THE FOLD / SPAWN / THE TOOLCHAIN / THE UNIT IN LAST PLACE
THE UNIT IN LAST PLACE
the ruler changes length as you walk
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The distance to the next representable number is not a constant. It doubles at every power of two, so precision is a function of where you are standing.
LIT verified live. At 1 the ulp is 2-52 and it doubles at every octave, so by 1016 the ulp exceeds 1 and
LIT verified live. At 1 the ulp is 2-52 and it doubles at every octave, so by 1016 the ulp exceeds 1 and
1e16 + 1 === 1e16 exactly — while 1e15 + 1 still moves. And 253 is the last integer whose every predecessor is also representable.2 HOW IT WAS WEAVED · AI + HUMAN
An ulp is the unit in the last place: the gap between a float and its neighbour. It is the natural unit for error in floating point, and the reason “accurate to six decimal places” is a claim that needs a magnitude attached.
AVAN (AI) measured the ruler at seven magnitudes rather than quoting the exponent rule, because the consequence is what matters: there is a specific decade where adding 1 stops changing a number, and it sits closer to everyday values than most people expect.
AVAN (AI) measured the ruler at seven magnitudes rather than quoting the exponent rule, because the consequence is what matters: there is a specific decade where adding 1 stops changing a number, and it sits closer to everyday values than most people expect.
3 ONE DIMENSION
One ulp, measured at seven magnitudes.
4 TWO DIMENSIONS · INTERACTIVE
Walk up the number line and watch the step grow.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a ruler that stretches as it travels.
AVAN’s addition (the inverse-companion): the forward reading is that precision degrades with magnitude. The inverse is that relative precision never degrades at all. The ulp grows exactly in step with the value, so the number of significant digits is constant everywhere — what changes is the absolute gap, and only code that mixes magnitudes ever notices. Read backwards, floating point is not losing accuracy as numbers grow; it is holding relative accuracy fixed, and the failures come from summing quantities that never belonged on the same scale.
LIT at 1 the ulp is 2^-52 and it doubles at every octave, so by 1e16 the ulp exceeds 1 and 1e16 plus 1 equals 1e16 exactly, while 1e15 plus 1 still moves - and 2^53 is the last integer whose every predecessor is also representable
FIG An ulp is the unit in the last place: the gap between a float and its neighbour. It is the natural unit for error in floating point, and the reason 'accurate to six decimal places' is a claim that needs a magnitude attached. AVAN measured the ruler at seven magnitudes rather than quoting the exponent rule, because the consequence is what matters: there is a specific decade where adding 1 stops changing a number, and it sits closer to everyday values than most people expect.
FIG An ulp is the unit in the last place: the gap between a float and its neighbour. It is the natural unit for error in floating point, and the reason 'accurate to six decimal places' is a claim that needs a magnitude attached. AVAN measured the ruler at seven magnitudes rather than quoting the exponent rule, because the consequence is what matters: there is a specific decade where adding 1 stops changing a number, and it sits closer to everyday values than most people expect.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN