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15. Floating Point Arithmetic: Issues and Limitations





15. Floating Point Arithmetic: Issues and Limitations¶


Floating-point numbers are represented in computer hardware as base 2 (binary)
fractions. For example, the decimal fraction 0.125
has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction 0.001
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.


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.


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:


0.3


or, better,


0.33


or, better,


0.333


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.


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


0.0001100110011001100110011001100110011001100110011...


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.


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


>>> 0.1
0.1000000000000000055511151231257827021181583404541015625


That is more digits than most people find useful, so Python keeps the number
of digits manageable by displaying a rounded value instead


>>> 1 / 10
0.1


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.


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 .


Historically, the Python prompt and built-in 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 .


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).


For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:


>>> format(math.pi, '.12g')  # give 12 significant digits
'3.14159265359'

>>> format(math.pi, '.2f') # give 2 digits after the point
'3.14'

>>> repr(math.pi)
'3.141592653589793'


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.


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:


>>> .1 + .1 + .1 == .3
False


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
round() function cannot help:


>>> round(.1, 1) + round(.1, 1) + round(.1, 1) == round(.3, 1)
False


Though the numbers cannot be made closer to their intended exact values,
the round() function can be useful for post-rounding so that results
with inexact values become comparable to one another:


>>> round(.1 + .1 + .1, 10) == round(.3, 10)
True


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
for a more complete account of other common surprises.


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.


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.
str() usually suffices, and for finer control see the str.format()
method’s format specifiers in Format String Syntax .


For use cases which require exact decimal representation, try using the
decimal module which implements decimal arithmetic suitable for
accounting applications and high-precision applications.


Another form of exact arithmetic is supported by the fractions module
which implements arithmetic based on rational numbers (so the numbers like
1/3 can be represented exactly).


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>.


Python provides tools that may help on those rare occasions when you really
do want to know the exact value of a float. The
float.as_integer_ratio() method expresses the value of a float as a
fraction:


>>> x = 3.14159
>>> x.as_integer_ratio()
(3537115888337719, 1125899906842624)


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


>>> x == 3537115888337719 / 1125899906842624
True


The float.hex() method expresses a float in hexadecimal (base
16), again giving the exact value stored by your computer:


>>> x.hex()
'0x1.921f9f01b866ep+1'


This precise hexadecimal representation can be used to reconstruct
the float value exactly:


>>> x == float.fromhex('0x1.921f9f01b866ep+1')
True


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).


Another helpful tool is the 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:


>>> sum([0.1] * 10) == 1.0
False
>>> math.fsum([0.1] * 10) == 1.0
True



15.1. Representation Error¶


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.


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.


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


1 / 10 ~= J / (2**N)


as


J ~= 2**N / 10


and recalling that J has exactly 53 bits (is >= 2**52 but < 2**53 ),
the best value for N is 56:


>>> 2**52 <=  2**56 // 10  < 2**53
True


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:


>>> q, r = divmod(2**56, 10)
>>> r
6


Since the remainder is more than half of 10, the best approximation is obtained
by rounding up:


>>> q+1
7205759403792794


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


7205759403792794 / 2 ** 56


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


3602879701896397 / 2 ** 55


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!


So the computer never “sees” 1/10: what it sees is the exact fraction given
above, the best 754 double approximation it can get:


>>> 0.1 * 2 ** 55
3602879701896397.0


If we multiply that fraction by 10**55, we can see the value out to
55 decimal digits:


>>> 3602879701896397 * 10 ** 55 // 2 ** 55
1000000000000000055511151231257827021181583404541015625


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:


>>> format(0.1, '.17f')
'0.10000000000000001'


The fractions and decimal modules make these calculations
easy:


>>> from decimal import Decimal
>>> from fractions import Fraction

>>> Fraction.from_float(0.1)
Fraction(3602879701896397, 36028797018963968)

>>> (0.1).as_integer_ratio()
(3602879701896397, 36028797018963968)

>>> Decimal.from_float(0.1)
Decimal('0.1000000000000000055511151231257827021181583404541015625')

>>> format(Decimal.from_float(0.1), '.17')
'0.10000000000000001'









16. Appendix





16. Appendix¶



16.1. Interactive Mode¶



16.1.1. Error Handling¶


When an error occurs, the interpreter prints an error message and a stack trace.
In interactive mode, it then returns to the primary prompt; when input came from
a file, it exits with a nonzero exit status after printing the stack trace.
(Exceptions handled by an except clause in a try statement
are not errors in this context.) Some errors are unconditionally fatal and
cause an exit with a nonzero exit; this applies to internal inconsistencies and
some cases of running out of memory. All error messages are written to the
standard error stream; normal output from executed commands is written to
standard output.


Typing the interrupt character (usually Control - C or Delete ) to the primary or
secondary prompt cancels the input and returns to the primary prompt. 1
Typing an interrupt while a command is executing raises the
KeyboardInterrupt exception, which may be handled by a try
statement.




16.1.2. Executable Python Scripts¶


On BSD’ish Unix systems, Python scripts can be made directly executable, like
shell scripts, by putting the line


#!/usr/bin/env python3.5


(assuming that the interpreter is on the user’s PATH ) at the beginning
of the script and giving the file an executable mode. The #! must be the
first two characters of the file. On some platforms, this first line must end
with a Unix-style line ending ( '\n' ), not a Windows ( '\r\n' ) line
ending. Note that the hash, or pound, character, '#' , is used to start a
comment in Python.


The script can be given an executable mode, or permission, using the
chmod command.


$ chmod +x myscript.py


On Windows systems, there is no notion of an “executable mode”. The Python
installer automatically associates .py files with python.exe so that
a double-click on a Python file will run it as a script. The extension can
also be .pyw , in that case, the console window that normally appears is
suppressed.




