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VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF ENGINEERING AND TECHNOLOGY NGUYEN BAO NGOC FINDING ROUND-OFF ERROR FOR JAVA PROGRAMS USING SYMBOLIC PATHFINDER MASTER THESIS OF INFORMATION TECHNOLOGY Hanoi - 2014 VIETNAM NATIONAL UNIVERSITY, HANOI UNIVERSITY OF ENGINEERING AND TECHNOLOGY NGUYEN BAO NGOC FINDING ROUND-OFF ERROR FOR JAVA PROGRAMS USING SYMBOLIC PATHFINDER Major: Computer science Code: 60 48 01 MASTER THESIS OF INFORMATION TECHNOLOGY SUPERVISOR: PhD Truong Anh Hoang CO-SUPERVISOR: PhD Le Trong Vinh Hanoi - 2014 ❖r✐❣✐♥❛❧✐t② ❙t❛t❡♠❡♥t ■ ❤❡r❡❜② ❞❡❝❧❛r❡ t❤❛t t❤✐s s✉❜♠✐ss✐♦♥ ✐s ♠② ♦✇♥ ✇♦r❦ ❛♥❞ t♦ t❤❡ ❜❡st ♦❢ ♠② ❦♥♦✇❧❡❞❣❡ ✐t ❝♦♥t❛✐♥s ♥♦ ♠❛t❡r✐❛❧s ♣r❡✈✐♦✉s❧② ♣✉❜❧✐s❤❡❞ ♦r ✇r✐tt❡♥ ❜② ❛♥♦t❤❡r ♣❡rs♦♥✱ ♦r s✉❜st❛♥t✐❛❧ ♣r♦♣♦rt✐♦♥s ♦❢ ♠❛t❡r✐❛❧ ✇❤✐❝❤ ❤❛✈❡ ❜❡❡♥ ❛❝❝❡♣t❡❞ ❢♦r t❤❡ ❛✇❛r❞ ♦❢ ❛♥② ♦t❤❡r ❞❡❣r❡❡ ♦r ❞✐♣❧♦♠❛ ❛t ❯♥✐✈❡rs✐t② ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✭❯❊❚✴❈♦❧t❡❝❤✮ ♦r ❛♥② ♦t❤❡r ❡❞✉❝❛t✐♦♥❛❧ ✐♥st✐t✉t✐♦♥✱ ❡①❝❡♣t ✇❤❡r❡ ❞✉❡ ❛❝❦♥♦✇❧✲ ❡❞❣♠❡♥t ✐s ♠❛❞❡ ✐♥ t❤❡ t❤❡s✐s✳ ❆♥② ❝♦♥tr✐❜✉t✐♦♥ ♠❛❞❡ t♦ t❤❡ r❡s❡❛r❝❤ ❜② ♦t❤❡rs✱ ✇✐t❤ ✇❤♦♠ ■ ❤❛✈❡ ✇♦r❦❡❞ ❛t ❯❊❚✴❈♦❧t❡❝❤ ♦r ❡❧s❡✇❤❡r❡✱ ✐s ❡①♣❧✐❝✐t❧② ❛❝❦♥♦✇❧✲ ❡❞❣❡❞ ✐♥ t❤❡ t❤❡s✐s✳ ■ ❛❧s♦ ❞❡❝❧❛r❡ t❤❛t t❤❡ ✐♥t❡❧❧❡❝t✉❛❧ ❝♦♥t❡♥t ♦❢ t❤✐s t❤❡s✐s ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ♠② ♦✇♥ ✇♦r❦✱ ❡①❝❡♣t t♦ t❤❡ ❡①t❡♥t t❤❛t ❛ss✐st❛♥❝❡ ❢r♦♠ ♦t❤❡rs ✐♥ t❤❡ ♣r♦❥❡❝t✬s ❞❡s✐❣♥ ❛♥❞ ❝♦♥❝❡♣t✐♦♥ ♦r ✐♥ st②❧❡✱ ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❧✐♥❣✉✐st✐❝ ❡①♣r❡ss✐♦♥ ✐s ❛❝❦♥♦✇❧❡❞❣❡❞✳ ❉❛t❡✿ ❙✐❣♥❡❞✿ ✐ ❆❜str❛❝t ◆♦✇❛❞❛②s✱ ✇✐t❤ t❤❡ ❡①♣❧♦s✐♦♥ ♦❢ ♠♦❜✐❧❡ ❛♥❞ ❡♠❜❡❞❞❡❞ ❞❡✈✐❝❡s✱ t❤❡ ♥❡❡❞ ❢♦r ❛♣✲ ♣❧✐❝❛t✐♦♥s ❢♦r t❤❡s❡ ❞❡✈✐❝❡s ❤❛s ❜❡❡♥ ✐♥❝r❡❛s✐♥❣ ♥♦♥✲st♦♣✳ ❉❡✈❡❧♦♣♠❡♥t ♣r♦❝❡ss ♦❢ t❤❡s❡ ♣r♦❣r❛♠s ✐s ❝♦♠♠♦♥❧② ❝♦♠♣r✐s❡❞ ♦❢ t✇♦ ♣❤❛s❡s✿ ❞❡✈❡❧♦♣✐♥❣ ♣❤❛s❡ t❛❦❡♥ ♦♥ ♣❡rs♦♥❛❧ ❝♦♠♣✉t❡rs ❛♥❞ ❞❡♣❧♦②♠❡♥t ♣❤❛s❡ t❛❦❡♥ ♦♥ ❛❝t✉❛❧ ❞❡✈✐❝❡s✳ ❚❤❡ ❧❛tt❡r ♣❤❛s❡ ✐♥❝❧✉❞❡s ♣♦rt✐♥❣ ✇❤✐❝❤ r❡q✉✐r❡s ❝❤❛♥❣✐♥❣ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ❛♥❞ ♦♣❡r❛✲ t✐♦♥ t♦ ✜①❡❞✲♣♦✐♥t ❛♥❞ ❤❡r❡ ♦❢t❡♥ ♦❝❝✉r r♦✉♥❞✲♦✛ ❡rr♦rs ❜❡t✇❡❡♥ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ ♣r♦❣r❛♠✳ ■♥ t❤✐s t❤❡s✐s✱ ✇❡ ♣r❡s❡♥t ❛ ♥♦✈❡❧ ❛♣♣r♦❛❝❤ t♦ ♣r♦❞✉❝❡ ❛ ♣r❡❝✐s❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ✉s✐♥❣ s②♠❜♦❧✐❝ ❝♦♠♣✉t❛t✐♦♥✳ ❲✐t❤ t❤✐s r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ❝❛♥ ❛♥❛❧②③❡ ✈❛r✐♦✉s ❛s♣❡❝ts ♦❢ t❤❡ ❡rr♦r s✉❝❤ ❛s ✜♥❞✐♥❣ ❧❛r❣❡st r♦✉♥❞✲♦✛ ❡rr♦r ✉s✐♥❣ ♦♣t✐♠✐③❛t✐♦♥ t♦♦❧s ❧✐❦❡ ▼❛t❤❡♠❛t✐❝❛✱ ♦r ❞♦✐♥❣ ❡rr♦r ❜♦✉♥❞✲❝❤❡❝❦✐♥❣ ✉s✐♥❣ ❙▼❚ s♦❧✈❡rs✳ ❲❡ ❛❧r❡❛❞② ✐♠♣❧❡♠❡♥t❡❞ ❛ t♦♦❧ ✉s✐♥❣ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r t♦ ❣❡♥❡r❛t❡ s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡ss✐♦♥ ♦❢ ❛ ❏❛✈❛ ♣r♦❣r❛♠✳ ❯s✐♥❣ t❤❛t s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡s✲ s✐♦♥✱ ✇❡ ❡①❡❝✉t❡ s②♠❜♦❧✐❝❛❧❧② r♦✉♥❞✲♦✛ ❡rr♦r ♣❛rt t♦ ❣❡t ♣r❡❝✐s❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❣r❛♠✳ ❊①♣❡r✐♠❡♥ts ❡①❡❝✉t❡❞ ✇✐t❤ s✐♠♣❧❡ ❏❛✈❛ ♣r♦❣r❛♠s ✇✐❧❧ ❞❡♠♦♥str❛t❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦✉r s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r✳ P✉❜❧✐❝❛t✐♦♥s✿ ✯ ❆♥❤✲❍♦❛♥❣ ❚r✉♦♥❣✱ ❍✉②✲❱✉ ❚r❛♥✱ ❇❛♦✲◆❣♦❝ ◆❣✉②❡♥✳ ❋✐♥❞✐♥❣ ❘♦✉♥❞✲❖✛ ❊rr♦r ❯s✐♥❣ ❙②♠✲ ❜♦❧✐❝ ❊①❡❝✉t✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❋✐❢t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❑♥♦✇❧❡❞❣❡ ❛♥❞ ❙②st❡♠s ❊♥❣✐♥❡❡r✐♥❣ ✭❑❙❊ ✷✵✶✸✮✳ ✐✐ ❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ❋✐rst ❛♥❞ ❢♦r❡♠♦st✱ ■ ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss ♠② ❣r❡❛t❡st ❛♣♣r❡❝✐❛t✐♦♥ t♦ ♠② s✉♣❡r✲ ✈✐s♦r ✲ ❉r✳ ❚r✉♦♥❣ ❆♥❤ ❍♦❛♥❣✳ ■t ✐s ♥♦ ❞♦✉❜t t❤❛t ✇✐t❤♦✉t ❤✐s ♣❛t✐❡♥t ❣✉✐❞❛♥❝❡✱ ❝♦♥t✐♥✉♦✉s ❝♦♥tr✐❜✉t✐✈❡ ✐♥✈♦❧✈❡♠❡♥ts ❛♥❞ ✉♥❝❡❛s✐♥❣ ❡✛♦rts t♦ ✐♥s♣✐r❡ ♠❡ t❤r♦✉❣❤✲ ♦✉t t❤❡ ♣r♦❝❡ss ♦❢ r❡s❡❛r❝❤✐♥❣✱ t❤✐s t❤❡s✐s ✇♦✉❧❞ ❤❛✈❡ ♥❡✈❡r ❜❡❡♥ ❛❝❝♦♠♣❧✐s❤❡❞✳ ■ ❛♠ ❛❧s♦ ❣r❛t❡❢✉❧ t♦ ♠② ❝♦✲s✉♣❡r✈✐s♦r✱ ❉r✳ ▲❡ ❚r♦♥❣ ❱✐♥❤✱ ❢♦r ❤✐s ❝♦❧❧❛❜♦r❛✲ t✐✈❡ ❛❞✈✐❝❡✱ ♠② ❧❡❝t✉r❡rs ✐♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣ ❉❡♣❛rt♠❡♥t ❢♦r t❤❡✐r s✉♣♣♦rt✐✈❡ ❛ss✐st❛♥❝❡✱ ❛♥❞ ♠② ❝♦❧❧❡❛❣✉❡s ❢r♦♠ ■♥❢♦r♠❛t✐♦♥ ❚❡❝❤♥♦❧♦❣② ❋❛❝✉❧t② ❢♦r t❤❡✐r ✉♥✲ ❝♦♥❞✐t✐♦♥❛❧ ❤❡❧♣ t♦ ✇❤✐❝❤ ■ ❛♠ ✐♥❞❡❜t❡❞✳ ❋✐♥❛❧❧②✱ ✐t ♠✉st ❜❡ ♠② s❤♦rt❝♦♠✐♥❣ ♥♦t t♦ ♠❡♥t✐♦♥ s✉♣♣♦rts ❢r♦♠ ♠② ❜❡❧♦✈❡❞ ❢❛♠✐❧② ❛♥❞ ❢r✐❡♥❞s✳ ❚❤❡② ❤❛✈❡ ❜❡❡♥ ❛❧✇❛②s ❡♥❝♦✉r❛❣✐♥❣ ♠❡ ❛ ❧♦t✳ ❚❤❛♥❦ ②♦✉✦ ✐✐✐ ❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts ❖r✐❣✐♥❛❧✐t② ❙t❛t❡♠❡♥t ✐ ❆❜str❛❝t ✐✐ ❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ✐✐✐ ▲✐st ♦❢ ❋✐❣✉r❡s ✈✐ ▲✐st ♦❢ ❚❛❜❧❡s ✈✐✐ ▲✐st ♦❢ ❆❜❜r❡✈✐❛t✐♦♥s ✈✐✐✐ ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✷ ❇❛❝❦❣r♦✉♥❞ ✸ ✷✳✶ ✷✳✷ ✷✳✸ ◆✉♠❜❡r r❡♣r❡s❡♥t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✶ ❋❧♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✷ ❋✐①❡❞✲♣♦✐♥t ♥✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❙②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ✇✐t❤ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✶ ❙②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✷ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❘❡❧❛t❡❞ ✇♦r❦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✐✈ ✸ ❙②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ✸✳✶ ❙②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ✸✳✷ ❈♦♥str❛✐♥ts ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✸ ❙②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❢♦r ❡①♣r❡ss✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✹ ❙②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❢♦r ♣r♦❣r❛♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✺ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❊①♣❡r✐♠❡♥ts ✹✳✶ ✹✳✷ ✹✳✸ ✶✻ ■♠♣❧❡♠❡♥t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✹✳✶✳✶ ✶✾ ❖✉r ❡①t❡♥❞❡❞ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷✳✶ ❊①♣❡r✐♠❡♥t ✇✐t❤ s✐♠♣❧❡ ♣r♦❣r❛♠ ✷✶ ✹✳✷✳✷ ❊①♣❡r✐♠❡♥t ✇✐t❤ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✸ ❊①♣❡r✐♠❡♥t ✇✐t❤ ❚❛②❧♦r s❡r✐❡s ♦❢ s✐♥❡ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❉✐s❝✉ss✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ✷✽ ❘❡❢❡r❡♥❝❡s ✷✾ ✈ ▲✐st ♦❢ ❋✐❣✉r❡s ✷✳✶ ❆ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ tr❡❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ♦✈❡r✈✐❡✇ ❬✶✻❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❈❆◆❆ s②st❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✸✳✶ ❆♥ ❡①❛♠♣❧❡ ♣r♦❣r❛♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✹✳✶ ❙②st❡♠ ❛r❝❤✐t❡❝t✉r❡ ♦✈❡r✈✐❡✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✷ ❏❛✈❛ P❛t❤❋✐♥❞❡r ❛r❝❤✐t❡❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✸ ▲✐st❡♥❡r ❛r❝❤✐t❡❝t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✹ ▼❛t❤❡♠❛t✐❝❛ ♣r♦❜❧❡♠ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❋✐❣✉r❡ ✸✳✹✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✈✐ ✻ ▲✐st ♦❢ ❚❛❜❧❡s ✷✳✶ ✸✷ ❜✐ts ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✢♦❛t ♥✉♠❜❡r ✷✳✷ ✶✻ ❜✐ts ✜①❡❞✲♣♦✐♥t ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ 118.625 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✹✳✶ ❚♦♣ r♦✉♥❞✲♦✛ ❡rr♦rs ✐♥ ✶✵✵✳✵✵✵✳✵✵✵ t❡sts ✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st ✳ ✷✻ ✹✳✷ ❚♦♣ r♦✉♥❞✲♦✛ ❡rr♦rs ✐♥ ✶✵✵✳✵✵✵✳✵✵✵ t❡sts ✇✐t❤ r♦✉♥❞ t♦✇❛r❞s ✈✐✐ 118.625 −∞ ✳ ✷✼ ▲✐st ♦❢ ❆❜❜r❡✈✐❛t✐♦♥s ❆■ ❆✣♥❡ ■♥t❡r✈❛❧✳ ✽✱ ✾ ❆▲❯ ❈■ ❆r✐t❤♠❡t✐❝ ▲♦❣✐❝ ❯♥✐t✳ ✺ ❈❧❛ss✐❝❛❧ ■♥t❡r✈❛❧✳ ✽✱ ✾ ❊❆■ ❋P❯ ❡①t❡♥❞❡❞ ❛✣♥❡ ✐♥t❡r✈❛❧✳ ✽✱ ✾ ❋❧♦❛t✐♥❣✲P♦✐♥t ❯♥✐t✳ ✶✱ ✺ ❏P❋ ❏❛✈❛ P❛t❤❋✐♥❞❡r✳ ✼ ❙■▼ ❙✉❜s❝r✐❜❡r ■❞❡♥t✐t② ▼♦❞✉❧❡✳ ✶ ❙▼❚ ❙P❋ ❙❛t✐s✜❛❜✐❧✐t② ▼♦❞✉❧♦ ❚❤❡♦r✐❡s✳ ✷✱ ✷✹ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r✳ ✸✱ ✻✱ ✼✱ ✶✽ ✈✐✐✐ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ❋✐❣✉r❡ ✹✳✶✿ ❙②st❡♠ ❛r❝❤✐t❡❝t✉r❡ ♦✈❡r✈✐❡✇ ✶✳ ❯s✐♥❣ ♦✉r ❡①t❡♥❞❡❞ ✈❡rs✐♦♥ ♦❢ ❙P❋ ✭s❡❡ ❙✉❜ ❙❡❝t✐♦♥ ✹✳✶✳✶✮ t♦ ♣❡r❢♦r♠ s②♠✲ ❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ♦♥ ❏❛✈❛ ✐♥♣✉t ❢✉♥❝t✐♦♥ ❛♥❞ r❡t✉r♥ ❛♥ ♦✉t♣✉t ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥ ✇✐t❤ ❝♦♥str❛✐♥ts ✭✐♥✐t✐❛❧ ✐♥♣✉t r❛♥❣❡s ❛♥❞ ♣❛t❤ ❝♦♥❞✐t✐♦♥s✮ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❡①♣r❡ss✐♦♥✳ ✷✳ P❛rs❡ t❤❡ ❡①♣r❡ss✐♦♥ ❛♥❞ ❣❡♥❡r❛t❡ ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✐♥ ❏❛✈❛✳ ❲❡ ✉s❡ t❤❡ ✶ ❆✉r♦❝❤s ♣❛rs❡r ❣❡♥❡r❛t♦r ❢♦r t❤✐s ♣✉r♣♦s❡✳ ✸✳ P❡r❢♦r♠ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ♦♥ t❤❡ ❡①♣r❡ss✐♦♥ tr❡❡ ✇✐t❤ +, −, × t♦ ♣r♦❞✉❝❡ ❛ r♦✉♥❞✲♦✛ ❡rr♦r s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ t♦❣❡t❤❡r ✇✐t❤ ✈❛r✐❛❜❧❡ ❝♦♥str❛✐♥ts✳ ■♥ t❤✐s st❡♣ ✇❡ ❝♦♥✈❡rt t❤❡ ❡①♣r❡ss✐♦♥ tr❡❡ ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s st❡♣ t♦ ❛ ♣♦st✲♦r❞❡r ❡①♣r❡ss✐♦♥✳ ✹✳ ❙✐♠♣❧✐❢② t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❛♥❞ ❝♦♥str❛✐♥ts ♦❢ ✈❛r✐❛❜❧❡s ✉s✐♥❣ t❤❡ ▼❛t❤❡♠❛t✐❝❛ ❢✉♥❝t✐♦♥ Simplify✳ ◆♦t❡ t❤❛t t❤❡ ❝♦♥st❛♥ts ❛♥❞ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ✐♥♣✉t ❡①♣r❡ss✐♦♥ ❛r❡ ❛❧s♦ s♣❧✐t ✐♥t♦ t✇♦ ♣❛rts✿ t❤❡ ✜①❡❞✲♣♦✐♥t ♣❛rt ❛♥❞ ✶ ❤tt♣✿✴✴❧❛♠❜❞❛✲❞✐♦❞❡✳❝♦♠✴s♦❢t✇❛r❡✴❛✉r♦❝❤s ✶✼ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ♣❛rt✱ ❜♦t❤ ♦❢ ✇❤✐❝❤ ❛r❡ ❝♦♥st❛♥ts✳ ❲❤❡♥ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ✐s ♥♦♥✲③❡r♦✱ t❤❡ s✐♠♣❧✐✜❝❛t✐♦♥ st❡♣ ❝❛♥ ♠❛❦❡ t❤❡ s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ♠♦r❡ ❝♦♠♣❛❝t✳ ✺✳ ❯s❡ ▼❛t❤❡♠❛t✐❝❛ t♦ ✜♥❞ ♦♣t✐♠✉♠ ♦❢ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ✇✐t❤ ❝♦♥str❛✐♥ts ♦❢ ✈❛r✐❛❜❧❡s✳ ❲❡ ✉s❡ t❤❡ ▼❛t❤❡♠❛t✐❝❛ ❢✉♥❝t✐♦♥ N M aximize[{f, constraints}, {variables}] ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❙✐♥❝❡ ▼❛t❤✲ ❡♠❛t✐❝❛ ❞♦❡s ♥♦t s✉♣♣♦rt ✜①❡❞✲♣♦✐♥t ❢♦r♠❛t✱ ✇❡ ♥❡❡❞❡❞ t♦ ❞❡✈❡❧♦♣❡❞ ❛ ♠♦❞✉❧❡ ✐♥ ▼❛t❤❡♠❛t✐❝❛ ❢♦r ❝♦♥✈❡rt✐♥❣ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs t♦ ✜①❡❞✲♣♦✐♥t ♥✉♠❜❡rs ❛♥❞ ❡♠✉❧❛t✐♥❣ ✜①❡❞✲♣♦✐♥t ♥✉♠❜❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✭s❡❡ ❆❧❣♦r✐t❤♠ ✶ ❛♥❞ ✷ ✭r♦✉♥❞ t♦ ♥❡❛r❡st✮✮✳ ■♥♣✉t ✿ ❆ ✢♦❛t✐♥❣✲♣♦✐♥t ✈❛❧✉❡ x ❖✉t♣✉t✿ ❚❤❡ ❝♦♥✈❡rt❡❞ ✜①❡❞✲♣♦✐♥t ✈❛❧✉❡ ♦❢ x ❉❛t❛ ✿ binx st♦r❡s t❤❡ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ x ❉❛t❛ ✿ f p ✐s t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ❉❛t❛ ✿ x1 ✐s t❤❡ r❡s✉❧t ♦❢ binx ❛❢t❡r ❧❡❢t s❤✐❢t❡❞ ❉❛t❛ ✿ ip ✐s t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ x1 ❉❛t❛ ✿ digit ✐s t❤❡ ✈❛❧✉❡ ♦❢ nth ❜✐t ❉❛t❛ ✿ f ixed✿ t❤❡ r❡s✉❧t ♦❢ ❝♦♥✈❡rt✐♥❣ Pr♦❝❡❞✉r❡ ❝♦♥✈❡rt❚♦❋✐①❡❞✭①✱ ❢♣ ✮❀ ❜❡❣✐♥ x t♦ ❜✐♥❛r② f p ❜✐ts❀ ❈♦♥✈❡rt ❛ ✢♦❛t✐♥❣✲♣♦✐♥t x1 ❂ ❙❤✐❢t ❧❡❢t binx ❜② ip ❂ ■♥t❡❣❡r ♣❛rt ♦❢ x1 ❀ th ❚❛❦❡ t❤❡ (f p + 1) ❜✐t ♦❢ binx ✐❢ digit ❡q✉❛❧s ✶ t❤❡♥ ✐❢ x > ✵ t❤❡♥ ip ❂ ip +l ✶❀ ❡♥❞ ❡❧s❡ ip ❡♥❞ ❡♥❞ f ixed ❂ ip −l ❛s ♥✉♠❜❡rs binx ❀ digit❀ ✶❀ ❂ ❙❤✐❢t r✐❣❤t ip ❜② fp ❜✐ts❀ r❡t✉r♥ f ixed ❡♥❞ ❆❧❣♦r✐t❤♠ ✶✿ ❘♦✉♥❞✐♥❣ ❛ ✢♦❛t✐♥❣✲♣♦✐♥t ✈❛❧✉❡ t♦ ✜①❡❞✲♣♦✐♥t ✈❛❧✉❡ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ✜①❡❞✲♣♦✐♥t ♦♥ ▼❛t❤❡♠❛t✐❝❛ ✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st ✐s ❞❡✜♥❡❞ ✐♥ ❆❧❣♦r✐t❤♠ 2✳ ❆ss✉♠❡ t❤❛t t❤❡ ✜①❡❞✲♣♦✐♥t ♥✉♠❜❡r ❤❛s ✶✽ fp ❜✐ts t♦ r❡♣r❡s❡♥t t❤❡ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ❢r❛❝t✐♦♥❛❧ ♣❛rt✳ ❚❤❡ ✐♥♣✉ts ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ t✇♦ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs b a ❛♥❞ ❛♥❞ ✇❡ ♥❡❡❞ t♦ ♣r♦❞✉❝❡ t❤❡ r❡s✉❧t ✐♥ ✜①❡❞✲♣♦✐♥t✳ ❋✐rst✱ ✇❡ s❤✐❢t ❧❡❢t ❡❛❝❤ ♥✉♠❜❡r ❜② t❛❦❡ t❤❡✐r ♣r♦❞✉❝t ❛♥❞ s❤✐❢t r✐❣❤t f p ❜✐ts t♦ ∗ f p ❜✐ts ❣❡t t✇♦ ✐♥t❡❣❡r ♥✉♠❜❡rs✳ ❚❤❡♥ ✇❡ t♦ ♣r♦❞✉❝❡ t❤❡ r❛✇ r❡s✉❧t ✇✐t❤♦✉t f p ❜✐ts✱ st♦r❡ ✐t i❴mul❴shr❴f p✳ r♦✉♥❞✐♥❣✳ ❲✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st✱ ✇❡ s❤✐❢t r✐❣❤t t❤❡ ♣r♦❞✉❝t ✐♥ i❴mul❴shr❴f p✱ ❛♥❞ t❛❦❡ t❤❡ ✐♥t❡❣❡r ❛♥❞ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢ ■❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢ i❴mul❴shr❴f p ✐s ❧❛r❣❡r t❤❛♥ ✵✳✺ t❤❡♥ t❤❡ ✐♥t❡❣❡r ♣❛rt i❴mul❴shr❴f p ✐s ✐♥❝r❡❛s❡❞ ❜② ✶✳ ❲❡ st♦r❡ t❤❡ r❡s✉❧t ❛❢t❡r r♦✉♥❞✐♥❣ ✐t ✐♥ i❴mul❴rounded✳ ❙❤✐❢t✐♥❣ ❧❡❢t i❴mul❴rounded f p ❜✐ts ♣r♦❞✉❝❡s t❤❡ r❡s✉❧t ♦❢ t❤❡ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ✜①❡❞✲♣♦✐♥t✳ ✹✳✶✳✶ ❖✉r ❡①t❡♥❞❡❞ ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ❚♦ ❛✉t♦♠❛t✐❝❛❧❧② ❣❡♥❡r❛t❡ s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡ss✐♦♥ ❢♦r ❏❛✈❛ ♣r♦❣r❛♠s✱ ✇❡ ❤❛✈❡ t♦ ♠♦❞✐❢② ❙P❋ t♦ r❡t✉r♥ s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡ss✐♦♥ ❢♦r ❡❛❝❤ ♣❛t❤ ❜② s②♠❜♦❧✐❝❛❧❧② ❡①❡❝✉t✐♥❣ ♣r♦❣r❛♠s✳ ❙P❋ ❤❛s ❤✐❣❤✲❧❡✈❡❧ ♦✈❡r✈✐❡✇ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳✷✳ ■ts ✐♥♣✉ts ✐♥❝❧✉❞❡s✿ • ❆ ❏❛✈❛ ❜②t❡❝♦❞❡ ♣r♦❣r❛♠ ✭✯✳❝❧❛ss ✜❧❡✮ • ❏P❋ ❝♦♥✜❣✉r❛t✐♦♥ ✜❧❡ s♣❡❝✐❢②✐♥❣ ✇❤✐❝❤ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❞♦♥❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥✳ • ❖t❤❡r ❝r✐t❡r✐❛✳ ❖r✐❣✐♥❛❧❧②✱ ❙P❋ ♦✉t♣✉ts ❛ t❡st s✉✐t❡ ✭t❡st ✐♥♣✉ts✴t❡st s❡q✉❡♥❝❡s✮ ❛♥❞ ♠❛②❜❡ t❤❡ ❝♦✉♥t❡r ❡①❛♠♣❧❡s ❢♦r