shSrSSKrSSKJr SSKJr SSKJr SSKJ r J r SSK J r J r JrJrJrJrJrJrJrJrJrJrJr SSKJrJrJrJr SS KJrJ r J!r!J"r" SS KJ#r#J$r$J%r% SS K&J'r' /S Qr(\)"5r*\+"S S9 \ "\+S S9r,SSKJ.r/ Sr1STSjr2Sr3SUSjr4SUSjr5SUSjr6\74Sjr8Sr9\9r:Sr;SrSr?SSK J@rA Sr@\?R\@l"S S!\B5rCS"rDS#rESVS$jrFS%rGS&rHS'rISUS(jrJSUS)jrKSWS+jrLSUS,jrMSXS-jrNS.S/.S0jrOSUS1jrPS2rQS3rRS4rSS5rTS6rUS7rVS8rWS9rXS:rYS;rZS<r[S=r\SYS>jr]S?r^S*S.S@jr_\'SA:aSSBK J`ra S*S.SCjr`O\_r`\_R\`lSDrbSErcSFrdSGre\f"\^"SH55rgSIrhSJriSKrjSLrkSMrl/SNQrm\ SO5rnSPro\R"5RFrqSQrrSRrsSSrtg!\-a \+r,GN^f=f!\0a Sr/GNgf=f!\0a \?r@GN%f=f)ZaImported from the recipes section of the itertools documentation. All functions taken from the recipes section of the itertools library docs [1]_. Some backward-compatible usability improvements have been made. .. [1] http://docs.python.org/library/itertools.html#recipes N)dequesuppress)Sized) lru_cachepartial) accumulatechain combinationscompresscountcyclegroupbyisliceproductrepeatstarmaptee zip_longest)prodcombisqrtgcd)mulnot_ itemgettergetitem) randrangesamplechoice) hexversion)1 all_equalbatchedbefore_and_afterconsumeconvolve dotproduct first_truefactorflattengrouperis_prime iter_except iter_indexloopsmatmul multinomialncyclesnthnth_combinationpadnonepad_nonepairwise partitionpolynomial_evalpolynomial_from_rootspolynomial_derivativepowersetprependquantifyreshape#random_combination_with_replacementrandom_combinationrandom_permutationrandom_product repeatfunc roundrobinsievesliding_window subslicessum_of_squarestabulatetailtaketotient transpose triplewiseuniqueunique_everseenunique_justseenTstrict)sumprodc[X5$N)r')xys j/Users/tiagomarins/Projetos/claudeai/copy_bank/venv/lib/python3.13/site-packages/more_itertools/recipes.pyr[is Jq,c*[[X55$)zReturn first *n* items of the *iterable* as a list. >>> take(3, range(10)) [0, 1, 2] If there are fewer than *n* items in the iterable, all of them are returned. >>> take(10, range(3)) [0, 1, 2] )listrniterables rZrLrLls x# $$r\c,[U[U55$)aReturn an iterator over the results of ``func(start)``, ``func(start + 1)``, ``func(start + 2)``... *func* should be a function that accepts one integer argument. If *start* is not specified it defaults to 0. It will be incremented each time the iterator is advanced. >>> square = lambda x: x ** 2 >>> iterator = tabulate(square, -3) >>> take(4, iterator) [9, 4, 1, 0] )mapr )functionstarts rZrJrJ|s xu &&r\c [U[5(a#[U[S[ U5U- 5S5$[ [ XS95$)zsReturn an iterator over the last *n* items of *iterable*. >>> t = tail(3, 'ABCDEFG') >>> list(t) ['E', 'F', 'G'] rNmaxlen) isinstancerrmaxleniterrr_s rZrKrKs@(E""hAs8}q'8 94@@E(-..r\cLUc [USS9 g[[XU5S5 g)aAdvance *iterable* by *n* steps. If *n* is ``None``, consume it entirely. Efficiently exhausts an iterator without returning values. Defaults to consuming the whole iterator, but an optional second argument may be provided to limit consumption. >>> i = (x for x in range(10)) >>> next(i) 0 >>> consume(i, 3) >>> next(i) 4 >>> consume(i) >>> next(i) Traceback (most recent call last): File "", line 1, in StopIteration If the iterator has fewer items remaining than the provided limit, the whole iterator will be consumed. >>> i = (x for x in range(3)) >>> consume(i, 5) >>> next(i) Traceback (most recent call last): File "", line 1, in StopIteration Nrrg)rnextr)iteratorr`s rZr%r%s'@ y hq! VH #T*r\c.