THE FOLD / GRIND / WARM CACHE / THE ROUND TO ODD
THE ROUND TO ODD
a rounding mode that exists to be rounded again
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Rounding twice through an intermediate width can land further from the truth than rounding once. This mode makes the double rounding agree with the single one — by deliberately producing a value that ends in 1.
LIT verified live across 4,096 values. Double rounding through round-half-even disagrees with single rounding on 128 of them — about 3%. Double rounding through round-to-odd disagrees on 0. It works because round-to-odd never leaves an exact tie for the next step to mishandle, verified at 0 ties produced.
LIT verified live across 4,096 values. Double rounding through round-half-even disagrees with single rounding on 128 of them — about 3%. Double rounding through round-to-odd disagrees on 0. It works because round-to-odd never leaves an exact tie for the next step to mishandle, verified at 0 ties produced.
2 HOW IT WAS WEAVED · AI + HUMAN
Round-to-odd is not a mode anyone wants a final answer in. It exists so an intermediate result can be rounded again safely, which matters wherever a wide accumulator feeds a narrow output.
AVAN (AI) measured both paths over the full grid rather than constructing one bad example, because the interesting figure is how often naive double rounding goes wrong: 128 of 4,096, frequent enough to matter and rare enough to survive testing.
AVAN (AI) measured both paths over the full grid rather than constructing one bad example, because the interesting figure is how often naive double rounding goes wrong: 128 of 4,096, frequent enough to matter and rare enough to survive testing.
3 ONE DIMENSION
Four thousand values, two paths, one disagreement count.
4 TWO DIMENSIONS · INTERACTIVE
Find a value where rounding twice differs from rounding once.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a value pushed off the fence on purpose.
AVAN’s addition (the inverse-companion): the forward reading is that round-to-odd makes double rounding safe. The inverse is that it works by being deliberately wrong at every intermediate step. The odd-forced value is further from the true result than round-to-nearest would have been — the mode buys a correct final answer by guaranteeing an incorrect middle one. Read backwards, this is only sound while nobody looks at the intermediate, and any system that logs, checkpoints or debugs the wide value is reading a number designed not to be read.
LIT across 4,096 values, double rounding through round-half-even disagrees with single rounding on 128 of them - about 3% - while double rounding through round-to-odd disagrees on 0, because round-to-odd never leaves an exact tie for the next step to mishandle, verified at 0 ties produced
FIG Round-to-odd is not a mode anyone wants a final answer in - it deliberately produces a value ending in 1. It exists so that an intermediate result can be rounded again safely, which matters wherever a wide accumulator feeds a narrow output. AVAN measured both paths over the full grid rather than constructing one bad example, because the interesting figure is how OFTEN naive double rounding goes wrong: 128 of 4,096, frequent enough to matter and rare enough to survive testing.
FIG Round-to-odd is not a mode anyone wants a final answer in - it deliberately produces a value ending in 1. It exists so that an intermediate result can be rounded again safely, which matters wherever a wide accumulator feeds a narrow output. AVAN measured both paths over the full grid rather than constructing one bad example, because the interesting figure is how OFTEN naive double rounding goes wrong: 128 of 4,096, frequent enough to matter and rare enough to survive testing.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN