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<div class="section" id="module-math">
<span id="math-mathematical-functions"></span><h1>9.2. <a class="reference internal" href="#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal"><span class="pre">math</span></code></a> — Mathematical functions<a class="headerlink" href="#module-math" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
<p>This module is always available. It provides access to the mathematical
functions defined by the C standard.</p>
<p>These functions cannot be used with complex numbers; use the functions of the
same name from the <a class="reference internal" href="cmath.html#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal"><span class="pre">cmath</span></code></a> module if you require support for complex
numbers. The distinction between functions which support complex numbers and
those which don’t is made since most users do not want to learn quite as much
mathematics as required to understand complex numbers. Receiving an exception
instead of a complex result allows earlier detection of the unexpected complex
number used as a parameter, so that the programmer can determine how and why it
was generated in the first place.</p>
<p>The following functions are provided by this module. Except when explicitly
noted otherwise, all return values are floats.</p>
<div class="section" id="number-theoretic-and-representation-functions">
<h2>9.2.1. Number-theoretic and representation functions<a class="headerlink" href="#number-theoretic-and-representation-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="math.ceil">
<code class="descclassname">math.</code><code class="descname">ceil</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.ceil" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the ceiling of <em>x</em>, the smallest integer greater than or equal to <em>x</em>.
If <em>x</em> is not a float, delegates to <code class="docutils literal"><span class="pre">x.__ceil__()</span></code>, which should return an
<a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal"><span class="pre">Integral</span></code></a> value.</p>
</dd></dl>
<dl class="function">
<dt id="math.copysign">
<code class="descclassname">math.</code><code class="descname">copysign</code><span class="sig-paren">(</span><em>x</em>, <em>y</em><span class="sig-paren">)</span><a class="headerlink" href="#math.copysign" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a float with the magnitude (absolute value) of <em>x</em> but the sign of
<em>y</em>. On platforms that support signed zeros, <code class="docutils literal"><span class="pre">copysign(1.0,</span> <span class="pre">-0.0)</span></code>
returns <em>-1.0</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.fabs">
<code class="descclassname">math.</code><code class="descname">fabs</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.fabs" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the absolute value of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.factorial">
<code class="descclassname">math.</code><code class="descname">factorial</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.factorial" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>x</em> factorial. Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal"><span class="pre">ValueError</span></code></a> if <em>x</em> is not integral or
is negative.</p>
</dd></dl>
<dl class="function">
<dt id="math.floor">
<code class="descclassname">math.</code><code class="descname">floor</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.floor" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the floor of <em>x</em>, the largest integer less than or equal to <em>x</em>.
If <em>x</em> is not a float, delegates to <code class="docutils literal"><span class="pre">x.__floor__()</span></code>, which should return an
<a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal"><span class="pre">Integral</span></code></a> value.</p>
</dd></dl>
<dl class="function">
<dt id="math.fmod">
<code class="descclassname">math.</code><code class="descname">fmod</code><span class="sig-paren">(</span><em>x</em>, <em>y</em><span class="sig-paren">)</span><a class="headerlink" href="#math.fmod" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">fmod(x,</span> <span class="pre">y)</span></code>, as defined by the platform C library. Note that the
Python expression <code class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> may not return the same result. The intent of the C
standard is that <code class="docutils literal"><span class="pre">fmod(x,</span> <span class="pre">y)</span></code> be exactly (mathematically; to infinite
precision) equal to <code class="docutils literal"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">n*y</span></code> for some integer <em>n</em> such that the result has
the same sign as <em>x</em> and magnitude less than <code class="docutils literal"><span class="pre">abs(y)</span></code>. Python’s <code class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code>
