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Re: Reverse factorial

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Paul Rubin <no.email@nospam.invalid> writes Re: Reverse factorial
[..]
>>> log10(205022!) = 999999.08681421082059069559626773741428638233100481270
>>> log10(205023!) = 1.00000439861679486551512255600216645545545382220e6

>>> Therefore 205023 as the "Smallest number with a factorial of at least 10^6
>>> digits" is correct.

> Of course 10**999999.08 has 1e6 digits just like 10^9 has 10 digits.

It is beyond my semantic capabability to understand what "Smallest 
number with a factorial of at least 10^x digits" means exactly. Nail the
constraint in terms of log(n!) and the program will probably be correct.

[..]
> I think for inverse factorial of large numbers, it probably is pretty
> accurate to use a numerical rootfinder on the first few terms of the
> Stirling series:

> http://en.wikipedia.org/wiki/Stirling%27s_approximation#Speed_of_convergence_and_error_estimates 

> Even just the first (1/12n) correction term makes Stirling's
> approximation pretty accurate.

Hmm, apparently that algorithm doesn't jump out of the Forth for you :-)

The upperbound of log(n!) is not strictly n (for low n where I was not 
intending to use the words), and so the binary search walked against the
bound before finding the root.

-marcel

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Reverse factorial "Ed" <invalid@nospam.com> - 2012-06-29 11:51 +1000
  Re: Reverse factorial mhx@iae.nl (Marcel Hendrix) - 2012-06-30 02:05 +0200
    Re: Reverse factorial "Ed" <invalid@nospam.com> - 2012-06-30 12:26 +1000
    Re: Reverse factorial "Ed" <invalid@nospam.com> - 2012-07-01 23:17 +1000
      Re: Reverse factorial mhx@iae.nl (Marcel Hendrix) - 2012-07-01 16:51 +0200
        Re: Reverse factorial "Ed" <invalid@nospam.com> - 2012-07-02 05:40 +1000
          Re: Reverse factorial Paul Rubin <no.email@nospam.invalid> - 2012-07-02 22:08 -0700
            Re: Reverse factorial "Ed" <invalid@nospam.com> - 2012-07-03 17:02 +1000
            Re: Reverse factorial mhx@iae.nl (Marcel Hendrix) - 2012-07-03 21:14 +0200
              Re: Reverse factorial Paul Rubin <no.email@nospam.invalid> - 2012-07-03 12:44 -0700
                Re: Reverse factorial mhx@iae.nl (Marcel Hendrix) - 2012-07-03 22:37 +0200
                Re: Reverse factorial "Ed" <invalid@nospam.com> - 2012-07-05 23:14 +1000

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