Rounding 1.42 to 4 significand bits, five ways
At its defaults it draws rounding 1.42 to 4 significand bits, five ways. A number line of the representable values in one octave, with the exact value marked between two of them and a row for each rounding mode showing which neighbour it returns.
rounding-line 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.
A number line of the representable values in one octave, with the exact value marked between two of them and a row for each rounding mode showing which neighbour it returns.
value: 1.05, bits: 4
The arguments are the ones A coin flip that fixes the average passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A number line of the representable values in one octave, with the exact value marked between two of them and a row for each rounding mode showing which neighbour it returns.
value: 1.42
The arguments are the ones The direction the error leans passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A number line of the representable values in one octave, with the exact value marked between two of them and a row for each rounding mode showing which neighbour it returns.
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.
8 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 value inside one octave, so the spacing is uniform across the picture
few enough bits for the representable numbers to be drawn apart
nearest lands on one of the two neighbours
the share of stochastic draws that round up is the value's position between the neighbours
the two directed modes bracket the value by one step
toward +∞ lands on one of the two neighbours
toward −∞ lands on one of the two neighbours
toward zero lands on one of the two neighbours
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 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.