[[XS5U5$)zmReturns the nth item or a default value. >>> l = range(10) >>> nth(l, 3) 3 >>> nth(l, 20, "zebra") 'zebra' N)rnr)rar`defaults rZr3r3s xD)7 33r\c@[X5nUH nUHn g g g)aw Returns ``True`` if all the elements are equal to each other. >>> all_equal('aaaa') True >>> all_equal('aaab') False A function that accepts a single argument and returns a transformed version of each input item can be specified with *key*: >>> all_equal('AaaA', key=str.casefold) True >>> all_equal([1, 2, 3], key=lambda x: x < 10) True FT)r)rakeyrofirstseconds rZr"r"s-$x%HF r\c*[[X55$)zWReturn the how many times the predicate is true. >>> quantify([True, False, True]) 2 )sumrc)rapreds rZr>r>s s4" ##r\c,[U[S55$)zReturns the sequence of elements and then returns ``None`` indefinitely. >>> take(5, pad_none(range(3))) [0, 1, 2, None, None] Useful for emulating the behavior of the built-in :func:`map` function. See also :func:`padded`. N)r rras rZr6r6s 6$< ((r\cT[R"[[U5U55$)zjReturns the sequence elements *n* times >>> list(ncycles(["a", "b"], 3)) ['a', 'b', 'a', 'b', 'a', 'b'] )r from_iterablertuplerar`s rZr2r2 s    veHoq9 ::r\c4[[[X55$)zReturns the dot product of the two iterables. >>> dotproduct([10, 15, 12], [0.65, 0.80, 1.25]) 33.5 >>> 10 * 0.65 + 15 * 0.80 + 12 * 1.25 33.5 In Python 3.12 and later, use ``math.sumprod()`` instead. )rwrcr)vec1vec2s rZr'r's s3# $$r\c.[R"U5$)zReturn an iterator flattening one level of nesting in a list of lists. >>> list(flatten([[0, 1], [2, 3]])) [0, 1, 2, 3] See also :func:`collapse`, which can flatten multiple levels of nesting. )r r|) listOfListss rZr*r*!s   { ++r\c\Uc[U[U55$[U[X!55$)a Call *func* with *args* repeatedly, returning an iterable over the results. If *times* is specified, the iterable will terminate after that many repetitions: >>> from operator import add >>> times = 4 >>> args = 3, 5 >>> list(repeatfunc(add, times, *args)) [8, 8, 8, 8] If *times* is ``None`` the iterable will not terminate: >>> from random import randrange >>> times = None >>> args = 1, 11 >>> take(6, repeatfunc(randrange, times, *args)) # doctest:+SKIP [2, 4, 8, 1, 8, 4] )rr)functimesargss rZrDrD-s,, }tVD\** 4, --r\cJ[U5up[US5 [X5$)zReturns an iterator of paired items, overlapping, from the original >>> take(4, pairwise(count())) [(0, 1), (1, 2), (2, 3), (3, 4)] On Python 3.10 and above, this is an alias for :func:`itertools.pairwise`. Nrrnzip)raabs rZ _pairwiserHs" x=DADM q9r\)r7c[U5$rW)itertools_pairwiserzs rZr7r7\s !(++r\c,^\rSrSrSU4SjjrSrU=r$)UnequalIterablesErroribcR>SnUbUSR"U6- n[TU] U5 g)Nz Iterables have different lengthsz/: index 0 has length {}; index {} has length {})formatsuper__init__)selfdetailsmsg __class__s rZrUnequalIterablesError.