Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The spacing between consecutive numbers at 53-bit precision

One function in the arith library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 6 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

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.

The spacing between consecutive numbers at 53-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numbera gap of one whole unitat 1: 2.2·10⁻¹⁶at a million: 1.2·10⁻¹⁰gap reaches 1: 153-bit significandthe gap follows the magnitude

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 What a float can hold passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The spacing between consecutive numbers at 24-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numberat 1: 1.2·10⁻⁷at a million: 0.063gap reaches 1: 124-bit significandthe gap follows the magnitude

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: 16

The arguments are the ones What a float can hold passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The spacing between consecutive numbers at 16-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numberat 1: 3.1·10⁻⁵at a million: 16gap reaches 1: 116-bit significandthe gap follows the magnitude

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: 20

The arguments are the ones What a float can hold passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The spacing between consecutive numbers at 20-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numberat 1: 1.9·10⁻⁶at a million: 1gap reaches 1: 120-bit significandthe gap follows the magnitude

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: 32

The arguments are the ones What a float can hold passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The spacing between consecutive numbers at 32-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numbera gap of one whole unitat 1: 4.7·10⁻¹⁰at a million: 2.4·10⁻⁴gap reaches 1: 132-bit significandthe gap follows the magnitude

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: 40

The arguments are the ones What a float can hold passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The spacing between consecutive numbers at 40-bit precisionA 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.110³10⁶10⁹10¹²10¹⁵10¹⁸10⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³110³magnitude of the numbergap to the next representable numbera gap of one whole unitat 1: 1.8·10⁻¹²at a million: 9.5·10⁻⁷gap reaches 1: 140-bit significandthe gap follows the magnitude

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 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.

6 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.

the gap at 2^0 agree — checked 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 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 arithmetic underneath

What 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 whole library · All essays · What must fail