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#
SOME DESCRIPTIVE TITLE.
#
Copyright (C) 2001 Python Software Foundation
#
This file is distributed under the same license as the Python package.
#
FIRST AUTHOR <EMAIL@ADDRESS>, YEAR.
#
#
Translators:
#
python-doc bot, 2025
#
#,
fuzzy
msgid ""
msgstr ""
"
Project-Id-Version
:
Python 3.15\n
"
"
Report-Msgid-Bugs-To
:
\n
"
"
POT-Creation-Date
:
2026-09-29 18:58+0000\n
"
"
PO-Revision-Date
:
2025-09-16 00:02+0000\n
"
"
Last-Translator
:
python-doc bot, 2025\n
"
"
Language-Team
:
Polish (https://app.transifex.com/python-doc/teams/5390/pl/)\n
"
"
MIME-Version
:
1.0\n
"
"
Content-Type
:
text/plain; charset=UTF-8\n
"
"
Content-Transfer-Encoding
:
8bit\n
"
"
Language
:
pl\n
"
"
Plural-Forms
:
nplurals=4; plural=(n==1 ? 0 : (n%10>=2 && n%10<=4) &&
"
"
(n%100<12 || n%100>14) ? 1 : n!=1 && (n%10>=0 && n%10<=1) || (n%10>=5 &&
"
"
n%10<=9) || (n%100>=12 && n%100<=14) ? 2 : 3);\n
"
msgid
"Floating-Point Arithmetic: Issues and Limitations"
msgstr
"Arytmetyka liczb zmiennoprzecinkowych: problemy i ograniczenia"
msgid
""
"Floating-point numbers are represented in computer hardware as base 2 "
"(binary) fractions. For example, the **decimal** fraction ``0.625`` has "
"value 6/10 + 2/100 + 5/1000, and in the same way the **binary** fraction "
"``0.101`` has value 1/2 + 0/4 + 1/8. These two fractions have identical "
"values, the only real difference being that the first is written in base 10 "
"fractional notation, and the second in base 2."
msgstr
""
"Liczby zmiennoprzecinkowe s reprezentowane w komputerze jako uamki o "
"podstawie 2 (binarne). Na przykad **dziesitny** uamek ``0.625`` ma "
"warto 6/10 + 2/100 + 5/1000 i analogicznie **binarny** uamek ``0.101`` ma "
"warto 1/2 + 0/4 + 1/8. Te dwa uamki maj identyczne wartoci, a jedyn "
"prawdziw rnic jest to, e pierwszy jest zapisany w notacji uamkowej o "
"podstawie 10, a drugi o podstawie 2."
msgid
""
"Unfortunately, most decimal fractions cannot be represented exactly as "
"binary fractions. A consequence is that, in general, the decimal floating-"
"point numbers you enter are only approximated by the binary floating-point "
"numbers actually stored in the machine."
msgstr
""
"Niestety, wikszoci uamkw dziesitnych nie mona przedstawi dokadnie "
"jako uamkw binarnych. Konsekwencj jest to, e oglnie wprowadzane "
"dziesitne liczby zmiennoprzecinkowe s jedynie przybliane przez binarne "
"liczby zmiennoprzecinkowe faktycznie przechowywane w maszynie."
msgid
""
"The problem is easier to understand at first in base 10. Consider the "
"fraction 1/3. You can approximate that as a base 10 fraction::"
msgstr
""
"Problem jest atwiejszy do zrozumienia na pocztku przy podstawie 10. Rozwa "
"uamek 1/3. Moesz go przybliy jako uamek o podstawie 10::"
msgid
"0.3"
msgstr
"0.3"
msgid
"or, better, ::"
msgstr
"albo lepiej::"
msgid
"0.33"
msgstr
"0.33"
msgid
"0.333"
msgstr
"0.333"
msgid
""
"and so on. No matter how many digits you're willing to write down, the "
"result will never be exactly 1/3, but will be an increasingly better "
"approximation of 1/3."
msgstr
""
"i tak dalej. Bez wzgldu na to, ile cyfr jeste w stanie zapisa, wynik "
"nigdy nie bdzie dokadnie 1/3, ale bdzie coraz lepszym przyblieniem 1/3."
msgid
""
"In the same way, no matter how many base 2 digits you're willing to use, the "
"decimal value 0.1 cannot be represented exactly as a base 2 fraction. In "
"base 2, 1/10 is the infinitely repeating fraction ::"
msgstr
""
"W ten sam sposb, bez wzgldu na to, ile cyfr o podstawie 2 chcesz uy, "
"warto dziesitna 0,1 nie moe by dokadnie przedstawiona jako uamek o "
"podstawie 2. W podstawie 2, 1/10 to uamek okresowy ::"
msgid
"0.0001100110011001100110011001100110011001100110011..."
msgstr
"0.0001100110011001100110011001100110011001100110011..."
msgid
""
"Stop at any finite number of bits, and you get an approximation. On most "
"machines today, floats are approximated using a binary fraction with the "
"numerator using the first 53 bits starting with the most significant bit and "
"with the denominator as a power of two. In the case of 1/10, the binary "
"fraction is ``3602879701896397 / 2 ** 55`` which is close to but not exactly "
"equal to the true value of 1/10."
msgstr
""
"Zatrzymaj si na dowolnej skoczonej liczbie bitw, a otrzymasz "
"przyblienie. Na wikszoci dzisiejszych maszyn liczby zmiennoprzecinkowe s "
"aproksymowane przy uyciu uamka binarnego z licznikiem wykorzystujcym "
"pierwsze 53 bity, zaczynajc od najbardziej znaczcego bitu i mianownikiem "
"jako potg dwjki. W przypadku 1/10 uamek binarny jest rwny "
"``3602879701896397 / 2 ** 55`` i jest zbliony do prawdziwej wartoci 1/10, "
"ale nie do koca jej rwny."
msgid
""
"Many users are not aware of the approximation because of the way values are "
"displayed. Python only prints a decimal approximation to the true decimal "
"value of the binary approximation stored by the machine. On most machines, "
"if Python were to print the true decimal value of the binary approximation "
"stored for 0.1, it would have to display::"
msgstr
""
"Wielu uytkownikw nie jest wiadomych przyblienia ze wzgldu na sposb "
"wywietlania wartoci. Python wypisuje tylko przyblienie dziesitne do "
"prawdziwej wartoci dziesitnej przyblienia binarnego zapisanego przez "
"maszyn. Na wikszoci maszyn, gdyby Python mia wydrukowa prawdziw "
"warto dziesitn przyblienia binarnego zapisanego dla 0,1 musiaby "
"wywietli::"
msgid
""
">>> 0.1\n"
"0.1000000000000000055511151231257827021181583404541015625"
msgstr
""
">>> 0.1\n"
"0.1000000000000000055511151231257827021181583404541015625"
msgid
""
"That is more digits than most people find useful, so Python keeps the number "
"of digits manageable by displaying a rounded value instead:"
msgstr
""
"To wicej cyfr, ni wikszo ludzi uwaa za przydatne, wic Python "
"utrzymuje liczb cyfr tak by byy one do opanowania, wywietlajc zamiast "
"tego zaokrglon warto:"
msgid
""
">>> 1 / 10\n"
"0.1"
msgstr
""
">>> 1 / 10\n"
"0.1"
msgid
""
"Just remember, even though the printed result looks like the exact value of "
"1/10, the actual stored value is the nearest representable binary fraction."
msgstr
""
"Pamitaj tylko, e chocia wydrukowany wynik wyglda jak dokadna warto "
"1/10, rzeczywista zapisana warto to najblisza reprezentatywna cz "
"binarna."
msgid
""
"Interestingly, there are many different decimal numbers that share the same "
"nearest approximate binary fraction. For example, the numbers ``0.1`` and "
"``0.10000000000000001`` and "
"``0.1000000000000000055511151231257827021181583404541015625`` are all "
"approximated by ``3602879701896397 / 2 ** 55``. Since all of these decimal "
"values share the same approximation, any one of them could be displayed "
"while still preserving the invariant ``eval(repr(x)) == x``."
msgstr
""
"Co ciekawe, istnieje wiele rnych liczb dziesitnych, ktre maj ten sam "
"najbliszy przybliony uamek binarny. Na przykad liczby ``0.1``, "
"``0.10000000000000001`` i "
"``0.1000000000000000055511151231257827021181583404541015625`` s wszystkie "
"przyblione przez ``3602879701896397 / 2 ** 55``. Poniewa wszystkie te "
"wartoci dziesitne maj to samo przyblienie, kada z nich moe zosta "
"wywietlona przy jednoczesnym zachowaniu niezmiennika ``eval(repr(x)) == x``."
