Tim Peters added the comment:
Adding one more version of the last code, faster by cutting the number of extra
digits used, and by playing "the usual" low-level CPython speed tricks.
I don't claim it's always correctly rounded - although I haven't found a
specific case where it isn't. I do claim it will return the exact result
whenever the exact result is representable as a float.
I also claim it's "fast enough" - this version does on the high end of 50 to
100 thousand roots per second on my box, pretty much independent of `n`.
import decimal
c = decimal.DefaultContext.copy()
c.prec = 25
def rootn(x, n,
D=decimal.Decimal,
pow=c.power,
mul=c.multiply,
add=c.add,
div=c.divide):
g = D(x**(1.0/n))
n1 = D(n-1)
g = div(add(mul(n1, g),
div(D(x), pow(g, n1))),
n)
return float(g)
del decimal, c
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