returns a result with the sign of <em>y</em> instead, and may not be exactly computable
for float arguments. For example, <code class="docutils literal"><span class="pre">fmod(-1e-100,</span> <span class="pre">1e100)</span></code> is <code class="docutils literal"><span class="pre">-1e-100</span></code>, but
the result of Python’s <code class="docutils literal"><span class="pre">-1e-100</span> <span class="pre">%</span> <span class="pre">1e100</span></code> is <code class="docutils literal"><span class="pre">1e100-1e-100</span></code>, which cannot be
represented exactly as a float, and rounds to the surprising <code class="docutils literal"><span class="pre">1e100</span></code>. For
this reason, function <a class="reference internal" href="#math.fmod" title="math.fmod"><code class="xref py py-func docutils literal"><span class="pre">fmod()</span></code></a> is generally preferred when working with
floats, while Python’s <code class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> is preferred when working with integers.</p>
</dd></dl>
<dl class="function">
<dt id="math.frexp">
<code class="descclassname">math.</code><code class="descname">frexp</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.frexp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the mantissa and exponent of <em>x</em> as the pair <code class="docutils literal"><span class="pre">(m,</span> <span class="pre">e)</span></code>. <em>m</em> is a float
and <em>e</em> is an integer such that <code class="docutils literal"><span class="pre">x</span> <span class="pre">==</span> <span class="pre">m</span> <span class="pre">*</span> <span class="pre">2**e</span></code> exactly. If <em>x</em> is zero,
returns <code class="docutils literal"><span class="pre">(0.0,</span> <span class="pre">0)</span></code>, otherwise <code class="docutils literal"><span class="pre">0.5</span> <span class="pre"><=</span> <span class="pre">abs(m)</span> <span class="pre"><</span> <span class="pre">1</span></code>. This is used to “pick
apart” the internal representation of a float in a portable way.</p>
</dd></dl>
<dl class="function">
<dt id="math.fsum">
<code class="descclassname">math.</code><code class="descname">fsum</code><span class="sig-paren">(</span><em>iterable</em><span class="sig-paren">)</span><a class="headerlink" href="#math.fsum" title="Permalink to this definition">¶</a></dt>
<dd><p>Return an accurate floating point sum of values in the iterable. Avoids
loss of precision by tracking multiple intermediate partial sums:</p>
<div class="highlight-python3"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">sum</span><span class="p">([</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">])</span>
<span class="go">0.9999999999999999</span>
<span class="gp">>>> </span><span class="n">fsum</span><span class="p">([</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">])</span>
<span class="go">1.0</span>
</pre></div>
</div>
<p>The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the
typical case where the rounding mode is half-even. On some non-Windows
builds, the underlying C library uses extended precision addition and may
occasionally double-round an intermediate sum causing it to be off in its
least significant bit.</p>
<p>For further discussion and two alternative approaches, see the <a class="reference external" href="https://code.activestate.com/recipes/393090/">ASPN cookbook
recipes for accurate floating point summation</a>.</p>
</dd></dl>
<dl class="function">
<dt id="math.gcd">
<code class="descclassname">math.</code><code class="descname">gcd</code><span class="sig-paren">(</span><em>a</em>, <em>b</em><span class="sig-paren">)</span><a class="headerlink" href="#math.gcd" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the greatest common divisor of the integers <em>a</em> and <em>b</em>. If either
<em>a</em> or <em>b</em> is nonzero, then the value of <code class="docutils literal"><span class="pre">gcd(a,</span> <span class="pre">b)</span></code> is the largest
positive integer that divides both <em>a</em> and <em>b</em>. <code class="docutils literal"><span class="pre">gcd(0,</span> <span class="pre">0)</span></code> returns
<code class="docutils literal"><span class="pre">0</span></code>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.5.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.isclose">
<code class="descclassname">math.</code><code class="descname">isclose</code><span class="sig-paren">(</span><em>a</em>, <em>b</em>, <em>*</em>, <em>rel_tol=1e-09</em>, <em>abs_tol=0.0</em><span class="sig-paren">)</span><a class="headerlink" href="#math.isclose" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if the values <em>a</em> and <em>b</em> are close to each other and
<code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
<p>Whether or not two values are considered close is determined according to
given absolute and relative tolerances.</p>
<p><em>rel_tol</em> is the relative tolerance – it is the maximum allowed difference
between <em>a</em> and <em>b</em>, relative to the larger absolute value of <em>a</em> or <em>b</em>.
