search-map
At its defaults it draws every box a search over [−3, 3]² leaves behind, at a stopping width of 0.05. A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
search-map is one function in lib/figures/branchbound.js —
from a verdict to a search — where the box is cut, and what that costs. 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
minWidth: 0.02
The arguments are the ones A bound that is proved 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
minWidth: 0.1
The arguments are the ones An answer that is known 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
minWidth: 0.05
The arguments are the ones Proving the answer is in the box 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
minWidth: 0.2
The arguments are the ones The exact answer to a nearby problem 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
minWidth: 0.3
The arguments are the ones The numbers below the smallest one 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.
A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.
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 6 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.
the root at (1.414, 1.414) is inside a verified box — asserted 2 times
a square that contains both roots
a stopping width the picture has room for
a subdivision point inside the box
and inside no box proved empty
LU is for square matrices
the search terminates
with both roots verified
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 70
of 151 generators —
55 print a residual and
15 are exempt with a published reason;
81 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 bound that is proved
Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
The arithmetic underneathProving the answer is in the box
Every other method here computes a number and estimates how wrong it is. This one returns a verdict: there is exactly one solution in this box, or there is none, or — the honest third outcome — nothing can be said. Two of the three are proofs about infinitely many points from finitely many operations.
Two errors, and whose fault they areThe exact answer to a nearby problem
A good algorithm does not give an approximate answer to your problem. It gives the exact answer to a problem very close to yours — and once that is the definition, a wrong result has two possible authors and they can be measured apart.
The arithmetic underneathThe numbers below the smallest one
Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.
The arithmetic underneathWhat a float can hold
The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.
The arithmetic underneathWhere the box is cut
A branch-and-bound with an interval operator settles a whole square — two roots proved unique, forty-two regions proved empty, nothing left undecided, in 87 evaluations. Move the roots so one lands on the first bisection and it proves nothing at all, at any depth. Cutting at 0.485 instead of 0.5 finds both, in a quarter of the work.