16.1.3. The Interactive Startup File¶


When you use Python interactively, it is frequently handy to have some standard
commands executed every time the interpreter is started. You can do this by
setting an environment variable named PYTHONSTARTUP to the name of a
file containing your start-up commands. This is similar to the .profile
feature of the Unix shells.


This file is only read in interactive sessions, not when Python reads commands
from a script, and not when /dev/tty is given as the explicit source of
commands (which otherwise behaves like an interactive session). It is executed
in the same namespace where interactive commands are executed, so that objects
that it defines or imports can be used without qualification in the interactive
session. You can also change the prompts sys.ps1 and sys.ps2 in this
file.


If you want to read an additional start-up file from the current directory, you
can program this in the global start-up file using code like if
os.path.isfile('.pythonrc.py'): exec(open('.pythonrc.py').read())
.
If you want to use the startup file in a script, you must do this explicitly
in the script:


import os
filename = os.environ.get('PYTHONSTARTUP')
if filename and os.path.isfile(filename):
with open(filename) as fobj:
startup_file = fobj.read()
exec(startup_file)




16.1.4. The Customization Modules¶


Python provides two hooks to let you customize it: sitecustomize and
usercustomize . To see how it works, you need first to find the location
of your user site-packages directory. Start Python and run this code:


>>> import site
>>> site.getusersitepackages()
'/home/user/.local/lib/python3.5/site-packages'


Now you can create a file named usercustomize.py in that directory and
put anything you want in it. It will affect every invocation of Python, unless
it is started with the -s option to disable the automatic import.


sitecustomize works in the same way, but is typically created by an
administrator of the computer in the global site-packages directory, and is
imported before usercustomize . See the documentation of the site
module for more details.


Footnotes



1

A problem with the GNU Readline package may prevent this.












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Python Setup and Usage





Python Setup and Usage¶


This part of the documentation is devoted to general information on the setup
of the Python environment on different platforms, the invocation of the
interpreter and things that make working with Python easier.




  • 1. Command line and environment

    • 1.1. Command line

      • 1.1.1. Interface options

      • 1.1.2. Generic options

      • 1.1.3. Miscellaneous options

      • 1.1.4. Options you shouldn’t use



    • 1.2. Environment variables

      • 1.2.1. Debug-mode variables





  • 2. Using Python on Unix platforms

    • 2.1. Getting and installing the latest version of Python

      • 2.1.1. On Linux

      • 2.1.2. On FreeBSD and OpenBSD

      • 2.1.3. On OpenSolaris



    • 2.2. Building Python

    • 2.3. Python-related paths and files

    • 2.4. Miscellaneous

    • 2.5. Custom OpenSSL



  • 3. Configure Python

    • 3.1. Configure Options

      • 3.1.1. General Options

      • 3.1.2. WebAssembly Options

      • 3.1.3. Install Options

      • 3.1.4. Performance options

      • 3.1.5. Python Debug Build

      • 3.1.6. Debug options

      • 3.1.7. Linker options

      • 3.1.8. Libraries options

      • 3.1.9. Security Options

      • 3.1.10. macOS Options

      • 3.1.11. Cross Compiling Options



    • 3.2. Python Build System

      • 3.2.1. Main files of the build system

      • 3.2.2. Main build steps

      • 3.2.3. Main Makefile targets

      • 3.2.4. C extensions



    • 3.3. Compiler and linker flags

      • 3.3.1. Preprocessor flags

      • 3.3.2. Compiler flags

      • 3.3.3. Linker flags





  • 4. Using Python on Windows

    • 4.1. The full installer

      • 4.1.1. Installation steps

      • 4.1.2. Removing the MAX_PATH Limitation

      • 4.1.3. Installing Without UI

      • 4.1.4. Installing Without Downloading

      • 4.1.5. Modifying an install



    • 4.2. The Microsoft Store package

      • 4.2.1. Known issues

        • 4.2.1.1. Redirection of local data, registry, and temporary paths





    • 4.3. The nuget.org packages

    • 4.4. The embeddable package

      • 4.4.1. Python Application

      • 4.4.2. Embedding Python



    • 4.5. Alternative bundles

    • 4.6. Configuring Python

      • 4.6.1. Excursus: Setting environment variables

      • 4.6.2. Finding the Python executable



    • 4.7. UTF-8 mode

    • 4.8. Python Launcher for Windows

      • 4.8.1. Getting started

        • 4.8.1.1. From the command-line

        • 4.8.1.2. Virtual environments

        • 4.8.1.3. From a script

        • 4.8.1.4. From file associations



      • 4.8.2. Shebang Lines

      • 4.8.3. Arguments in shebang lines

      • 4.8.4. Customization

        • 4.8.4.1. Customization via INI files

        • 4.8.4.2. Customizing default Python versions



      • 4.8.5. Diagnostics

      • 4.8.6. Dry Run

      • 4.8.7. Install on demand

      • 4.8.8. Return codes



    • 4.9. Finding modules

    • 4.10. Additional modules

      • 4.10.1. PyWin32

      • 4.10.2. cx_Freeze



    • 4.11. Compiling Python on Windows

    • 4.12. Other Platforms



  • 5. Using Python on a Mac

    • 5.1. Getting and Installing MacPython

      • 5.1.1. How to run a Python script

      • 5.1.2. Running scripts with a GUI

      • 5.1.3. Configuration



    • 5.2. The IDE

    • 5.3. Installing Additional Python Packages

    • 5.4. GUI Programming on the Mac

    • 5.5. Distributing Python Applications on the Mac

    • 5.6. Other Resources



  • 6. Editors and IDEs









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