Generator

precision-threshold

One function in the mixed library, called 4 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 35 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 how far short of a double solve refinement finishes, and where each format stops. A log-log plot of how far short of a double-precision solve iterative refinement finishes, against the condition number, for three low-precision formats. Each curve sits flat at one and then climbs steeply past a vertical mark showing that format's threshold.

precision-threshold is one function in lib/figures/mixed.js — mixed precision — refinement, and the threshold at 1/u. 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 plot of how far short of a double-precision solve iterative refinement finishes, against the condition number, for three low-precision formats. Each curve sits flat at one and then climbs steeply past a vertical mark showing that format's threshold.

`n: 24`

The arguments are the ones A bound that is proved 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.

`n: 30`

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.

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.

35 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.

the constructed matrix has κ = 10 — asserted 10 times

fp32 lands on the double answer below its threshold, at κ = 10 — asserted 3 times

fp16 / tf32 lands on the double answer below its threshold, at κ = 10 — asserted 2 times

and finishes hopelessly short of it above, at κ = 10¹⁰

and finishes hopelessly short of it above, at κ = 10⁴

and finishes hopelessly short of it above, at κ = 10⁵

and finishes hopelessly short of it above, at κ = 10⁶

and finishes hopelessly short of it above, at κ = 10⁷

and finishes hopelessly short of it above, at κ = 10⁸

and finishes hopelessly short of it above, at κ = 10⁹

bfloat16 has a region below its threshold in this sweep

bfloat16 has a region past its threshold in this sweep

bfloat16 lands on the double answer below its threshold, at κ = 10

fp16 / tf32 has a higher threshold than bfloat16

fp16 / tf32 has a region below its threshold in this sweep

fp16 / tf32 has a region past its threshold in this sweep

fp32 has a higher threshold than fp16 / tf32

fp32 has a region below its threshold in this sweep

fp32 has a region past its threshold in this sweep

fp32 lands on the double answer below its threshold, at κ = 10⁴

fp32 lands on the double answer below its threshold, at κ = 10⁵

LU is for square matrices

matmul shapes agree

Against the rule

It draws a decomposition and prints its residual. It calls luFactor, refine, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

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.

The arithmetic underneath

A bound that is proved

Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.

The arithmetic underneath

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 underneath

Where 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.

The whole library · All essays · What must fail