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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.)."><tt class="xref py py-mod docutils literal"><span class="pre">math</span></tt></a> — Mathematical functions<a class="headerlink" href="#module-math" title="Permalink to this headline">¶</a></h1>
<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."><tt class="xref py py-mod docutils literal"><span class="pre">cmath</span></tt></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">
<tt class="descclassname">math.</tt><tt class="descname">ceil</tt><big>(</big><em>x</em><big>)</big><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 <tt class="docutils literal"><span class="pre">x.__ceil__()</span></tt>, which should return an
<tt class="xref py py-class docutils literal"><span class="pre">Integral</span></tt> value.</p>
</dd></dl>
<dl class="function">
<dt id="math.copysign">
<tt class="descclassname">math.</tt><tt class="descname">copysign</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#math.copysign" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <em>x</em> with the sign of <em>y</em>. On a platform that supports
signed zeros, <tt class="docutils literal"><span class="pre">copysign(1.0,</span> <span class="pre">-0.0)</span></tt> returns <em>-1.0</em>.</p>
</dd></dl>
<dl class="function">
<dt id="math.fabs">
<tt class="descclassname">math.</tt><tt class="descname">fabs</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">factorial</tt><big>(</big><em>x</em><big>)</big><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"><tt class="xref py py-exc docutils literal"><span class="pre">ValueError</span></tt></a> if <em>x</em> is not integral or
is negative.</p>
</dd></dl>
<dl class="function">
<dt id="math.floor">
<tt class="descclassname">math.</tt><tt class="descname">floor</tt><big>(</big><em>x</em><big>)</big><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 <tt class="docutils literal"><span class="pre">x.__floor__()</span></tt>, which should return an
<tt class="xref py py-class docutils literal"><span class="pre">Integral</span></tt> value.</p>
</dd></dl>
<dl class="function">
<dt id="math.fmod">
<tt class="descclassname">math.</tt><tt class="descname">fmod</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#math.fmod" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">fmod(x,</span> <span class="pre">y)</span></tt>, as defined by the platform C library. Note that the
Python expression <tt class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></tt> may not return the same result. The intent of the C
standard is that <tt class="docutils literal"><span class="pre">fmod(x,</span> <span class="pre">y)</span></tt> be exactly (mathematically; to infinite
precision) equal to <tt class="docutils literal"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">n*y</span></tt> for some integer <em>n</em> such that the result has
the same sign as <em>x</em> and magnitude less than <tt class="docutils literal"><span class="pre">abs(y)</span></tt>. Python’s <tt class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></tt>
returns a result with the sign of <em>y</em> instead, and may not be exactly computable
for float arguments. For example, <tt class="docutils literal"><span class="pre">fmod(-1e-100,</span> <span class="pre">1e100)</span></tt> is <tt class="docutils literal"><span class="pre">-1e-100</span></tt>, but
the result of Python’s <tt class="docutils literal"><span class="pre">-1e-100</span> <span class="pre">%</span> <span class="pre">1e100</span></tt> is <tt class="docutils literal"><span class="pre">1e100-1e-100</span></tt>, which cannot be
represented exactly as a float, and rounds to the surprising <tt class="docutils literal"><span class="pre">1e100</span></tt>. For
this reason, function <a class="reference internal" href="#math.fmod" title="math.fmod"><tt class="xref py py-func docutils literal"><span class="pre">fmod()</span></tt></a> is generally preferred when working with
floats, while Python’s <tt class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></tt> is preferred when working with integers.</p>
</dd></dl>
<dl class="function">
<dt id="math.frexp">
<tt class="descclassname">math.</tt><tt class="descname">frexp</tt><big>(</big><em>x</em><big>)</big><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 <tt class="docutils literal"><span class="pre">(m,</span> <span class="pre">e)</span></tt>. <em>m</em> is a float
and <em>e</em> is an integer such that <tt 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></tt> exactly. If <em>x</em> is zero,
returns <tt class="docutils literal"><span class="pre">(0.0,</span> <span class="pre">0)</span></tt>, otherwise <tt 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></tt>. 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">
<tt class="descclassname">math.</tt><tt class="descname">fsum</tt><big>(</big><em>iterable</em><big>)</big><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 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="http://code.activestate.com/recipes/393090/">ASPN cookbook
recipes for accurate floating point summation</a>.</p>
</dd></dl>
<dl class="function">
<dt id="math.isfinite">
<tt class="descclassname">math.</tt><tt class="descname">isfinite</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.isfinite" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="xref docutils literal"><span class="pre">True</span></tt> if <em>x</em> is neither an infinity nor a NaN, and
<tt class="xref docutils literal"><span class="pre">False</span></tt> otherwise. (Note that <tt class="docutils literal"><span class="pre">0.0</span></tt> <em>is</em> considered finite.)</p>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</dd></dl>
<dl class="function">
<dt id="math.isinf">
<tt class="descclassname">math.</tt><tt class="descname">isinf</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.isinf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="xref docutils literal"><span class="pre">True</span></tt> if <em>x</em> is a positive or negative infinity, and
<tt class="xref docutils literal"><span class="pre">False</span></tt> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="math.isnan">
<tt class="descclassname">math.</tt><tt class="descname">isnan</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.isnan" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="xref docutils literal"><span class="pre">True</span></tt> if <em>x</em> is a NaN (not a number), and <tt class="xref docutils literal"><span class="pre">False</span></tt> otherwise.</p>
</dd></dl>
<dl class="function">
<dt id="math.ldexp">
<tt class="descclassname">math.</tt><tt class="descname">ldexp</tt><big>(</big><em>x</em>, <em>i</em><big>)</big><a class="headerlink" href="#math.ldexp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">(2**i)</span></tt>. This is essentially the inverse of function
<a class="reference internal" href="#math.frexp" title="math.frexp"><tt class="xref py py-func docutils literal"><span class="pre">frexp()</span></tt></a>.</p>
</dd></dl>
<dl class="function">
<dt id="math.modf">
<tt class="descclassname">math.</tt><tt class="descname">modf</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">trunc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.trunc" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <tt class="xref py py-class docutils literal"><span class="pre">Real</span></tt> value <em>x</em> truncated to an <tt class="xref py py-class docutils literal"><span class="pre">Integral</span></tt> (usually
an integer). Delegates to <tt class="docutils literal"><span class="pre">x.__trunc__()</span></tt>.</p>
</dd></dl>
<p>Note that <a class="reference internal" href="#math.frexp" title="math.frexp"><tt class="xref py py-func docutils literal"><span class="pre">frexp()</span></tt></a> and <a class="reference internal" href="#math.modf" title="math.modf"><tt class="xref py py-func docutils literal"><span class="pre">modf()</span></tt></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"><tt class="xref py py-func docutils literal"><span class="pre">ceil()</span></tt></a>, <a class="reference internal" href="#math.floor" title="math.floor"><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt></a>, and <a class="reference internal" href="#math.modf" title="math.modf"><tt class="xref py py-func docutils literal"><span class="pre">modf()</span></tt></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 <tt class="docutils literal"><span class="pre">abs(x)</span> <span class="pre">>=</span> <span class="pre">2**52</span></tt>
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">
<tt class="descclassname">math.</tt><tt class="descname">exp</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.exp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">e**x</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="math.expm1">
<tt class="descclassname">math.</tt><tt class="descname">expm1</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.expm1" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">e**x</span> <span class="pre">-</span> <span class="pre">1</span></tt>. For small floats <em>x</em>, the subtraction in <tt class="docutils literal"><span class="pre">exp(x)</span> <span class="pre">-</span> <span class="pre">1</span></tt>