For example, to set a tolerance of 5%, pass <code class="docutils literal"><span class="pre">rel_tol=0.05</span></code>. The default
tolerance is <code class="docutils literal"><span class="pre">1e-09</span></code>, which assures that the two values are the same
within about 9 decimal digits. <em>rel_tol</em> must be greater than zero.</p>
<p><em>abs_tol</em> is the minimum absolute tolerance – useful for comparisons near
zero. <em>abs_tol</em> must be at least zero.</p>
<p>If no errors occur, the result will be:
<code class="docutils literal"><span class="pre">abs(a-b)</span> <span class="pre"><=</span> <span class="pre">max(rel_tol</span> <span class="pre">*</span> <span class="pre">max(abs(a),</span> <span class="pre">abs(b)),</span> <span class="pre">abs_tol)</span></code>.</p>
<p>The IEEE 754 special values of <code class="docutils literal"><span class="pre">NaN</span></code>, <code class="docutils literal"><span class="pre">inf</span></code>, and <code class="docutils literal"><span class="pre">-inf</span></code> will be
handled according to IEEE rules. Specifically, <code class="docutils literal"><span class="pre">NaN</span></code> is not considered
close to any other value, including <code class="docutils literal"><span class="pre">NaN</span></code>. <code class="docutils literal"><span class="pre">inf</span></code> and <code class="docutils literal"><span class="pre">-inf</span></code> are only
considered close to themselves.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.5.</span></p>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><span class="target" id="index-0"></span><a class="pep reference external" href="https://www.python.org/dev/peps/pep-0485"><strong>PEP 485</strong></a> – A function for testing approximate equality</p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.isfinite">
<code class="descclassname">math.</code><code class="descname">isfinite</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.isfinite" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if <em>x</em> is neither an infinity nor a NaN, and
<code class="docutils literal"><span class="pre">False</span></code> otherwise. (Note that <code class="docutils literal"><span class="pre">0.0</span></code> <em>is</em> considered finite.)</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.isinf">
<code class="descclassname">math.</code><code class="descname">isinf</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.isinf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if <em>x</em> is a positive or negative infinity, and
<code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="math.isnan">
<code class="descclassname">math.</code><code class="descname">isnan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.isnan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">True</span></code> if <em>x</em> is a NaN (not a number), and <code class="docutils literal"><span class="pre">False</span></code> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="math.ldexp">
<code class="descclassname">math.</code><code class="descname">ldexp</code><span class="sig-paren">(</span><em>x</em>, <em>i</em><span class="sig-paren">)</span><a class="headerlink" href="#math.ldexp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">(2**i)</span></code>. This is essentially the inverse of function
<a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal"><span class="pre">frexp()</span></code></a>.</p>
</dd></dl>
<dl class="function">
<dt id="math.modf">
<code class="descclassname">math.</code><code class="descname">modf</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.modf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the fractional and integer parts of <em>x</em>. Both results carry the sign
of <em>x</em> and are floats.</p>
</dd></dl>
<dl class="function">
<dt id="math.trunc">
<code class="descclassname">math.</code><code class="descname">trunc</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.trunc" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <a class="reference internal" href="numbers.html#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal"><span class="pre">Real</span></code></a> value <em>x</em> truncated to an
<a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal"><span class="pre">Integral</span></code></a> (usually an integer). Delegates to
<code class="docutils literal"><span class="pre">x.__trunc__()</span></code>.</p>
</dd></dl>
<p>Note that <a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal"><span class="pre">frexp()</span></code></a> and <a class="reference internal" href="#math.modf" title="math.modf"><code class="xref py py-func docutils literal"><span class="pre">modf()</span></code></a> have a different call/return pattern
than their C equivalents: they take a single argument and return a pair of
values, rather than returning their second return value through an ‘output
parameter’ (there is no such thing in Python).</p>
<p>For the <a class="reference internal" href="#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal"><span class="pre">ceil()</span></code></a>, <a class="reference internal" href="#math.floor" title="math.floor"><code class="xref py py-func docutils literal"><span class="pre">floor()</span></code></a>, and <a class="reference internal" href="#math.modf" title="math.modf"><code class="xref py py-func docutils literal"><span class="pre">modf()</span></code></a> functions, note that <em>all</em>
floating-point numbers of sufficiently large magnitude are exact integers.
