Edit File: floatingpoint.html
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>14. Floating Point Arithmetic: Issues and Limitations — Python v2.6.6 documentation</title> <link rel="stylesheet" href="../_static/default.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../', VERSION: '2.6.6', COLLAPSE_MODINDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: true }; </script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python v2.6.6 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="top" title="Python v2.6.6 documentation" href="../index.html" /> <link rel="up" title="The Python Tutorial" href="index.html" /> <link rel="next" title="Using Python" href="../using/index.html" /> <link rel="prev" title="13. Interactive Input Editing and History Substitution" href="interactive.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> </head> <body> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../modindex.html" title="Global Module Index" accesskey="M">modules</a> |</li> <li class="right" > <a href="../using/index.html" title="Using Python" accesskey="N">next</a> |</li> <li class="right" > <a href="interactive.html" title="13. Interactive Input Editing and History Substitution" accesskey="P">previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="../index.html">Python v2.6.6 documentation</a> »</li> <li><a href="index.html" accesskey="U">The Python Tutorial</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <div class="section" id="floating-point-arithmetic-issues-and-limitations"> <span id="tut-fp-issues"></span><h1>14. Floating Point Arithmetic: Issues and Limitations<a class="headerlink" href="#floating-point-arithmetic-issues-and-limitations" title="Permalink to this headline">¶</a></h1> <p>Floating-point numbers are represented in computer hardware as base 2 (binary) fractions. For example, the decimal fraction</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.125</span> </pre></div> </div> <p>has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.001</span> </pre></div> </div> <p>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.</p> <p>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.</p> <p>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:</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.3</span> </pre></div> </div> <p>or, better,</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.33</span> </pre></div> </div> <p>or, better,</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.333</span> </pre></div> </div> <p>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.</p> <p>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</p> <div class="highlight-python"><pre>0.0001100110011001100110011001100110011001100110011...</pre> </div> <p>Stop at any finite number of bits, and you get an approximation. This is why you see things like:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mf">0.1</span> <span class="go">0.10000000000000001</span> </pre></div> </div> <p>On most machines today, that is what you’ll see if you enter 0.1 at a Python prompt. You may not, though, because the number of bits used by the hardware to store floating-point values can vary across machines, and 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</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mf">0.1</span> <span class="go">0.1000000000000000055511151231257827021181583404541015625</span> </pre></div> </div> <p>instead! The Python prompt uses the built-in <a title="repr" class="reference external" href="../library/functions.html#repr"><tt class="xref docutils literal"><span class="pre">repr()</span></tt></a> function to obtain a string version of everything it displays. For floats, <tt class="docutils literal"><span class="pre">repr(float)</span></tt> rounds the true decimal value to 17 significant digits, giving</p> <div class="highlight-python"><div class="highlight"><pre><span class="mf">0.10000000000000001</span> </pre></div> </div> <p><tt class="docutils literal"><span class="pre">repr(float)</span></tt> produces 17 significant digits because it turns out that’s enough (on most machines) so that <tt class="docutils literal"><span class="pre">eval(repr(x))</span> <span class="pre">==</span> <span class="pre">x</span></tt> exactly for all finite floats <em>x</em>, but rounding to 16 digits is not enough to make that true.</p> <p>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 <em>display</em> the difference by default, or in all output modes).</p> <p>Python’s built-in <a title="str" class="reference external" href="../library/functions.html#str"><tt class="xref docutils literal"><span class="pre">str()</span></tt></a> function produces only 12 significant digits, and you may wish to use that instead. It’s unusual for <tt class="docutils literal"><span class="pre">eval(str(x))</span></tt> to reproduce <em>x</em>, but the output may be more pleasant to look at:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="k">print</span> <span class="nb">str</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span> <span class="go">0.1</span> </pre></div> </div> <p>It’s important to realize that this is, in a real sense, an illusion: the value in the machine is not exactly 1/10, you’re simply rounding the <em>display</em> of the true machine value.</p> <p>Other surprises follow from this one. For example, after seeing</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mf">0.1</span> <span class="go">0.10000000000000001</span> </pre></div> </div> <p>you may be tempted to use the <a title="round" class="reference external" href="../library/functions.html#round"><tt class="xref docutils literal"><span class="pre">round()</span></tt></a> function to chop it back to the single digit you expect. But that makes no difference:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="nb">round</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="go">0.10000000000000001</span> </pre></div> </div> <p>The problem is that the binary floating-point value stored for “0.1” was already the best possible binary approximation to 1/10, so trying to round it again can’t make it better: it was already as good as it gets.</p> <p>Another consequence is that since 0.1 is not exactly 1/10, summing ten values of 0.1 may not yield exactly 1.0, either:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="nb">sum</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="gp">>>> </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span> <span class="gp">... </span> <span class="nb">sum</span> <span class="o">+=</span> <span class="mf">0.1</span> <span class="gp">...