pythoncheckins at python
May 13, 2012, 1:04 PM
Post #1 of 1
(54 views)
Permalink

cpython (merge 3.2 > default): Issue #14245: Merge changes from 3.2.


http://hg.python.org/cpython/rev/a79b07e05d0d changeset: 76909:a79b07e05d0d parent: 76907:93748e2d64e3 parent: 76908:2b2a7861255d user: Mark Dickinson <mdickinson [at] enthought> date: Sun May 13 21:02:22 2012 +0100 summary: Issue #14245: Merge changes from 3.2. files: Doc/faq/design.rst  63 ++++++++++++++ 1 files changed, 26 insertions(+), 37 deletions() diff git a/Doc/faq/design.rst b/Doc/faq/design.rst  a/Doc/faq/design.rst +++ b/Doc/faq/design.rst @@ 43,56 +43,45 @@ See the next question. Why are floating point calculations so inaccurate? +Why are floatingpoint 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. +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 floatingpoint +numbers. 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. +The :class:`float` type in CPython uses a C ``double`` for storage. A +:class:`float` object's value is stored in binary floatingpoint 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 floatingpoint +operations. This means that as far as floatingpoint operations are concerned, +Python behaves like many popular languages including C and Java. 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....... +Many numbers that can be written easily in decimal notation cannot be expressed +exactly in binary floatingpoint. For example, after:: 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... + >>> x = 1.2 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. +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:: 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:: + 1.0011001100110011001100110011001100110011001100110011 (binary)  >>> 1.1  0.9  0.20000000000000007  >>> print(1.1  0.9)  0.2 +which is exactly:: One of the consequences of this is that it is errorprone 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:: + 1.1999999999999999555910790149937383830547332763671875 (decimal)  epsilon = 0.0000000000001 # Tiny allowed error  expected_result = 0.4 +The typical precision of 53 bits provides Python floats with 1516 +decimal digits of accuracy.  if expected_resultepsilon <= computation() <= expected_result+epsilon:  ...  Please see the chapter on :ref:`floating point arithmetic <tutfpissues>` in the Python tutorial for more information. +For a fuller explanation, please see the :ref:`floating point arithmetic +<tutfpissues>` chapter in the Python tutorial. Why are Python strings immutable?  Repository URL: http://hg.python.org/cpython
