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  198.   <section id="module-math">

  199. <span id="math-mathematical-functions"></span><h1><a class="reference internal" href="#module-math" title="math: Mathematical functions (sin() etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code></a> — Mathematical functions<a class="headerlink" href="#module-math" title="Link to this heading">¶</a></h1>

  200. <hr class="docutils" />

  201. <p>This module provides access to the mathematical functions defined by the C

  202. standard.</p>

  203. <p>These functions cannot be used with complex numbers; use the functions of the

  204. 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 notranslate"><span class="pre">cmath</span></code></a> module if you require support for complex

  205. numbers.  The distinction between functions which support complex numbers and

  206. those which don’t is made since most users do not want to learn quite as much

  207. mathematics as required to understand complex numbers.  Receiving an exception

  208. instead of a complex result allows earlier detection of the unexpected complex

  209. number used as a parameter, so that the programmer can determine how and why it

  210. was generated in the first place.</p>

  211. <p>The following functions are provided by this module.  Except when explicitly

  212. noted otherwise, all return values are floats.</p>

  213. <section id="number-theoretic-and-representation-functions">

  214. <h2>Number-theoretic and representation functions<a class="headerlink" href="#number-theoretic-and-representation-functions" title="Link to this heading">¶</a></h2>

  215. <dl class="py function">

  216. <dt class="sig sig-object py" id="math.ceil">

  217. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">ceil</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.ceil" title="Link to this definition">¶</a></dt>

  218. <dd><p>Return the ceiling of <em>x</em>, the smallest integer greater than or equal to <em>x</em>.

  219. If <em>x</em> is not a float, delegates to <a class="reference internal" href="../reference/datamodel.html#object.__ceil__" title="object.__ceil__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">x.__ceil__</span></code></a>,

  220. which should return an <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">Integral</span></code></a> value.</p>

  221. </dd></dl>

  222. 

  223. <dl class="py function">

  224. <dt class="sig sig-object py" id="math.comb">

  225. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">comb</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.comb" title="Link to this definition">¶</a></dt>

  226. <dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items without repetition

  227. and without order.</p>

  228. <p>Evaluates to <code class="docutils literal notranslate"><span class="pre">n!</span> <span class="pre">/</span> <span class="pre">(k!</span> <span class="pre">*</span> <span class="pre">(n</span> <span class="pre">-</span> <span class="pre">k)!)</span></code> when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&lt;=</span> <span class="pre">n</span></code> and evaluates

  229. to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&gt;</span> <span class="pre">n</span></code>.</p>

  230. <p>Also called the binomial coefficient because it is equivalent

  231. to the coefficient of k-th term in polynomial expansion of

  232. <code class="docutils literal notranslate"><span class="pre">(1</span> <span class="pre">+</span> <span class="pre">x)ⁿ</span></code>.</p>

  233. <p>Raises <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> if either of the arguments are not integers.

  234. Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if either of the arguments are negative.</p>

  235. <div class="versionadded">

  236. <p><span class="versionmodified added">New in version 3.8.</span></p>

  237. </div>

  238. </dd></dl>

  239. 

  240. <dl class="py function">

  241. <dt class="sig sig-object py" id="math.copysign">

  242. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">copysign</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.copysign" title="Link to this definition">¶</a></dt>

  243. <dd><p>Return a float with the magnitude (absolute value) of <em>x</em> but the sign of

  244. <em>y</em>.  On platforms that support signed zeros, <code class="docutils literal notranslate"><span class="pre">copysign(1.0,</span> <span class="pre">-0.0)</span></code>

  245. returns <em>-1.0</em>.</p>

  246. </dd></dl>

  247. 

  248. <dl class="py function">

  249. <dt class="sig sig-object py" id="math.fabs">

  250. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">fabs</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.fabs" title="Link to this definition">¶</a></dt>

  251. <dd><p>Return the absolute value of <em>x</em>.</p>

  252. </dd></dl>

  253. 

  254. <dl class="py function">

  255. <dt class="sig sig-object py" id="math.factorial">

  256. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">factorial</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.factorial" title="Link to this definition">¶</a></dt>

  257. <dd><p>Return <em>n</em> factorial as an integer.  Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if <em>n</em> is not integral or

  258. is negative.</p>

  259. <div class="deprecated">

  260. <p><span class="versionmodified deprecated">Deprecated since version 3.9: </span>Accepting floats with integral values (like <code class="docutils literal notranslate"><span class="pre">5.0</span></code>) is deprecated.</p>

  261. </div>

  262. </dd></dl>

  263. 

  264. <dl class="py function">

  265. <dt class="sig sig-object py" id="math.floor">

  266. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">floor</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.floor" title="Link to this definition">¶</a></dt>

  267. <dd><p>Return the floor of <em>x</em>, the largest integer less than or equal to <em>x</em>.  If

  268. <em>x</em> is not a float, delegates to <a class="reference internal" href="../reference/datamodel.html#object.__floor__" title="object.__floor__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">x.__floor__</span></code></a>, which

  269. should return an <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">Integral</span></code></a> value.</p>

  270. </dd></dl>

  271. 

  272. <dl class="py function">

  273. <dt class="sig sig-object py" id="math.fmod">

  274. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">fmod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.fmod" title="Link to this definition">¶</a></dt>

  275. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">fmod(x,</span> <span class="pre">y)</span></code>, as defined by the platform C library. Note that the

  276. Python expression <code class="docutils literal notranslate"><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

  277. standard is that <code class="docutils literal notranslate"><span class="pre">fmod(x,</span> <span class="pre">y)</span></code> be exactly (mathematically; to infinite

  278. precision) equal to <code class="docutils literal notranslate"><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

  279. the same sign as <em>x</em> and magnitude less than <code class="docutils literal notranslate"><span class="pre">abs(y)</span></code>.  Python’s <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code>

  280. returns a result with the sign of <em>y</em> instead, and may not be exactly computable

  281. for float arguments. For example, <code class="docutils literal notranslate"><span class="pre">fmod(-1e-100,</span> <span class="pre">1e100)</span></code> is <code class="docutils literal notranslate"><span class="pre">-1e-100</span></code>, but

  282. the result of Python’s <code class="docutils literal notranslate"><span class="pre">-1e-100</span> <span class="pre">%</span> <span class="pre">1e100</span></code> is <code class="docutils literal notranslate"><span class="pre">1e100-1e-100</span></code>, which cannot be

  283. represented exactly as a float, and rounds to the surprising <code class="docutils literal notranslate"><span class="pre">1e100</span></code>.  For

  284. this reason, function <a class="reference internal" href="#math.fmod" title="math.fmod"><code class="xref py py-func docutils literal notranslate"><span class="pre">fmod()</span></code></a> is generally preferred when working with

  285. floats, while Python’s <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> is preferred when working with integers.</p>

  286. </dd></dl>

  287. 

