e4m3-values
At its defaults it draws every value e4m3 can hold, and the one it cannot. A logarithmic axis with a vertical tick at every representable magnitude, crowding together towards the left and thinning to the right, with the top of the range marked.
e4m3-values 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 logarithmic axis with a vertical tick at every representable magnitude, crowding together towards the left and thinning to the right, with the top of the range marked.
expBits: 4
The arguments are the ones Eight bits, and a format that breaks the rules 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 logarithmic axis with a vertical tick at every representable magnitude, crowding together towards the left and thinning to the right, with the top of the range marked.
expBits: 5
The arguments are the ones Eight bits, and a format that breaks the rules 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 logarithmic axis with a vertical tick at every representable magnitude, crowding together towards the left and thinning to the right, with the top of the range marked.
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.
5 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.
and the largest of them is the format's largest finite value
every value drawn is one the format holds exactly
one of the eight-bit formats, which are small enough to draw entire
one of the two eight-bit splits
the format has enough values to be worth drawing
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.
Eight 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 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.
The arithmetic underneathWhere the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.