msgid
""
"Historically, the Python prompt and built-in :func:`repr` function would "
"choose the one with 17 significant digits, ``0.10000000000000001``. "
"Starting with Python 3.1, Python (on most systems) is now able to choose the "
"shortest of these and simply display ``0.1``."
msgstr
""
"W przeszoci, interaktywny prompt oraz wbudowana funkcja Pythona :func:"
"`repr` wybieray t z 17 cyframi znaczcymi, ``0.10000000000000001``. "
"Poczwszy od Pythona 3.1, Python (w wikszoci systemw) moe teraz wybra "
"najkrtszy z nich i po prostu wywietli ``0.1``."
msgid
""
"Note that this is in the very nature of binary floating point: this is not a "
"bug in Python, and it is not a bug in your code either. You'll see the same "
"kind of thing in all languages that support your hardware's floating-point "
"arithmetic (although some languages may not *display* the difference by "
"default, or in all output modes)."
msgstr
""
"Zauwa, e ley to w samej naturze binarnej liczby zmiennoprzecinkowej: nie "
"jest to bd w Pythonie ani w twoim kodzie. Zobaczysz to samo we wszystkich "
"jzykach obsugujcych arytmetyk zmiennoprzecinkow twojego sprztu "
"(chocia niektre jzyki mog nie *wywietla* rnicy domylnie lub we "
"wszystkich trybach wyjciowych)."
msgid
""
"For more pleasant output, you may wish to use string formatting to produce a "
"limited number of significant digits:"
msgstr
""
"Aby uzyska przyjemniejszy wynik, moesz uy formatowania cigu znakw w "
"celu uzyskania ograniczonej liczby cyfr znaczcych:"
msgid
""
">>> format(math.pi, '.12g') # give 12 significant digits\n"
"'3.14159265359'\n"
"\n"
">>> format(math.pi, '.2f') # give 2 digits after the point\n"
"'3.14'\n"
"\n"
">>> repr(math.pi)\n"
"'3.141592653589793'"
msgstr
""
">>> format(math.pi, '.12g') # daj 12 znaczcych cyfr\n"
"'3.14159265359'\n"
"\n"
">>> format(math.pi, '.2f') # daj 2 cyfry po kropce\n"
"'3.14'\n"
"\n"
">>> repr(math.pi)\n"
"'3.141592653589793'"
msgid
""
"It's important to realize that this is, in a real sense, an illusion: you're "
"simply rounding the *display* of the true machine value."
msgstr
""
"Wane jest, aby zda sobie spraw, e w rzeczywistoci jest to zudzenie: po "
"prostu zaokrglasz *wywietlanie* prawdziwej wartoci zapisanej w komputerze."
msgid
""
"One illusion may beget another. For example, since 0.1 is not exactly 1/10, "
"summing three values of 0.1 may not yield exactly 0.3, either:"
msgstr
""
"Jedna iluzja moe zrodzi kolejn. Na przykad, z powodu e 0,1 nie jest "
"dokadnie 1/10, zsumowanie trzech wartoci 0,1 moe nie da dokadnie 0,3:"
msgid
""
">>> 0.1 + 0.1 + 0.1 == 0.3\n"
"False"
msgstr
""
">>> 0.1 + 0.1 + 0.1 == 0.3\n"
"False"
msgid
""
"Also, since the 0.1 cannot get any closer to the exact value of 1/10 and 0.3 "
"cannot get any closer to the exact value of 3/10, then pre-rounding with :"
"func:`round` function cannot help:"
msgstr
""
"Ponadto, poniewa 0,1 nie moe zbliy si do dokadnej wartoci 1/10, a 0,3 "
"nie moe zbliy si do dokadnej wartoci 3/10, to wstpne zaokrglenie za "
"pomoc funkcji :func:`round` nie moe pomc:"
msgid
""
">>> round(0.1, 1) + round(0.1, 1) + round(0.1, 1) == round(0.3, 1)\n"
"False"
msgstr
""
">>> round(0.1, 1) + round(0.1, 1) + round(0.1, 1) == round(0.3, 1)\n"
"False"
msgid
""
"Though the numbers cannot be made closer to their intended exact values, "
"the :func:`math.isclose` function can be useful for comparing inexact values:"
msgstr
""
"Chocia liczb nie mona ju bardziej przybliy do ich dokadnych wartoci, "
