[issue34376] Improve accuracy of math.hypot() and math.dist()

[Tim Peters]

Tim Peters <tim at python.org> added the comment:

Not that it matters:  "ulp" is a measure of absolute error, but the script is computing some notion of relative error and _calling_ that "ulp".  It can understate the true ulp error by up to a factor of 2 (the "wobble" of base 2 fp).

Staying away from denorms, this is an easy way to get one ulp with respect to a specific 754 double:

    def ulp(x):
        import math
        mant, exp = math.frexp(x)
        return math.ldexp(0.5, exp - 52)

Then, e.g.,

>>> x
1.9999999999999991
>>> y
1.9999999999999996
>>> y - x
4.440892098500626e-16
>>> oneulp = ulp(x)
>>> oneulp # the same as sys.float_info.epsilon for this x
2.220446049250313e-16
>>> (y - x) / oneulp
2.0

which is the true absolute error of y wrt x.

>>> x + 2 * oneulp == y
True

But:

>>> (y - x) / x
2.220446049250314e-16
>>> _ / oneulp
1.0000000000000004

understates the true ulp error by nearly a factor of 2, while the mathematically (but not numerically) equivalent spelling used in the script:

>>> (y/x - 1.0) / oneulp
1.0

understates it by exactly a factor of 2.

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Python tracker <report at bugs.python.org>
<https://bugs.python.org/issue34376>
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