Python floats typically carry no more than 53 bits of precision (the same as the
platform C double type), in which case any float <em>x</em> with <code class="docutils literal"><span class="pre">abs(x)</span> <span class="pre">>=</span> <span class="pre">2**52</span></code>
necessarily has no fractional bits.</p>
</div>
<div class="section" id="power-and-logarithmic-functions">
<h2>9.2.2. Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="math.exp">
<code class="descclassname">math.</code><code class="descname">exp</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.exp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">e**x</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="math.expm1">
<code class="descclassname">math.</code><code class="descname">expm1</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.expm1" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">e**x</span> <span class="pre">-</span> <span class="pre">1</span></code>. For small floats <em>x</em>, the subtraction in <code class="docutils literal"><span class="pre">exp(x)</span> <span class="pre">-</span> <span class="pre">1</span></code>
can result in a <a class="reference external" href="https://en.wikipedia.org/wiki/Loss_of_significance">significant loss of precision</a>; the <a class="reference internal" href="#math.expm1" title="math.expm1"><code class="xref py py-func docutils literal"><span class="pre">expm1()</span></code></a>
function provides a way to compute this quantity to full precision:</p>
<div class="highlight-python3"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">expm1</span>
<span class="gp">>>> </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># gives result accurate to 11 places</span>
<span class="go">1.0000050000069649e-05</span>
<span class="gp">>>> </span><span class="n">expm1</span><span class="p">(</span><span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span> <span class="c1"># result accurate to full precision</span>
<span class="go">1.0000050000166668e-05</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.log">
<code class="descclassname">math.</code><code class="descname">log</code><span class="sig-paren">(</span><em>x</em><span class="optional">[</span>, <em>base</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#math.log" title="Permalink to this definition">¶</a></dt>
<dd><p>With one argument, return the natural logarithm of <em>x</em> (to base <em>e</em>).</p>
<p>With two arguments, return the logarithm of <em>x</em> to the given <em>base</em>,
calculated as <code class="docutils literal"><span class="pre">log(x)/log(base)</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="math.log1p">
<code class="descclassname">math.</code><code class="descname">log1p</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.log1p" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the natural logarithm of <em>1+x</em> (base <em>e</em>). The
result is calculated in a way which is accurate for <em>x</em> near zero.</p>
</dd></dl>
<dl class="function">
<dt id="math.log2">
<code class="descclassname">math.</code><code class="descname">log2</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.log2" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the base-2 logarithm of <em>x</em>. This is usually more accurate than
<code class="docutils literal"><span class="pre">log(x,</span> <span class="pre">2)</span></code>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.3.</span></p>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="stdtypes.html#int.bit_length" title="int.bit_length"><code class="xref py py-meth docutils literal"><span class="pre">int.bit_length()</span></code></a> returns the number of bits necessary to represent
an integer in binary, excluding the sign and leading zeros.</p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.log10">
<code class="descclassname">math.</code><code class="descname">log10</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.log10" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the base-10 logarithm of <em>x</em>. This is usually more accurate
than <code class="docutils literal"><span class="pre">log(x,</span> <span class="pre">10)</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="math.pow">
<code class="descclassname">math.</code><code class="descname">pow</code><span class="sig-paren">(</span><em>x</em>, <em>y</em><span class="sig-paren">)</span><a class="headerlink" href="#math.pow" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">x</span></code> raised to the power <code class="docutils literal"><span class="pre">y</span></code>. Exceptional cases follow
Annex ‘F’ of the C99 standard as far as possible. In particular,
<code class="docutils literal"><span class="pre">pow(1.0,</span> <span class="pre">x)</span></code> and <code class="docutils literal"><span class="pre">pow(x,</span> <span class="pre">0.0)</span></code> always return <code class="docutils literal"><span class="pre">1.0</span></code>, even
when <code class="docutils literal"><span class="pre">x</span></code> is a zero or a NaN. If both <code class="docutils literal"><span class="pre">x</span></code> and <code class="docutils literal"><span class="pre">y</span></code> are finite,
