Yintoni ekufuneka isazi sonke ngekhompyutha malunga ne-arithmetic yendawo edadayo (1991) [pdf]

I-Trap echanekileyo engabonakaliyo: Kutheni yonke iProgrammer idinga le PDF ka-1991

Kwihlabathi elichanekileyo, elinengqiqo yesayensi yekhompyutha, amaxwebhu ambalwa anefuthe elihlala lihleli, isiseko sephepha likaDavid Goldberg lika-1991, "Yintoni yonke iNzululwazi yeKhompyutha emele iyazi malunga ne-Floating-Point Arithmetic." Ngaphezu kwamashumi amathathu eminyaka kamva, isihloko sayo sisengubizo olucacileyo, isilumkiso, kunye nenxalenye ebalulekileyo yobulumko. Nabani na obhala ikhowudi ejongene namanani okwenene-ukusuka kukulinganisa kwezenzululwazi kunye neenkqubo zemali ukuya kwiinjini zomdlalo kunye nohlalutyo lwedatha-ukungahoywa izifundo zayo kukuhlawula iintsilelo ezifihlakeleyo, ezixabisa kakhulu, kwaye zihlala zixaka. Kwixesha apho imisebenzi yoshishino ixhaswe ngamandla yisoftware entsonkothileyo, edityanisiweyo, ukuqonda isiseko sokubala kwamanani akusiyo imfundo; yimfuneko yokusebenza. Oku kuyinyani ngakumbi xa usebenzisa i-OS yeshishini lemodyuli efana neMewayz, apho imfezeko yedatha kuzo zonke iimodyuli-ukusuka kuhlalutyo ukuya kwi-automated billing-ixhomekeke ekuqikelelekeni, ukubala okuthembekileyo.

Ingxaki enguNdoqo: Awunako ukuMela okuNgapheleliyo kwiiBits eziPheleleyo

Umba ongundoqo ulula kodwa unzulu. Iikhompyuter zethu zinememori elinganiselweyo, kodwa sihlala sidinga ukusebenza ngokuqhubekeka okungapheliyo kwamanani okwenyani (njengo π okanye 0.1). I-arithmetic yeendawo ezidadayo yimilinganiselo esemgangathweni, inkqubo ekrelekrele yokumela uluhlu olubanzi lwamanani ngokuchaneka okulinganiselweyo. Nangona kunjalo, oku kuhambelana kuthetha ukuba amanani amaninzi aqikelelweyo, awagcinwanga ngokuthe ngqo. Iphepha leGoldberg licacisa ngokucokisekileyo umgangatho we-IEEE 754, ozise ukuhambelana okufunekayo kolu qhushululu. Unika iinkcukacha ngendlela amanani afakwa ngayo ikhowudi kumqondiso, i-exponent, kunye ne-fraction bits, ukudala imbonakalo yomhlaba eqikelelwayo kodwa engaqondakaliyo yamaxabiso ameleyo, ukuziphatha okurhangqwayo, kunye nemibutho ekhethekileyo efana ne-NaN (Hayi inani) kunye nokungapheliyo. Kubaphuhlisi abakha imifuziselo yezemali kwi-Mewayz, impazamo ejikelezayo ebonakala ngathi i-microscopic inokuphumela kwiiyantlukwano ezinkulu kwiingxelo okanye kwintengiselwano, ijongela phantsi ukuthembana kuyo yonke inkqubo.

