drift-growth
At its defaults it draws relative error of a sum against the number of terms, at 24 significand bits. Five curves of relative error against the number of terms, both axes logarithmic. Three rise with unit slope; two rise with about half that.
drift-growth is one function in lib/figures/rounding.js —
rounding — the mode, not the precision, that decides the exponent of the error. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Five curves of relative error against the number of terms, both axes logarithmic. Three rise with unit slope; two rise with about half that.
bits: 11
The arguments are the ones A coin flip that fixes the average passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Five curves of relative error against the number of terms, both axes logarithmic. Three rise with unit slope; two rise with about half that.
bits: 24
The arguments are the ones The direction the error leans passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Five curves of relative error against the number of terms, both axes logarithmic. Three rise with unit slope; two rise with about half that.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
9 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
a precision inside the range this comparison is drawn over
and by ten thousand terms they are more than a decade apart
and the biased modes are worse still
at a precision this low the sum has stagnated and even nearest grows linearly
down accumulates linearly
nearest accumulates like a random walk
stochastic accumulates like a random walk
up accumulates linearly
zero accumulates linearly
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A coin flip that fixes the average
Add 0.1 to 256 a thousand times at eight significand bits and the answer is 256. Not approximately — the total never moves, not once, and no error bound says so. Round up one time in twenty instead of never, and it arrives at 348 against a true 356.
The arithmetic underneathThe direction the error leans
The size of one rounding error is set by the precision. How ten thousand of them combine is set by something else entirely — the rounding mode — and the fitted exponents are 0.47 for round-to-nearest and 1.01 for round-toward-infinity, on identical data at identical precision.
The arithmetic underneathThe order they are added in
Addition is associative in the algebra and is not associative in the arithmetic. The same million numbers, added in a different order, give answers that differ in the third significant figure — and the fix is not a wider float, it is a different order.
Least squares, and the road not to takeThe road that squares the problem
The normal equations are the first method every course teaches and the method no library uses. Forming AᵀA squares the condition number, and below ε = √u it does not degrade — it produces a matrix that is exactly singular, from data that was perfectly usable.