format-map
At its defaults it draws a 16-bit budget, split between range and precision. Two curves against the width of the exponent field. One rises steeply and one falls in a straight line. Vertical lines mark the splits real hardware formats use.
format-map is one function in lib/figures/range.js —
range — overflow, the subnormal band, and the norm that dies before it is one. 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.
Two curves against the width of the exponent field. One rises steeply and one falls in a straight line. Vertical lines mark the splits real hardware formats use.
width: 16
The arguments are the ones A norm that overflows before it is a norm 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.
Two curves against the width of the exponent field. One rises steeply and one falls in a straight line. Vertical lines mark the splits real hardware formats use.
width: 32
The arguments are the ones The other half of a format 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.
Two curves against the width of the exponent field. One rises steeply and one falls in a straight line. Vertical lines mark the splits real hardware formats use.
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.
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.
fp16 is on the curve this figure draws — asserted 2 times
fp16's significand comes out of the budget agree — asserted 2 times
a format width the hardware actually uses
bfloat16 is on the curve this figure draws
bfloat16's significand comes out of the budget agree
there is a range of splits to draw
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.
A norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
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 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 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.