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Author tim.peters mark.dickinson, ned.deily, rhettinger, serhiy.storchaka, steven.daprano, tim.peters, vstinner 2016-09-18.01:19:10 -1.0 Yes <1474161551.86.0.0964193189725.issue27761@psf.upfronthosting.co.za>
Content
```Let me clarify something about the extended algorithm:  the starting guess is no longer the most significant source of error.  It's the `mul(D(x), pow(D2, e))` part.  `D(x)` is exact, but `pow(D2, e)` may not be exactly representable with 26 decimal digits, and the multiplication may also need to round.  Those 2 operations may each suffer a half ulp error (but "ulp" in this case is 1 in the 26th decimal digit).

The rough error analysis is straightforward to extend, but I used wxMaxima to do the Taylor expansions and to ensure I didn't drop any terms.

So suppose that instead of exactly x*2**e we get x*2**e*(1+r1) for some small |r1| (on the order of 10**-25), and instead of the exact n'th root `t` the starting guess g = t*(1+r2) for some small |r2| (on the order of 2**-50).  Then the mathematical value of the corrected `g` is `t` times (dropping terms higher than quadratic in `r1` and `r2`):

1   + r1/n   - (n-1)/n * r1*r2   + (n-1)/2 * r2**2

The largest term (besides 1) by far is now `r1/n` (assuming a not-insanely-large `n`).  But that's still "way at the end" of the extended-precision decimal format, which _should be_ ;-) intuitively obvious.  It's too small to have a direct effect on the last bit of the 53-bit result (but can certainly affect "close call" rounding cases).```
History
Date User Action Args
2016-09-18 01:19:11tim.peterssetrecipients: + tim.peters, rhettinger, mark.dickinson, vstinner, ned.deily, steven.daprano, serhiy.storchaka