</span> <span class="gp">>>> </span><span class="nb">sum</span> <span class="go">0.99999999999999989</span> </pre></div> </div> <p>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 <a class="reference external" href="http://www.lahey.com/float.htm">The Perils of Floating Point</a> for a more complete account of other common surprises.</p> <p>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.</p> <p>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. <a title="str" class="reference external" href="../library/functions.html#str"><tt class="xref docutils literal"><span class="pre">str()</span></tt></a> usually suffices, and for finer control see the <a title="str.format" class="reference external" href="../library/stdtypes.html#str.format"><tt class="xref docutils literal"><span class="pre">str.format()</span></tt></a> method’s format specifiers in <a class="reference external" href="../library/string.html#formatstrings"><em>Format String Syntax</em></a>.</p> <div class="section" id="representation-error"> <span id="tut-fp-error"></span><h2>14.1. Representation Error<a class="headerlink" href="#representation-error" title="Permalink to this headline">¶</a></h2> <p>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.</p> <p><em>Representation error</em> 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:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mf">0.1</span> <span class="go">0.10000000000000001</span> </pre></div> </div> <p>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 <em>J</em>/2**<em>N</em> where <em>J</em> is an integer containing exactly 53 bits. Rewriting</p> <div class="highlight-python"><pre>1 / 10 ~= J / (2**N)</pre> </div> <p>as</p> <div class="highlight-python"><pre>J ~= 2**N / 10</pre> </div> <p>and recalling that <em>J</em> has exactly 53 bits (is <tt class="docutils literal"><span class="pre">>=</span> <span class="pre">2**52</span></tt> but <tt class="docutils literal"><span class="pre"><</span> <span class="pre">2**53</span></tt>), the best value for <em>N</em> is 56:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mi">2</span><span class="o">**</span><span class="mi">52</span> <span class="go">4503599627370496L</span> <span class="gp">>>> </span><span class="mi">2</span><span class="o">**</span><span class="mi">53</span> <span class="go">9007199254740992L</span> <span class="gp">>>> </span><span class="mi">2</span><span class="o">**</span><span class="mi">56</span><span class="o">/</span><span class="mi">10</span> <span class="go">7205759403792793L</span> </pre></div> </div> <p>That is, 56 is the only value for <em>N</em> that leaves <em>J</em> with exactly 53 bits. The best possible value for <em>J</em> is then that quotient rounded:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">56</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="gp">>>> </span><span class="n">r</span> <span class="go">6L</span> </pre></div> </div> <p>Since the remainder is more than half of 10, the best approximation is obtained by rounding up:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">q</span><span class="o">+</span><span class="mi">1</span> <span class="go">7205759403792794L</span> </pre></div> </div> <p>Therefore the best possible approximation to 1/10 in 754 double precision is that over 2**56, or</p> <div class="highlight-python"><div class="highlight"><pre><span class="mi">7205759403792794</span> <span class="o">/</span> <span class="mi">72057594037927936</span> </pre></div> </div> <p>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 <em>exactly</em> 1/10!</p> <p>So the computer never “sees” 1/10: what it sees is the exact fraction given above, the best 754 double approximation it can get:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="o">.</span><span class="mi">1</span> <span class="o">*</span> <span class="mi">2</span><span class="o">**</span><span class="mi">56</span> <span class="go">7205759403792794.0</span> </pre></div> </div> <p>If we multiply that fraction by 10**30, we can see the (truncated) value of its 30 most significant decimal digits:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="mi">7205759403792794</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**</span><span class="mi">30</span> <span class="o">/</span> <span class="mi">2</span><span class="o">**</span><span class="mi">56</span> <span class="go">100000000000000005551115123125L</span> </pre></div> </div> <p>meaning that the exact number stored in the computer is approximately equal to the decimal value 0.100000000000000005551115123125. Rounding that to 17 significant digits gives the 0.10000000000000001 that Python displays (well, will display on any 754-conforming platform that does best-possible input and output conversions in its C library — yours may not!).</p> </div> </div> </div> </div> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table Of Contents</a></h3> <ul> <li><a class="reference external" href="#">14. Floating Point Arithmetic: Issues and Limitations</a><ul> <li><a class="reference external" href="#representation-error">14.1. Representation Error</a></li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="interactive.html" title="previous chapter">13. Interactive Input Editing and History Substitution</a></p> <h4>Next topic</h4> <p class="topless"><a href="../using/index.html" title="next chapter">Using Python</a></p> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../bugs.html">Report a Bug</a></li> <li><a href="../_sources/tutorial/floatingpoint.txt" rel="nofollow">Show Source</a></li> </ul> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../search.html" method="get"> <input type="text" name="q" size="18" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> <p class="searchtip" style="font-size: 90%"> Enter search terms or a module, class or function name. </p> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../modindex.html" title="Global Module Index" >modules</a> |</li> <li class="right" > <a href="../using/index.html" title="Using Python" >next</a> |</li> <li class="right" > <a href="interactive.html" title="13. Interactive Input Editing and History Substitution" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="../index.html">Python v2.6.6 documentation</a> »</li> <li><a href="index.html" >The Python Tutorial</a> »</li> </ul> </div> <div class="footer"> © <a href="../copyright.html">Copyright</a> 1990-2011, Python Software Foundation. <br /> The Python Software Foundation is a non-profit corporation. <a href="http://www.python.org/psf/donations/">Please donate.</a> <br /> Last updated on Jul 20, 2011. <a href="../bugs.html">Found a bug</a>? <br /> Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 0.6.6. </div> </body> </html>
Back to File Manager