t❤❡ ❢❛✐❧❡❞ ♣r♦♣❡rt✐❡s✳ ❆♥❞ ❙P❋ ✉t✐❧✐③❡ t❤❡s❡ ❢♦❧❧♦✇✐♥❣ ❡①t❡♥✲ s✐♦♥ ♠❡❝❤❛♥✐s♠s ♦❢ ❏P❋ ✭s❡❡ ❋✐❣✉r❡ ✹✳✷✮ t♦ ❛❝❤✐❡✈❡ t❤❛t✿ • ❈❤♦✐❝❡ ●❡♥❡r❛t♦rs✿ st❛t❡ s♣❛❝❡ ❜r❛♥❝❤ ♣♦✐♥ts ✉s❡❞ t♦ ❣❡♥❡r❛t❡ ❝❤♦✐❝❡s • ■♥str✉❝t✐♦♥ ❋❛❝t♦r✐❡s✿ ✐♥str✉❝t✐♦♥ s❡t s❡♠❛♥t✐❝s ✇❤✐❝❤ ♣r♦✈✐❞❡ t❤❡ ❛❜✐❧✐t② t♦ r❡♣❧❛❝❡ t❤❡ st❛♥❞❛r❞ ❡①❡❝✉t✐♦♥ s❡♠❛♥t✐❝s ✇✐t❤ ❛ s②♠❜♦❧✐❝ s❡♠❛♥t✐❝s✳ • ❆ttr✐❜✉t❡ ❖❜❥❡❝ts✿ ✉s❡❞ t♦ st♦r❡ ❛♥❞ ♣r♦♣❛❣❛t❡ s②♠❜♦❧✐❝ ✐♥❢♦r♠❛t✐♦♥ ❛s ♠❡t❛❞❛t❛ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❝♦♥❝r❡t❡ ✈❛❧✉❡s ❛♥❞ ♦❜❥❡❝ts✳ ✶✾ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts • ◆❛t✐✈❡ P❡❡rs✿ ❧✐❜r❛r② ❛❜str❛❝t✐♦♥s • ▲✐st❡♥❡rs✿ ✉s❡❞ ❢♦r ❡①❡❝✉t✐♦♥ ♠♦♥✐t♦r✐♥❣ ❋✐❣✉r❡ ✹✳✷✿ ❏❛✈❛ P❛t❤❋✐♥❞❡r ❛r❝❤✐t❡❝t✉r❡ ❙P❋ ❝✉rr❡♥t❧② ♣r♦✈✐❞❡s t✇♦ t②♣❡s ♦❢ ❧✐st❡♥❡r ✇❤✐❝❤ ❣❛t❤❡r ❛♥❞ ❞✐s♣❧❛② ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♣❛t❤ ❝♦♥❞✐t✐♦♥s ❣❡♥❡r❛t❡❞ ❞✉r✐♥❣ t❤❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥✳ • ❙②♠❜♦❧✐❝▲✐st❡♥❡r✿ s❤♦✇s ❛♥❛❧②③❡❞ ♣❛t❤ ❝♦♥❞✐t✐♦♥s ❛♥❞ ❛❧s♦ t❤❡ t❡st ❝❛s❡s ❣❡♥❡r❛t❡❞ ❜② s②♠❜♦❧✐❝❛❧ ❡①❡❝✉t✐♥❣ ✐♥❝❧✉❞❡s✿ t❡st ✐♥♣✉ts ❛♥❞ t❤❡✐r ❡①♣❡❝t❡❞ ♦✉t♣✉t ✐♥ ♣❧❛✐♥ t❡①t ♦r ❍❚▼▲ ❢♦r♠❛t✳ • ❙②♠❜♦❧✐❝❙❡q✉❡♥❝❡▲✐st❡♥❡r✿ s❤♦✇s t❡st s❡q✉❡♥❝❡s ✭✐✳❡✳ s❡q✉❡♥❝❡s ♦❢ ♠❡t❤♦❞ ❝❛❧❧s✮✱ ❢♦r ❡❛❝❤ ♣❛t❤ ❝♦♥❞✐t✐♦♥ ❡♥❝♦✉♥t❡r❡❞✳ ❖✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ r❡q✉✐r❡s s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡ss✐♦♥ ♦❢ ❡❛❝❤ ♣❛t❤ ❛♥❞ ✐ts r❡s♣❡❝t✐✈❡ ♣❛t❤ ❝♦♥❞✐t✐♦♥ s♦ t❤❛t ✇❡ ♠♦❞✐✜❡❞ ❙②♠❜♦❧✐❝▲✐st❡♥❡r ❝❧❛ss t♦ ❛❝❤✐❡✈❡ ❞❡s✐r❡ r❡s✉❧ts✳ ❈♦♥s✐❞❡r ❧✐st❡♥❡r ❛r❝❤✐t❡❝t✉r❡ ✐♥ ❋✐❣✉r❡ ✹✳✸✳ ❲❡ ❤♦♦❦❡❞ ✐♥t♦ ❧✐st❡♥❡r✬s ✐♥str✉❝✲ t✐♦♥❊①❡❝✉t❡❞✭✮ ♠❡t❤♦❞ t♦ ❝❛♣t✉r❡ ♥♦t✐✜❝❛t✐♦♥ ✇❤❡♥ ❛ s②♠❜♦❧✐❝ ♠❡t❤♦❞ ✐s ❝❛❧❧❡❞✱ ❛♥❞ ❣❡♥❡r❛t❡ ♣❛t❤ ❝♦♥❞✐t✐♦♥ ❛♥❞ s②♠❜♦❧✐❝ ♦✉t♣✉t ❡①♣r❡ss✐♦♥ ❢♦r t❤❛t ♣❛t❤ ✇❤❡♥ ❛ r❡t✉r♥ ❜②t❡❝♦❞❡ ✐♥str✉❝t✐♦♥ ❣♦t ❡①❡❝✉t❡❞✳ ❋✐♥❛❧❧②✱ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✐s ♣✉❜❧✐s❤❡❞ ✇❤❡♥ t❤❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ s❡❛r❝❤ ✐s ❞♦♥❡ ✈✐❛ P✉❜❧✐s❤❡r❊①t❡♥s✐♦♥✳ ✷✵ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ❋✐❣✉r❡ ✹✳✸✿ ▲✐st❡♥❡r ❛r❝❤✐t❡❝t✉r❡ ✹✳✷ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❋♦r ❝♦♠♣❛r✐s♦♥✱ ✇❡ ✉s❡ t✇♦ ❡①❛♠♣❧❡s ❢r♦♠ ❬✶✷❪ ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳✹ ❛♥❞ t❤❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ✺ ✐♥ ❛ ✺✻ ❜✐ts ✢♦❛t✐♥❣✲♣♦✐♥t ❝♦♠♣✉t❡r✳ ❚❤❡♥ ✇❡ ❡①♣❡r✐♠❡♥t❡❞ ✇✐t❤ ❛ ❚❛②❧♦r s❡r✐❡s ♦❢ ❛ s✐♥❡ ❢✉♥❝t✐♦♥ t♦ s❡❡ ❤♦✇ ❢❛r ✇❡ ❝❛♥ ❣♦ ✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡s✱ ✇❡ tr✐❡❞ t♦ ❞❡t❡r♠✐♥❡ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥ts t❤❛t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ✜①❡❞ ♣♦✐♥t✱ s♦ t❤❡② ❞♦♥✬t ❝❛✉s❡ ❛♥② r♦✉♥❞✲♦✛ ❡rr♦r✱ t♦ s✐♠♣❧✐❢② t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❢♦r t❡st✐♥❣ ♣✉r♣♦s❡✳ ✹✳✷✳✶ ❊①♣❡r✐♠❡♥t ✇✐t❤ s✐♠♣❧❡ ♣r♦❣r❛♠ ❋♦r t❤❡ ♣r♦❣r❛♠ ✐♥ ❋✐❣✉r❡ ✸✳✹✱ ✐t ✐s ❡❛s② t♦ ❝♦♠♣✉t❡ ✐ts s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ❢♦r (x > ∧ x × x − y) ❛♥❞ (x < ∧ × x − y)✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ✜rst ♦♥❡✳ ❈♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ✐♥♣✉t r❛♥❣❡ ♦❢ x ∈ [0, 3] ✇❡ ❣❡t x > ∧ −1 ≤ x ≤ ✇❤✐❝❤ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ t♦ < x ≤ 3✳ ❙♦ ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ❢♦r x × x − y ✇❤❡r❡ < x ≤ ❛♥❞ −10 ≤ y ≤ 10✳ t❤❡ t✇♦ ♣♦ss✐❜❧❡ r✉♥s✿ ❆♣♣❧②✐♥❣ t❤❡ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ✇✐t❤ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ♣❛rt✱ ✇❡ ❣❡t t❤❡ s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ❢♦r r♦✉♥❞✲♦✛ ❡rr♦r✿ (xi ×l xi ) −l (xi ×i xi ) +l ×l xi ×l xe +l x2e −l ye ✷✶ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ◆▼❛①✐♠✐③❡❬④❛✶ ✰ ✷✯①❢✯①r ✰ ①r❫✷ ✲ ②r✱ ❏♦✐♥❬④①❢ ❂❂ ❙✉♠❬✭❙②♠❜♦❧❬✧①✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮✯✭✷❫✭✶✶ ✲ ✐✮✮✱ ④✐✱ ✶✱ ✶✺⑥❪⑥✱ ❚❛❜❧❡❬✵ ❁❂ ✭❙②♠❜♦❧❬✧①✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮ ❁❂ ✶ ✫✫ ❊❧❡♠❡♥t❬✭❙②♠❜♦❧❬✧①✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮✱ ■♥t❡❣❡rs❪✱ ④✐✱ ✶✱ ✶✺⑥❪✱ ④②❢ ❂❂ ❙✉♠❬✭❙②♠❜♦❧❬✧②✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮✯✭✷❫✭✶✶ ✲ ✐✮✮✱ ④✐✱ ✶✱ ✶✺⑥❪⑥✱ ❚❛❜❧❡❬✵ ❁❂ ✭❙②♠✲ ❜♦❧❬✧②✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮ ❁❂ ✶ ✫✫ ❊❧❡♠❡♥t❬✭❙②♠❜♦❧❬✧②✧❁❃ ❚♦❙tr✐♥❣❬✐❪❪✮✱ ■♥t❡✲ ❣❡rs❪✱ ④✐✱ ✶✱ ✶✺⑥❪✱ ④❛✶ ❂❂ ✭①❢✯①❢ ✮ ✲ ♠✉❧t✐♣❧②❋✐①❡❞❬①❢✱①❢❪⑥✱ ④✲❜♦✉♥❞❛r② ❁❂ ②r ❁❂ ❜♦✉♥❞❛r②✱ ✲❜♦✉♥❞❛r② ❁❂ ①r ❁❂ ❜♦✉♥❞❛r②✱ ✵ ❁❂ ①❢ ❁❂ ✸✱ ✲✶✵ ❁❂ ②❢ ❁❂ ✶✵⑥❪⑥✱ ✭❏♦✐♥❬④②r✱❛✶✱①r✱①❢✱②❢⑥✱ ❚❛❜❧❡❬❙②♠❜♦❧❬✧①✧ ❁❃ ❚♦❙tr✐♥❣❬✐❪❪✱ ④✐✱ ✶✱ ✶✺⑥❪✱ ❚❛✲ ❜❧❡❬❙②♠❜♦❧❬✧②✧ ❁❃ ❚♦❙tr✐♥❣❬✐❪❪✱ ④✐✱ ✶✱ ✶✺⑥❪❪✮❪ ❋✐❣✉r❡ ✹✳✹✿ ▼❛t❤❡♠❛t✐❝❛ ♣r♦❜❧❡♠ ❢♦r ❡①❛♠♣❧❡ ✐♥ ❋✐❣✉r❡ ✸✳✹✳ ❛♥❞ t❤❡ ❝♦♥str❛✐♥ts ♦❢ ✈❛r✐❛❜❧❡s ✭✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st✮ ❛r❡ 15 j=1 dj 211−j dj ∈ {0, 1} −1 ≤ xi ≤ xi ≥ −0.03125 ≤ xe , ye ≤ 0.03125 xi = ◆❡①t ✇❡ ❝♦♥✈❡rt t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r s②♠❜♦❧✐❝ ❡①♣r❡ss✐♦♥ ❛♥❞ ❝♦♥str❛✐♥ts t♦ ▼❛t❤✲ ❡♠❛t✐❝❛ s②♥t❛①✳ ❚❤❡ r❡s✉❧t ✐s ✐♥ ❋✐❣✉r❡ ✹✳✹✳ ▼❛t❤❡♠❛t✐❝❛ ❢♦✉♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠❛ ❢♦r t❤❡ ♣r♦❜❧❡♠ ✐♥ ❋✐❣✉r❡ ✹✳✹✿ • ❲✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st✱ t❤❡ ♠❛①✐♠❛❧ ❡rr♦r ❜❡t✇❡❡♥ ✢♦❛t✐♥❣✲♣♦✐♥t ❛♥❞ 0.2275390625✳ ❚❤❡ ✐♥♣✉ts t❤❛t ❝❛✉s❡ t❤❡ ♠❛①✐♠❛❧ r♦✉♥❞✲♦✛ ❡rr♦r ❛r❡✿ xi = 2.875; xe = 0.03125 s♦ x = xi +l xe = 2.90625 ❛♥❞ yi = 4.125; ye = − 0.03125 s♦ y = yi +l ye = 4.09375✳ ✜①❡❞✲♣♦✐♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♣r♦❣r❛♠ ✐s • −∞ ✭■❊❊❊ ✼✺✹ ❬✼❪✮✿ t❤❡ ❡rr♦r ✐s 0.4531245939250891 − 24 j ✇✐t❤ xi = 2.8125; xe = j=− → x = xi +l xe = 2.874999940395355225 − 24 j ❛♥❞ yi = 4; ye = − j=− → y = yi +l ye = 3.937500059604644775✳ ❲✐t❤ r♦✉♥❞ t♦✇❛r❞s ❈♦♠♣❛r✐♥❣ t♦ ❬✶✷❪ ✇❡ ✜♥❞ t❤❛t ✉s✐♥❣ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st t❤❡ ❡rr♦r ✐s ✐♥ [−0.250976, 0.250976] s♦ ♦✉r r❡s✉❧t ✐s ♠♦r❡ ♣r❡❝✐s❡✳ ❚♦ ✈❡r✐❢② ♦✉r r❡s✉❧t✱ ✇❡ ✇r♦t❡ ❛ t❡st ♣r♦❣r❛♠ ❢♦r ❜♦t❤ r♦✉♥❞✐♥❣ ♠❡t❤♦❞s t❤❛t ❣❡♥❡r❛t❡s ✶✵✵✳✵✵✵✳✵✵✵ r❛♥❞♦♠ t❡st ❝❛s❡s ❢♦r −1 ≤ x ≤ ❛♥❞ −10 ≤ y ≤ 10 ❛♥❞ ❞✐r❡❝t❧② ❝♦♠♣✉t❡s t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ❜❡t✇❡❡♥ ✢♦❛t✐♥❣✲♣♦✐♥t ❛♥❞ ✜①❡❞✲♣♦✐♥t r❡s✉❧ts✳ ❙♦♠❡ ♦❢ t❤❡ ❧❛r❣❡st r♦✉♥❞✲♦✛ ❡rr♦r r❡s✉❧ts ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹✳✶ ❛♥❞ ✐♥ ❚❛❜❧❡ ✹✳✷✳ ❚❤❡ t❡sts ✇❡r❡ r✉♥ ♠❛♥② t✐♠❡s✱ ❜✉t ✇❡ ❞✐❞ ♥♦t ✜♥❞ ❛♥② ✐♥♣✉ts t❤❛t ❝❛✉s❡❞ ❧❛r❣❡r r♦✉♥❞✲♦✛ ❡rr♦r t❤❛♥ ♣r❡❞✐❝t❡❞ ❜② ♦✉r ❛♣♣r♦❛❝❤✳ ✷✷ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ✹✳✷✳✷ ❊①♣❡r✐♠❡♥t ✇✐t❤ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ✺ ❖✉r s❡❝♦♥❞ ❡①♣❡r✐♠❡♥t ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ ✺ t❛❦❡♥ ❢r♦♠ ❬✶✷❪✿ P 5(x) = − x + 3x2 − 2x3 + x4 − 5x5 ✇❤❡r❡ ✜①❡❞✲♣♦✐♥t ❢♦r♠❛t ✐s (2, 11, 8) ❛♥❞ x ∈ [0, 0.2]✳ ❆❢t❡r s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥✱ t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ✐s✿ +l ×l x2i −l ×l x3i −l ×l x5i −l xe +l ×l x2i ×l xe +l ×l x3i ×l xe −l 25 ×l x4i ×l xe +l ×l x2e −l ×l xi ×l x2e +l ×l x2i ×l x2e −l 50 ×l x3i ×l x2e −l ×l x3e +l ×l xi ×l x3e −l 50 ×l x2i ×l x3e +l x4e −l 25 ×l xi ×l x4e −l ×l x5e −l ×l iM ul[xi , xi ] +l ×l iM ul[iM ul[xi , xi ], xi ]−l iM ul[iM ul[iM ul[xi , xi ], xi ], xi ]+l × iM ul[iM ul[iM ul[iM ul[xi , xi ], xi ], xi ], xi ] ❛♥❞ t❤❡ ❝♦♥str❛✐♥ts ♦❢ ✈❛r✐❛❜❧❡s ✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st ❛r❡ xi = 19 j=1 dj 211−j dj ∈ {0, 1} ≤ xi ≤ 0.2 −0.001953125 ≤ xe ≤ 0.001953125 ≤ xi +l xe ≤ 0.2 0.0625 ≤ yr ≤ 0.0625 0.007244555 ✇✐t❤ xi = 0.12890625; xe = −0.001953125 s♦ x = xi +l xe = 0.126953125✳ ■♥ ❬✶✷❪✱ t❤❡✐r r❡❛❧ ❡rr♦r ✐s ✐♥ [−0.01909, 0.01909] ✇❤❡♥ ✉s✐♥❣ r♦✉♥❞ t♦ ♥❡❛r❡st✱ s♦ ♦✉r ❋♦r t❤✐s ♣r♦❜❧❡♠✱ ▼❛t❤❡♠❛t✐❝❛ ❢♦✉♥❞ t❤❡ ♠❛①✐♠✉♠ r❡s✉❧t ✐s ♠♦r❡ ♣r❡❝✐s❡✳ ❲❡ ✈❡r✐❢② ♦✉r r❡s✉❧ts ✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st ❜② ❞✐r❡❝t❧② ❝♦♠♣✉t✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ✷✸ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ❜❡t✇❡❡♥ t❤❡ ✜①❡❞✲♣♦✐♥t ❛♥❞ ✢♦❛t✐♥❣✲♣♦✐♥t ✇✐t❤ ✶✵✵✳✵✵✵✳✵✵✵ r❛♥❞♦♠ t❡st ❝❛s❡s ❢♦r ≤ x ≤ 0.2✳ ❚❤❡ ❧❛r❣❡st ❡rr♦r ✇❡ ❢♦✉♥❞ ✐s ✵✳✵✵✼✶✺✼✼✸✺✹✽✼✺✺ ✇❤✐❝❤ ✐s ✈❡r② ❝❧♦s❡ ❜✉t st✐❧❧ ✉♥❞❡r ♦✉r ❜♦✉♥❞✳ ✹✳✷✳✸ ❊①♣❡r✐♠❡♥t ✇✐t❤ ❚❛②❧♦r s❡r✐❡s ♦❢ s✐♥❡ ❢✉♥❝t✐♦♥ ■♥ t❤❡ ❧❛st ❡①♣❡r✐♠❡♥t✱ ✇❡ ✇❛♥t t♦ s❡❡ ❤♦✇ ❢❛r ✇❡ ❝❛♥ ❣♦ ✇✐t❤ ♦✉r ❛♣♣r♦❛❝❤✱ s♦ ✇❡ ✉s❡ ❛ ❚❛②❧♦r s❡r✐❡s ♦❢ s✐♥❡ ❢✉♥❝t✐♦♥✳ P 7(x) = x − 0.166667x3 + 0.00833333x5 − 0.000198413x7 ✇❤❡r❡ t❤❡ ✜①❡❞✲♣♦✐♥t ❢♦r♠❛t ✐s ❚❤❡ ❧❛r❣❡st r♦✉♥❞✲♦✛ ❡rr♦r ✐s (2, 11, 8) ❛♥❞ 0.00195312 x ∈ [0, 0.2]✳ ✇✐t❤ xi = 0; xe = 0.00195313 xi +l