A 6.25-bit block format against 8-bit E4M3, read two ways
At its defaults it draws a 6.25-bit block format against 8-bit e4m3, read two ways. Four curves of relative error against the spread of the data. Two, for the block format, rise steeply when read as a median and gently when read as a norm; the other two are nearly flat.
block-crossing is one function in lib/figures/lowprec.js —
low precision — eight bits, the format that breaks the rules, and one shared exponent. 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.
Four curves of relative error against the spread of the data. Two, for the block format, rise steeply when read as a median and gently when read as a norm; the other two are nearly flat.
bits: 4
The arguments are the ones A bit buys an octave passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves of relative error against the spread of the data. Two, for the block format, rise steeply when read as a median and gently when read as a norm; the other two are nearly flat.
bits: 8
The arguments are the ones A bit buys an octave passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves of relative error against the spread of the data. Two, for the block format, rise steeply when read as a median and gently when read as a norm; the other two are nearly flat.
bits: 6
The arguments are the ones An ordering that does not wait for the numbers passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Four curves of relative error against the spread of the data. Two, for the block format, rise steeply when read as a median and gently when read as a norm; the other two are nearly flat.
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.
3 distinct claims across 4 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 block significand of four to eight bits
at the wide end a great many entries are gone outright
the two readings of the same data disagree about what happened
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 bit buys an octave
The outlier a block survives is exactly two raised to its significand width — 8 at three bits, 32 at five, 128 at seven, 512 at nine. Each extra bit doubles the range the block tolerates and halves the ordinary entry's error. Reordering the same numbers buys every octave at once and costs nothing.
Sparsity, and what elimination costsAn ordering that does not wait for the numbers
A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.
The arithmetic underneathOne exponent for thirty-two numbers
Share the exponent across a block and the cost per value drops from eight bits to 6.25, and the accuracy improves — up to about three octaves of spread inside a block. Past that a single outlier deletes the thirty-one values beside it, and the 2-norm barely notices.