Tim Peters <[email protected]> added the comment:
My apologies if nobody cares about this ;-) I just think it's helpful if we all
understand what really helps here.
> Pretty much the whole trick relies on computing
> "sumsq - result**2" to greater than basic machine
> precision.
But why is that necessary at all? Because we didn't compute the sqrt of the
sum of the squares to begin with - the input to sqrt was a _rounded_-to-double
approximation to the sum of squares. The correction step tries to account for
the sum-of-squares bits sqrt() didn't see to begin with.
So, e.g., instead of doing
result = sqrt(to_float(parts))
a, b = split(result)
parts = add_on(-a*a, parts)
parts = add_on(-b*b, parts)
parts = add_on(-2.0*a*b, parts)
x = to_float(parts)
result += x / (2.0 * result)
it works just as well to do just
result = float((D(parts[0] - 1.0) + D(parts[1])).sqrt())
where D is the default-precision decimal.Decimal constructor. Then sqrt sees
all the sum-of-squares bits we have (although the "+" rounds away those
following the 28th decimal digit).
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Python tracker <[email protected]>
<https://bugs.python.org/issue41513>
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