The same problem solved at eight precisions
At its defaults it draws the same problem solved at eight precisions. Relative error against the number of significand bits, on a logarithmic vertical axis. The error runs from 460 at 8 bits to 5.53·10⁹ at 53 — it rises with the precision. The horizontal line is the floor the four filters reach, 0.1406, which no precision approaches.
precision-nonknob is one function in lib/figures/compose.js —
four knobs — one floor under them, and a fifth that is not a knob at all. 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.
Relative error against the number of significand bits, on a logarithmic vertical axis. The error runs from 460 at 8 bits to 5.53·10⁹ at 53 — it rises with the precision. The horizontal line is the floor the four filters reach, 0.1406, which no precision approaches.
noise: 0.01
The arguments are the ones A rule that is correct and unusable passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Relative error against the number of significand bits, on a logarithmic vertical axis. The error runs from 460 at 8 bits to 5.53·10⁹ at 53 — it rises with the precision. The horizontal line is the floor the four filters reach, 0.1406, which no precision approaches.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
6 distinct claims across 2 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 noise level small enough to be noise
a size the dense solve is affordable at
and the most precise arithmetic is the worst of them
LU is for square matrices
matmul shapes agree
no precision comes near the floor the filters reach
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 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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 rule that is correct and unusable
Cramer's rule gives every component of the solution in closed form, in terms of determinants, and it is a theorem. On two-by-two systems whose rows are nearly parallel it returns an answer with a backward error of 458 units of roundoff where elimination returns 1.3 — on a matrix whose condition number is 32,000 and which elimination solved perfectly.
Methods that were designed apartFour knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.