"funkcja :func:`math.isclose` moe by przydatna do porwnywania "
"niedokadnych wartoci:"
msgid
""
">>> math.isclose(0.1 + 0.1 + 0.1, 0.3)\n"
"True"
msgstr
""
">>> math.isclose(0.1 + 0.1 + 0.1, 0.3)\n"
"True"
msgid
""
"Alternatively, the :func:`round` function can be used to compare rough "
"approximations:"
msgstr
""
"Alternatywnie do porwnania zgrubnych przyblie mona uy funkcji :func:"
"`round`:"
msgid
""
">>> round(math.pi, ndigits=2) == round(22 / 7, ndigits=2)\n"
"True"
msgstr
""
">>> round(math.pi, ndigits=2) == round(22 / 7, ndigits=2)\n"
"True"
msgid
""
"Binary floating-point arithmetic holds many surprises like this. The "
"problem with
\"
0.1
\"
is explained in precise detail below, in the "
"
\"
Representation Error
\"
section. See `Examples of Floating Point Problems "
"<https://jvns.ca/blog/2023/01/13/examples-of-floating-point-problems/>`_ for "
"a pleasant summary of how binary floating point works and the kinds of "
"problems commonly encountered in practice. Also see `The Perils of Floating "
"Point <http://www.indowsway.com/floatingpoint.htm>`_ for a more complete "
"account of other common surprises."
msgstr
""
"Binarna arytmetyka zmiennoprzecinkowa kryje w sobie wiele takich "
"niespodzianek. Problem z 0,1 zosta dokadnie wyjaniony poniej, w sekcji "
"Bd reprezentacji. Zobacz `Przykady problemw zmiennoprzecinkowych "
"<https://jvns.ca/blog/2023/01/13/examples-of-floating-point-problems/>`_ dla "
"przyjemnego podsumowania jak dziaa binarna arytmetyka zmiennoprzecinkowa i "
"jakie rodzaje problemw czsto spotyka si w praktyce. Zobacz take `The "
"Perils of Floating Point <http://www.indowsway.com/floatingpoint.htm>`_ dla "
"bardziej kompletnego opisu innych typowych niespodzianek."
msgid
""
"As that says near the end,
\"
there are no easy answers.
\"
Still, don't be "
"unduly wary of floating point! The errors in Python float operations are "
"inherited from the floating-point hardware, and on most machines are on the "
"order of no more than 1 part in 2
\\
*
\\
*53 per operation. That's more than "
"adequate for most tasks, but you do need to keep in mind that it's not "
"decimal arithmetic and that every float operation can suffer a new rounding "
"error."
msgstr
""
"Jak jest tam napisane pod koniec, nie ma atwych odpowiedzi. Mimo to nie "
"naley nadmiernie uwaa na liczby zmiennoprzecinkowe! Bdy w operacjach "
"zmiennoprzecinkowych Pythona s dziedziczone z budowy komputera i na "
"wikszoci maszyn s rzdu nie wicej ni 1 przez 2
\\
*
\\
*53 na operacj. "
"Jest to wicej ni wystarczajce dla wikszoci zada, ale naley pamita, "
"e nie jest to arytmetyka dziesitna i e kada operacja zmiennoprzecinkowa "
"moe napotka nowy bd zaokrglenia."
msgid
""
"While pathological cases do exist, for most casual use of floating-point "
"arithmetic you'll see the result you expect in the end if you simply round "
"the display of your final results to the number of decimal digits you "
"expect. :func:`str` usually suffices, and for finer control see the :meth:"
"`str.format` method's format specifiers in :ref:`formatstrings`."
msgstr
""
"Chocia istniej przypadki skrajne, w wikszoci przypadkowych zastosowa "
"arytmetyki zmiennoprzecinkowej zobaczysz oczekiwany wynik, jeli po prostu "
"zaokrglisz wywietlanie kocowych wynikw do oczekiwanej liczby cyfr "
"dziesitnych. :func:`str` zwykle wystarcza, lecz dla dokadniejszej kontroli "
"moesz zobaczy opis formatowania tekstu za pomoc metody :meth:`str.format` "
"w :ref:`formatstrings`."
msgid
""
"For use cases which require exact decimal representation, try using the :mod:"
"`decimal` module which implements decimal arithmetic suitable for accounting "
"applications and high-precision applications."
msgstr
""
"W przypadkach uycia, ktre wymagaj dokadnej reprezentacji dziesitnej, "
"sprbuj uy moduu :mod:`decimal`, ktry implementuje arytmetyk dziesitn "
"odpowiedni dla aplikacji ksigowych i aplikacji o wysokiej precyzji."
msgid
""
"Another form of exact arithmetic is supported by the :mod:`fractions` module "
"which implements arithmetic based on rational numbers (so the numbers like "
"1/3 can be represented exactly)."
msgstr
""
"Inn form wspierajc arytmetyk dokadn jest modu :mod:`fractions` "
"realizujcy arytmetyk opart na liczbach wymiernych (aby liczby takie jak "
"1/3 mogy by reprezentowane dokadnie)."
msgid
""
"If you are a heavy user of floating-point operations you should take a look "
"at the NumPy package and many other packages for mathematical and "
"statistical operations supplied by the SciPy project. See <https://scipy."
"org>."
msgstr
""
"Jeli czsto korzystasz z operacji zmiennoprzecinkowych, powiniene rzuci "
"okiem na pakiet NumPy i wiele innych pakietw do operacji matematycznych i "
"statystycznych dostarczonych przez projekt SciPy. Zobacz <https://scipy.org>."
msgid
""
"Python provides tools that may help on those rare occasions when you really "
"*do* want to know the exact value of a float. The :meth:`float."
"as_integer_ratio` method expresses the value of a float as a fraction:"
msgstr
""
"Python udostpnia narzdzia, ktre mog pomc w tych rzadkich przypadkach, "
"gdy naprawd *musisz* pozna dokadn warto liczby zmiennoprzecinkowej. "
"Metoda :meth:`float.as_integer_ratio` wyraa warto float jako uamek:"
msgid
""
">>> x = 3.14159\n"
">>> x.as_integer_ratio()\n"
"(3537115888337719, 1125899906842624)"
msgstr
""
">>> x = 3.14159\n"
">>> x.as_integer_ratio()\n"
"(3537115888337719, 1125899906842624)"
msgid
""
"Since the ratio is exact, it can be used to losslessly recreate the original "
"value:"
msgstr
""
"Poniewa stosunek jest dokadny, mona go uy do bezstratnego odtworzenia "
"oryginalnej wartoci:"
msgid
""
">>> x == 3537115888337719 / 1125899906842624\n"
"True"
msgstr
""
">>> x == 3537115888337719 / 1125899906842624\n"
"True"
msgid
""
"The :meth:`float.hex` method expresses a float in hexadecimal (base 16), "
"again giving the exact value stored by your computer:"
msgstr
""
"Metoda :meth:`float.hex` wyraa liczb zmiennoprzecinkow w systemie "
"hexadecymalnym (podstawa 16), ponownie podajc dokadn warto "
"przechowywan przez komputer:"
msgid
""
">>> x.hex()\n"
"'0x1.921f9f01b866ep+1'"
msgstr
""
">>> x.hex()\n"
"'0x1.921f9f01b866ep+1'"
msgid
""
"This precise hexadecimal representation can be used to reconstruct the float "
"value exactly:"
msgstr
""
"Ta precyzyjna reprezentacja szesnastkowa moe by wykorzystana do dokadnego "
"zrekonstruowania wartoci liczby zmiennoprzecinkowej:"
msgid
""
">>> x == float.fromhex('0x1.921f9f01b866ep+1')\n"
"True"
msgstr
""
">>> x == float.fromhex('0x1.921f9f01b866ep+1')\n"
"True"
msgid
""
"Since the representation is exact, it is useful for reliably porting values "
"across different versions of Python (platform independence) and exchanging "
"data with other languages that support the same format (such as Java and "
"C99)."
msgstr
""
"Poniewa ta reprezentacja jest dokadna, przydatna jest ona do niezawodnego "
"przenoszenia wartoci midzy rnymi wersjami Pythona (niezaleno od "
"platformy) i wymiany danych z innymi jzykami obsugujcymi ten sam format "
"(takimi jak Java czy C99)."
msgid
""
"Another helpful tool is the :func:`sum` function which helps mitigate loss-"
"of-precision during summation. It uses extended precision for intermediate "
"rounding steps as values are added onto a running total. That can make a "
"difference in overall accuracy so that the errors do not accumulate to the "
"point where they affect the final total:"
msgstr
""
"Kolejnym pomocnym narzdziem jest funkcja :func:`sum`, ktra pomaga "
"zagodzi utrat precyzji podczas sumowania. Uywa rozszerzonej precyzji dla "
"porednich krokw zaokrglania, gdy wartoci s dodawane do biecej sumy. "
"Moe to mie wpyw na ogln dokadno, aby bdy nie kumuloway si do "
"punktu, w ktrym wpywaj na ostateczny wynik:"
msgid
""
">>> 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.0\n"
"False\n"
">>> sum([0.1] * 10) == 1.0\n"
"True"
msgstr
""
">>> 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.0\n"
"False\n"
">>> sum([0.1] * 10) == 1.0\n"
"True"
msgid
""
"The :func:`math.fsum` goes further and tracks all of the
\"
lost digits
\"
as "
"values are added onto a running total so that the result has only a single "
"rounding. This is slower than :func:`sum` but will be more accurate in "
"uncommon cases where large magnitude inputs mostly cancel each other out "
"leaving a final sum near zero:"
msgstr
""
":func:`math.fsum` idzie dalej i ledzi wszystkie utracone cyfry, gdy "
"wartoci s dodawane do biecej sumy, tak aby wynik mia tylko jedno "
"zaokrglenie. Jest to wolniejsze ni :func:`sum`, ale bdzie dokadniejsze w "
"rzadkich przypadkach, w ktrych due wartoci wejciowe w wikszoci si "
"znosz, pozostawiajc kocow sum blisk zeru:"
msgid
""
">>> arr = [-0.10430216751806065, -266310978.67179024, 143401161448607.16,\n"
"... -143401161400469.7, 266262841.31058735, -0.003244936839808227]\n"
">>> float(sum(map(Fraction, arr))) # Exact summation with single rounding\n"
"8.042173697819788e-13\n"
">>> math.fsum(arr) # Single rounding\n"
"8.042173697819788e-13\n"
">>> sum(arr) # Multiple roundings in extended "
"precision\n"
"8.042178034628478e-13\n"
">>> total = 0.0\n"
">>> for x in arr:\n"
"... total += x # Multiple roundings in standard "
"precision\n"
"...\n"
">>> total # Straight addition has no correct "
"digits!\n"
"-0.0051575902860057365"
msgstr
""
">>> arr = [-0.10430216751806065, -266310978.67179024, 143401161448607.16,\n"
"... -143401161400469.7, 266262841.31058735, -0.003244936839808227]\n"
">>> float(sum(map(Fraction, arr))) # Dokadne sumowanie z pojedynczym "
"zaokrgleniem\n"
"8.042173697819788e-13\n"
">>> math.fsum(arr) # Pojedyncze zaokrglenie\n"
"8.042173697819788e-13\n"
">>> sum(arr) # Wielokrotne zaokrglenia w "
"rozszerzonej precyzji\n"
"8.042178034628478e-13\n"
">>> total = 0.0\n"
">>> for x in arr:\n"
"... total += x # Wielokrotne zaokrglenia w "
"standardowej precyzji\n"
"...\n"
">>> total # Proste dodawanie nie ma poprawnych "
"cyfr!\n"
"-0.0051575902860057365"
msgid
"Representation Error"
msgstr
"Bd reprezentacji"
msgid
""
"This section explains the
\"
0.1
\"
example in detail, and shows how you can "
"perform an exact analysis of cases like this yourself. Basic familiarity "
"with binary floating-point representation is assumed."
msgstr
""
"Ta sekcja wyjania szczegowo przykad 0,1 i pokazuje, jak samodzielnie "
"przeprowadzi dokadn analiz takich przypadkw. Zakadamy e masz "
"podstawow znajomo binarnej reprezentacji liczb zmiennoprzecinkowych."
msgid
""
":dfn:`Representation error` refers to the fact that some (most, actually) "
"decimal fractions cannot be represented exactly as binary (base 2) "
"fractions. This is the chief reason why Python (or Perl, C, C++, Java, "
"Fortran, and many others) often won't display the exact decimal number you "
"expect."
msgstr
""
":dfn:`Bd reprezentacji` odnosi si do faktu, e niektre (waciwie "
"wikszo) uamkw dziesitnych nie moe by reprezentowane dokadnie jako "
"uamki binarne (o podstawie 2). Jest to gwny powd, dla ktrego Python "
"(lub Perl, C, C++, Java, Fortran i wiele innych) czsto nie wywietla "
"dokadnie takiej liczby dziesitnej, jakiej oczekujesz."
msgid
""
"Why is that? 1/10 is not exactly representable as a binary fraction. Since "
"at least 2000, almost all machines use IEEE 754 binary floating-point "
"arithmetic, and almost all platforms map Python floats to IEEE 754 binary64 "
"
\"
double precision
\"
values. IEEE 754 binary64 values contain 53 bits of "
"precision, so on input the computer strives to convert 0.1 to the closest "
"fraction it can of the form *J*/2**
\\
*N* where *J* is an integer containing "
"exactly 53 bits. Rewriting ::"
msgstr
""
"Dlaczego? 1/10 nie jest dokadnie reprezentowalna jako uamek binarny. Od co "
"najmniej 2000 roku, prawie wszystkie dzisiejsze maszyny uywaj binarnej "
"arytmetyki zmiennoprzecinkowej IEEE-754 i prawie wszystkie platformy "
"odwzorowuj liczby zmiennoprzecinkowe Pythona na podwjn precyzj "
"IEEE-754 binary64. Wartoci IEEE 754 binary64 zawieraj 53 bity dokadnoci, "
"wic na wejciu komputer stara si zamieni 0,1 na najbliszy moliwy uamek "
"w postaci *J*/2**
\\
*N*, gdzie *J* jest liczb cakowit zawierajc "
"dokadnie 53 bity. Zapisujc ::"
msgid
"1 / 10 ~= J / (2**N)"
msgstr
"1 / 10 ~= J / (2**N)"
msgid
"as ::"
msgstr
"jako ::"
msgid
"J ~= 2**N / 10"
msgstr
"J ~= 2**N / 10"
msgid
""
"and recalling that *J* has exactly 53 bits (is ``>= 2**52`` but ``< "
"2**53``), the best value for *N* is 56:"
msgstr
""
"i pamitajc, e *J* ma dokadnie 53 bity (jest ``>= 2**52`` ale ``< "
"2**53``), najlepsz wartoci dla *N* jest 56:"
msgid
""
">>> 2**52 <= 2**56 // 10 < 2**53\n"
"True"
msgstr
""
">>> 2**52 <= 2**56 // 10 < 2**53\n"
"True"
msgid
""
"That is, 56 is the only value for *N* that leaves *J* with exactly 53 bits. "
"The best possible value for *J* is then that quotient rounded:"
msgstr
""
"Oznacza to, e 56 jest jedyn wartoci dla *N*, ktra pozostawia *J* "
"dokadnie 53 bity. Najlepsz moliw wartoci dla *J* jest zatem "
"zaokrglony iloraz:"
msgid
""
">>> q, r = divmod(2**56, 10)\n"
">>> r\n"
"6"
msgstr
""
">>> q, r = divmod(2**56, 10)\n"
">>> r\n"
"6"
msgid
""
"Since the remainder is more than half of 10, the best approximation is "
"obtained by rounding up:"
msgstr
""
"Poniewa reszta jest wiksza ni poowa z 10, najlepsze przyblienie "
"uzyskuje si zaokrglajc w gr:"
msgid
""
">>> q+1\n"
"7205759403792794"
msgstr
""
">>> q+1\n"
"7205759403792794"
msgid
""
"Therefore the best possible approximation to 1/10 in IEEE 754 double "
"precision is::"
msgstr
""
"Dlatego najlepszym moliwym przyblieniem do 1/10 w arytmetyce podwjnej "
"precyzji typu IEEE 754 jest::"
msgid
"7205759403792794 / 2 ** 56"
msgstr
"7205759403792794 / 2 ** 56"
msgid
""
"Dividing both the numerator and denominator by two reduces the fraction to::"
msgstr
"Podzielenie licznika i mianownika przez dwa zmniejsza uamek do::"
msgid
"3602879701896397 / 2 ** 55"
msgstr
"3602879701896397 / 2 ** 55"
msgid
""
"Note that since we rounded up, this is actually a little bit larger than "
"1/10; if we had not rounded up, the quotient would have been a little bit "
"smaller than 1/10. But in no case can it be *exactly* 1/10!"