__init__cs80   EMM C r\rW)__name__ __module__ __qualname____firstlineno__r__static_attributes__ __classcell__)rs@rZrrbs r\rc#x# [US[06H#nUHnU[LdM[5e Uv M% g7f)N fillvalue)r_markerr) iterablescombovals rZ_zip_equal_generatorrms:i;7;Cg~+-- >> list(grouper('ABCDEF', 3)) [('A', 'B', 'C'), ('D', 'E', 'F')] The keyword arguments *incomplete* and *fillvalue* control what happens for iterables whose length is not a multiple of *n*. When *incomplete* is `'fill'`, the last group will contain instances of *fillvalue*. >>> list(grouper('ABCDEFG', 3, incomplete='fill', fillvalue='x')) [('A', 'B', 'C'), ('D', 'E', 'F'), ('G', 'x', 'x')] When *incomplete* is `'ignore'`, the last group will not be emitted. >>> list(grouper('ABCDEFG', 3, incomplete='ignore', fillvalue='x')) [('A', 'B', 'C'), ('D', 'E', 'F')] When *incomplete* is `'strict'`, a subclass of `ValueError` will be raised. >>> iterator = grouper('ABCDEFG', 3, incomplete='strict') >>> list(iterator) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... UnequalIterablesError fillrrTignorez Expected fill, strict, or ignore)rlrrr ValueError)rar` incompleter iteratorss rZr+r+s^:h 1$IVI;;;X9%%XI;<>> list(roundrobin('ABC', 'D', 'EF')) ['A', 'D', 'E', 'B', 'F', 'C'] This function produces the same output as :func:`interleave_longest`, but may perform better for some inputs (in particular when the number of iterables is small). rN)rcrlrangerkrrrn)rr num_actives rZrErEsLD)$IC NAr2 &78 tY'''3'sAAA AcUc[n[US5up#n[[X55upV[U[[U55[X654$)aw Returns a 2-tuple of iterables derived from the input iterable. The first yields the items that have ``pred(item) == False``. The second yields the items that have ``pred(item) == True``. >>> is_odd = lambda x: x % 2 != 0 >>> iterable = range(10) >>> even_items, odd_items = partition(is_odd, iterable) >>> list(even_items), list(odd_items) ([0, 2, 4, 6, 8], [1, 3, 5, 7, 9]) If *pred* is None, :func:`bool` is used. >>> iterable = [0, 1, False, True, '', ' '] >>> false_items, true_items = partition(None, iterable) >>> list(false_items), list(true_items) ([0, False, ''], [1, True, ' ']) )boolrrcr r)rxrat1t2pp1p2s rZr8r8sK( |Ha IBA T FB RT2 '")9 ::r\c^[U5m[R"U4Sj[[ T5S-555$)aYields all possible subsets of the iterable. >>> list(powerset([1, 2, 3])) [(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)] :func:`powerset` will operate on iterables that aren't :class:`set` instances, so repeated elements in the input will produce repeated elements in the output. >>> seq = [1, 1, 0] >>> list(powerset(seq)) [(), (1,), (1,), (0,), (1, 1), (1, 0), (1, 0), (1, 1, 0)] For a variant that efficiently yields actual :class:`set` instances, see :func:`powerset_of_sets`. c3<># UHn[TU5v M g7frW)r ).0rss rZ powerset..sM;La|Aq11;Lr)r^r r|rrk)rars @rZr<r<s2" XA   M5Q!;LM MMr\c## [5nURn/nURnUSLnUH(nU(aU"U5OUnX;aU"U5 Uv M(M* g![a X;aU"U5 Uv MKMNf=f7f)aJ Yield unique elements, preserving order. >>> list(unique_everseen('AAAABBBCCDAABBB')) ['A', 'B', 'C', 'D'] >>> list(unique_everseen('ABBCcAD', str.lower)) ['A', 'B', 'C', 'D'] Sequences with a mix of hashable and unhashable items can be used. The function will be slower (i.e., `O(n^2)`) for unhashable items. Remember that ``list`` objects are unhashable - you can use the *key* parameter to transform the list to a tuple (which is hashable) to avoid a slowdown. >>> iterable = ([1, 2], [2, 3], [1, 2]) >>> list(unique_everseen(iterable)) # Slow [[1, 2], [2, 3]] >>> list(unique_everseen(iterable, key=tuple)) # Faster [[1, 2], [2, 