At tiny magnitudes ROUNDUP may add no unit, and the two signs differ

FN-ROUNDUP-003: At tiny magnitudes ROUNDUP may add no unit, and the two signs differ

Excel differs from the mathematical answer · measured · recorded 2026-11-04

ROUNDUP adds no unit when the unit itself is below the normal range, whichever the sign. For a positive number it also adds none when the 15-digit reading, as binary64, lies less than 2^-1026 above the truncation; negative numbers have no such distance. So ROUNDUP(1.0001086461844975E-307,307) is 1E-307 and ROUNDUP(-1.0001086461844975E-307,307) is -2E-307, with the same digits dropped. At -1.2225E-307 and 308 places the unit, 1E-308, is subnormal, and the answer stays at -1.2E-307.

Correct value: The number has a nonzero digit past 307 places, so rounding away from zero at 307 places gives 2E-307; nearest binary64.

Correct value: The number has a nonzero digit past 308 places, so rounding away from zero at 308 places gives -1.3E-307; nearest binary64.

Basis: ROUNDUP rows match under these rules as counted in FN-ROUND-001. In the 1,148 recorded rows nearest the 2^-1026 boundary this quantity has 1 violation, itself an artefact of how the row was classified; the exact value, the reading and x minus the truncation give 54 to 547. The opposite-sign pair is a reading from Excel build 20527.

Open: 14 ROUNDUP rows of tiny magnitude stay unmatched, in the near-tie class of FN-ROUND-002.