Ukuziphatha okumangalisayo kunye nokungaphumeleli kwentlekele

Iphepha lidume ngokubonisa imigibe ephikisayo eyaphula uqikelelo lwemathematika olusisiseko. Umzekelo, ngenxa yokurhangqa, ukongezwa kwendawo edadayo akunxulumanisi; `(a + b) + c` ayisoloko ilingana `a + (b + c)`. Oku kunokukhokelela kwiziphumo ezingamiselwanga kubalo oluhambelanayo. Ukuthabatha phantse amanani alinganayo kunokukhokelela ekurhoxisweni okuyintlekele, apho amanani abalulekileyo athi anyamalale, okushiya ubukhulu becala impazamo yokusondeza. Mhlawumbi esona sifundo sidumileyo sisidingo sokungaze uthelekise amanani eendawo ezidadayo ngokulingana ngqo (`==`) kodwa endaweni yoko jonga ukuba umahluko uphakathi konyamezelo oluncinci. Ezi ayizozinto nje zethiyori. Baye babangela iintlekele zehlabathi lokwenene, ukusuka ekuqhumeni kwe-Ariane 5 rocket ukuya kwizinto ezingachanekanga kwiinkqubo zokuqala ze-missile ye-Patriot. Kwimeko yeshishini, iimpazamo ezinjalo kubalo lwe-inventri, i-algorithms yamaxabiso, okanye i-metrics yokusebenza inokukhokelela ekukhohliseni inkohlakalo yedatha, ukwenza iiplatifomu eziqinileyo ezifana Mewayz zibalulekile ekunyanzeliseni ukuqinisekiswa kwedatha kunye nokuhlolwa kokuhambelana kuzo zonke iimodyuli.

"Ukucudisa amanani okwenene amaninzi kwinani eliqingqiweyo lamasuntswana kufuna umboniso oqikelelweyo."

Izinto eziphambili zokuThatha uMvelisi wangoku

Iphepha likaGoldberg alinikezi nje izilumkiso kodwa nesikhokelo esisebenzayo. Oyena ndoqo wokuthatha kukuhlakulela "ukwazi kwamanani" -ukwazi rhoqo ukuba amanani ahambayo aqikelelwa. Le ngcamango kufuneka yazise ukhetho ukusuka ekukhethweni kwesakhiwo sedatha ukuya kuyilo lwe-algorithm. Umsebenzi wakhe ugxininisa ukuba kutheni ukusebenzisa `i-double` (i-64-bit) phantse isoloko ikhethwa kwi `float` (32-bit) kumsebenzi obalulekileyo ochanekileyo, kwaye kutheni iialgorithms ezithile zizinzile ngokwamanani ngelixa ezinye zingezizo. Xa uyila okanye udibanisa iimodyuli ngaphakathi kweMewayzimeko-imeko-nokuba ngaba ngumatshini wokufunda kwangaphambili okanye umcwangcisi wemithombo-oku kuqonda kuqinisekisa ukuba imisebenzi yesiseko yamanani iphathwa ngentlonipho abayifunayo, ukuthintela ama-glitches anzima kakhulu ukulandelela umva kwingcambu yawo.

Wonke umdwelisi wenkqubo kufuneka aziqhelanise nala magama abalulekileyo asuka kwiphepha:

• Impazamo yokujikeleza: Ukungachaneki okungenakuthintelwa ukusuka ekufakeni inani kwixabiso elikufutshane elimeleyo.
• Amanani Oonogada: Amanani awongezelelweyo asetyenziswa kubalo oluphakathi ukunciphisa impazamo yokusondeza.
• I-IEEE 754 Standard: Iplani yendalo yonke yobalo lwamanqaku adadayo, ukuchaza iifomathi, imithetho yokurhangqa, kunye nezinto ezingaphandle.
• NaN kunye ne-Infinity: Amaxabiso akhethekileyo avumela imisebenzi ukuba isasaze iimpazamo ngobubele kunokuba ingqubeke.
• Uzinzo lwamanani: Ipropathi ye-algorithm yokulawula ukukhulisa impazamo kwimisebenzi emininzi.

Uxwebhu oluphilayo lwehlabathi leDijithali

Ngeli xesha libhalwe ngo-1991, ukufaneleka kwephepha kuye kwakhula. Imigaqo ye-IEEE 754 ixhasa yonke i-CPU yangoku, i-GPU, kunye nolwimi lwenkqubo. Njengoko sityhala kwimida efana ne-AI, uhlalutyo olukhulu lwedatha, kunye nokulinganisa inkqubo entsonkothileyo, ukuchaneka kokubala kwethu kuya kuba baluleke ngakumbi. Kumaqela asebenzisa inkqubo yokusebenza yemodyuli efana neMewayzukulungelelanisa ingqiqo yeshishini labo, ukubethelela oku kungqongqo kwamanani kwiimodyuli zabo zesiko yindlela efanelekileyo yokuthintela udidi lweebhugi kwinqanaba elibalulekileyo. Umsebenzi wobugcisa weGoldberg ungaphezulu kwephepha; yinxalenye esisigxina yesiseko sobunjineli besoftware ethembekileyo. Ukungayihoyi kukwakha phezu kwesanti, ubeka emngciphekweni imfezeko yolwakhiwo lwedijithali luphela, nokuba siskripthi esilula okanye i-OS yeshishini loshishino.

