Lokho wonke usosayensi wekhompyutha okufanele akwazi mayelana ne-floating-point arithmetic (1991) [pdf]

I-Invisible Precision Trap: Kungani Wonke Umhleli Edinga Le PDF Yango-1991

Ezweni elinembayo, elinengqondo lesayensi yekhompiyutha, imibhalo embalwa ibe nomthelela ohlala njalo, oyisisekelo wephepha lika-David Goldberg lango-1991, elithi, "Lokho Okufanele Ukwazi Wonke Usosayensi Wekhompyutha Nge-Floating-Point Arithmetic." Eminyakeni engaphezu kwamashumi amathathu kamuva, isihloko salo sisalokhu siwubizo lokucacisa, isexwayiso, nocezu lokuhlakanipha olubalulekile. Kunoma ubani obhala ikhodi ephathelene nezinombolo zangempela—kusuka ekulingiseni kwesayensi nezinhlelo zezimali kuya ezinjinini zegeyimu nokuhlaziya idatha—ukuziba izifundo zayo kuwukuphikisa ukwehluleka okucashile, okumba eqolo, futhi okuvame ukudida. Enkathini lapho imisebenzi yebhizinisi inikwa amandla ngamandla isofthiwe eyinkimbinkimbi, exhumene, ukuqonda umsuka wokubala izinombolo akuyona imfundo; kuyisidingo sokusebenza. Lokhu kuyiqiniso ikakhulukazi uma usebenzisa i-modular ye-OS yebhizinisi efana ne-Mewayz, lapho ubuqotho bedatha kuwo wonke amamojula—kusuka kuzibalo kuya ekukhokheni okuzenzakalelayo—kuncike ekubaleni okubikezelwe, okuthembekile.

Inkinga Eyinhloko: Awukwazi Ukumela Okungapheli Ku-Finite Bits

Indaba ebalulekile ilula kodwa ijulile. Amakhompyutha ethu anenani elilinganiselwe lenkumbulo, kodwa ngokuvamile sidinga ukusebenza ngokuqhubekayo okungapheli kwezinombolo zangempela (njengo-π noma 0.1). I-arithmetic yephuzu elintantayo iwukuvumelana okujwayelekile, isistimu ehlakaniphile yokumela izinombolo eziningi ngokunemba okulinganiselwe. Nokho, lokhu kuvumelana kusho ukuthi izinombolo eziningi ziyalinganiselwa, azigcinwanga kahle. Iphepha likaGoldberg lichaza ngokucophelela izinga le-IEEE 754, elilethe ukungaguquguquki obekudingeka kakhulu kule siphithiphithi. Uchaza ukuthi izinombolo zifakwa kanjani ngekhodi zibe uphawu, i-eksponenti, nama-fraction bits, okwenza isimo sezwe esibikezelwe kodwa esixakile samanani amelelekayo, ukuziphatha okuzungezayo, nezinhlangano ezikhethekile ezifana ne-NaN (Akuyona Inombolo) kanye nokungapheli. Konjiniyela abakha amamodeli ezezimali ku-Mewayz, iphutha elihlanganisayo elibonakala lincane kakhulu lingadlulela ekuhlukeni okukhulu emibikweni noma kokwenziwe, licekele phansi ukwethenjwa kulo lonke uhlelo.

Ukuziphatha Okumangazayo kanye Nokwehluleka Okuyinhlekelele

Iphepha lidume ngokuveza izingibe eziphikisayo ezephula ukuqagela okuyisisekelo kwezibalo. Isibonelo, ngenxa yokuzungezisa, ukungezwa kwephuzu elintantayo akuhlanganisi; `(a + b) + c` ayihlali ilingana `a + (b + c)`. Lokhu kungaholela emiphumeleni enganqunyelwe ekubaleni okufanayo. Ukukhipha izinombolo ezicishe zilingane kungaholela ekukhanselweni okuyinhlekelele, lapho amadijithi abalulekile enyamalala khona, okushiya ngokuvamile iphutha lokufinyeza. Mhlawumbe isifundo esidume kakhulu isidingo sokungalokothi uqhathanise izinombolo zephoyinti elintantayo ukuze uthole ukulingana ncamashi (`==`) kodwa esikhundleni salokho hlola ukuthi ingabe umehluko wabo ungaphakathi kokubekezelelana okuncane. Lezi akuzona nje izinselele zetiyori. Badale izinhlekelele zomhlaba wangempela, kusukela ekuqhumeni kwerokhethi ye-Ariane 5 kuya ekungalumbeni kahle ezinhlelweni ezicitshwayo ze-Patriot zakuqala. Kumongo webhizinisi, amaphutha anjalo ekubalweni kwempahla, ama-algorithm entengo, noma amamethrikhi okusebenza angaholela ekukhohlakaleni kwedatha buthule, okwenza izinkundla eziqinile ezifana ne-Mewayz zibaluleke kakhulu ekuphoqeleleni ukuqinisekiswa kwedatha nokuhlola ukungaguquguquki kuwo wonke amamojula.