can result in a <a class="reference external" href="http://en.wikipedia.org/wiki/Loss_of_significance">significant loss of precision</a>; the <a class="reference internal" href="#math.expm1" title="math.expm1"><tt class="xref py py-func docutils literal"><span class="pre">expm1()</span></tt></a>
function provides a way to compute this quantity to full precision:</p>
<div class="highlight-python3"><div class="highlight"><pre><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="c"># 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="c"># result accurate to full precision</span>
<span class="go">1.0000050000166668e-05</span>
</pre></div>
</div>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</dd></dl>
<dl class="function">
<dt id="math.log">
<tt class="descclassname">math.</tt><tt class="descname">log</tt><big>(</big><em>x</em><span class="optional">[</span>, <em>base</em><span class="optional">]</span><big>)</big><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 <tt class="docutils literal"><span class="pre">log(x)/log(base)</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="math.log1p">
<tt class="descclassname">math.</tt><tt class="descname">log1p</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">log2</tt><big>(</big><em>x</em><big>)</big><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
<tt class="docutils literal"><span class="pre">log(x,</span> <span class="pre">2)</span></tt>.</p>
<p class="versionadded">
<span class="versionmodified">New in version 3.3.</span> </p>
<div class="admonition-see-also 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"><tt class="xref py py-meth docutils literal"><span class="pre">int.bit_length()</span></tt></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">
<tt class="descclassname">math.</tt><tt class="descname">log10</tt><big>(</big><em>x</em><big>)</big><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 <tt class="docutils literal"><span class="pre">log(x,</span> <span class="pre">10)</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="math.pow">
<tt class="descclassname">math.</tt><tt class="descname">pow</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#math.pow" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">x</span></tt> raised to the power <tt class="docutils literal"><span class="pre">y</span></tt>. Exceptional cases follow
Annex ‘F’ of the C99 standard as far as possible. In particular,
<tt class="docutils literal"><span class="pre">pow(1.0,</span> <span class="pre">x)</span></tt> and <tt class="docutils literal"><span class="pre">pow(x,</span> <span class="pre">0.0)</span></tt> always return <tt class="docutils literal"><span class="pre">1.0</span></tt>, even
when <tt class="docutils literal"><span class="pre">x</span></tt> is a zero or a NaN. If both <tt class="docutils literal"><span class="pre">x</span></tt> and <tt class="docutils literal"><span class="pre">y</span></tt> are finite,
<tt class="docutils literal"><span class="pre">x</span></tt> is negative, and <tt class="docutils literal"><span class="pre">y</span></tt> is not an integer then <tt class="docutils literal"><span class="pre">pow(x,</span> <span class="pre">y)</span></tt>
is undefined, and raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><tt class="xref py py-exc docutils literal"><span class="pre">ValueError</span></tt></a>.</p>
</dd></dl>
<dl class="function">
<dt id="math.sqrt">
<tt class="descclassname">math.</tt><tt class="descname">sqrt</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">acos</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">asin</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">atan</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">atan2</tt><big>(</big><em>y</em>, <em>x</em><big>)</big><a class="headerlink" href="#math.atan2" title="Permalink to this definition">¶</a></dt>
<dd><p>Return <tt class="docutils literal"><span class="pre">atan(y</span> <span class="pre">/</span> <span class="pre">x)</span></tt>, in radians. The result is between <tt class="docutils literal"><span class="pre">-pi</span></tt> and <tt class="docutils literal"><span class="pre">pi</span></tt>.
The vector in the plane from the origin to point <tt class="docutils literal"><span class="pre">(x,</span> <span class="pre">y)</span></tt> makes this angle
with the positive X axis. The point of <a class="reference internal" href="#math.atan2" title="math.atan2"><tt class="xref py py-func docutils literal"><span class="pre">atan2()</span></tt></a> is that the signs of both
inputs are known to it, so it can compute the correct quadrant for the angle.