<code class="docutils literal"><span class="pre">x</span></code> is negative, and <code class="docutils literal"><span class="pre">y</span></code> is not an integer then <code class="docutils literal"><span class="pre">pow(x,</span> <span class="pre">y)</span></code>
is undefined, and raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal"><span class="pre">ValueError</span></code></a>.</p>
<p>Unlike the built-in <code class="docutils literal"><span class="pre">**</span></code> operator, <a class="reference internal" href="#math.pow" title="math.pow"><code class="xref py py-func docutils literal"><span class="pre">math.pow()</span></code></a> converts both
its arguments to type <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal"><span class="pre">float</span></code></a>. Use <code class="docutils literal"><span class="pre">**</span></code> or the built-in
<a class="reference internal" href="functions.html#pow" title="pow"><code class="xref py py-func docutils literal"><span class="pre">pow()</span></code></a> function for computing exact integer powers.</p>
</dd></dl>
<dl class="function">
<dt id="math.sqrt">
<code class="descclassname">math.</code><code class="descname">sqrt</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.sqrt" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the square root of <em>x</em>.</p>
</dd></dl>
</div>
<div class="section" id="trigonometric-functions">
<h2>9.2.3. Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="math.acos">
<code class="descclassname">math.</code><code class="descname">acos</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.acos" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc cosine of <em>x</em>, in radians.</p>
</dd></dl>
<dl class="function">
<dt id="math.asin">
<code class="descclassname">math.</code><code class="descname">asin</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.asin" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc sine of <em>x</em>, in radians.</p>
</dd></dl>
<dl class="function">
<dt id="math.atan">
<code class="descclassname">math.</code><code class="descname">atan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.atan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the arc tangent of <em>x</em>, in radians.</p>
</dd></dl>
<dl class="function">
<dt id="math.atan2">
<code class="descclassname">math.</code><code class="descname">atan2</code><span class="sig-paren">(</span><em>y</em>, <em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.atan2" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <code class="docutils literal"><span class="pre">atan(y</span> <span class="pre">/</span> <span class="pre">x)</span></code>, in radians. The result is between <code class="docutils literal"><span class="pre">-pi</span></code> and <code class="docutils literal"><span class="pre">pi</span></code>.
The vector in the plane from the origin to point <code class="docutils literal"><span class="pre">(x,</span> <span class="pre">y)</span></code> makes this angle
with the positive X axis. The point of <a class="reference internal" href="#math.atan2" title="math.atan2"><code class="xref py py-func docutils literal"><span class="pre">atan2()</span></code></a> is that the signs of both
inputs are known to it, so it can compute the correct quadrant for the angle.
For example, <code class="docutils literal"><span class="pre">atan(1)</span></code> and <code class="docutils literal"><span class="pre">atan2(1,</span> <span class="pre">1)</span></code> are both <code class="docutils literal"><span class="pre">pi/4</span></code>, but <code class="docutils literal"><span class="pre">atan2(-1,</span>
<span class="pre">-1)</span></code> is <code class="docutils literal"><span class="pre">-3*pi/4</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="math.cos">
<code class="descclassname">math.</code><code class="descname">cos</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.cos" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the cosine of <em>x</em> radians.</p>
</dd></dl>
<dl class="function">
<dt id="math.hypot">
<code class="descclassname">math.</code><code class="descname">hypot</code><span class="sig-paren">(</span><em>x</em>, <em>y</em><span class="sig-paren">)</span><a class="headerlink" href="#math.hypot" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the Euclidean norm, <code class="docutils literal"><span class="pre">sqrt(x*x</span> <span class="pre">+</span> <span class="pre">y*y)</span></code>. This is the length of the vector
from the origin to point <code class="docutils literal"><span class="pre">(x,</span> <span class="pre">y)</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="math.sin">
<code class="descclassname">math.</code><code class="descname">sin</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.sin" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the sine of <em>x</em> radians.</p>
</dd></dl>
<dl class="function">
<dt id="math.tan">
<code class="descclassname">math.</code><code class="descname">tan</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.tan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the tangent of <em>x</em> radians.</p>
</dd></dl>
</div>
<div class="section" id="angular-conversion">
<h2>9.2.4. Angular conversion<a class="headerlink" href="#angular-conversion" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="math.degrees">