"Ukucindezela izinombolo zangempela eziningi ngokungenamkhawulo zibe yinani elilinganiselwe lamabhithi kudinga ukumelwa okulinganiselwe."

Okubalulekile Okuthathayo Konjiniyela Wesimanje

Iphepha le-Goldberg alinikezi nje izexwayiso kodwa neziqondiso ezisebenzayo. Okubalulekile okuthathayo ukuhlakulela "ukwazi ngezinombolo" -ukuqaphela njalo ukuthi izinombolo zephoyinti elintantayo ziyizilinganiso. Lesi simo sengqondo kufanele sazise ukukhetha kusuka ekukhethweni kwesakhiwo sedatha kuya kumklamo we-algorithm. Umsebenzi wakhe ugcizelela ukuthi kungani ukusebenzisa `okukabili` (i-64-bit) cishe njalo kungcono kune `float` (32-bit) ngomsebenzi obaluleke kakhulu, nokuthi kungani ama-algorithms athile ezinzile ngokwezinombolo kuyilapho amanye engenzi njalo. Lapho udizayina noma uhlanganisa amamojula ngaphakathi kwendawo ye-Mewayz—noma ngabe isibikezelo sokufunda komshini noma isihleli sezinsiza—lokhu kwazi kuqinisekisa ukuthi imisebenzi eyisisekelo yezinombolo isingathwa ngenhlonipho eyifunayo, kuvinjwe izinkinga ezaziwa ngokuba nzima ukulandelela emuva kumsuka wazo.

Wonke umhleli kufanele ajwayele le miqondo ebalulekile evela ephepheni:

• Iphutha Lokuzungeza: Ukunemba okungenakugwenywa kusukela ekufakeni inombolo enanini elimele eliseduze.
• Amadijithi Onogada: Amadijithi engeziwe asetshenziswa ezibalweni eziphakathi nendawo ukuze kuncishiswe iphutha lokusondeza.
• I-IEEE 754 Standard: Ipulani yendawo yonke yokubala amaphuzu antantayo, amafomethi achazayo, imithetho yokusondeza, nokunye.
• I-NaN ne-Infinity: Amanani akhethekile avumela imisebenzi ukuthi isakaze amaphutha ngomusa kunokuphahlazeka.
• Ukuzinza Kwezinombolo: Impahla ye-algorithm yokulawula ukukhuliswa kwamaphutha emisebenzini eminingi.

Idokhumenti Ephilayo Yomhlaba Wedijithali

Ngenkathi libhalwa ngo-1991, ukuhambisana kwephepha kukhule kuphela. Izimiso ze-IEEE 754 zisekela yonke i-CPU yesimanje, i-GPU, nolimi lokuhlela. Njengoba singena emingceleni efana ne-AI, ukuhlaziywa kwedatha okukhulu, kanye nokulingiswa kwesistimu okuyinkimbinkimbi, ukunemba kwezibalo zethu kuba bucayi kakhulu. Kumathimba asebenzisa isistimu yokusebenza eyimojuli efana ne-Mewayz ukuze aqondise ingqondo yebhizinisi lawo, ukushumeka lokhu kuqina kwezinombolo kumamojula awo angokwezifiso kuwumkhuba ongcono kakhulu ovimbela isigaba seziphazamisi ezingeni elibaluleke kakhulu. Umsebenzi wobuciko kaGoldberg ungaphezu kwephepha; kuyingxenye engunaphakade yesisekelo sobunjiniyela besofthiwe obuthembekile. Ukukuziba kuwukwakha phezu kwesihlabathi, ubeka engcupheni ubuqotho baso sonke isakhiwo sedijithali, noma ngabe umbhalo olula noma i-OS yebhizinisi lezinga lebhizinisi.