▁functions . ▁However , ▁determ ining ▁a ▁limit ▁for ▁a ▁given ▁precision ▁on ▁how ▁accurate ▁results ▁need ▁to ▁be ▁computed , ▁before ▁a ▁correctly ▁rounded ▁result ▁can ▁be ▁guaranteed , ▁may ▁demand ▁a ▁lot ▁of ▁computation ▁time ▁or ▁may ▁be ▁out ▁of ▁reach . ▁In ▁practice , ▁when ▁this ▁limit ▁is ▁not ▁known ▁or ▁too ▁large , ▁some ▁decision ▁has ▁to ▁be ▁made ▁in ▁the ▁implementation ▁( see ▁below ), ▁but ▁according ▁to ▁a ▁probabil istic ▁model , ▁correct ▁round ing ▁can ▁be ▁satisfied ▁with ▁a ▁very ▁high ▁probability . ▁ ▁Some ▁programming ▁packages ▁offer ▁correct ▁round ing . ▁The ▁GNU ▁M PF R ▁package ▁gives ▁correctly ▁rounded ▁arbitrary ▁precision ▁results . ▁Some ▁other ▁libraries ▁implement ▁elementary ▁functions ▁with ▁correct ▁round ing ▁in ▁double ▁precision : ▁IBM ' s ▁lib ult im , ▁in ▁round ing ▁to ▁nearest ▁only . ▁This ▁library ▁uses ▁up ▁to ▁ 7 6 8 ▁bits ▁of ▁working ▁precision . ▁Sun ▁Mic ros ystem s ' s ▁lib m cr , ▁in ▁the ▁ 4 ▁round ing ▁modes . ▁For ▁the ▁difficult ▁cases , ▁this ▁library ▁also ▁uses ▁multiple ▁precision , ▁and ▁the ▁number ▁of ▁words ▁is ▁increased ▁by ▁ 2 ▁each ▁time ▁the ▁Table - maker ' s ▁d ile m ma ▁occurs ▁( with ▁undefined ▁behavior ▁in ▁the ▁very ▁unlikely ▁event ▁that ▁some ▁limit ▁of ▁the ▁machine ▁is ▁reached ). ▁CR lib m , ▁written ▁in ▁the ▁old ▁Ar é naire ▁team ▁( LI P , ▁E NS ▁Lyon ). ▁It ▁supports ▁the ▁ 4 ▁round ing