  288. <dl class="py function">

  289. <dt class="sig sig-object py" id="math.frexp">

  290. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">frexp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.frexp" title="Link to this definition">¶</a></dt>

  291. <dd><p>Return the mantissa and exponent of <em>x</em> as the pair <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">e)</span></code>.  <em>m</em> is a float

  292. and <em>e</em> is an integer such that <code class="docutils literal notranslate"><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,

  293. returns <code class="docutils literal notranslate"><span class="pre">(0.0,</span> <span class="pre">0)</span></code>, otherwise <code class="docutils literal notranslate"><span class="pre">0.5</span> <span class="pre">&lt;=</span> <span class="pre">abs(m)</span> <span class="pre">&lt;</span> <span class="pre">1</span></code>.  This is used to “pick

  294. apart” the internal representation of a float in a portable way.</p>

  295. </dd></dl>

  296. 

  297. <dl class="py function">

  298. <dt class="sig sig-object py" id="math.fsum">

  299. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">fsum</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">iterable</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.fsum" title="Link to this definition">¶</a></dt>

  300. <dd><p>Return an accurate floating point sum of values in the iterable.  Avoids

  301. loss of precision by tracking multiple intermediate partial sums.</p>

  302. <p>The algorithm’s accuracy depends on IEEE-754 arithmetic guarantees and the

  303. typical case where the rounding mode is half-even.  On some non-Windows

  304. builds, the underlying C library uses extended precision addition and may

  305. occasionally double-round an intermediate sum causing it to be off in its

  306. least significant bit.</p>

  307. <p>For further discussion and two alternative approaches, see the <a class="reference external" href="https://code.activestate.com/recipes/393090/">ASPN cookbook

  308. recipes for accurate floating point summation</a>.</p>

  309. </dd></dl>

  310. 

  311. <dl class="py function">

  312. <dt class="sig sig-object py" id="math.gcd">

  313. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">gcd</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">integers</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.gcd" title="Link to this definition">¶</a></dt>

  314. <dd><p>Return the greatest common divisor of the specified integer arguments.

  315. If any of the arguments is nonzero, then the returned value is the largest

  316. positive integer that is a divisor of all arguments.  If all arguments

  317. are zero, then the returned value is <code class="docutils literal notranslate"><span class="pre">0</span></code>.  <code class="docutils literal notranslate"><span class="pre">gcd()</span></code> without arguments

  318. returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p>

  319. <div class="versionadded">

  320. <p><span class="versionmodified added">New in version 3.5.</span></p>

  321. </div>

  322. <div class="versionchanged">

  323. <p><span class="versionmodified changed">Changed in version 3.9: </span>Added support for an arbitrary number of arguments. Formerly, only two

  324. arguments were supported.</p>

  325. </div>

  326. </dd></dl>

  327. 

  328. <dl class="py function">

  329. <dt class="sig sig-object py" id="math.isclose">

  330. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">isclose</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rel_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-09</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">abs_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.isclose" title="Link to this definition">¶</a></dt>

  331. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the values <em>a</em> and <em>b</em> are close to each other and

  332. <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  333. <p>Whether or not two values are considered close is determined according to

  334. given absolute and relative tolerances.</p>

  335. <p><em>rel_tol</em> is the relative tolerance – it is the maximum allowed difference

  336. between <em>a</em> and <em>b</em>, relative to the larger absolute value of <em>a</em> or <em>b</em>.

  337. For example, to set a tolerance of 5%, pass <code class="docutils literal notranslate"><span class="pre">rel_tol=0.05</span></code>.  The default

  338. tolerance is <code class="docutils literal notranslate"><span class="pre">1e-09</span></code>, which assures that the two values are the same

  339. within about 9 decimal digits.  <em>rel_tol</em> must be greater than zero.</p>

  340. <p><em>abs_tol</em> is the minimum absolute tolerance – useful for comparisons near

  341. zero. <em>abs_tol</em> must be at least zero.</p>

  342. <p>If no errors occur, the result will be:

  343. <code class="docutils literal notranslate"><span class="pre">abs(a-b)</span> <span class="pre">&lt;=</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>

  344. <p>The IEEE 754 special values of <code class="docutils literal notranslate"><span class="pre">NaN</span></code>, <code class="docutils literal notranslate"><span class="pre">inf</span></code>, and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> will be

  345. handled according to IEEE rules.  Specifically, <code class="docutils literal notranslate"><span class="pre">NaN</span></code> is not considered

  346. close to any other value, including <code class="docutils literal notranslate"><span class="pre">NaN</span></code>.  <code class="docutils literal notranslate"><span class="pre">inf</span></code> and <code class="docutils literal notranslate"><span class="pre">-inf</span></code> are only

  347. considered close to themselves.</p>

  348. <div class="versionadded">

  349. <p><span class="versionmodified added">New in version 3.5.</span></p>

  350. </div>

  351. <div class="admonition seealso">

  352. <p class="admonition-title">See also</p>

  353. <p><span class="target" id="index-0"></span><a class="pep reference external" href="https://peps.python.org/pep-0485/"><strong>PEP 485</strong></a> – A function for testing approximate equality</p>

  354. </div>

  355. </dd></dl>

  356. 

  357. <dl class="py function">

  358. <dt class="sig sig-object py" id="math.isfinite">

  359. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">isfinite</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.isfinite" title="Link to this definition">¶</a></dt>

  360. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if <em>x</em> is neither an infinity nor a NaN, and

  361. <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.  (Note that <code class="docutils literal notranslate"><span class="pre">0.0</span></code> <em>is</em> considered finite.)</p>

  362. <div class="versionadded">

  363. <p><span class="versionmodified added">New in version 3.2.</span></p>

  364. </div>

  365. </dd></dl>

  366. 

  367. <dl class="py function">

  368. <dt class="sig sig-object py" id="math.isinf">

  369. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">isinf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.isinf" title="Link to this definition">¶</a></dt>

  370. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if <em>x</em> is a positive or negative infinity, and

  371. <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  372. </dd></dl>

  373. 

  374. <dl class="py function">

  375. <dt class="sig sig-object py" id="math.isnan">

  376. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">isnan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.isnan" title="Link to this definition">¶</a></dt>

  377. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if <em>x</em> is a NaN (not a number), and <code class="docutils literal notranslate"><span class="pre">False</span></code> otherwise.</p>

  378. </dd></dl>

  379. 

  380. <dl class="py function">

  381. <dt class="sig sig-object py" id="math.isqrt">

  382. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">isqrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.isqrt" title="Link to this definition">¶</a></dt>

  383. <dd><p>Return the integer square root of the nonnegative integer <em>n</em>. This is the

  384. floor of the exact square root of <em>n</em>, or equivalently the greatest integer

  385. <em>a</em> such that <em>a</em>² ≤ <em>n</em>.</p>

  386. <p>For some applications, it may be more convenient to have the least integer

  387. <em>a</em> such that <em>n</em> ≤ <em>a</em>², or in other words the ceiling of

  388. the exact square root of <em>n</em>. For positive <em>n</em>, this can be computed using

  389. <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">1</span> <span class="pre">+</span> <span class="pre">isqrt(n</span> <span class="pre">-</span> <span class="pre">1)</span></code>.</p>

  390. <div class="versionadded">

  391. <p><span class="versionmodified added">New in version 3.8.</span></p>

  392. </div>

  393. </dd></dl>

  394. 

  395. <dl class="py function">

  396. <dt class="sig sig-object py" id="math.lcm">

  397. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">lcm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">integers</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.lcm" title="Link to this definition">¶</a></dt>

  398. <dd><p>Return the least common multiple of the specified integer arguments.

  399. If all arguments are nonzero, then the returned value is the smallest

  400. positive integer that is a multiple of all arguments.  If any of the arguments

  401. is zero, then the returned value is <code class="docutils literal notranslate"><span class="pre">0</span></code>.  <code class="docutils literal notranslate"><span class="pre">lcm()</span></code> without arguments

  402. returns <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p>

  403. <div class="versionadded">

  404. <p><span class="versionmodified added">New in version 3.9.</span></p>

  405. </div>

  406. </dd></dl>

  407. 

  408. <dl class="py function">

  409. <dt class="sig sig-object py" id="math.ldexp">

  410. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">ldexp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">i</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.ldexp" title="Link to this definition">¶</a></dt>

  411. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">*</span> <span class="pre">(2**i)</span></code>.  This is essentially the inverse of function

  412. <a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal notranslate"><span class="pre">frexp()</span></code></a>.</p>

  413. </dd></dl>

  414. 

  415. <dl class="py function">

  416. <dt class="sig sig-object py" id="math.modf">

  417. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">modf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.modf" title="Link to this definition">¶</a></dt>

  418. <dd><p>Return the fractional and integer parts of <em>x</em>.  Both results carry the sign

  419. of <em>x</em> and are floats.</p>

  420. </dd></dl>

  421. 

  422. <dl class="py function">

  423. <dt class="sig sig-object py" id="math.nextafter">

  424. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">nextafter</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">steps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.nextafter" title="Link to this definition">¶</a></dt>

  425. <dd><p>Return the floating-point value <em>steps</em> steps after <em>x</em> towards <em>y</em>.</p>

  426. <p>If <em>x</em> is equal to <em>y</em>, return <em>y</em>, unless <em>steps</em> is zero.</p>

  427. <p>Examples:</p>

  428. <ul class="simple">

  429. <li><p><code class="docutils literal notranslate"><span class="pre">math.nextafter(x,</span> <span class="pre">math.inf)</span></code> goes up: towards positive infinity.</p></li>

  430. <li><p><code class="docutils literal notranslate"><span class="pre">math.nextafter(x,</span> <span class="pre">-math.inf)</span></code> goes down: towards minus infinity.</p></li>

  431. <li><p><code class="docutils literal notranslate"><span class="pre">math.nextafter(x,</span> <span class="pre">0.0)</span></code> goes towards zero.</p></li>

  432. <li><p><code class="docutils literal notranslate"><span class="pre">math.nextafter(x,</span> <span class="pre">math.copysign(math.inf,</span> <span class="pre">x))</span></code> goes away from zero.</p></li>

  433. </ul>

  434. <p>See also <a class="reference internal" href="#math.ulp" title="math.ulp"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.ulp()</span></code></a>.</p>

  435. <div class="versionadded">

  436. <p><span class="versionmodified added">New in version 3.9.</span></p>

  437. </div>

  438. <div class="versionchanged">

  439. <p><span class="versionmodified changed">Changed in version 3.12: </span>Added the <em>steps</em> argument.</p>

  440. </div>

  441. </dd></dl>

  442. 

  443. <dl class="py function">

  444. <dt class="sig sig-object py" id="math.perm">

  445. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">perm</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.perm" title="Link to this definition">¶</a></dt>

  446. <dd><p>Return the number of ways to choose <em>k</em> items from <em>n</em> items

  447. without repetition and with order.</p>

  448. <p>Evaluates to <code class="docutils literal notranslate"><span class="pre">n!</span> <span class="pre">/</span> <span class="pre">(n</span> <span class="pre">-</span> <span class="pre">k)!</span></code> when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&lt;=</span> <span class="pre">n</span></code> and evaluates

  449. to zero when <code class="docutils literal notranslate"><span class="pre">k</span> <span class="pre">&gt;</span> <span class="pre">n</span></code>.</p>

  450. <p>If <em>k</em> is not specified or is None, then <em>k</em> defaults to <em>n</em>

  451. and the function returns <code class="docutils literal notranslate"><span class="pre">n!</span></code>.</p>

  452. <p>Raises <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a> if either of the arguments are not integers.

  453. Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if either of the arguments are negative.</p>

  454. <div class="versionadded">

  455. <p><span class="versionmodified added">New in version 3.8.</span></p>

  456. </div>

  457. </dd></dl>

  458. 

  459. <dl class="py function">

  460. <dt class="sig sig-object py" id="math.prod">

  461. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">prod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">iterable</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">start</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.prod" title="Link to this definition">¶</a></dt>

  462. <dd><p>Calculate the product of all the elements in the input <em>iterable</em>.

  463. The default <em>start</em> value for the product is <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p>

  464. <p>When the iterable is empty, return the start value.  This function is

  465. intended specifically for use with numeric values and may reject

  466. non-numeric types.</p>

  467. <div class="versionadded">

  468. <p><span class="versionmodified added">New in version 3.8.</span></p>

  469. </div>

  470. </dd></dl>

  471. 

  472. <dl class="py function">

  473. <dt class="sig sig-object py" id="math.remainder">

  474. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">remainder</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.remainder" title="Link to this definition">¶</a></dt>

  475. <dd><p>Return the IEEE 754-style remainder of <em>x</em> with respect to <em>y</em>.  For

  476. finite <em>x</em> and finite nonzero <em>y</em>, this is the difference <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">n*y</span></code>,

  477. where <code class="docutils literal notranslate"><span class="pre">n</span></code> is the closest integer to the exact value of the quotient <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span>

  478. <span class="pre">y</span></code>.  If <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code> is exactly halfway between two consecutive integers, the

  479. nearest <em>even</em> integer is used for <code class="docutils literal notranslate"><span class="pre">n</span></code>.  The remainder <code class="docutils literal notranslate"><span class="pre">r</span> <span class="pre">=</span> <span class="pre">remainder(x,</span>

  480. <span class="pre">y)</span></code> thus always satisfies <code class="docutils literal notranslate"><span class="pre">abs(r)</span> <span class="pre">&lt;=</span> <span class="pre">0.5</span> <span class="pre">*</span> <span class="pre">abs(y)</span></code>.</p>

  481. <p>Special cases follow IEEE 754: in particular, <code class="docutils literal notranslate"><span class="pre">remainder(x,</span> <span class="pre">math.inf)</span></code> is

  482. <em>x</em> for any finite <em>x</em>, and <code class="docutils literal notranslate"><span class="pre">remainder(x,</span> <span class="pre">0)</span></code> and

  483. <code class="docutils literal notranslate"><span class="pre">remainder(math.inf,</span> <span class="pre">x)</span></code> raise <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> for any non-NaN <em>x</em>.

  484. If the result of the remainder operation is zero, that zero will have

  485. the same sign as <em>x</em>.</p>

  486. <p>On platforms using IEEE 754 binary floating-point, the result of this

  487. operation is always exactly representable: no rounding error is introduced.</p>

  488. <div class="versionadded">

  489. <p><span class="versionmodified added">New in version 3.7.</span></p>

  490. </div>

  491. </dd></dl>

  492. 

  493. <dl class="py function">

  494. <dt class="sig sig-object py" id="math.sumprod">

  495. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">sumprod</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">q</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.sumprod" title="Link to this definition">¶</a></dt>

  496. <dd><p>Return the sum of products of values from two iterables <em>p</em> and <em>q</em>.</p>

  497. <p>Raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if the inputs do not have the same length.</p>

  498. <p>Roughly equivalent to:</p>

  499. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="nb">sum</span><span class="p">(</span><span class="n">itertools</span><span class="o">.</span><span class="n">starmap</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">mul</span><span class="p">,</span> <span class="nb">zip</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="kc">True</span><span class="p">)))</span>

  500. </pre></div>

  501. </div>

  502. <p>For float and mixed int/float inputs, the intermediate products

  503. and sums are computed with extended precision.</p>

  504. <div class="versionadded">

  505. <p><span class="versionmodified added">New in version 3.12.</span></p>

  506. </div>

  507. </dd></dl>

  508. 

  509. <dl class="py function">

  510. <dt class="sig sig-object py" id="math.trunc">

  511. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">trunc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.trunc" title="Link to this definition">¶</a></dt>

  512. <dd><p>Return <em>x</em> with the fractional part

  513. removed, leaving the integer part.  This rounds toward 0: <code class="docutils literal notranslate"><span class="pre">trunc()</span></code> is

  514. equivalent to <a class="reference internal" href="#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate"><span class="pre">floor()</span></code></a> for positive <em>x</em>, and equivalent to <a class="reference internal" href="#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate"><span class="pre">ceil()</span></code></a>

  515. for negative <em>x</em>. If <em>x</em> is not a float, delegates to <a class="reference internal" href="../reference/datamodel.html#object.__trunc__" title="object.__trunc__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">x.__trunc__</span></code></a>, which should return an <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">Integral</span></code></a> value.</p>

  516. </dd></dl>

  517. 

  518. <dl class="py function">

  519. <dt class="sig sig-object py" id="math.ulp">

  520. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">ulp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.ulp" title="Link to this definition">¶</a></dt>

  521. <dd><p>Return the value of the least significant bit of the float <em>x</em>:</p>

  522. <ul class="simple">

  523. <li><p>If <em>x</em> is a NaN (not a number), return <em>x</em>.</p></li>

  524. <li><p>If <em>x</em> is negative, return <code class="docutils literal notranslate"><span class="pre">ulp(-x)</span></code>.</p></li>

  525. <li><p>If <em>x</em> is a positive infinity, return <em>x</em>.</p></li>

  526. <li><p>If <em>x</em> is equal to zero, return the smallest positive

  527. <em>denormalized</em> representable float (smaller than the minimum positive

  528. <em>normalized</em> float, <a class="reference internal" href="sys.html#sys.float_info" title="sys.float_info"><code class="xref py py-data docutils literal notranslate"><span class="pre">sys.float_info.min</span></code></a>).</p></li>

  529. <li><p>If <em>x</em> is equal to the largest positive representable float,

  530. return the value of the least significant bit of <em>x</em>, such that the first

  531. float smaller than <em>x</em> is <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">ulp(x)</span></code>.</p></li>

  532. <li><p>Otherwise (<em>x</em> is a positive finite number), return the value of the least

  533. significant bit of <em>x</em>, such that the first float bigger than <em>x</em>

  534. is <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">ulp(x)</span></code>.</p></li>

  535. </ul>

  536. <p>ULP stands for “Unit in the Last Place”.</p>

  537. <p>See also <a class="reference internal" href="#math.nextafter" title="math.nextafter"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.nextafter()</span></code></a> and <a class="reference internal" href="sys.html#sys.float_info" title="sys.float_info"><code class="xref py py-data docutils literal notranslate"><span class="pre">sys.float_info.epsilon</span></code></a>.</p>

  538. <div class="versionadded">

  539. <p><span class="versionmodified added">New in version 3.9.</span></p>

  540. </div>

  541. </dd></dl>

  542. 

  543. <p>Note that <a class="reference internal" href="#math.frexp" title="math.frexp"><code class="xref py py-func docutils literal notranslate"><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 notranslate"><span class="pre">modf()</span></code></a> have a different call/return pattern

  544. than their C equivalents: they take a single argument and return a pair of

  545. values, rather than returning their second return value through an ‘output

  546. parameter’ (there is no such thing in Python).</p>

  547. <p>For the <a class="reference internal" href="#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate"><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 notranslate"><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 notranslate"><span class="pre">modf()</span></code></a> functions, note that <em>all</em>

  548. floating-point numbers of sufficiently large magnitude are exact integers.

  549. Python floats typically carry no more than 53 bits of precision (the same as the

  550. platform C double type), in which case any float <em>x</em> with <code class="docutils literal notranslate"><span class="pre">abs(x)</span> <span class="pre">&gt;=</span> <span class="pre">2**52</span></code>

  551. necessarily has no fractional bits.</p>

  552. </section>

  553. <section id="power-and-logarithmic-functions">

  554. <h2>Power and logarithmic functions<a class="headerlink" href="#power-and-logarithmic-functions" title="Link to this heading">¶</a></h2>

  555. <dl class="py function">

  556. <dt class="sig sig-object py" id="math.cbrt">

  557. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">cbrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.cbrt" title="Link to this definition">¶</a></dt>

  558. <dd><p>Return the cube root of <em>x</em>.</p>

  559. <div class="versionadded">

  560. <p><span class="versionmodified added">New in version 3.11.</span></p>

  561. </div>

  562. </dd></dl>

  563. 

  564. <dl class="py function">

  565. <dt class="sig sig-object py" id="math.exp">

  566. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">exp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.exp" title="Link to this definition">¶</a></dt>

  567. <dd><p>Return <em>e</em> raised to the power <em>x</em>, where <em>e</em> = 2.718281… is the base

  568. of natural logarithms.  This is usually more accurate than <code class="docutils literal notranslate"><span class="pre">math.e</span> <span class="pre">**</span> <span class="pre">x</span></code>

  569. or <code class="docutils literal notranslate"><span class="pre">pow(math.e,</span> <span class="pre">x)</span></code>.</p>

  570. </dd></dl>

  571. 

  572. <dl class="py function">

  573. <dt class="sig sig-object py" id="math.exp2">

  574. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">exp2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.exp2" title="Link to this definition">¶</a></dt>

  575. <dd><p>Return <em>2</em> raised to the power <em>x</em>.</p>

  576. <div class="versionadded">

  577. <p><span class="versionmodified added">New in version 3.11.</span></p>

  578. </div>

  579. </dd></dl>

  580. 

  581. <dl class="py function">

  582. <dt class="sig sig-object py" id="math.expm1">

  583. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">expm1</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.expm1" title="Link to this definition">¶</a></dt>

  584. <dd><p>Return <em>e</em> raised to the power <em>x</em>, minus 1.  Here <em>e</em> is the base of natural

  585. logarithms.  For small floats <em>x</em>, the subtraction in <code class="docutils literal notranslate"><span class="pre">exp(x)</span> <span class="pre">-</span> <span class="pre">1</span></code>

  586. 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 notranslate"><span class="pre">expm1()</span></code></a>

  587. function provides a way to compute this quantity to full precision:</p>

  588. <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">expm1</span>

  589. <span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mf">1e-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>

  590. <span class="go">1.0000050000069649e-05</span>

  591. <span class="gp">&gt;&gt;&gt; </span><span class="n">expm1</span><span class="p">(</span><span class="mf">1e-5</span><span class="p">)</span>    <span class="c1"># result accurate to full precision</span>

  592. <span class="go">1.0000050000166668e-05</span>

  593. </pre></div>

  594. </div>

  595. <div class="versionadded">

  596. <p><span class="versionmodified added">New in version 3.2.</span></p>

  597. </div>

  598. </dd></dl>

  599. 

  600. <dl class="py function">

  601. <dt class="sig sig-object py" id="math.log">

  602. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">log</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="optional">[</span>, <em class="sig-param"><span class="n"><span class="pre">base</span></span></em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#math.log" title="Link to this definition">¶</a></dt>

  603. <dd><p>With one argument, return the natural logarithm of <em>x</em> (to base <em>e</em>).</p>

  604. <p>With two arguments, return the logarithm of <em>x</em> to the given <em>base</em>,

  605. calculated as <code class="docutils literal notranslate"><span class="pre">log(x)/log(base)</span></code>.</p>

  606. </dd></dl>

  607. 

  608. <dl class="py function">

  609. <dt class="sig sig-object py" id="math.log1p">

  610. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">log1p</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.log1p" title="Link to this definition">¶</a></dt>

  611. <dd><p>Return the natural logarithm of <em>1+x</em> (base <em>e</em>). The

  612. result is calculated in a way which is accurate for <em>x</em> near zero.</p>

  613. </dd></dl>

  614. 

  615. <dl class="py function">

  616. <dt class="sig sig-object py" id="math.log2">

  617. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">log2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.log2" title="Link to this definition">¶</a></dt>

  618. <dd><p>Return the base-2 logarithm of <em>x</em>. This is usually more accurate than

  619. <code class="docutils literal notranslate"><span class="pre">log(x,</span> <span class="pre">2)</span></code>.</p>

  620. <div class="versionadded">

  621. <p><span class="versionmodified added">New in version 3.3.</span></p>

  622. </div>

  623. <div class="admonition seealso">

  624. <p class="admonition-title">See also</p>

  625. <p><a class="reference internal" href="stdtypes.html#int.bit_length" title="int.bit_length"><code class="xref py py-meth docutils literal notranslate"><span class="pre">int.bit_length()</span></code></a> returns the number of bits necessary to represent

  626. an integer in binary, excluding the sign and leading zeros.</p>

  627. </div>

  628. </dd></dl>

  629. 

  630. <dl class="py function">

  631. <dt class="sig sig-object py" id="math.log10">

  632. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">log10</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.log10" title="Link to this definition">¶</a></dt>

  633. <dd><p>Return the base-10 logarithm of <em>x</em>.  This is usually more accurate

  634. than <code class="docutils literal notranslate"><span class="pre">log(x,</span> <span class="pre">10)</span></code>.</p>

  635. </dd></dl>

  636. 

  637. <dl class="py function">

  638. <dt class="sig sig-object py" id="math.pow">

  639. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">pow</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.pow" title="Link to this definition">¶</a></dt>

  640. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">x</span></code> raised to the power <code class="docutils literal notranslate"><span class="pre">y</span></code>.  Exceptional cases follow

  641. the IEEE 754 standard as far as possible.  In particular,

  642. <code class="docutils literal notranslate"><span class="pre">pow(1.0,</span> <span class="pre">x)</span></code> and <code class="docutils literal notranslate"><span class="pre">pow(x,</span> <span class="pre">0.0)</span></code> always return <code class="docutils literal notranslate"><span class="pre">1.0</span></code>, even

  643. when <code class="docutils literal notranslate"><span class="pre">x</span></code> is a zero or a NaN.  If both <code class="docutils literal notranslate"><span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> are finite,

  644. <code class="docutils literal notranslate"><span class="pre">x</span></code> is negative, and <code class="docutils literal notranslate"><span class="pre">y</span></code> is not an integer then <code class="docutils literal notranslate"><span class="pre">pow(x,</span> <span class="pre">y)</span></code>

  645. is undefined, and raises <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a>.</p>

  646. <p>Unlike the built-in <code class="docutils literal notranslate"><span class="pre">**</span></code> operator, <a class="reference internal" href="#math.pow" title="math.pow"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.pow()</span></code></a> converts both

  647. its arguments to type <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.  Use <code class="docutils literal notranslate"><span class="pre">**</span></code> or the built-in

  648. <a class="reference internal" href="functions.html#pow" title="pow"><code class="xref py py-func docutils literal notranslate"><span class="pre">pow()</span></code></a> function for computing exact integer powers.</p>

  649. <div class="versionchanged">

  650. <p><span class="versionmodified changed">Changed in version 3.11: </span>The special cases <code class="docutils literal notranslate"><span class="pre">pow(0.0,</span> <span class="pre">-inf)</span></code> and <code class="docutils literal notranslate"><span class="pre">pow(-0.0,</span> <span class="pre">-inf)</span></code> were

  651. changed to return <code class="docutils literal notranslate"><span class="pre">inf</span></code> instead of raising <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a>,

  652. for consistency with IEEE 754.</p>

  653. </div>

  654. </dd></dl>

  655. 

  656. <dl class="py function">

  657. <dt class="sig sig-object py" id="math.sqrt">

  658. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">sqrt</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.sqrt" title="Link to this definition">¶</a></dt>

  659. <dd><p>Return the square root of <em>x</em>.</p>

  660. </dd></dl>

  661. 

  662. </section>

  663. <section id="trigonometric-functions">

  664. <h2>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">¶</a></h2>

  665. <dl class="py function">

  666. <dt class="sig sig-object py" id="math.acos">

  667. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">acos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.acos" title="Link to this definition">¶</a></dt>

  668. <dd><p>Return the arc cosine of <em>x</em>, in radians. The result is between <code class="docutils literal notranslate"><span class="pre">0</span></code> and

  669. <code class="docutils literal notranslate"><span class="pre">pi</span></code>.</p>

  670. </dd></dl>

  671. 

  672. <dl class="py function">

  673. <dt class="sig sig-object py" id="math.asin">

  674. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">asin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.asin" title="Link to this definition">¶</a></dt>

  675. <dd><p>Return the arc sine of <em>x</em>, in radians. The result is between <code class="docutils literal notranslate"><span class="pre">-pi/2</span></code> and

  676. <code class="docutils literal notranslate"><span class="pre">pi/2</span></code>.</p>

  677. </dd></dl>

  678. 

  679. <dl class="py function">

  680. <dt class="sig sig-object py" id="math.atan">

  681. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">atan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.atan" title="Link to this definition">¶</a></dt>

  682. <dd><p>Return the arc tangent of <em>x</em>, in radians. The result is between <code class="docutils literal notranslate"><span class="pre">-pi/2</span></code> and

  683. <code class="docutils literal notranslate"><span class="pre">pi/2</span></code>.</p>

  684. </dd></dl>

  685. 

  686. <dl class="py function">

  687. <dt class="sig sig-object py" id="math.atan2">

  688. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">atan2</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.atan2" title="Link to this definition">¶</a></dt>

  689. <dd><p>Return <code class="docutils literal notranslate"><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 notranslate"><span class="pre">-pi</span></code> and <code class="docutils literal notranslate"><span class="pre">pi</span></code>.

  690. The vector in the plane from the origin to point <code class="docutils literal notranslate"><span class="pre">(x,</span> <span class="pre">y)</span></code> makes this angle

  691. 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 notranslate"><span class="pre">atan2()</span></code></a> is that the signs of both

  692. inputs are known to it, so it can compute the correct quadrant for the angle.

  693. For example, <code class="docutils literal notranslate"><span class="pre">atan(1)</span></code> and <code class="docutils literal notranslate"><span class="pre">atan2(1,</span> <span class="pre">1)</span></code> are both <code class="docutils literal notranslate"><span class="pre">pi/4</span></code>, but <code class="docutils literal notranslate"><span class="pre">atan2(-1,</span>

  694. <span class="pre">-1)</span></code> is <code class="docutils literal notranslate"><span class="pre">-3*pi/4</span></code>.</p>

  695. </dd></dl>

  696. 

  697. <dl class="py function">

  698. <dt class="sig sig-object py" id="math.cos">

  699. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">cos</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.cos" title="Link to this definition">¶</a></dt>

  700. <dd><p>Return the cosine of <em>x</em> radians.</p>

  701. </dd></dl>

  702. 

  703. <dl class="py function">

  704. <dt class="sig sig-object py" id="math.dist">

  705. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">dist</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">q</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.dist" title="Link to this definition">¶</a></dt>

  706. <dd><p>Return the Euclidean distance between two points <em>p</em> and <em>q</em>, each

  707. given as a sequence (or iterable) of coordinates.  The two points

  708. must have the same dimension.</p>

  709. <p>Roughly equivalent to:</p>

  710. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="n">sqrt</span><span class="p">(</span><span class="nb">sum</span><span class="p">((</span><span class="n">px</span> <span class="o">-</span> <span class="n">qx</span><span class="p">)</span> <span class="o">**</span> <span class="mf">2.0</span> <span class="k">for</span> <span class="n">px</span><span class="p">,</span> <span class="n">qx</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)))</span>

  711. </pre></div>

  712. </div>

  713. <div class="versionadded">

  714. <p><span class="versionmodified added">New in version 3.8.</span></p>

  715. </div>

  716. </dd></dl>

  717. 

  718. <dl class="py function">

  719. <dt class="sig sig-object py" id="math.hypot">

  720. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">hypot</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">coordinates</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.hypot" title="Link to this definition">¶</a></dt>

  721. <dd><p>Return the Euclidean norm, <code class="docutils literal notranslate"><span class="pre">sqrt(sum(x**2</span> <span class="pre">for</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">coordinates))</span></code>.

  722. This is the length of the vector from the origin to the point

  723. given by the coordinates.</p>

  724. <p>For a two dimensional point <code class="docutils literal notranslate"><span class="pre">(x,</span> <span class="pre">y)</span></code>, this is equivalent to computing

  725. the hypotenuse of a right triangle using the Pythagorean theorem,

  726. <code class="docutils literal notranslate"><span class="pre">sqrt(x*x</span> <span class="pre">+</span> <span class="pre">y*y)</span></code>.</p>

  727. <div class="versionchanged">

  728. <p><span class="versionmodified changed">Changed in version 3.8: </span>Added support for n-dimensional points. Formerly, only the two

  729. dimensional case was supported.</p>

  730. </div>

  731. <div class="versionchanged">

  732. <p><span class="versionmodified changed">Changed in version 3.10: </span>Improved the algorithm’s accuracy so that the maximum error is

  733. under 1 ulp (unit in the last place).  More typically, the result

  734. is almost always correctly rounded to within 1/2 ulp.</p>

  735. </div>

  736. </dd></dl>

  737. 

  738. <dl class="py function">

  739. <dt class="sig sig-object py" id="math.sin">

  740. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">sin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.sin" title="Link to this definition">¶</a></dt>

  741. <dd><p>Return the sine of <em>x</em> radians.</p>

  742. </dd></dl>

  743. 

  744. <dl class="py function">

  745. <dt class="sig sig-object py" id="math.tan">

  746. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">tan</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.tan" title="Link to this definition">¶</a></dt>

  747. <dd><p>Return the tangent of <em>x</em> radians.</p>

  748. </dd></dl>

  749. 