<code class="descclassname">math.</code><code class="descname">degrees</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.degrees" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert angle <em>x</em> from radians to degrees.</p>
</dd></dl>
<dl class="function">
<dt id="math.radians">
<code class="descclassname">math.</code><code class="descname">radians</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.radians" title="Permalink to this definition">¶</a></dt>
<dd><p>Convert angle <em>x</em> from degrees to radians.</p>
</dd></dl>
</div>
<div class="section" id="hyperbolic-functions">
<h2>9.2.5. Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h2>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Hyperbolic_function">Hyperbolic functions</a>
are analogs of trigonometric functions that are based on hyperbolas
instead of circles.</p>
<dl class="function">
<dt id="math.acosh">
<code class="descclassname">math.</code><code class="descname">acosh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.acosh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic cosine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.asinh">
<code class="descclassname">math.</code><code class="descname">asinh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.asinh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic sine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.atanh">
<code class="descclassname">math.</code><code class="descname">atanh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.atanh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the inverse hyperbolic tangent of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.cosh">
<code class="descclassname">math.</code><code class="descname">cosh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.cosh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic cosine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.sinh">
<code class="descclassname">math.</code><code class="descname">sinh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.sinh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic sine of <em>x</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.tanh">
<code class="descclassname">math.</code><code class="descname">tanh</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.tanh" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the hyperbolic tangent of <em>x</em>.</p>
</dd></dl>
</div>
<div class="section" id="special-functions">
<h2>9.2.6. Special functions<a class="headerlink" href="#special-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="math.erf">
<code class="descclassname">math.</code><code class="descname">erf</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.erf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Error_function">error function</a> at
<em>x</em>.</p>
<p>The <a class="reference internal" href="#math.erf" title="math.erf"><code class="xref py py-func docutils literal"><span class="pre">erf()</span></code></a> function can be used to compute traditional statistical
functions such as the <a class="reference external" href="https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function">cumulative standard normal distribution</a>:</p>
<div class="highlight-python3"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">phi</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="s1">'Cumulative distribution function for the standard normal distribution'</span>
<span class="k">return</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">+</span> <span class="n">erf</span><span class="p">(</span><span class="n">x</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)))</span> <span class="o">/</span> <span class="mf">2.0</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.erfc">
<code class="descclassname">math.</code><code class="descname">erfc</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.erfc" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the complementary error function at <em>x</em>. The <a class="reference external" href="https://en.wikipedia.org/wiki/Error_function">complementary error
function</a> is defined as
<code class="docutils literal"><span class="pre">1.0</span> <span class="pre">-</span> <span class="pre">erf(x)</span></code>. It is used for large values of <em>x</em> where a subtraction
from one would cause a <a class="reference external" href="https://en.wikipedia.org/wiki/Loss_of_significance">loss of significance</a>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.gamma">
<code class="descclassname">math.</code><code class="descname">gamma</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.gamma" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Gamma_function">Gamma function</a> at
<em>x</em>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="math.lgamma">
<code class="descclassname">math.</code><code class="descname">lgamma</code><span class="sig-paren">(</span><em>x</em><span class="sig-paren">)</span><a class="headerlink" href="#math.lgamma" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the natural logarithm of the absolute value of the Gamma
function at <em>x</em>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.2.</span></p>
</div>
