The spacing of the numbers below the smallest normal, at 11 significand bits
At its defaults it draws the spacing of the numbers below the smallest normal, at 11 significand bits. A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
subnormal-gap 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.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
bits: 11
The arguments are the ones The numbers below the smallest one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
bits: 5
The arguments are the ones The numbers below the smallest one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
bits: 4
The arguments are the ones The numbers below the smallest one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
bits: 7
The arguments are the ones The numbers below the smallest one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
bits: 9
The arguments are the ones The numbers below the smallest one passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A staircase of spacing against magnitude on logarithmic axes. It descends in steps from the right and then flattens to a horizontal line at the left. A dashed line drops away instead.
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.
23 distinct claims across 6 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.
with subnormals, 1 bits below the smallest normal is still a number — checked 10 times
a barrier depth between 0 and 60 octaves
a depth the dial draws
a depth the sweep draws
a significand between four and eleven bits
and flush-to-zero loses every one of them
each arithmetic survives a deeper barrier than the one before it
flush to zero never inflates it
past the barrier every component carries the error the dip made
rounding to nearest at the bottom of a format inflates the far well before it empties it, with subnormals and without
the componentwise condition number does not move with the depth
the normwise backward error stays below the unit roundoff at every depth, in both arithmetics
the subnormal band is one octave per significand bit, less one agree
there are neighbours to test
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.
The 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 well on the far side of the band
Gradual underflow was said to buy a predicate and not an answer, because a quantity that has decayed into the subnormal range is already lost. A quantity that passes through the band on its way somewhere else is not. The stationary distribution of a two-well chain, computed in half precision across a barrier whose top is two to the minus sixteen of the near well, keeps its far well's probability of 0.2454 to three digits with subnormals and returns exactly zero without them. The normwise backward error calls both answers exact.