xe = 0.00195313✳ ❈♦♠♣❛r✐♥❣ t♦ t❤❡ r❡s✉❧ts ✐♥ ❬✶✷❪✱ ✉s✐♥❣ r♦✉♥❞ ♥❡❛r❡st t❤❡ ❡rr♦r ✐s ✐♥ [−0.00647, 0.00647] s♦ ♦✉r r❡s✉❧t ✐s ♠✉❝❤ ❜❡tt❡r✳ s♦ x = t♦ t❤❡ ❲❡ tr✐❡❞ ✇✐t❤ ❧♦♥❣❡r ❚❛②❧♦r s❡r✐❡s ❜✉t ▼❛t❤❡♠❛t✐❝❛ ❝♦✉❧❞ ♥♦t s♦❧✈❡ t❤❡ ❣❡♥❡r❛t❡❞ ♣r♦❜❧❡♠✳ ❲❡ ❛r❡ ❛✇❛r❡ ♦❢ t❤❡ s❝❛❧❛❜✐❧✐t② ♦❢ t❤✐s ❛♣♣r♦❛❝❤ ❛♥❞ ♣❧❛♥ t♦ tr② ✇✐t❤ ♠♦r❡ r❡❝❡♥t s♦❧✈❡rs s✉❝❤ ❛s r❛❙❆❚ ❬✶✾❪ ✐♥ ♦✉r ❢✉t✉r❡ ✇♦r❦✳ ✹✳✸ ❉✐s❝✉ss✐♦♥ ❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r t❡❝❤♥✐q✉❡ t❤❛t ♣r❡❝✐s❡❧② r❡♣r❡s❡♥ts r♦✉♥❞✲♦✛ ❡rr♦r ❜❡t✇❡❡♥ ✢♦❛t✐♥❣✲♣♦✐♥t ❛♥❞ ✜①❡❞✲♣♦✐♥t ✈❡rs✐♦♥s ♦❢ ❛ ♣r♦❣r❛♠✳ ❚❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❡♥❛❜❧❡s s❡✈❡r❛❧ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❡rr♦r ❛♥❛❧②s✐s ❛♥❞ t❡st ❝❛s❡s ❣❡♥❡r❛t✐♦♥✳ ❲❡ ❛r❡ ❛✇❛r❡ ♦❢ s❡✈❡r❛❧ ❛❞✈❛♥t❛❣❡s ❛♥❞ ❞r❛✇❜❛❝❦s ✇✐t❤ t❤✐s ❛♣♣r♦❛❝❤✿ • ❋✐rst✱ ♦✉r ❛♣♣r♦❛❝❤ ❛ss✉♠❡s t❤❛t ▼❛t❤❡♠❛t✐❝❛ ❞♦❡s ♥♦t ♦✈❡r ❛♣♣r♦①✐♠❛t❡ t❤❡ ♦♣t✐♠❛✳ ❍♦✇❡✈❡r✱ ❡✈❡♥ ✐❢ t❤❡ ♦♣t✐♠❛ ✐s ♦✈❡r ❛♣♣r♦①✐♠❛t❡❞✱ t❤❡ ♣♦✐♥t t❤❛t ♣r♦❞✉❝❡s t❤❡ ♦♣t✐♠✉♠ ✐s st✐❧❧ ❧✐❦❡❧② t♦ ❜❡ t❤❡ t❡st ❝❛s❡ ✇❡ ♥❡❡❞ t♦ ✐❞❡♥t✐❢②✳ ❲❡ ❝❛♥ r❡❝♦♠♣✉t❡ t❤❡ ❛❝t✉❛❧ r♦✉♥❞✲♦✛ ❡rr♦r ✇❤❡♥ t❤✐s ♦❝❝✉rs✳ ✷✹ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts • ❙❡❝♦♥❞✱ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t ♦✉r ❛♣♣r♦❛❝❤ ♠❛② ♥♦t ❜❡ s❝❛❧❛❜❧❡ ❢♦r ♠♦r❡ ❝♦♠♣❧❡① ♣r♦❣r❛♠s✳ ❚❤❡ r♦✉♥❞✲♦✛ ❡rr♦r r❡♣r❡s❡♥t❛t✐♦♥ ✇✐❧❧ ❣r♦✇ ✈❡r② ❧❛r❣❡ ✐♥ ❝♦♠♣❧❡① ♣r♦❣r❛♠s✳ ❙♦♠❡ s✐♠♣❧✐✜❝❛t✐♦♥ str❛t❡❣② ♠❛② ❜❡ ♥❡❡❞❡❞✱ s✉❝❤ ❛s s♦rt✐♥❣ t❤❡ ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ♦❢ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r ❡①♣r❡ss✐♦♥ ❛♥❞ r❡♠♦✈✐♥❣ ❝♦♠♣♦♥❡♥ts t❤❛t ❛r❡ ❝♦♠♣❧❡① ❜✉t ❝♦♥tr✐❜✉t❡ ✐♥s✐❣♥✐✜❝❛♥t❧② t♦ t❤❡ ❡rr♦r✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝❛♥ ❞✐✈✐❞❡ t❤❡ ❡①✲ ♣r❡ss✐♦♥ ✐♥t♦ ♠✉❧t✐♣❧❡ ✐♥❞❡♣❡♥❞❡♥t ♣❛rts t♦ s❡♥❞ s♠❛❧❧❡r ♣r♦❜❧❡♠s t♦ ▼❛t❤✲ ❡♠❛t✐❝❛✳ • ❚❤✐r❞✱ ✐❢ ❛ t❤r❡s❤♦❧❞ ✐s ❣✐✈❡♥✱ ✇❡ ❝❛♥ ❝♦♠❜✐♥❡ ✐t ✇✐t❤ t❡st✐♥❣ t♦ ✜♥❞ ❛ s♦❧✉t✐♦♥ ❢♦r t❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠✱ ♦r ✇❡ ❝❛♥ ✉s❡ ❛♥ ❙▼❚ s♦❧✈❡r ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❲❡ ♣❧❛♥ t♦ ✉s❡ r❛❙❆❚ ❬✶✾❪ ❢♦r t❤✐s ❛♣♣❧✐❝❛t✐♦♥ ✇❤❡♥ t❤❡ t♦♦❧ ✐s ❛✈❛✐❧❛❜❧❡ ❢♦r ✉s❡✳ • ❋✐♥❛❧❧②✱ ✇❡ ❝❛♥ ❝♦♠❜✐♥❡ ♦✉r ❛♣♣r♦❛❝❤ ✇✐t❤ ✐♥t❡r✈❛❧ ❛♥❛❧②s✐s✳ ❚❤❡ ✐♥t❡r✈❛❧ ❛♥❛❧②s✐s ✇✐❧❧ ❜❡ ✉s❡❞ ❢♦r ❝♦♠♣❧❡① ♣❛rts ♦❢ t❤❡ ♣r♦❣r❛♠✱ ✇❤✐❧❡ ♦t❤❡r✱ s✐♠♣❧❡r ♣❛rts ❝❛♥ ❤❛✈❡ ♣r❡❝✐s❡ r♦✉♥❞✲♦✛ ❡rr♦rs ❞❡t❡r♠✐♥❡❞✳ ◆♦t❡ t❤❛t t❤❡ ❧❛r❣❡st r♦✉♥❞✲♦✛ ❡rr♦r ✐s ♦♥❧② ♦♥❡ ♦❢ t❤❡ ♠❡tr✐❝s ❢♦r t❤❡ ♣r❡❝✐s❡♥❡ss ♦❢ t❤❡ ✜①❡❞✲♣♦✐♥t ❢✉♥❝t✐♦♥ ✈❡rs✉s ✐ts ✢♦❛t✐♥❣✲♣♦✐♥t ♦♥❡✳ P❤❛♠ ❡t ❛❧✳ ❬✶✺❪ ♣r♦♣♦s❡❞ s❡✈❡r❛❧ ♠❡tr✐❝s ❛♥❞ t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r s❡❡♠s ❝♦♥✈❡♥✐❡♥t t♦ ❝♦♠♣✉t❡ t❤❡s❡ ♠❡tr✐❝s ❛s ✐t ❝♦♥t❛✐♥s r✐❝❤ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r✳ ✷✺ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ■♥♣✉t ✿ ❆ ✢♦❛t✐♥❣ ✈❛❧✉❡ a ■♥♣✉t ✿ ❆ ✢♦❛t✐♥❣ ✈❛❧✉❡ b ❖✉t♣✉t✿ ❚❤❡ ♣r♦❞✉❝t ♦❢ a ×i b ❉❛t❛ ✿ a❴shl❴f p ✐s t❤❡ r❡s✉❧t ❛❢t❡r a ❛❢t❡r ❧❡❢t s❤✐❢t❡❞ f p ❜✐ts ❉❛t❛ ✿ i❴a❴shl❴f p ✐s t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ a❴shl❴f p ❉❛t❛ ✿ b❴shl❴f p ✐s t❤❡ r❡s✉❧t ♦❢ b ❛❢t❡r ❧❡❢t s❤✐❢t❡❞ f p ❜✐ts ❉❛t❛ ✿ i❴b❴shl❴f p ✐s t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ b❴shl❴f p ❉❛t❛ ✿ i❴mul ✐s t❤❡ ♣r♦❞✉❝t ♦❢ i❴a❴shl❴f p ❛♥❞ i❴b❴shl❴f p ❉❛t❛ ✿ i❴mul❴shr❴f p ✐s t❤❡ r❡s✉❧t ♦❢ i❴mul ❛❢t❡r r✐❣❤t s❤✐❢t❡❞ f p ❜✐ts ❉❛t❛ ✿ ipart❴i❴mul ✐s t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ i❴mul❴shr❴f p ❉❛t❛ ✿ f part❴i❴mul ✐s t❤❡ ❢r❛❝t✐♦♥ ♣❛rt ♦❢ i❴mul❴shr❴f p ❉❛t❛ ✿ truncate❴part ✐s r❡s✉❧t ♦❢ f part❴i❴mul ❛❢t❡r ❧❡❢t s❤✐❢t❡❞ ❜✐t ❉❛t❛ ✿ round❴part ✐s t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ truncate❴part ❉❛t❛ ✿ i❴mul❴rounded ✐s t❤❡ r❡s✉❧t ❛❢t❡r r♦✉♥❞✐♥❣ ❉❛t❛ ✿ result ✐s t❤❡ ♣r♦❞✉❝t ♦❢ a ❛♥❞ b ✐♥ ✜①❡❞✲♣♦✐♥t Pr♦❝❡❞✉r❡ ✐▼✉❧✭❛✱ ❜ ✮❀ ❜❡❣✐♥ ❈♦♥✈❡rt ❛ ✢♦❛t✐♥❣✲♣♦✐♥t x t♦ ❜✐♥❛r② ♥✉♠❜❡rs binx ❀ a❴shl❴f p ❂ ❙❤✐❢t ❧❡❢t a ❜② f p ❜✐ts❀ i❴a❴shl❴f p ❂ ■♥t❡❣❡r ♣❛rt ♦❢ a❴shl❴f p❀ b❴shl❴f p ❂ ❙❤✐❢t ❧❡❢t b ❜② f p ❜✐ts❀ i❴b❴shl❴f p ❂ ■♥t❡❣❡r ♣❛rt ♦❢ b❴shl❴f p❀ i❴mul ❂ ♠✉❧t✐♣❧② t✇♦ ✐♥t❡❣❡rs i❴a❴shl❴f p ❛♥❞ i❴b❴shl❴f p❀ i❴mul❴shr❴f p ❂ ❙❤✐❢t r✐❣❤t i❴mul ❜② f p ❜✐ts ❛♥❞ t❤❡♥ t❛❦❡ t❤❡ ✐♥t❡❣❡r ❛♥❞ t❤❡ ❢r❛❝t✐♦♥❛❧ ♣❛rt ♦❢ im ul ✐s t❤❛t ipart❴i❴mul ❛♥❞ f part❴i❴mul ❀ truncate❴part ❂ ❙❤✐❢t ❧❡❢t ✶ ❜✐t f part❴i❴mul❀ round❴part ❂ ❚❛❦❡ ■♥t❡❣❡r ♣❛rt ♦❢ truncate❴part❀ i❴mul❴rounded ❂ ipart❴i❴mul ✰ round❴part❀ ✇✐t❤ r♦✉♥❞✐♥❣ t♦ t❤❡ ♥❡❛r❡st result ❂ ❙❤✐❢t r✐❣❤t i❴mul❴rounded ❜② fp ❜✐ts❀ r❡t✉r♥ result ❡♥❞ ❆❧❣♦r✐t❤♠ ✷✿ ❋✐①❡❞✲♣♦✐♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❡♠✉❧❛t✐♦♥ ✐♥ ▼❛t❤❡♠❛t✐❝❛ ❚❛❜❧❡ ✹✳✶✿ ❚♦♣ r♦✉♥❞✲♦✛ ❡rr♦rs ✐♥ ✶✵✵✳✵✵✵✳✵✵✵ t❡sts ✇✐t❤ r♦✉♥❞ t♦ t❤❡ ♥❡❛r❡st ◆♦✳ x y err ✶ ✷✳✾✵✻✶✽✹✻✺✾✺✾✼✾✼✻✸ ✲✻✳✺✸✵✽✸✵✼✺✷✺✷✺✻✼✹ ✵✳✷✷✻✼✹✵✵✷✽✷✵✽✷✼✾✻✺ ✷ ✷✳✾✵✻✶✸✹✶✷✹✺✾✵✹✻✸✺ ✲✹✳✹✵✻✶✸✽✺✹✵✹✸✸✵✵✹✺ ✵✳✷✷✻✼✺✹✵✾✵✺✹✷✶✽✸✷ ✸ ✷✳✾✵✻✷✷✷✸✾✸✹✼✷✺✶✵✼ ✲✸✳✷✶✽✹✾✵✷✺✾✻✾✺✷✾✹✼ ✵✳✷✷✼✶✶✽✽✻✵✵✶✻✸✽✷✹✽ ✷✻ ❈❤❛♣t❡r ✹✿ ❊①♣❡r✐♠❡♥ts ❚❛❜❧❡ ✹✳✷✿ ❚♦♣ r♦✉♥❞✲♦✛ ❡rr♦rs ✐♥ ✶✵✵✳✵✵✵✳✵✵✵ t❡sts ✇✐t❤ r♦✉♥❞ t♦✇❛r❞s ◆♦✳ x ✶ ✷ ✸ −∞ y err ✷✳✽✼✹✾✻✾✹✸✵✹✸✺✼✶✵✸ ✲✵✳✽✼✹✸✻✶✷✾✾✽✷✼✹✷✷ ✵✳✹✺✷✸✶✵✺✷✺✼✻✼✷✺✹✹ ✷✳✽✼✹✾✻✹✼✾✺✺✷✶✵✽✺ ✲✺✳✹✸✼✶✻✸✸✺✺✺✽✽✽✵✺✺ ✵✳✹✺✷✺✽✺✾✸✶✵✼✹✸✾✽✾✸ ✷✳✽✼✹✾✹✸✼✹✻✵✹✻✷✹✹✹ ✲✶✳✷✹✾✼✽✺✷✽✾✻✻✸✾✺✻ ✵✳✹✺✷✺✽✻✽✸✷✺✾✹✸✻✽✼ ✷✼ ❈❤❛♣t❡r ✺ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❲❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❛♥❞ ✐♥st❛♥t✐❛t❡❞ t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲ ♦✛ ❡rr♦r ❜❡t✇❡❡♥ ❛ ✢♦❛t✐♥❣✲♣♦✐♥t ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✜①❡❞✲♣♦✐♥t ✈❡rs✐♦♥✳ ❚❤❡ s②♠✲ ❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ✐s ❜❛s❡❞ ♦♥ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❡①t❡♥❞❡❞ ❢♦r t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r s♦ t❤❛t ✇❡ ❝❛♥ ♣r♦❞✉❝❡ ❛ ♣r❡❝✐s❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r✳ ■t ❛❧❧♦✇s ✉s t♦ ❞❡t❡r♠✐♥❡ ❛ ♣r❡❝✐s❡ ♠❛①✐♠❛❧ r♦✉♥❞✲♦✛ ❡rr♦r ❛♥❞ t♦ ♣r♦❞✉❝❡ t❤❡ t❡st ❝❛s❡ ❢♦r t❤❡ ✇♦rst ❡rr♦r✳ ❲❡ ❛❧s♦ ❜✉✐❧t ❛ t♦♦❧ t❤❛t ✉s❡s ❙②♠❜♦❧✐❝ P❛t❤❋✐♥❞❡r ❛♥❞ ▼❛t❤❡♠❛t✐❝❛ t♦ ✜♥❞ t❤❡ ✇♦rst ❡rr♦r ❢r♦♠ ❏❛✈❛ ♣r♦❣r❛♠s✳ ❚❤❡ ✐♥✐t✐❛❧ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❛r❡ ✈❡r② ♣r♦♠✐s✐♥❣✳ ■♥ t❤❡ ❢✉t✉r❡✱ ✇❡ ♣❧❛♥ t♦ ✐♥✈❡st✐❣❛t❡ ♣♦ss✐❜✐❧✐t✐❡s t♦ r❡❞✉❝❡ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r ❜❡❢♦r❡ s❡♥❞✐♥❣ ✐t t♦ t❤❡ s♦❧✈❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ♠✐❣❤t ✐♥tr♦❞✉❝❡ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ s②♠❜♦❧✐❝ r♦✉♥❞✲♦✛ ❡rr♦r✳ ❋♦r r❡❛❧ ✇♦r❧❞ ♣r♦❣r❛♠s✱ ❡s♣❡❝✐❛❧❧② t❤❡ ♦♥❡s ✇✐t❤ ❧♦♦♣s✱ ✇❡ ❜❡❧✐❡✈❡ t❤❛t ❝♦♠❜✐♥✐♥❣ ✐♥t❡r✈❛❧ ❛♥❛❧②s✐s ✇✐t❤ ♦✉r ❛♣♣r♦❛❝❤ ♠❛② ❛❧❧♦✇ ✉s t♦ ✜♥❞ ❛ ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥ ♣r❡❝✐s❡♥❡ss ❛♥❞ s❝❛❧❛❜✐❧✐t②✳ ❘❡♣❧❛❝✐♥❣ ▼❛t❤❡♠❛t✐❝❛ ✇✐t❤ ♦t❤❡r s♦❧✈❡rs s✉❝❤ ❛s r❛❙❆❚ ❬✶✾❪ ♦r ❩✸ ❬✺❪ ✐s ❛❧s♦ ❛♥♦t❤❡r t❤♦✉❣❤t✳ ❆s ❩✸ s✉♣♣♦rts ✜①❡❞✲♣♦✐♥t ♥✉♠❜❡r r❡♣r❡s❡♥t❛t✐♦♥✱ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♠✐❣❤t ❜❡ ❡❛s②✳ ❲❡ ❛❧s♦ ♣❧❛♥ t♦ st✉❞② t❤❡ r♦✉♥❞✲♦✛ ❡rr♦r t❤❛t ♠❛② ❝❛✉s❡ t❤❡ t✇♦ ✈❡rs✐♦♥s ♦❢ t❤❡ ♣r♦❣r❛♠ ❢♦❧❧♦✇✐♥❣ ❞✐✛❡r❡♥t ❡①❡❝✉t✐♦♥ ♣❛t❤s✳ ✷✽ ❘❡❢❡r❡♥❝❡s ❬✶❪ ▼✐❝❤❡❧ ❘✉❡❤❡r ❆❧❡①❛♥❞r❡ ●♦❧❞s③t❡❥♥✱ ❉❛✈✐❞ ❉❛♥❡② ❛♥❞ P❛tr✐❝❦ ❚❛✐❧❧✐❜❡rt✳ ▼♦❞❛❧ ✐♥t❡r✈❛❧s r❡✈✐s✐t❡❞✿ ❛ ♠❡❛♥✲✈❛❧✉❡ ❡①t❡♥s✐♦♥ t♦ ❣❡♥❡r❛❧✐③❡❞ ✐♥t❡r✈❛❧s✳ Pr♦❝✳ ♦❢ t❤❡ ✶st ■♥t❡r♥❛t✐♦♥❛❧ ❲♦r❦s❤♦♣ ♦♥ ◗✉❛♥t✐✜❝❛t✐♦♥ ✐♥ ❈♦♥str❛✐♥t Pr♦❣r❛♠♠✐♥❣✱ ✷✵✵✺✳ ■♥ ❬✷❪ ❘♦❜❡rt ❙ ❇♦②❡r✱ ❇❡r♥❛r❞ ❊❧s♣❛s✱ ❛♥❞ ❑❛r❧ ◆ ▲❡✈✐tt✳ ❙❡❧❡❝t✖❛ ❢♦r♠❛❧ s②st❡♠ ❢♦r t❡st✐♥❣ ❛♥❞ ❞❡❜✉❣❣✐♥❣ ♣r♦❣r❛♠s ❜② s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥✳ ◆♦t✐❝❡s✱ ✶✵✭✻✮✿✷✸✹✕✷✹✺✱ ✶✾✼✺✳ ❆❈▼ ❙✐❣P❧❛♥ ❬✸❪ ❈✳ ❈❛❞❛r✱ P✳ ●♦❞❡❢r♦✐❞✱ ❙✳ ❑❤✉rs❤✐❞✱ ❈✳❙✳ P❛s❛r❡❛♥✉✱ ❑♦✉s❤✐❦ ❙❡♥✱ ◆✳ ❚✐❧❧✲ ♠❛♥♥✱ ❛♥❞ ❲✳ ❱✐ss❡r✳ ♣r❡❧✐♠✐♥❛r② ❛ss❡ss♠❡♥t✳ ❙②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❢♦r s♦❢t✇❛r❡ t❡st✐♥❣ ✐♥ ♣r❛❝t✐❝❡✿ ■♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣ ✭■❈❙❊✮✱ ✷✵✶✶ ✸✸r❞ ■♥t❡r✲ ♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥✱ ♣❛❣❡s ✶✵✻✻✕✶✵✼✶✱ ▼❛② ✷✵✶✶✳ ❞♦✐✿ ✶✵✳✶✶✹✺✴✶✾✽✺✼✾✸✳ ✶✾✽✺✾✾✺✳ ❬✹❪ ❈r✐st✐❛♥ ❈❛❞❛r ❛♥❞ ❑♦✉s❤✐❦ ❙❡♥✳ t❤r❡❡ ❞❡❝❛❞❡s ❧❛t❡r✳ ❙②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ❢♦r s♦❢t✇❛r❡ t❡st✐♥❣✿ ❈♦♠♠✉♥✐❝❛t✐♦♥s ♦❢ t❤❡ ❆❈▼✱ ✺✻✭✷✮✿✽✷✕✾✵✱ ✷✵✶✸✳ ❬✺❪ ▲❡♦♥❛r❞♦ ❞❡ ▼♦✉r❛ ❛♥❞ ◆✐❦♦❧❛❥ ❇❥ør♥❡r✳ ❩✸✿ ❆♥ ❡✣❝✐❡♥t s♠t s♦❧✈❡r✳ ■♥ ❈✳❘✳ ❚♦♦❧s ❛♥❞ ❆❧❣♦r✐t❤♠s ❢♦r t❤❡ ❈♦♥✲ str✉❝t✐♦♥ ❛♥❞ ❆♥❛❧②s✐s ♦❢ ❙②st❡♠s✱ ✈♦❧✉♠❡ ✹✾✻✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✸✼✕✸✹✵✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✽✳ ■❙❇◆ ✾✼✽✲✸✲✺✹✵✲ ❘❛♠❛❦r✐s❤♥❛♥ ❛♥❞ ❏❛❦♦❜ ❘❡❤♦❢✱ ❡❞✐t♦rs✱ ✼✽✼✾✾✲✵✳ ❞♦✐✿ ✶✵✳✶✵✵✼✴✾✼✽✲✸✲✺✹✵✲✼✽✽✵✵✲✸❴✷✹✳ ❯❘▲ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴ ✶✵✳✶✵✵✼✴✾✼✽✲✸✲✺✹✵✲✼✽✽✵✵✲✸❴✷✹✳ ❬✻❪ ❍❡r❞❡ ❈✳ ❚❡✐❣❡ ❚✳ ❘❛ts❝❤❛♥ ❙✳ ❋r❛♥③❧❡✱ ▼✳ ❛♥❞ ❚✳ ❙❝❤✉❜❡rt✳ ❊✣❝✐❡♥t s♦❧✈✲ ✐♥❣ ♦❢ ❧❛r❣❡ ♥♦♥✲❧✐♥❡❛r ❛r✐t❤♠❡t✐❝ ❝♦♥str❛✐♥t s②st❡♠s ✇✐t❤ ❝♦♠♣❧❡① ❜♦♦❧❡❛♥ str✉❝t✉r❡✳ ■♥ ❏♦✉r♥❛❧ ♦♥ ❙❛t✐s✜❛❜✐❧✐t②✱ ❇♦♦❧❡❛♥ ▼♦❞❡❧✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ♣❛❣❡s ✷✵✾ ✕ ✷✸✻✱ ✷✵✵✼✳ ✷✾ ❘❊❋❊❘❊◆❈❊❙ ❬✼❪ ❉❛✈✐❞ ●♦❧❞❜❡r❣✳ ❲❤❛t ❊✈❡r② ❈♦♠♣✉t❡r ❙❝✐❡♥t✐st ❙❤♦✉❧❞ ❑♥♦✇ ❆❜♦✉t ❋❧♦❛t✐♥❣✲P♦✐♥t ❆r✐t❤♠❡t✐❝✳ ■♥ ❆❈▼ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s✱ ♣❛❣❡s ✺ ✕ ✹✽✱ ✶✾✾✶✳ ❞♦✐✿ ✶✵✳✶✶✹✺✴✶✵✸✶✻✷✳✶✵✸✶✻✸✳ ❬✽❪ ◆✐❝❤♦❧❛s ❏✳ ❍✐❣❤❛♠✳ ❆❝❝✉r❛❝② ❛♥❞ ❙t❛❜✐❧✐t② ♦❢ ◆✉♠❡r✐❝❛❧ ❆❧❣♦r✐t❤♠s✳ ❙■❆▼✿ ❙♦❝✐❡t② ❢♦r ■♥❞✉str✐❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✷✵✵✷✳ ■❙❇◆ ✾✼✽✲✵✽✾✽✼✶✺✷✶✼✳ ❬✾❪ ❏❛♠❡s ❈✳ ❑✐♥❣ ❛♥❞ ❏✳ ❲❛ts♦♥✳ ❙②♠❜♦❧✐❝ ❊①❡❝✉t✐♦♥ ❛♥❞ Pr♦❣r❛♠ ❚❡st✐♥❣✳ ■♥ ❈♦♠♠✉♥✐❝❛t✐♦♥s ♦❢ t❤❡ ❆❈▼✱ ♣❛❣❡s ✸✽✺ ✕ ✸✾✹✱ ✶✾✼✻✳ ❞♦✐✿ ✶✵✳✶✶✹✺✴✸✻✵✷✹✽✳ ✸✻✵✷✺✷✳ ❬✶✵❪ ▼✳ ▼❛rt❡❧✳ ❙❡♠❛♥t✐❝s ♦❢ r♦✉♥❞♦✛ ❡rr♦r ♣r♦♣❛❣❛t✐♦♥ ✐♥ ✜♥✐t❡ ♣r❡❝✐s✐♦♥ ❝❛❧❝✉✲ ❧❛t✐♦♥s✳ ■♥ ❍✐❣❤❡r✲❖r❞❡r ❛♥❞ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t❛t✐♦♥✱ ♣❛❣❡s ✼ ✕ ✸✵✱ ✷✵✵✻✳ ❬✶✶❪ ❘❛♠♦♥ ❊ ▼♦♦r❡✳ ■♥t❡r✈❛❧ ❛♥❛❧②s✐s✱ ✈♦❧✉♠❡ ✹✳ Pr❡♥t✐❝❡✲❍❛❧❧ ❊♥❣❧❡✇♦♦❞ ❈❧✐✛s✱ ✶✾✻✻✳ ❬✶✷❪ ❉♦ ❚❤✐ ❇✐❝❤ ◆❣♦❝ ❛♥❞ ▼✐③✉❤✐t♦ ❖❣❛✇❛✳ ❖✈❡r✢♦✇ ❛♥❞ ❘♦✉♥❞♦✛ ❊rr♦r ❆♥❛❧✲ ②s✐s ✈✐❛ ▼♦❞❡❧ ❈❤❡❝❦✐♥❣✳ ■♥ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❋♦r♠❛❧ ▼❡t❤♦❞s✱ ♣❛❣❡s ✶✵✺✕✶✶✹✱ ✷✵✵✾✳ ❞♦✐✿ ✶✵✳✶✶✵✾✴❙❊❋▼✳✷✵✵✾✳✸✷✳ ❬✶✸❪ ❉♦ ❚❤✐ ❇✐❝❤ ◆❣♦❝ ❛♥❞ ▼✐③✉❤✐t♦ ❖❣❛✇❛✳ ❈♦♠❜✐♥✐♥❣ ❚❡st✐♥❣ ❛♥❞ ❙t❛t✐❝ ❆♥❛❧✲ ②s✐s t♦ ❖✈❡r✢♦✇ ❛♥❞ ❘♦✉♥❞♦✛ ❊rr♦r ❉❡t❡❝t✐♦♥✳ ■♥ ❏❆■❙❚ ❘❡s❡❛r❝❤ ❘❡♣♦rts✱ ♣❛❣❡s ✶✵✺✕✶✶✹✱ ✷✵✶✵✳ ❬✶✹❪ ❈♦r✐♥❛ ❙ P➔s➔r❡❛♥✉ ❛♥❞ ◆❡❤❛ ❘✉♥❣t❛✳ ❙②♠❜♦❧✐❝ ♣❛t❤✜♥❞❡r✿ s②♠❜♦❧✐❝ ❡①✲ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■❊❊❊✴❆❈▼ ✐♥t❡r♥❛t✐♦♥❛❧ ❝♦♥❢❡r❡♥❝❡ ♦♥ ❆✉t♦♠❛t❡❞ s♦❢t✇❛r❡ ❡♥❣✐♥❡❡r✐♥❣✱ ♣❛❣❡s ✶✼✾✕✶✽✵✳ ❆❈▼✱ ✷✵✶✵✳ ❡❝✉t✐♦♥ ♦❢ ❥❛✈❛ ❜②t❡❝♦❞❡✳ ❬✶✺❪ ❚✉❛♥✲❍✉♥❣ P❤❛♠✱ ■♥ ❆♥❤✲❍♦❛♥❣ ❚r✉♦♥❣✱ ❲❡✐✲◆❣❛♥ ❈❤✐♥✱ ❛♥❞ ❚❛❦❡♥♦❜✉ ❆♦s❤✐♠❛✳ ❚❡st ❈❛s❡ ●❡♥❡r❛t✐♦♥ ❢♦r ❆❞❡q✉❛❝② ♦❢ ❋❧♦❛t✐♥❣✲♣♦✐♥t t♦ ❋✐①❡❞✲♣♦✐♥t ❈♦♥✈❡rs✐♦♥✳ ❊❧❡❝tr♦♥✐❝ ◆♦t❡s ✐♥ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✷✻✻✭✵✮✿✹✾ ✕ ✻✶✱ ❤tt♣✿✴✴✇✇✇✳ s❝✐❡♥❝❡❞✐r❡❝t✳❝♦♠✴s❝✐❡♥❝❡✴❛rt✐❝❧❡✴♣✐✐✴❙✶✺✼✶✵✻✻✶✶✵✵✵✶✷✼✶✳ Pr♦❝❡❡❞✲ ✷✵✶✵✳ ■❙❙◆ ✶✺✼✶✲✵✻✻✶✳ ❞♦✐✿ ✶✵✳✶✵✶✻✴❥✳❡♥t❝s✳✷✵✶✵✳✵✽✳✵✹✽✳ ❯❘▲ ✐♥❣s ♦❢ t❤❡ ✸r❞ ■♥t❡r♥❛t✐♦♥❛❧ ❲♦r❦s❤♦♣ ♦♥ ❍❛r♥❡ss✐♥❣ ❚❤❡♦r✐❡s ❢♦r ❚♦♦❧ ❙✉♣✲ ♣♦rt ✐♥ ❙♦❢t✇❛r❡ ✭❚❚❙❙✮✳ ✸✵ ❘❊❋❊❘❊◆❈❊❙ ❬✶✻❪ ❈♦r✐♥❛❙✳ P➔s➔r❡❛♥✉✱ ❲✐❧❧❡♠ ❱✐ss❡r✱ ❉❛✈✐❞ ❇✉s❤♥❡❧❧✱ ❏❛❝♦ ●❡❧❞❡♥❤✉②s✱ P❡✲ t❡r ▼❡❤❧✐t③✱ ❛♥❞ ◆❡❤❛ ❘✉♥❣t❛✳ ❜♦❧✐❝ ❡①❡❝✉t✐♦♥ ✇✐t❤ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ t♦♠❛t❡❞ ❙♦❢t✇❛r❡ ❊♥❣✐♥❡❡r✐♥❣✱ ❞♦✐✿ ❙②♠❜♦❧✐❝ ♣❛t❤✜♥❞❡r✿ ❢♦r ❥❛✈❛ ✷✵✭✸✮✿✸✾✶✕✹✷✺✱ ✶✵✳✶✵✵✼✴s✶✵✺✶✺✲✵✶✸✲✵✶✷✷✲✷✳ ❯❘▲ ✐♥t❡❣r❛t✐♥❣ s②♠✲ ❜②t❡❝♦❞❡ ✷✵✶✸✳ ❛♥❛❧②s✐s✳ ■❙❙◆ ❆✉✲ ✵✾✷✽✲✽✾✶✵✳ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴ s✶✵✺✶✺✲✵✶✸✲✵✶✷✷✲✷✳ ❬✶✼❪ ❲✐❧❧✐❛♠ ❙t❛❧❧✐♥❣s✳ ❈♦♠♣✉t❡r ❖r❣❛♥✐③❛t✐♦♥ ❛♥❞ ❆r❝❤✐t❡❝t✉r❡✳ ▼❛❝♠✐❧❧❛♥ P✉❜✲ ❧✐s❤✐♥❣ ❈♦♠♣❛♥②✱ ✷✵✵✵✳ ■❙❇◆ ✵✲✵✷✲✹✶✺✹✽✵✲✻✳ ❬✶✽❪ ❏✳ ❙t♦❧✜ ❛♥❞ ▲✳❍✳ ❞❡ ❋✐❣✉❡✐r❡❞♦✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❛✣♥❡ ❛r✐t❤♠❡t✐❝✳ ■♥ ❚❡♥❞❡♥❝✐❛s ❡♠ ▼❛t❡♠❛t✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✱ ✷✵✵✺✳ ❬✶✾❪ ❱❛♥✲❑❤❛♥❤ ❚♦ ❛♥❞ ▼✐③✉❤✐t♦ ❖❣❛✇❛✳ r❛❙❆❚✿ ❙▼❚ ❢♦r P♦❧②♥♦♠✐❛❧ ■♥❡q✉❛❧✲ ✐t②✳ ❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱ ❘❡s❡❛r❝❤ r❡♣♦rt ✭❙❝❤♦♦❧ ♦❢ ■♥❢♦r♠❛t✐♦♥ ❙❝✐❡♥❝❡✱ ❏❛♣❛♥ ❆❞✈❛♥❝❡❞ ■♥st✐t✉t❡ ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✮✱ ✷✵✶✸✳ ❬✷✵❪ ❆♥❤✲❍♦❛♥❣ ❚r✉♦♥❣✱ ❍✉②✲❱✉ ❚r❛♥✱ ❛♥❞ ❇❛♦✲◆❣♦❝ ◆❣✉②❡♥✳ ❋✐♥❞✐♥❣ r♦✉♥❞✲ ❑♥♦✇❧❡❞❣❡ ❛♥❞ ❙②st❡♠s ❊♥❣✐♥❡❡r✲ ❆❞✈❛♥❝❡s ✐♥ ■♥t❡❧❧✐❣❡♥t ❙②st❡♠s ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ♣❛❣❡s ♦✛ ❡rr♦r ✉s✐♥❣ s②♠❜♦❧✐❝ ❡①❡❝✉t✐♦♥✳ ✐♥❣✱ ✈♦❧✉♠❡ ✷✹✹ ♦❢ ■♥ ✹✶✺✕✹✷✽✳ ❙♣r✐♥❣❡r ■♥t❡r♥❛t✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ✷✵✶✹✳ ■❙❇◆ ✾✼✽✲✸✲✸✶✾✲✵✷✼✹✵✲✶✳ ❞♦✐✿ ✶✵✳✶✵✵✼✴✾✼✽✲✸✲✸✶✾✲✵✷✼✹✶✲✽❴✸✺✳ ❯❘▲ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴ ✾✼✽✲✸✲✸✶✾✲✵✷✼✹✶✲✽❴✸✺✳ ❬✷✶❪ ❙t❡♣❤❡♥ ❲♦❧❢r❛♠✳ ♣✉t❡r✳ ▼❛t❤❡♠❛t✐❝❛✿ ❆ ❙②st❡♠ ❢♦r ❉♦✐♥❣ ▼❛t❤❡♠❛t✐❝s ❜② ❈♦♠✲ ❆❞❞✐s♦♥✲❲❡s❧❡②✱ ✶✾✾✶✳ ■❙❇◆ ✾✼✽✲✵✷✵✶✺✶✺✵✷✷✳ ❬✷✷❪ ❘❛♥❞② ❨❛t❡s✳ ❋✐①❡❞✲♣♦✐♥t ❛r✐t❤♠❡t✐❝✿ ❆♥ ✐♥tr♦❞✉❝t✐♦♥✳ ✽✶✭✽✸✮✿✶✾✽✱ ✷✵✵✾✳ ✸✶ ❉✐❣✐t❛❧ ❙✐❣♥❛❧ ▲❛❜s✱
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