[racket] Implementation of Simpson's Rule (SICP Exercise 1.29)

In racket/base, you can also simply use bignums instead:

> (define janus
    '(31.0 2e+34 -1.2345678901235e+80 2749.0 -2939234.0 -2e+33
           3.2e+270 17.0 -2.4e+270 4.2344294738446e+170 1.0 -8e+269 0.0
99.0))

> (exact->inexact (apply + (map inexact->exact janus)))
4.2344294738446e+170

No need for the math lib, even racket/base is awesome per se :)

On Thu, Nov 7, 2013 at 12:24 PM, Stephan Houben <stephanh42 at gmail.com>wrote:

> The competition does it better, see Python's fsum:
>
> import math
>
> janus = [31.0, 2e+34, -1.2345678901235e+80, 2749.0, -2939234.0, -2e+33,
>          3.2e+270, 17.0, -2.4e+270, 4.2344294738446e+170, 1.0, -8e+269,
> 0.0, 99.0]
>
> print(math.fsum(janus))
> # 4.2344294738446e+170
>
> The fsum algorithm is interesting, it essentially emulates
> unlimited-precision FP
> by using a list of limited-precision FP numbers.
>
> Stephan
>
>
> 2013/11/7 Todd O'Bryan <toddobryan at gmail.com>
>
>> I just found a lovely Java expression to emphasize the inexactness of
>> doubles to my AP students. The problem--which I think is from
>> HtDP/1e--is to find the value of a bag of coins given the number of
>> pennies, nickels, dimes, and quarters. In BlueJ's code pad (or similar
>> in DrJava, jGrasp, etc.)
>>
>> > 0.01 + 0.05 + 0.10 + 0.25
>> 0.410000000000000000003
>>
>> (my number of zeroes may be off)
>>
>> As one of my students said--"You can do that in your head. What's the
>> computer's problem?"
>>
>> Todd
>>
>> On Wed, Nov 6, 2013 at 1:30 PM, Neil Toronto <neil.toronto at gmail.com>
>> wrote:
>> > On 11/06/2013 09:24 AM, Matthias Felleisen wrote:
>> >>
>> >>
>> >> On Nov 6, 2013, at 7:13 AM, Ben Duan <yfefyf at gmail.com> wrote:
>> >>
>> >>> Thank you, Jens. I didn't know that the inexactness of floating point
>> >>> numbers could make such a big difference.
>> >>
>> >>
>> >>
>> >>  From HtDP/1e:
>> >>
>> >> (define JANUS
>> >>    (list #i31
>> >>          #i2e+34
>> >>          #i-1.2345678901235e+80
>> >>          #i2749
>> >>          #i-2939234
>> >>          #i-2e+33
>> >>          #i3.2e+270
>> >>          #i17
>> >>          #i-2.4e+270
>> >>          #i4.2344294738446e+170
>> >>          #i1
>> >>          #i-8e+269
>> >>          #i0
>> >>          #i99))
>> >>
>> >>
>> >>
>> >> ;; [List-of Number] -> Number
>> >> ;; add numbers from left to right
>> >> (check-expect (sumlr '(1 2 3)) 6)
>> >> (define (sumlr l)
>> >>    (foldl + 0 l))
>> >>
>> >> ;; [List-of Number] -> Number
>> >> ;; add numbers from right to left
>> >> (check-expect (sumrl '(1 2 3)) 6)
>> >> (define (sumrl l) (foldr + 0 l))
>> >>
>> >> Then apply the two functions to JANUS. Enjoy -- Matthias
>> >
>> >
>> > Nice example!
>> >
>> > You could also (require math) and apply its `sum' or `flsum' to JANUS.
>> Then
>> > *really* enjoy. :D
>> >
>> >> (sumlr JANUS)
>> > 99.0
>> >
>> >> (sumrl JANUS)
>> > -1.2345678901235e+80
>> >
>> >> (sum JANUS)
>> > 4.2344294738446e+170
>> >
>> >> (exact->inexact (sumlr (map inexact->exact JANUS)))
>> > 4.2344294738446e+170
>> >
>> > On my computer, using `sum' is about 20x faster than converting JANUS to
>> > exact numbers.
>> >
>> > You can also sort by absolute value before summing, which is a little
>> faster
>> > still but loses some precision. Do not trust Teh Internets on this one.
>> > Popular Q-and-A sites say to sort ascending, which makes intuitive
>> sense:
>> > adding a big number to two small numbers in turn might do nothing, but
>> > adding a big number to their *sum* might result in something larger.
>> >
>> >> (expt 2 53.0)
>> > 9007199254740992.0
>> >
>> >> (sumlr (list (expt 2 53.0) 1.0 1.0))
>> > 9007199254740992.0
>> >
>> >> (sumlr (list 1.0 1.0 (expt 2 53.0)))
>> > 9007199254740994.0
>> >
>> > But JANUS shows that sorting ascending doesn't work when summing huge
>> > numbers with alternating signs:
>> >
>> >> (sumlr (sort JANUS (λ (x y) (< (abs x) (abs y)))))
>> > 0.0
>> >
>> >> (sumlr (sort JANUS (λ (x y) (> (abs x) (abs y)))))
>> > 4.2344294738446e+170
>> >
>> > All the research papers on summation by sorting sort descending,
>> contrary to
>> > the wisdom of Teh Internets. So either do that, or use `sum' or `flsum'
>> when
>> > you want an accurate sum of flonums.
>> >
>> > Neil ⊥
>> >
>> >
>> > ____________________
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>> >  http://lists.racket-lang.org/users
>>
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>
>
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