Pathological floating point problems

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This page is a solution to the task Pathological floating point problems in the Rosetta Code, written in Fōrmulæ.

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, straight-forward 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:

v1 = 2
v2 = -4
vn = 111 - 1130 / vn-1 + 3000 / (vn-1 * vn-2)

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 ...

  • after 1 year, your balance will be multiplied by 1 and $1 will be removed for service charges.
  • after 2 years your balance will be doubled and $1 removed.
  • after 3 years your balance will be tripled and $1 removed.
  • ...
  • after 10 years, multiplied by 10 and $1 removed, and so on.

What will your balance be after 25 years?

   Starting balance: $e-1
   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.

f(a,b) = 333.75b6 + a2( 11a2b2 - b6 - 121b4 - 2 ) + 5.5b8 + a/(2b)

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

PathologicalFP0.png

Task 1. A sequence that seems to converge to a wrong limit

The correct value converges to 6.

PathologicalFP1.png

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.

PathologicalFP2.png

Task 3. Extra credit. Siegfried Rump's example

Correct value is about -0.827396059946821.

PathologicalFP3.png

PathologicalFP4.png