Numerical issue with `foldl` and `foldr` in Haskell
I have the following Haskell script which computes the function f(x) = (2- x) - (2^3 - x^3/12)
calc x = (x - (x ^ 3) / 12)
calc2 x = (calc 2) - (calc x)
calcList1 :: [Float] -> Float
calcList1 l = foldl (+) 0.0 (map calc2 l)
calcList2 :: [Float] -> Float
calcList2 l = foldr (+) 0.0 (map calc2 l)
test1 :: Float -> Float
test1 step = (calcList1 l) - (calcList2 l)
where
l = [0.0,step..2.0]
Function calcList1 and calcList2 run calc2 function on each of list and then uses foldl and foldr respectively to sum the list. I was expecting both function to return the same answer but it does not.
*Main> test1 0.1
9.536743e-7
*Main> test1 0.01
2.2888184e-5
*Main> test1 0.001
2.4414063e-4
*Main> test1 0.0001
-3.7109375e-2
*Main>
Now I am confused. I can't see why numerical issues has to be involved here. Fold are essentially how ones collect each element which should be same in both cases, right?
In general, the order in which floating point values are added is important. An entry point for own research could be http://en.wikipedia.org/wiki/Loss_of_significance . To summarize the basic caveat, in an oversimplified form:
Due to the limited number of significant bits, you have to assume something like
100000000000000000.0 + 1.0 = 100000000000000000.0
in floating-point computations. Consequently, when computing
100000000000000000.0
+ 1.0
- 100000000000000000.0
the result will be 0.0 - and thus, be different from
100000000000000000.0
- 100000000000000000.0
+ 1.0
where the result will be 1.0 .
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