ulp-staircase
At its defaults it draws the spacing between consecutive numbers at 53-bit precision. A log–log staircase of the gap between neighbouring representable numbers against magnitude. The gap doubles at every power of two and reaches one whole unit partway along.
ulp-staircase is one function in lib/figures/arith.js —
arithmetic — what a float holds, and what it loses holding it. 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 log–log staircase of the gap between neighbouring representable numbers against magnitude. The gap doubles at every power of two and reaches one whole unit partway along.
bits: 24
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 log–log staircase of the gap between neighbouring representable numbers against magnitude. The gap doubles at every power of two and reaches one whole unit partway along.
bits: 11
The arguments are the ones The numbers below the smallest one 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 log–log staircase of the gap between neighbouring representable numbers against magnitude. The gap doubles at every power of two and reaches one whole unit partway along.
bits: 53
The arguments are the ones The swap that is not optional 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 log–log staircase of the gap between neighbouring representable numbers against magnitude. The gap doubles at every power of two and reaches one whole unit partway along.
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.
6 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.
the gap at 2^0 agree — asserted 4 times
and beyond 2^(p−1) the gap is at least one whole unit
the gap above 1 is machine epsilon
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 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.
Elimination, and the swapThe swap that is not optional
Run elimination without a row interchange on a matrix that needs one and nothing announces a failure. There is no division by zero, no warning, and an answer of the right shape. It is simply wrong, and how wrong depends on a number you did not look at.
The arithmetic underneathWhat a float can hold
The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.
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.