▁half ▁an ▁ul p ▁like ▁S Q RT ? ▁Because ▁nobody ▁knows ▁how ▁much ▁computation ▁it ▁would ▁cost ... ▁No ▁general ▁way ▁exists ▁to ▁predict ▁how ▁many ▁extra ▁digits ▁will ▁have ▁to ▁be ▁carried ▁to ▁compute ▁a ▁trans c endent al ▁expression ▁and ▁round ▁it ▁ ▁to ▁some ▁pre ass igned ▁number ▁of ▁digits . ▁Even ▁the ▁fact ▁( if ▁true ) ▁that ▁a ▁finite ▁number ▁of ▁extra ▁digits ▁will ▁ultimately ▁suff ice ▁may ▁be ▁a ▁deep ▁theorem ." ▁ ▁The ▁IE EE ▁ 7 5 4 ▁floating - point ▁standard ▁guarante es ▁that ▁add , ▁subtract , ▁multiply , ▁divide , ▁f used ▁multiply – add , ▁square ▁root , ▁and ▁floating - point ▁remainder ▁will ▁give ▁the ▁correctly ▁rounded ▁result ▁of ▁the ▁infinite - prec ision ▁operation . ▁No ▁such ▁guarantee ▁was ▁given ▁in ▁the ▁ 1 9 8 5 ▁standard ▁for ▁more ▁complex ▁functions ▁and ▁they ▁are ▁typically ▁only ▁accurate ▁to ▁within ▁the ▁last ▁bit ▁at ▁best . ▁However , ▁the ▁ 2 0 0 8 ▁standard ▁guarante es ▁that ▁conform ing ▁implementations ▁will ▁give ▁correctly ▁rounded ▁results ▁which ▁respect ▁the ▁active ▁round ing ▁mode ; ▁implementation ▁of ▁the ▁functions , ▁however , ▁is ▁optional . ▁ ▁Using ▁the ▁G elf ond – Sch ne ider ▁theorem ▁and ▁Lin dem ann – We ier str ass ▁theorem ▁many ▁of ▁the ▁standard ▁elementary ▁functions ▁can ▁be ▁proved ▁to ▁return ▁trans c endent al ▁results ▁when ▁given ▁rational ▁non - zero ▁arguments ; ▁therefore ▁it ▁is ▁always ▁possible ▁to ▁correctly ▁round ▁such