# [PD] weird behavior of factorial in expr

oscar pablo di liscia odiliscia at gmail.com
Thu Sep 17 23:10:03 CEST 2020

```Hello Albert:
work on some combinatorial stuff.
I just wanted to check if I was doing something wrong with the use of
of  "expr" is that I can have "packed" in just one object a complete
formula including
operator precedences.
Best

Oscar Pablo Di Liscia

El jue., 17 sept. 2020 a las 4:24, Albert Rafetseder (<
albert.rafetseder+pd at univie.ac.at>) escribiÃ³:

> Hi Oscar,
>
> > the "fact" (factorial) function does not seem to work properly in the
> > "expr" external when called with an argument greater than 12.
>
> the problem in [expr fact(...)] looks like an integer overflow. See [1]
> for conceptual details, TL;DR: Factorials produce huge numbers very
> quickly, but the implementation of `fact` reserves too little space to
> store the result's digits [2], and thus truncates the result, producing
> garbage:
>
> [expr fact(12)] is 4.79002e+08, just about right
> [expr fact(13)] is 1.93205e+09, clearly *not* the above times 13
> [expr fact(14)] is 1.27895e+09, even smaller than the previous result
> (...)
> [expr fact(17)] is a negative number altogether
>
> I can't comment on the efficiency your implementation as I'm not too
> well versed in Pd. I'd speculate it won't suffer [expr fact]'s numerical
> problems since AFAIK, patches use floats as the default number format,
> basically allowing for larger numbers to be stored.
>
> The usual suggestion for avoiding numerical problems with factorials is
> to re-think what the numbers are used for -- Taylor series?
> combinatorials of n-choose-k kind? something else? -- and use an
> appropriate alternative such as:
>
> * Stirling's approximation [3]
> * the Gamma function [4]
> * binomial coefficient without factorials [5]
>
> Cheers,
>   Albert.
>
> [1] https://en.wikipedia.org/wiki/Integer_overflow
> [2]
>
> https://github.com/pure-data/pure-data/blob/2af4b5d/src/x_vexp_fun.c#L913-L928
> [3] https://en.wikipedia.org/wiki/Stirling%27s_approximation
> [4] https://en.wikipedia.org/wiki/Gamma_function
> [5]
>
> https://en.wikipedia.org/wiki/Binomial_coefficient#Binomial_coefficient_in_programming_languages
>
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