AngouriMath
ApproachScales
Field
Summary
How close in the approach is sampled when asking whether a function stays real on
the way to its destination. Powers of ten rather than a fixed step, because what
matters is near and not evenly spaced:sqrt(x - 1) is real just to the left
of 2 and not just to the left of 1, and only the nearer samples tell those apart.
the way to its destination. Powers of ten rather than a fixed step, because what
matters is near and not evenly spaced:
of 2 and not just to the left of 1, and only the nearer samples tell those apart.
Remarks
It stops at a thousandth, and the reason is cost rather than principle. The sample
point goes into the expression, and an expression may put it in an exponent:
(1 + x)^(1/x) at 1e-9 is 1.000000001 raised to a billion, evaluated
at a hundred digits, and asking that six times on every limit the machinery takes
turned a 20 ms limit into a timeout.
destination than this reaches, and a miss leaves the limit answered exactly as it
was before. The guard withdraws on evidence and never on the absence of it, so the
cheap end of that trade is the safe one.
point goes into the expression, and an expression may put it in an exponent:
at a hundred digits, and asking that six times on every limit the machinery takes
turned a 20 ms limit into a timeout.
destination than this reaches, and a miss leaves the limit answered exactly as it
was before. The guard withdraws on evidence and never on the absence of it, so the
cheap end of that trade is the safe one.
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