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<div class="section" id="floating-point-arithmetic-issues-and-limitations">
<span id="tut-fp-issues"></span><h1>15. 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-python3 notranslate"><div class="highlight"><pre><span></span><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-python3 notranslate"><div class="highlight"><pre><span></span><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-python3 notranslate"><div class="highlight"><pre><span></span><span class="mf">0.3</span>
</pre></div>
</div>
<p>or, better,</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="mf">0.33</span>
</pre></div>
</div>
<p>or, better,</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><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-python3 notranslate"><div class="highlight"><pre><span></span><span class="mf">0.0001100110011001100110011001100110011001100110011</span><span class="o">...</span>
</pre></div>
</div>
<p>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 <code class="docutils literal notranslate"><span class="pre">3602879701896397</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">**</span> <span class="pre">55</span></code> which is close to but not exactly
equal to the true value of 1/10.</p>
<p>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</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="mf">0.1</span>
<span class="go">0.1000000000000000055511151231257827021181583404541015625</span>
</pre></div>
</div>
<p>That is more digits than most people find useful, so Python keeps the number
of digits manageable by displaying a rounded value instead</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span> <span class="o">/</span> <span class="mi">10</span>
<span class="go">0.1</span>
</pre></div>
</div>
<p>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.</p>
<p>Interestingly, there are many different decimal numbers that share the same
nearest approximate binary fraction. For example, the numbers <code class="docutils literal notranslate"><span class="pre">0.1</span></code> and
<code class="docutils literal notranslate"><span class="pre">0.10000000000000001</span></code> and
<code class="docutils literal notranslate"><span class="pre">0.1000000000000000055511151231257827021181583404541015625</span></code> are all
approximated by <code class="docutils literal notranslate"><span class="pre">3602879701896397</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">**</span> <span class="pre">55</span></code>. Since all of these decimal
values share the same approximation, any one of them could be displayed
while still preserving the invariant <code class="docutils literal notranslate"><span class="pre">eval(repr(x))</span> <span class="pre">==</span> <span class="pre">x</span></code>.</p>
<p>Historically, the Python prompt and built-in <a class="reference internal" href="../library/functions.html#repr" title="repr"><code class="xref py py-func docutils literal notranslate"><span class="pre">repr()</span></code></a> function would choose
the one with 17 significant digits, <code class="docutils literal notranslate"><span class="pre">0.10000000000000001</span></code>. Starting with
Python 3.1, Python (on most systems) is now able to choose the shortest of
these and simply display <code class="docutils literal notranslate"><span class="pre">0.1</span></code>.</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>For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="s1">&#39;.12g&#39;</span><span class="p">)</span> <span class="c1"># give 12 significant digits</span>
<span class="go">&#39;3.14159265359&#39;</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="s1">&#39;.2f&#39;</span><span class="p">)</span> <span class="c1"># give 2 digits after the point</span>
<span class="go">&#39;3.14&#39;</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">repr</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="go">&#39;3.141592653589793&#39;</span>
</pre></div>
</div>
<p>It’s important to realize that this is, in a real sense, an illusion: you’re
simply rounding the <em>display</em> of the true machine value.</p>
<p>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:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="o">.</span><span class="mi">1</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span> <span class="o">==</span> <span class="o">.</span><span class="mi">3</span>
<span class="go">False</span>
</pre></div>
</div>
<p>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
<a class="reference internal" href="../library/functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate"><span class="pre">round()</span></code></a> function cannot help:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
<p>Though the numbers cannot be made closer to their intended exact values,
the <a class="reference internal" href="../library/functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate"><span class="pre">round()</span></code></a> function can be useful for post-rounding so that results
with inexact values become comparable to one another:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">1</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="nb">round</span><span class="p">(</span><span class="o">.</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">True</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 class="reference internal" href="../library/stdtypes.html#str" title="str"><code class="xref py py-func docutils literal notranslate"><span class="pre">str()</span></code></a> usually suffices, and for finer control see the <a class="reference internal" href="../library/stdtypes.html#str.format" title="str.format"><code class="xref py py-meth docutils literal notranslate"><span class="pre">str.format()</span></code></a>
method’s format specifiers in <a class="reference internal" href="../library/string.html#formatstrings"><span class="std std-ref">Format String Syntax</span></a>.</p>
<p>For use cases which require exact decimal representation, try using the
<a class="reference internal" href="../library/decimal.html#module-decimal" title="decimal: Implementation of the General Decimal Arithmetic Specification."><code class="xref py py-mod docutils literal notranslate"><span class="pre">decimal</span></code></a> module which implements decimal arithmetic suitable for
accounting applications and high-precision applications.</p>
<p>Another form of exact arithmetic is supported by the <a class="reference internal" href="../library/fractions.html#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> module
which implements arithmetic based on rational numbers (so the numbers like
1/3 can be represented exactly).</p>
<p>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 &lt;<a class="reference external" href="https://scipy.org">https://scipy.org</a>&gt;.</p>
<p>Python provides tools that may help on those rare occasions when you really
<em>do</em> want to know the exact value of a float. The
