Note that expressions of second column (exact values) are not in scientific notation. e is the mathematical constant that is the base of natural logarithms.
Task 3. Extra credit. Siegfried Rump's example
Correct value is about 0.827396059946821.