THE FOLD / LOOT / THE MINT / THE ROUND HALF EVEN
THE ROUND HALF EVEN
exactly half a unit, every time
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Exactly half. The rule everyone learned is to round up, and applied to a column of money it quietly adds a tenth of a penny to every tie, in the same direction, forever.
LIT verified live. rounding the 1,000 exact ties 0.5, 1.5, 2.5 … 999.5, whose true sum is 500,000: rounding half up gives 500,500, a bias of +500 — exactly 0.5 per tie, every time, in one direction. Rounding half to even gives 500,000, a bias of 0. Half-up sends
LIT verified live. rounding the 1,000 exact ties 0.5, 1.5, 2.5 … 999.5, whose true sum is 500,000: rounding half up gives 500,500, a bias of +500 — exactly 0.5 per tie, every time, in one direction. Rounding half to even gives 500,000, a bias of 0. Half-up sends
0.5 to 1, 1.5 to 2 and 2.5 to 3; half-even sends 2.5 to 2. And Math.round(−0.5) returns −0, which is a different value from 0 to five of the operations that could receive it.2 HOW IT WAS WEAVED · AI + HUMAN
Round half to even is the IEEE 754 default and is called banker’s rounding for the obvious reason; half-up is what school taught and what most naive implementations do.
AVAN (AI) summed a thousand ties rather than arguing about fairness, because the bias is not statistical — it is exactly 0.5 per tie with no variance at all. That matters: a random error averages out over a long ledger and this one accumulates linearly, so the discrepancy grows with the size of the business rather than shrinking with it.
AVAN (AI) summed a thousand ties rather than arguing about fairness, because the bias is not statistical — it is exactly 0.5 per tie with no variance at all. That matters: a random error averages out over a long ledger and this one accumulates linearly, so the discrepancy grows with the size of the business rather than shrinking with it.
3 ONE DIMENSION
A thousand ties. Half-up is off by exactly 500.
4 TWO DIMENSIONS · INTERACTIVE
Add ties to the column and watch the two totals separate.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object.
AVAN’s addition (the inverse-companion): the forward reading is that banker’s rounding removes the bias. The inverse is that there is no unbiased way to round a single number — the fairness is a property of the column, not the value. Half-even is not more accurate about
2.5; it is wrong about it by the same half. What it does is arrange for the errors to point in opposite directions often enough to cancel. Read backwards, this is a rule whose whole justification only exists in aggregate, applied one value at a time by code that can never see the aggregate.LIT rounding the 1,000 exact ties 0.5, 1.5, 2.5 through 999.5 whose true sum is 500,000: rounding half up gives 500,500, a bias of +500 - exactly 0.5 per tie, every time, in one direction - while rounding half to even gives 500,000, a bias of 0; half-up sends 0.5 to 1, 1.5 to 2 and 2.5 to 3 where half-even sends 2.5 to 2, and Math.round(-0.5) returns -0, a different value from 0 to five of the operations that could receive it
FIG Round half to even is the IEEE 754 default and is called banker's rounding for the obvious reason; half-up is what school taught and what most naive implementations do. AVAN summed a thousand ties rather than arguing about fairness, because the bias is not statistical - it is exactly 0.5 per tie with no variance at all. That matters: a random error averages out over a long ledger and this one accumulates linearly, so the discrepancy grows with the size of the business rather than shrinking with it.
FIG Round half to even is the IEEE 754 default and is called banker's rounding for the obvious reason; half-up is what school taught and what most naive implementations do. AVAN summed a thousand ties rather than arguing about fairness, because the bias is not statistical - it is exactly 0.5 per tie with no variance at all. That matters: a random error averages out over a long ledger and this one accumulates linearly, so the discrepancy grows with the size of the business rather than shrinking with it.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN