AngouriMath
GrewTooMuch(AngouriMath.Entity,AngouriMath.Entity)
Method (no overloads)
Summary
Whether differentiating both parts has left a quotient bigger than the one it came
from by more than a step on the way to an answer accounts for. Each step asks what
the two parts of its quotient tend to, and a bigger quotient makes those two
questions harder than the one being answered, so a step that grows without bound is
a step away from an answer rather than towards one. One step of
x^(3/2) * sqrt(1 + 1/x^2) / x^2 adds twelve nodes and the next adds twenty-six,
which is that shape.
from by more than a step on the way to an answer accounts for. Each step asks what
the two parts of its quotient tend to, and a bigger quotient makes those two
questions harder than the one being answered, so a step that grows without bound is
a step away from an answer rather than towards one. One step of
x^(3/2) * sqrt(1 + 1/x^2) / x^2 adds twelve nodes and the next adds twenty-six,
which is that shape.
Remarks
The room was a flat eight nodes, which is what growth looks like when the divisor is
a power of the variable: differentiating that shrinks it, so the only thing that
grows is the dividend and it grows by addition. A divisor that is a *product* of
vanishing factors grows by the product rule instead, which is multiplication --
x^2 * sin(x)^2 differentiates into a sum of two products, each about the size of the
whole. Such a chain still terminates, since the order at which the divisor vanishes
falls by one at every step, but it grows for as long as it takes to get there and
only then collapses: 1/x^2 - 1/sin(x)^2 at 0 goes 17 -> 26 -> 33 -> 46 nodes before
four steps settle it at -1/3. A flat budget cannot say that, and turned the first of
those steps away, leaving the descent to answer +oo - +oo -- NaN, the claim that the
limit does not exist -- for a limit that exists.
proportional where the growth is proportional and unchanged for the small quotients
the flat budget was measured on. It is wider than the old rule everywhere, so no
chain that reached an answer before is turned away now. It costs nothing on the
shape that motivated the flat budget either: nineteen nodes to thirty-one is above
three fifths again as surely as it is above eight more, so that chain still stops at
the same step, and the limit still answers 0 in the same 240 ms. The corpus is
unchanged at 112/117 and, in total time, within noise of where it was.
a power of the variable: differentiating that shrinks it, so the only thing that
grows is the dividend and it grows by addition. A divisor that is a *product* of
vanishing factors grows by the product rule instead, which is multiplication --
x^2 * sin(x)^2 differentiates into a sum of two products, each about the size of the
whole. Such a chain still terminates, since the order at which the divisor vanishes
falls by one at every step, but it grows for as long as it takes to get there and
only then collapses: 1/x^2 - 1/sin(x)^2 at 0 goes 17 -> 26 -> 33 -> 46 nodes before
four steps settle it at -1/3. A flat budget cannot say that, and turned the first of
those steps away, leaving the descent to answer +oo - +oo -- NaN, the claim that the
limit does not exist -- for a limit that exists.
proportional where the growth is proportional and unchanged for the small quotients
the flat budget was measured on. It is wider than the old rule everywhere, so no
chain that reached an answer before is turned away now. It costs nothing on the
shape that motivated the flat budget either: nineteen nodes to thirty-one is above
three fifths again as surely as it is above eight more, so that chain still stops at
the same step, and the limit still answers 0 in the same 240 ms. The corpus is
unchanged at 112/117 and, in total time, within noise of where it was.
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