format-plane
At its defaults it draws precision against range, with the 4-bit-exponent formats marked. A scatter of formats with unit roundoff across and largest finite value up, both logarithmic. The two eight-bit formats sit at the lower left, joined to each other and to a marked position below one of them.
format-plane 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.
A scatter of formats with unit roundoff across and largest finite value up, both logarithmic. The two eight-bit formats sit at the lower left, joined to each other and to a marked position below one of them.
expBits: 4
The arguments are the ones Buying the accuracy back 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 scatter of formats with unit roundoff across and largest finite value up, both logarithmic. The two eight-bit formats sit at the lower left, joined to each other and to a marked position below one of them.
expBits: 5
The arguments are the ones One exponent for thirty-two numbers 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 scatter of formats with unit roundoff across and largest finite value up, both logarithmic. The two eight-bit formats sit at the lower left, joined to each other and to a marked position below one of them.
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.
7 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.
an exponent width one of these formats uses
and E5M2 reaches 57344
and its range is 128 times wider
E4M3 reaches 448
E5M2's steps are twice as coarse
some format uses that exponent width
under IEEE's rules the same bits would reach only 240
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.
Buying the accuracy back
Factorise in single precision, then correct the answer using residuals computed in double, and the result is what a full double-precision solve would have given. Compute those residuals in single instead and the identical algorithm, at identical cost, recovers nothing.
The arithmetic underneathEight bits, and a format that breaks the rules
E4M3 reuses the exponent code IEEE reserves for infinities, so it reaches 448 where the same bits under IEEE's rules would reach 240 — and has no infinity left to signal an overflow with. The same computation is a NaN on one conforming device and 448 on another.
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.
The arithmetic underneathThe other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.