For example, <tt class="docutils literal"><span class="pre">atan(1)</span></tt> and <tt class="docutils literal"><span class="pre">atan2(1,</span> <span class="pre">1)</span></tt> are both <tt class="docutils literal"><span class="pre">pi/4</span></tt>, but <tt class="docutils literal"><span class="pre">atan2(-1,</span>
<span class="pre">-1)</span></tt> is <tt class="docutils literal"><span class="pre">-3*pi/4</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="math.cos">
<tt class="descclassname">math.</tt><tt class="descname">cos</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">hypot</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#math.hypot" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the Euclidean norm, <tt class="docutils literal"><span class="pre">sqrt(x*x</span> <span class="pre">+</span> <span class="pre">y*y)</span></tt>. This is the length of the vector
from the origin to point <tt class="docutils literal"><span class="pre">(x,</span> <span class="pre">y)</span></tt>.</p>
</dd></dl>
<dl class="function">
<dt id="math.sin">
<tt class="descclassname">math.</tt><tt class="descname">sin</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">tan</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">degrees</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.degrees" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts angle <em>x</em> from radians to degrees.</p>
</dd></dl>
<dl class="function">
<dt id="math.radians">
<tt class="descclassname">math.</tt><tt class="descname">radians</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.radians" title="Permalink to this definition">¶</a></dt>
<dd><p>Converts 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="http://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">
<tt class="descclassname">math.</tt><tt class="descname">acosh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">asinh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">atanh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">cosh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">sinh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">tanh</tt><big>(</big><em>x</em><big>)</big><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">
<tt class="descclassname">math.</tt><tt class="descname">erf</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.erf" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <a class="reference external" href="http://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"><tt class="xref py py-func docutils literal"><span class="pre">erf()</span></tt></a> function can be used to compute traditional statistical
functions such as the <a class="reference external" href="http://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function">cumulative standard normal distribution</a>:</p>
<div class="highlight-python3"><div class="highlight"><pre><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="s">'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>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</dd></dl>
<dl class="function">
<dt id="math.erfc">
<tt class="descclassname">math.</tt><tt class="descname">erfc</tt><big>(</big><em>x</em><big>)</big><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="http://en.wikipedia.org/wiki/Error_function">complementary error
function</a> is defined as
<tt class="docutils literal"><span class="pre">1.0</span> <span class="pre">-</span> <span class="pre">erf(x)</span></tt>. It is used for large values of <em>x</em> where a subtraction
from one would cause a <a class="reference external" href="http://en.wikipedia.org/wiki/Loss_of_significance">loss of significance</a>.</p>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</dd></dl>
<dl class="function">
<dt id="math.gamma">
<tt class="descclassname">math.</tt><tt class="descname">gamma</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#math.gamma" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the <a class="reference external" href="http://en.wikipedia.org/wiki/Gamma_function">Gamma function</a> at
<em>x</em>.</p>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</dd></dl>
<dl class="function">
<dt id="math.lgamma">
<tt class="descclassname">math.</tt><tt class="descname">lgamma</tt><big>(</big><em>x</em><big>)</big><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>
<p class="versionadded">
<span class="versionmodified">New in version 3.2.</span> </p>
</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">
<tt class="descclassname">math.</tt><tt class="descname">pi</tt><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">
<tt class="descclassname">math.</tt><tt class="descname">e</tt><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>
<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.)."><tt class="xref py py-mod docutils literal"><span class="pre">math</span></tt></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"><tt class="xref py py-exc docutils literal"><span class="pre">ValueError</span></tt></a> for invalid operations like <tt class="docutils literal"><span class="pre">sqrt(-1.0)</span></tt> or <tt class="docutils literal"><span class="pre">log(0.0)</span></tt>
(where C99 Annex F recommends signaling invalid operation or divide-by-zero),
and <a class="reference internal" href="exceptions.html#OverflowError" title="OverflowError"><tt class="xref py py-exc docutils literal"><span class="pre">OverflowError</span></tt></a> for results that overflow (for example,
<tt class="docutils literal"><span class="pre">exp(1000.0)</span></tt>). 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 <tt class="docutils literal"><span class="pre">pow(float('nan'),</span> <span class="pre">0.0)</span></tt> or
<tt class="docutils literal"><span class="pre">hypot(float('nan'),</span> <span class="pre">float('inf'))</span></tt>.</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-see-also 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."><tt class="xref py py-mod docutils literal"><span class="pre">cmath</span></tt></a></dt>
<dd>Complex number versions of many of these functions.</dd>
</dl>
</div>
</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 internal" href="#">9.2. <tt class="docutils literal"><span class="pre">math</span></tt> — 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>
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