msgstr
""
"Zauwa, e poniewa zaokrglilimy w gr, jest to w rzeczywistoci troch "
"wicej ni 1/10; gdybymy nie zaokrglili w gr, iloraz byby nieco "
"mniejszy ni 1/10. Ale w adnym wypadku nie moe to by *dokadnie* 1/10!"
msgid
""
"So the computer never
\"
sees
\"
1/10: what it sees is the exact fraction "
"given above, the best IEEE 754 double approximation it can get:"
msgstr
""
"Tak wic komputer nigdy nie widzi 1/10: to, co widzi, to dokadny uamek "
"podany powyej, najlepsze przyblienie w standardzie podwjnej precyzji IEEE "
"754, jakie moe uzyska:"
msgid
""
">>> 0.1 * 2 ** 55\n"
"3602879701896397.0"
msgstr
""
">>> 0.1 * 2 ** 55\n"
"3602879701896397.0"
msgid
""
"If we multiply that fraction by 10
\\
*
\\
*55, we can see the value out to 55 "
"decimal digits:"
msgstr
""
"Jeli pomnoymy ten uamek przez 10
\\
*
\\
*55, otrzymamy warto z "
"dokadnoci do 55 cyfr dziesitnych:"
msgid
""
">>> 3602879701896397 * 10 ** 55 // 2 ** 55\n"
"1000000000000000055511151231257827021181583404541015625"
msgstr
""
">>> 3602879701896397 * 10 ** 55 // 2 ** 55\n"
"1000000000000000055511151231257827021181583404541015625"
msgid
""
"meaning that the exact number stored in the computer is equal to the decimal "
"value 0.1000000000000000055511151231257827021181583404541015625. Instead of "
"displaying the full decimal value, many languages (including older versions "
"of Python), round the result to 17 significant digits:"
msgstr
""
"co oznacza, e dokadna liczba zapisana w komputerze jest rwna wartoci "
"dziesitnej 0,1000000000000000055511151231257827021181583404541015625. "
"Zamiast wywietla pen warto dziesitn, wiele jzykw (w tym starsze "
"wersje Pythona) zaokrgla wynik do 17 cyfr znaczcych:"
msgid
""
">>> format(0.1, '.17f')\n"
"'0.10000000000000001'"
msgstr
""
">>> format(0.1, '.17f')\n"
"'0.10000000000000001'"
msgid
""
"The :mod:`fractions` and :mod:`decimal` modules make these calculations easy:"
msgstr
""
"Moduy :mod:`fractions` oraz :mod:`decimal` uatwiaj tego typu obliczenia:"
msgid
""
">>> from decimal import Decimal\n"
">>> from fractions import Fraction\n"
"\n"
">>> Fraction.from_float(0.1)\n"
"Fraction(3602879701896397, 36028797018963968)\n"
"\n"
">>> (0.1).as_integer_ratio()\n"
"(3602879701896397, 36028797018963968)\n"
"\n"
">>> Decimal.from_float(0.1)\n"
"Decimal('0.1000000000000000055511151231257827021181583404541015625')\n"
"\n"
">>> format(Decimal.from_float(0.1), '.17')\n"
"'0.10000000000000001'"
msgstr
""
">>> from decimal import Decimal\n"
">>> from fractions import Fraction\n"
"\n"
">>> Fraction.from_float(0.1)\n"
"Fraction(3602879701896397, 36028797018963968)\n"
"\n"
">>> (0.1).as_integer_ratio()\n"
"(3602879701896397, 36028797018963968)\n"
"\n"
">>> Decimal.from_float(0.1)\n"
"Decimal('0.1000000000000000055511151231257827021181583404541015625')\n"
"\n"
">>> format(Decimal.from_float(0.1), '.17')\n"
"'0.10000000000000001'"
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