Commit a9d9d17f authored by Mark Dickinson's avatar Mark Dickinson

Issue #14245: Merge changes from 3.2.

parents 9a234903 ba3b0d84
......@@ -43,56 +43,45 @@ Why am I getting strange results with simple arithmetic operations?
See the next question.
Why are floating point calculations so inaccurate?
Why are floating-point calculations so inaccurate?
--------------------------------------------------
People are often very surprised by results like this::
Users are often surprised by results like this::
>>> 1.2 - 1.0
0.199999999999999996
>>> 1.2 - 1.0
0.199999999999999996
and think it is a bug in Python. It's not. This has nothing to do with Python,
but with how the underlying C platform handles floating point numbers, and
ultimately with the inaccuracies introduced when writing down numbers as a
string of a fixed number of digits.
The internal representation of floating point numbers uses a fixed number of
binary digits to represent a decimal number. Some decimal numbers can't be
represented exactly in binary, resulting in small roundoff errors.
and think it is a bug in Python. It's not. This has little to do with Python,
and much more to do with how the underlying platform handles floating-point
numbers.
In decimal math, there are many numbers that can't be represented with a fixed
number of decimal digits, e.g. 1/3 = 0.3333333333.......
The :class:`float` type in CPython uses a C ``double`` for storage. A
:class:`float` object's value is stored in binary floating-point with a fixed
precision (typically 53 bits) and Python uses C operations, which in turn rely
on the hardware implementation in the processor, to perform floating-point
operations. This means that as far as floating-point operations are concerned,
Python behaves like many popular languages including C and Java.
In base 2, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, etc. .2 equals 2/10 equals 1/5,
resulting in the binary fractional number 0.001100110011001...
Many numbers that can be written easily in decimal notation cannot be expressed
exactly in binary floating-point. For example, after::
Floating point numbers only have 32 or 64 bits of precision, so the digits are
cut off at some point, and the resulting number is 0.199999999999999996 in
decimal, not 0.2.
>>> x = 1.2
A floating point number's ``repr()`` function prints as many digits are
necessary to make ``eval(repr(f)) == f`` true for any float f. The ``str()``
function prints fewer digits and this often results in the more sensible number
that was probably intended::
the value stored for ``x`` is a (very good) approximation to the decimal value
``1.2``, but is not exactly equal to it. On a typical machine, the actual
stored value is::
>>> 1.1 - 0.9
0.20000000000000007
>>> print(1.1 - 0.9)
0.2
1.0011001100110011001100110011001100110011001100110011 (binary)
One of the consequences of this is that it is error-prone to compare the result
of some computation to a float with ``==``. Tiny inaccuracies may mean that
``==`` fails. Instead, you have to check that the difference between the two
numbers is less than a certain threshold::
which is exactly::
epsilon = 0.0000000000001 # Tiny allowed error
expected_result = 0.4
1.1999999999999999555910790149937383830547332763671875 (decimal)
if expected_result-epsilon <= computation() <= expected_result+epsilon:
...
The typical precision of 53 bits provides Python floats with 15-16
decimal digits of accuracy.
Please see the chapter on :ref:`floating point arithmetic <tut-fp-issues>` in
the Python tutorial for more information.
For a fuller explanation, please see the :ref:`floating point arithmetic
<tut-fp-issues>` chapter in the Python tutorial.
Why are Python strings immutable?
......
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