Imibuzo Ebuzwa Rhoqo

I-Trap echanekileyo engabonakaliyo: Kutheni yonke iProgrammer idinga le PDF ka-1991

Kwihlabathi elichanekileyo, elinengqiqo yesayensi yekhompyutha, amaxwebhu ambalwa anefuthe elihlala lihleli, isiseko sephepha likaDavid Goldberg lika-1991, "Yintoni yonke iNzululwazi yeKhompyutha emele iyazi malunga ne-Floating-Point Arithmetic." Ngaphezu kwamashumi amathathu eminyaka kamva, isihloko sayo sisengubizo olucacileyo, isilumkiso, kunye nenxalenye ebalulekileyo yobulumko. Nabani na obhala ikhowudi ejongene namanani okwenene-ukusuka kukulinganisa kwezenzululwazi kunye neenkqubo zemali ukuya kwiinjini zomdlalo kunye nohlalutyo lwedatha-ukungahoywa izifundo zayo kukuhlawula iintsilelo ezifihlakeleyo, ezixabisa kakhulu, kwaye zihlala zixaka. Kwixesha apho imisebenzi yoshishino ixhaswe ngamandla yisoftware entsonkothileyo, edityanisiweyo, ukuqonda isiseko sokubala kwamanani akusiyo imfundo; yimfuneko yokusebenza. Oku kuyinyani ngakumbi xa usebenzisa i-OS yemodyuli yeshishini elifana ne-Mewayz, apho ukuthembeka kwedatha kuzo zonke iimodyuli-ukusuka kuhlalutyo ukuya kwi-automated billing-ixhomekeke kuqikelelo oluqikelelwayo, ukubala okuthembekileyo.

Ingxaki enguNdoqo: Awunako ukumela i-Infinity kwi-Finite Bits

Umba ongundoqo ulula kodwa unzulu. Iikhompyuter zethu zinememori elinganiselweyo, kodwa sihlala sidinga ukusebenza ngokuqhubekeka okungapheliyo kwamanani okwenyani (njengo π okanye 0.1). I-arithmetic yeendawo ezidadayo yimilinganiselo esemgangathweni, inkqubo ekrelekrele yokumela uluhlu olubanzi lwamanani ngokuchaneka okulinganiselweyo. Nangona kunjalo, oku kuhambelana kuthetha ukuba amanani amaninzi aqikelelweyo, awagcinwanga ngokuthe ngqo. Iphepha leGoldberg licacisa ngokucokisekileyo umgangatho we-IEEE 754, ozise ukuhambelana okufunekayo kolu qhushululu. Unika iinkcukacha ngendlela amanani afakwa ngayo ikhowudi kumqondiso, i-exponent, kunye ne-fraction bits, ukudala imbonakalo yomhlaba eqikelelwayo kodwa engaqondakaliyo yamaxabiso ameleyo, ukuziphatha okurhangqwayo, kunye nemibutho ekhethekileyo efana ne-NaN (Hayi inani) kunye nokungapheliyo. Kubaphuhlisi abakha imifuziselo yezemali kwi-Mewayz, impazamo ejikelezayo ebonakala ngathi i-microscopic inokuphumela kwiiyantlukwano ezinkulu kwiingxelo okanye kwintengiselwano, ijongela phantsi ukuthembana kuyo yonke inkqubo.

Ukuziphatha okumangalisayo kunye nokungaphumeleli kwentlekele

Iphepha lidume ngokubonisa imigibe ephikisayo eyaphula uqikelelo lwemathematika olusisiseko. Umzekelo, ngenxa yokurhangqa, ukongezwa kwendawo edadayo akunxulumanisi; `(a + b) + c` ayisoloko ilingana `a + (b + c)`. Oku kunokukhokelela kwiziphumo ezingamiselwanga kubalo oluhambelanayo. Ukuthabatha phantse amanani alinganayo kunokukhokelela ekurhoxisweni okuyintlekele, apho amanani abalulekileyo athi anyamalale, okushiya ubukhulu becala impazamo yokusondeza. Mhlawumbi esona sifundo sidumileyo sisidingo sokungaze uthelekise amanani eendawo ezidadayo ngokulingana ngqo (`==`) kodwa endaweni yoko jonga ukuba umahluko uphakathi konyamezelo oluncinci. Ezi ayizozinto nje zethiyori. Baye babangela iintlekele zehlabathi lokwenene, ukusuka ekuqhumeni kwe-Ariane 5 rocket ukuya kwizinto ezingachanekanga kwiinkqubo zokuqala ze-missile ye-Patriot. Kwimeko yeshishini, iimpazamo ezinjalo kubalo lwe-inventri, i-algorithms yamaxabiso, okanye i-metrics yokusebenza inokukhokelela ekukhohliseni kwedatha ethuleyo, ukwenza amaqonga aqinileyo afana ne-Mewayz abaluleke kakhulu ekunyanzeliseni ukuqinisekiswa kwedatha kunye nokuhlolwa kokuhambelana kuzo zonke iimodyuli.

Izinto eziphambili zokuThatha uMvelisi wangoku

Iphepha likaGoldberg alinikezi nje izilumkiso kodwa nesikhokelo esisebenzayo. Oyena ndoqo wokuthatha kukuhlakulela "ukwazi kwamanani" -ukwazi rhoqo ukuba amanani ahambayo aqikelelwa. Le ngcamango kufuneka yazise ukhetho ukusuka ekukhethweni kwesakhiwo sedatha ukuya kuyilo lwe-algorithm. Umsebenzi wakhe ugxininisa ukuba kutheni ukusebenzisa `i-double` (i-64-bit) phantse isoloko ikhethwa kwi `float` (32-bit) kumsebenzi obalulekileyo ochanekileyo, kwaye kutheni iialgorithms ezithile zizinzile ngokwamanani ngelixa ezinye zingezizo. Xa uyila okanye udibanisa iimodyuli ngaphakathi kwendawo ye-Mewayz-ingaba yinto yokufunda umatshini okanye umcwangcisi wemithombo-oku kuqonda kuqinisekisa ukuba imisebenzi yamanani esisiseko iphathwa ngentlonipho abayifunayo, ukuthintela i-glitches ekunzima kakhulu ukuyilandelela emva kwingcambu yazo.

Uxwebhu oluPhila lweLizwe leDijithali

Ngeli xesha libhalwe ngo-1991, ukufaneleka kwephepha kuye kwakhula. Imigaqo ye-IEEE 754 ixhasa yonke i-CPU yangoku, i-GPU, kunye nolwimi lwenkqubo. Njengoko sityhala kwimida efana ne-AI, uhlalutyo olukhulu lwedatha, kunye nokulinganisa inkqubo entsonkothileyo, ukuchaneka kokubala kwethu kuya kuba baluleke ngakumbi. Kumaqela asebenzisa inkqubo yokusebenza eyimodyuli efana neMewayz ukulungelelanisa ingqiqo yeshishini labo, ukubethelela obu bungqongqo bamanani kwiimodyuli zabo zesiko yeyona ndlela ilungileyo ethintela udidi lweebugs kwelona nqanaba lisisiseko. Umsebenzi wobugcisa weGoldberg ungaphezulu kwephepha; yinxalenye esisigxina yesiseko sobunjineli besoftware ethembekileyo. Ukungayihoyi kukwakha phezu kwesanti, ubeka emngciphekweni imfezeko yolwakhiwo lwedijithali luphela, nokuba siskripthi esilula okanye i-OS yeshishini loshishino.