Dan Bron wrote:
> How can I estimate the number of digits in   !200e6  ?

Try Stirling's approximation.

stirling=:(* ^.)-]
log10=:(^. 10) %~ stirling

   10^. !200x
374.897
   log10 200
373.347
   log10 200e6
1.57335e9

Best wishes,

John


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