<a class="reference internal" href="../library/stdtypes.html#float.as_integer_ratio" title="float.as_integer_ratio"><code class="xref py py-meth docutils literal notranslate"><span class="pre">float.as_integer_ratio()</span></code></a> method expresses the value of a float as a
fraction:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">3.14159</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">as_integer_ratio</span><span class="p">()</span>
<span class="go">(3537115888337719, 1125899906842624)</span>
</pre></div>
</div>
<p>Since the ratio is exact, it can be used to losslessly recreate the
original value:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">==</span> <span class="mi">3537115888337719</span> <span class="o">/</span> <span class="mi">1125899906842624</span>
<span class="go">True</span>
</pre></div>
</div>
<p>The <a class="reference internal" href="../library/stdtypes.html#float.hex" title="float.hex"><code class="xref py py-meth docutils literal notranslate"><span class="pre">float.hex()</span></code></a> method expresses a float in hexadecimal (base
16), again giving the exact value stored by your computer:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">hex</span><span class="p">()</span>
<span class="go">&#39;0x1.921f9f01b866ep+1&#39;</span>
</pre></div>
</div>
<p>This precise hexadecimal representation can be used to reconstruct
the float value exactly:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">==</span> <span class="nb">float</span><span class="o">.</span><span class="n">fromhex</span><span class="p">(</span><span class="s1">&#39;0x1.921f9f01b866ep+1&#39;</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<p>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).</p>
<p>Another helpful tool is the <a class="reference internal" href="../library/math.html#math.fsum" title="math.fsum"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.fsum()</span></code></a> 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:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">([</span><span class="mf">0.1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mf">1.0</span>
<span class="go">False</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">math</span><span class="o">.</span><span class="n">fsum</span><span class="p">([</span><span class="mf">0.1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mf">1.0</span>
<span class="go">True</span>
</pre></div>
</div>
<div class="section" id="representation-error">
<span id="tut-fp-error"></span><h2>15.1. 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 class="dfn">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>
<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-python3 notranslate"><div class="highlight"><pre><span></span><span class="mi">1</span> <span class="o">/</span> <span class="mi">10</span> <span class="o">~=</span> <span class="n">J</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">N</span><span class="p">)</span>
</pre></div>
</div>
<p>as</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="n">J</span> <span class="o">~=</span> <span class="mi">2</span><span class="o">**</span><span class="n">N</span> <span class="o">/</span> <span class="mi">10</span>
</pre></div>
</div>
<p>and recalling that <em>J</em> has exactly 53 bits (is <code class="docutils literal notranslate"><span class="pre">&gt;=</span> <span class="pre">2**52</span></code> but <code class="docutils literal notranslate"><span class="pre">&lt;</span> <span class="pre">2**53</span></code>),
the best value for <em>N</em> is 56:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">**</span><span class="mi">52</span> <span class="o">&lt;=</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="o">&lt;</span> <span class="mi">2</span><span class="o">**</span><span class="mi">53</span>
<span class="go">True</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-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </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">&gt;&gt;&gt; </span><span class="n">r</span>
<span class="go">6</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-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">+</span><span class="mi">1</span>
<span class="go">7205759403792794</span>
</pre></div>
</div>
<p>Therefore the best possible approximation to 1/10 in 754 double precision is:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="mi">7205759403792794</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">**</span> <span class="mi">56</span>
</pre></div>
</div>
<p>Dividing both the numerator and denominator by two reduces the fraction to:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="mi">3602879701896397</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">**</span> <span class="mi">55</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-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="mf">0.1</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">**</span> <span class="mi">55</span>
<span class="go">3602879701896397.0</span>
</pre></div>
</div>
<p>If we multiply that fraction by 10**55, we can see the value out to
55 decimal digits:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="mi">3602879701896397</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">**</span> <span class="mi">55</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">**</span> <span class="mi">55</span>
<span class="go">1000000000000000055511151231257827021181583404541015625</span>
</pre></div>
</div>
<p>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:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="s1">&#39;.17f&#39;</span><span class="p">)</span>
<span class="go">&#39;0.10000000000000001&#39;</span>
</pre></div>
</div>
<p>The <a class="reference internal" href="../library/fractions.html#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> and <a class="reference internal" href="../library/decimal.html#module-decimal" title="decimal: Implementation of the General Decimal Arithmetic Specification."><code class="xref py py-mod docutils literal notranslate"><span class="pre">decimal</span></code></a> modules make these calculations
easy:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="o">.</span><span class="n">from_float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
<span class="go">Fraction(3602879701896397, 36028797018963968)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span><span class="o">.</span><span class="n">as_integer_ratio</span><span class="p">()</span>
<span class="go">(3602879701896397, 36028797018963968)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">Decimal</span><span class="o">.</span><span class="n">from_float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
<span class="go">Decimal(&#39;0.1000000000000000055511151231257827021181583404541015625&#39;)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="n">Decimal</span><span class="o">.</span><span class="n">from_float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">),</span> <span class="s1">&#39;.17&#39;</span><span class="p">)</span>
<span class="go">&#39;0.10000000000000001&#39;</span>
</pre></div>
</div>
</div>
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