Re: Reverse factorial

From	Paul Rubin <no.email@nospam.invalid>
Newsgroups	comp.lang.forth
Subject	Re: Reverse factorial
References	<jspim5$hcn$1@speranza.aioe.org> <92841902968435@frunobulax.edu> <jsq93l$983$1@speranza.aioe.org>
Date	2012-07-02 22:08 -0700
Message-ID	<7xhatpig84.fsf@ruckus.brouhaha.com> (permalink)
Organization	Nightsong/Fort GNOX

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"Ed" <invalid@nospam.com> writes:
>> 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.

I calculated 205022! exactly with GHC (uses GMP underneath).  This took
about 8 seconds including the output conversion (which used around 300MB
of memory, doesn't seem good).  The first digits are:

 1221277091680869925277501633213930559467939034445151838141073953...

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.

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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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