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# SOME DESCRIPTIVE TITLE.

# This file is distributed under the same license as the Python package.

# FIRST AUTHOR <EMAIL@ADDRESS>, YEAR.

# Translators:

# Ciarbin Ciarbin <mrciarbin@gmail.com>, 2021

#, fuzzy

msgid ""

msgstr ""

"Project-Id-Version: Python 3.9\n"

"Report-Msgid-Bugs-To: \n"

"POT-Creation-Date: 2021-01-01 05:02+0000\n"

"PO-Revision-Date: 2017-02-16 23:40+0000\n"

"Last-Translator: Ciarbin Ciarbin <mrciarbin@gmail.com>, 2021\n"

"Language-Team: Polish (https://www.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 ::"

msgstr ""

"Liczby zmiennoprzecinkowe s reprezentowane w systemie komputerowym jako "

"uamki o podstawie 2 (binarne). Na przykad, uamek dziesitny ::"

msgid ""

"has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction ::"

msgstr ""

msgid ""

"has value 0/2 + 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 ""

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 ""

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 ""

msgid "or, better, ::"

msgstr ""

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 ""

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 ""

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 ""

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 ""

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 ""

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 ""

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 ""

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 ""

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 ""

msgid ""

"For more pleasant output, you may wish to use string formatting to produce a "

"limited number of significant digits::"

msgstr ""

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 ""

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 ""

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 ""

msgid ""

"Though the numbers cannot be made closer to their intended exact values, "

"the :func:`round` function can be useful for post-rounding so that results "

"with inexact values become comparable to one another::"

msgstr ""

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 `The Perils of Floating Point <http://"

"www.lahey.com/float.htm>`_ for a more complete account of other common "

"surprises."

msgstr ""

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 ""

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 ""

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 ""

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 ""

msgid ""

"If you are a heavy user of floating point operations you should take a look "

"at the Numerical Python package and many other packages for mathematical and "

"statistical operations supplied by the SciPy project. See <https://scipy."

"org>."

msgstr ""

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 ""

msgid ""

"Since the ratio is exact, it can be used to losslessly recreate the original "

"value::"

msgstr ""

msgid ""

"The :meth:`float.hex` method expresses a float in hexadecimal (base 16), "

"again giving the exact value stored by your computer::"

msgstr ""

msgid ""

"This precise hexadecimal representation can be used to reconstruct the float "

"value exactly::"

msgstr ""

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 ""

msgid ""

"Another helpful tool is the :func:`math.fsum` function which helps mitigate "

"loss-of-precision during summation. It tracks \"lost digits\" 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 ""

msgid "Representation Error"

msgstr ""

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 ""

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 ""

msgid ""

"Why is that? 1/10 is not exactly representable as a binary fraction. Almost "

"all machines today (November 2000) use IEEE-754 floating point arithmetic, "

"and almost all platforms map Python floats to IEEE-754 \"double "

"precision\". 754 doubles 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 ""

msgid "as ::"

msgstr ""

msgid ""

"and recalling that *J* has exactly 53 bits (is ``>= 2**52`` but ``< "

"2**53``), the best value for *N* is 56::"

msgstr ""

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 ""

msgid ""

"Since the remainder is more than half of 10, the best approximation is "

"obtained by rounding up::"

msgstr ""

msgid ""

"Therefore the best possible approximation to 1/10 in 754 double precision "

"is::"

msgstr ""

msgid ""

"Dividing both the numerator and denominator by two reduces the fraction to::"

msgstr ""

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 ""

msgid ""

"So the computer never \"sees\" 1/10: what it sees is the exact fraction "

"given above, the best 754 double approximation it can get::"

msgstr ""

msgid ""

"If we multiply that fraction by 10\\*\\*55, we can see the value out to 55 "

"decimal digits::"

msgstr ""

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 ""

msgid ""

"The :mod:`fractions` and :mod:`decimal` modules make these calculations "

"easy::"

msgstr ""

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