  750. </section>

  751. <section id="angular-conversion">

  752. <h2>Angular conversion<a class="headerlink" href="#angular-conversion" title="Link to this heading">¶</a></h2>

  753. <dl class="py function">

  754. <dt class="sig sig-object py" id="math.degrees">

  755. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">degrees</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.degrees" title="Link to this definition">¶</a></dt>

  756. <dd><p>Convert angle <em>x</em> from radians to degrees.</p>

  757. </dd></dl>

  758. 

  759. <dl class="py function">

  760. <dt class="sig sig-object py" id="math.radians">

  761. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">radians</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.radians" title="Link to this definition">¶</a></dt>

  762. <dd><p>Convert angle <em>x</em> from degrees to radians.</p>

  763. </dd></dl>

  764. 

  765. </section>

  766. <section id="hyperbolic-functions">

  767. <h2>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Link to this heading">¶</a></h2>

  768. <p><a class="reference external" href="https://en.wikipedia.org/wiki/Hyperbolic_functions">Hyperbolic functions</a>

  769. are analogs of trigonometric functions that are based on hyperbolas

  770. instead of circles.</p>

  771. <dl class="py function">

  772. <dt class="sig sig-object py" id="math.acosh">

  773. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">acosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.acosh" title="Link to this definition">¶</a></dt>

  774. <dd><p>Return the inverse hyperbolic cosine of <em>x</em>.</p>

  775. </dd></dl>

  776. 

  777. <dl class="py function">

  778. <dt class="sig sig-object py" id="math.asinh">

  779. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">asinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.asinh" title="Link to this definition">¶</a></dt>

  780. <dd><p>Return the inverse hyperbolic sine of <em>x</em>.</p>

  781. </dd></dl>

  782. 

  783. <dl class="py function">

  784. <dt class="sig sig-object py" id="math.atanh">

  785. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">atanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.atanh" title="Link to this definition">¶</a></dt>

  786. <dd><p>Return the inverse hyperbolic tangent of <em>x</em>.</p>

  787. </dd></dl>

  788. 

  789. <dl class="py function">

  790. <dt class="sig sig-object py" id="math.cosh">

  791. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">cosh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.cosh" title="Link to this definition">¶</a></dt>

  792. <dd><p>Return the hyperbolic cosine of <em>x</em>.</p>

  793. </dd></dl>

  794. 

  795. <dl class="py function">

  796. <dt class="sig sig-object py" id="math.sinh">

  797. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">sinh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.sinh" title="Link to this definition">¶</a></dt>

  798. <dd><p>Return the hyperbolic sine of <em>x</em>.</p>

  799. </dd></dl>

  800. 

  801. <dl class="py function">

  802. <dt class="sig sig-object py" id="math.tanh">

  803. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">tanh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.tanh" title="Link to this definition">¶</a></dt>

  804. <dd><p>Return the hyperbolic tangent of <em>x</em>.</p>

  805. </dd></dl>

  806. 

  807. </section>

  808. <section id="special-functions">

  809. <h2>Special functions<a class="headerlink" href="#special-functions" title="Link to this heading">¶</a></h2>

  810. <dl class="py function">

  811. <dt class="sig sig-object py" id="math.erf">

  812. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">erf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.erf" title="Link to this definition">¶</a></dt>

  813. <dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Error_function">error function</a> at

  814. <em>x</em>.</p>

  815. <p>The <a class="reference internal" href="#math.erf" title="math.erf"><code class="xref py py-func docutils literal notranslate"><span class="pre">erf()</span></code></a> function can be used to compute traditional statistical

  816. functions such as the <a class="reference external" href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">cumulative standard normal distribution</a>:</p>

  817. <div class="highlight-python3 notranslate"><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>

  818.     <span class="s1">&#39;Cumulative distribution function for the standard normal distribution&#39;</span>

  819.     <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>

  820. </pre></div>

  821. </div>

  822. <div class="versionadded">

  823. <p><span class="versionmodified added">New in version 3.2.</span></p>

  824. </div>

  825. </dd></dl>

  826. 

  827. <dl class="py function">

  828. <dt class="sig sig-object py" id="math.erfc">

  829. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">erfc</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.erfc" title="Link to this definition">¶</a></dt>

  830. <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

  831. function</a> is defined as

  832. <code class="docutils literal notranslate"><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

  833. from one would cause a <a class="reference external" href="https://en.wikipedia.org/wiki/Loss_of_significance">loss of significance</a>.</p>

  834. <div class="versionadded">

  835. <p><span class="versionmodified added">New in version 3.2.</span></p>

  836. </div>

  837. </dd></dl>

  838. 

  839. <dl class="py function">

  840. <dt class="sig sig-object py" id="math.gamma">

  841. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">gamma</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.gamma" title="Link to this definition">¶</a></dt>

  842. <dd><p>Return the <a class="reference external" href="https://en.wikipedia.org/wiki/Gamma_function">Gamma function</a> at

  843. <em>x</em>.</p>

  844. <div class="versionadded">

  845. <p><span class="versionmodified added">New in version 3.2.</span></p>

  846. </div>

  847. </dd></dl>

  848. 

  849. <dl class="py function">

  850. <dt class="sig sig-object py" id="math.lgamma">

  851. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">lgamma</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#math.lgamma" title="Link to this definition">¶</a></dt>

  852. <dd><p>Return the natural logarithm of the absolute value of the Gamma

  853. function at <em>x</em>.</p>

  854. <div class="versionadded">

  855. <p><span class="versionmodified added">New in version 3.2.</span></p>

  856. </div>

  857. </dd></dl>

  858. 

  859. </section>

  860. <section id="constants">

  861. <h2>Constants<a class="headerlink" href="#constants" title="Link to this heading">¶</a></h2>

  862. <dl class="py data">

  863. <dt class="sig sig-object py" id="math.pi">

  864. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">pi</span></span><a class="headerlink" href="#math.pi" title="Link to this definition">¶</a></dt>

  865. <dd><p>The mathematical constant <em>π</em> = 3.141592…, to available precision.</p>

  866. </dd></dl>

  867. 

  868. <dl class="py data">

  869. <dt class="sig sig-object py" id="math.e">

  870. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">e</span></span><a class="headerlink" href="#math.e" title="Link to this definition">¶</a></dt>

  871. <dd><p>The mathematical constant <em>e</em> = 2.718281…, to available precision.</p>

  872. </dd></dl>

  873. 

  874. <dl class="py data">

  875. <dt class="sig sig-object py" id="math.tau">

  876. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">tau</span></span><a class="headerlink" href="#math.tau" title="Link to this definition">¶</a></dt>

  877. <dd><p>The mathematical constant <em>τ</em> = 6.283185…, to available precision.

