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Comment by psyklic | Hacker News Reader

There seems to be a clear pattern for > 5 digits after the decimal:

  5 digits - Valid: <=0.39794, Invalid: >=0.39795
    10^0.39794 < 2.5
    10^0.39795 > 2.5

  6 digits - Valid: <=0.043429, Invalid: >=0.043431
  7 digits - Valid: <=0.0043429, Invalid: >=0.0043431
  8 digits - Valid: <=0.00043429, Invalid: >=0.00043431

The pattern seems to continue as the number of decimal places increases.

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10^0.043429 ~= e^0.1

10^0.0043429 ~= e^0.01

10^0.00043429 ~= e^0.001

So that gives some insight into the algorithm that is being used

Using this we can actually predict the smallest exponent that fails for other bases:

  base	exponent (6 digits)
  10	0.043429
  11	0.041703
  12	0.040243
  2	0.144271
  3	0.091024
  4	0.072135
  1000	0.014476

(the calculated exponent for two is actually 0.144270, but since it ends in zero the first failure occurs at 0.144271)

Just checked, calc fails at 11^0.41703 (curiously not at 41703, but 41704, maybe a floating point thing). Maybe it changes base to e and uses the Taylor series for e^x, but fails if x is too precise for what it expects?

Fantastic, that confirms it! And kudos to barbegal for recognizing the base.

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