</dd></dl>
</div>
<div class="section" id="constants">
<h2>9.2.7. Constants<a class="headerlink" href="#constants" title="Permalink to this headline">¶</a></h2>
<dl class="data">
<dt id="math.pi">
<code class="descclassname">math.</code><code class="descname">pi</code><a class="headerlink" href="#math.pi" title="Permalink to this definition">¶</a></dt>
<dd><p>The mathematical constant π = 3.141592..., to available precision.</p>
</dd></dl>
<dl class="data">
<dt id="math.e">
<code class="descclassname">math.</code><code class="descname">e</code><a class="headerlink" href="#math.e" title="Permalink to this definition">¶</a></dt>
<dd><p>The mathematical constant e = 2.718281..., to available precision.</p>
</dd></dl>
<dl class="data">
<dt id="math.inf">
<code class="descclassname">math.</code><code class="descname">inf</code><a class="headerlink" href="#math.inf" title="Permalink to this definition">¶</a></dt>
<dd><p>A floating-point positive infinity. (For negative infinity, use
<code class="docutils literal"><span class="pre">-math.inf</span></code>.) Equivalent to the output of <code class="docutils literal"><span class="pre">float('inf')</span></code>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.5.</span></p>
</div>
</dd></dl>
<dl class="data">
<dt id="math.nan">
<code class="descclassname">math.</code><code class="descname">nan</code><a class="headerlink" href="#math.nan" title="Permalink to this definition">¶</a></dt>
<dd><p>A floating-point “not a number” (NaN) value. Equivalent to the output of
<code class="docutils literal"><span class="pre">float('nan')</span></code>.</p>
<div class="versionadded">
<p><span class="versionmodified">New in version 3.5.</span></p>
</div>
</dd></dl>
<div class="impl-detail compound">
<p class="compound-first"><strong>CPython implementation detail:</strong> The <a class="reference internal" href="#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal"><span class="pre">math</span></code></a> module consists mostly of thin wrappers around the platform C
math library functions. Behavior in exceptional cases follows Annex F of
the C99 standard where appropriate. The current implementation will raise
<a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal"><span class="pre">ValueError</span></code></a> for invalid operations like <code class="docutils literal"><span class="pre">sqrt(-1.0)</span></code> or <code class="docutils literal"><span class="pre">log(0.0)</span></code>
(where C99 Annex F recommends signaling invalid operation or divide-by-zero),
and <a class="reference internal" href="exceptions.html#OverflowError" title="OverflowError"><code class="xref py py-exc docutils literal"><span class="pre">OverflowError</span></code></a> for results that overflow (for example,
<code class="docutils literal"><span class="pre">exp(1000.0)</span></code>). A NaN will not be returned from any of the functions
above unless one or more of the input arguments was a NaN; in that case,
most functions will return a NaN, but (again following C99 Annex F) there
are some exceptions to this rule, for example <code class="docutils literal"><span class="pre">pow(float('nan'),</span> <span class="pre">0.0)</span></code> or
<code class="docutils literal"><span class="pre">hypot(float('nan'),</span> <span class="pre">float('inf'))</span></code>.</p>
<p class="compound-last">Note that Python makes no effort to distinguish signaling NaNs from
quiet NaNs, and behavior for signaling NaNs remains unspecified.
Typical behavior is to treat all NaNs as though they were quiet.</p>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<dl class="last docutils">
<dt>Module <a class="reference internal" href="cmath.html#module-cmath" title="cmath: Mathematical functions for complex numbers."><code class="xref py py-mod docutils literal"><span class="pre">cmath</span></code></a></dt>
<dd>Complex number versions of many of these functions.</dd>
</dl>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
<div class="sphinxsidebarwrapper">
<h3><a href="../contents.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">9.2. <code class="docutils literal"><span class="pre">math</span></code> — Mathematical functions</a><ul>
<li><a class="reference internal" href="#number-theoretic-and-representation-functions">9.2.1. Number-theoretic and representation functions</a></li>
<li><a class="reference internal" href="#power-and-logarithmic-functions">9.2.2. Power and logarithmic functions</a></li>
<li><a class="reference internal" href="#trigonometric-functions">9.2.3. Trigonometric functions</a></li>
<li><a class="reference internal" href="#angular-conversion">9.2.4. Angular conversion</a></li>
<li><a class="reference internal" href="#hyperbolic-functions">9.2.5. Hyperbolic functions</a></li>
<li><a class="reference internal" href="#special-functions">9.2.6. Special functions</a></li>
<li><a class="reference internal" href="#constants">9.2.7. Constants</a></li>
</ul>
</li>
</ul>
<h4>Previous topic</h4>
<p class="topless"><a href="numbers.html"
title="previous chapter">9.1. <code class="docutils literal"><span class="pre">numbers</span></code> — Numeric abstract base classes</a></p>
<h4>Next topic</h4>
<p class="topless"><a href="cmath.html"
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