  878. Tau is a circle constant equal to 2<em>π</em>, the ratio of a circle’s circumference to

  879. its radius. To learn more about Tau, check out Vi Hart’s video <a class="reference external" href="https://www.youtube.com/watch?v=jG7vhMMXagQ">Pi is (still)

  880. Wrong</a>, and start celebrating

  881. <a class="reference external" href="https://tauday.com/">Tau day</a> by eating twice as much pie!</p>

  882. <div class="versionadded">

  883. <p><span class="versionmodified added">New in version 3.6.</span></p>

  884. </div>

  885. </dd></dl>

  886. 

  887. <dl class="py data">

  888. <dt class="sig sig-object py" id="math.inf">

  889. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">inf</span></span><a class="headerlink" href="#math.inf" title="Link to this definition">¶</a></dt>

  890. <dd><p>A floating-point positive infinity.  (For negative infinity, use

  891. <code class="docutils literal notranslate"><span class="pre">-math.inf</span></code>.)  Equivalent to the output of <code class="docutils literal notranslate"><span class="pre">float('inf')</span></code>.</p>

  892. <div class="versionadded">

  893. <p><span class="versionmodified added">New in version 3.5.</span></p>

  894. </div>

  895. </dd></dl>

  896. 

  897. <dl class="py data">

  898. <dt class="sig sig-object py" id="math.nan">

  899. <span class="sig-prename descclassname"><span class="pre">math.</span></span><span class="sig-name descname"><span class="pre">nan</span></span><a class="headerlink" href="#math.nan" title="Link to this definition">¶</a></dt>

  900. <dd><p>A floating-point “not a number” (NaN) value. Equivalent to the output of

  901. <code class="docutils literal notranslate"><span class="pre">float('nan')</span></code>. Due to the requirements of the <a class="reference external" href="https://en.wikipedia.org/wiki/IEEE_754">IEEE-754 standard</a>, <code class="docutils literal notranslate"><span class="pre">math.nan</span></code> and <code class="docutils literal notranslate"><span class="pre">float('nan')</span></code> are

  902. not considered to equal to any other numeric value, including themselves. To check

  903. whether a number is a NaN, use the <a class="reference internal" href="#math.isnan" title="math.isnan"><code class="xref py py-func docutils literal notranslate"><span class="pre">isnan()</span></code></a> function to test

  904. for NaNs instead of <code class="docutils literal notranslate"><span class="pre">is</span></code> or <code class="docutils literal notranslate"><span class="pre">==</span></code>.

  905. Example:</p>

  906. <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">math</span>

  907. <span class="gp">&gt;&gt;&gt; </span><span class="n">math</span><span class="o">.</span><span class="n">nan</span> <span class="o">==</span> <span class="n">math</span><span class="o">.</span><span class="n">nan</span>

  908. <span class="go">False</span>

  909. <span class="gp">&gt;&gt;&gt; </span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">)</span>

  910. <span class="go">False</span>

  911. <span class="gp">&gt;&gt;&gt; </span><span class="n">math</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">nan</span><span class="p">)</span>

  912. <span class="go">True</span>

  913. <span class="gp">&gt;&gt;&gt; </span><span class="n">math</span><span class="o">.</span><span class="n">isnan</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">))</span>

  914. <span class="go">True</span>

  915. </pre></div>

  916. </div>

  917. <div class="versionadded">

  918. <p><span class="versionmodified added">New in version 3.5.</span></p>

  919. </div>

  920. <div class="versionchanged">

  921. <p><span class="versionmodified changed">Changed in version 3.11: </span>It is now always available.</p>

  922. </div>

  923. </dd></dl>

  924. 

  925. <div class="impl-detail compound">

  926. <p><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 notranslate"><span class="pre">math</span></code></a> module consists mostly of thin wrappers around the platform C

  927. math library functions.  Behavior in exceptional cases follows Annex F of

  928. the C99 standard where appropriate.  The current implementation will raise

  929. <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> for invalid operations like <code class="docutils literal notranslate"><span class="pre">sqrt(-1.0)</span></code> or <code class="docutils literal notranslate"><span class="pre">log(0.0)</span></code>

  930. (where C99 Annex F recommends signaling invalid operation or divide-by-zero),

  931. and <a class="reference internal" href="exceptions.html#OverflowError" title="OverflowError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">OverflowError</span></code></a> for results that overflow (for example,

  932. <code class="docutils literal notranslate"><span class="pre">exp(1000.0)</span></code>).  A NaN will not be returned from any of the functions

  933. above unless one or more of the input arguments was a NaN; in that case,

  934. most functions will return a NaN, but (again following C99 Annex F) there

  935. are some exceptions to this rule, for example <code class="docutils literal notranslate"><span class="pre">pow(float('nan'),</span> <span class="pre">0.0)</span></code> or

  936. <code class="docutils literal notranslate"><span class="pre">hypot(float('nan'),</span> <span class="pre">float('inf'))</span></code>.</p>

  937. <p>Note that Python makes no effort to distinguish signaling NaNs from

  938. quiet NaNs, and behavior for signaling NaNs remains unspecified.

  939. Typical behavior is to treat all NaNs as though they were quiet.</p>

  940. </div>

  941. <div class="admonition seealso">

  942. <p class="admonition-title">See also</p>

  943. <dl class="simple">

  944. <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 notranslate"><span class="pre">cmath</span></code></a></dt><dd><p>Complex number versions of many of these functions.</p>

  945. </dd>

  946. </dl>

  947. </div>

  948. </section>

  949. </section>

  950. 

  951. 

  952.             <div class="clearer"></div>

  953.           </div>

  954.         </div>

  955.       </div>

  956.       <div class="sphinxsidebar" role="navigation" aria-label="main navigation">

  957.         <div class="sphinxsidebarwrapper">

  958.   <div>

  959.     <h3><a href="../contents.html">Table of Contents</a></h3>

  960.     <ul>

  961. <li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">math</span></code> — Mathematical functions</a><ul>

  962. <li><a class="reference internal" href="#number-theoretic-and-representation-functions">Number-theoretic and representation functions</a></li>

  963. <li><a class="reference internal" href="#power-and-logarithmic-functions">Power and logarithmic functions</a></li>

  964. <li><a class="reference internal" href="#trigonometric-functions">Trigonometric functions</a></li>

  965. <li><a class="reference internal" href="#angular-conversion">Angular conversion</a></li>

  966. <li><a class="reference internal" href="#hyperbolic-functions">Hyperbolic functions</a></li>

  967. <li><a class="reference internal" href="#special-functions">Special functions</a></li>

  968. <li><a class="reference internal" href="#constants">Constants</a></li>

  969. </ul>

  970. </li>

  971. </ul>

  972. 

  973.   </div>

  974.   <div>

  975.     <h4>Previous topic</h4>

  976.     <p class="topless"><a href="numbers.html"

  977.                           title="previous chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> — Numeric abstract base classes</a></p>

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