Tim Peters added the comment: 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). ---------- _______________________________________ Python tracker <rep...@bugs.python.org> <http://bugs.python.org/issue27761> _______________________________________ _______________________________________________ Python-bugs-list mailing list Unsubscribe: https://mail.python.org/mailman/options/python-bugs-list/archive%40mail-archive.com