3]] Similarly, you may want to convert unhashable ``set`` objects with ``key=frozenset``. For ``dict`` objects, ``key=lambda x: frozenset(x.items())`` can be used. N)setaddappendr) rarsseenset seenset_addseenlist seenlist_adduse_keyelementks rZrQrQs6eG++KH??LoG#CL A     Q ! s)?A?AA?A<5A?;A<<A?c Uc[[S5[U55$[[[[S5[X555$)zYields elements in order, ignoring serial duplicates >>> list(unique_justseen('AAAABBBCCDAABBB')) ['A', 'B', 'C', 'D', 'A', 'B'] >>> list(unique_justseen('ABBCcAD', str.lower)) ['A', 'B', 'C', 'A', 'D'] rr)rcrrrn)rarss rZrRrRs< {:a='("344 tSA(>? @@r\Fc([XUS9n[X1S9$)aYields unique elements in sorted order. >>> list(unique([[1, 2], [3, 4], [1, 2]])) [[1, 2], [3, 4]] *key* and *reverse* are passed to :func:`sorted`. >>> list(unique('ABBcCAD', str.casefold)) ['A', 'B', 'c', 'D'] >>> list(unique('ABBcCAD', str.casefold, reverse=True)) ['D', 'c', 'B', 'A'] The elements in *iterable* need not be hashable, but they must be comparable for sorting to work. )rsreverse)rs)sortedrR)rarsr sequenceds rZrPrP,s x':I 9 ..r\c#t# [U5 Ub U"5v U"5v M !,(df  g=f7f)aYields results from a function repeatedly until an exception is raised. Converts a call-until-exception interface to an iterator interface. Like ``iter(func, sentinel)``, but uses an exception instead of a sentinel to end the loop. >>> l = [0, 1, 2] >>> list(iter_except(l.pop, IndexError)) [2, 1, 0] Multiple exceptions can be specified as a stopping condition: >>> l = [1, 2, 3, '...', 4, 5, 6] >>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError))) [7, 6, 5] >>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError))) [4, 3, 2] >>> list(iter_except(lambda: 1 + l.pop(), (IndexError, TypeError))) [] Nr)r exceptionrts rZr-r-@s3, )   'M&L  s 8' 58c,[[X 5U5$)ad Returns the first true value in the iterable. If no true value is found, returns *default* If *pred* is not None, returns the first item for which ``pred(item) == True`` . >>> first_true(range(10)) 1 >>> first_true(range(10), pred=lambda x: x > 5) 6 >>> first_true(range(10), default='missing', pred=lambda x: x > 9) 'missing' )rnfilter)rarqrxs rZr(r(]s" t& 00r\rrcnUVs/sHn[U5PM snU-n[SU55$s snf)aDraw an item at random from each of the input iterables. >>> random_product('abc', range(4), 'XYZ') # doctest:+SKIP ('c', 3, 'Z') If *repeat* is provided as a keyword argument, that many items will be drawn from each iterable. >>> random_product('abcd', range(4), repeat=2) # doctest:+SKIP ('a', 2, 'd', 3) This equivalent to taking a random selection from ``itertools.product(*args, **kwarg)``. c38# UHn[U5v M g7frW)r )rpools rZr!random_product..s0%$%s)r})rrrpoolss rZrCrCqs7 &* *TTU4[T *V 3E 0%0 00 +s2c`[U5nUc [U5OUn[[X!55$)aFReturn a random *r* length permutation of the elements in *iterable*. If *r* is not specified or is ``None``, then *r* defaults to the length of *iterable*. >>> random_permutation(range(5)) # doctest:+SKIP (3, 4, 0, 1, 2) This equivalent to taking a random selection from ``itertools.permutations(iterable, r)``. )r}rkr)rarrs rZrBrBs+ ?DYD AA  !!r\c^[U5m[T5n[[[ U5U55n[U4SjU55$)zReturn a random *r* length subsequence of the elements in *iterable*. >>> random_combination(range(5), 3) # doctest:+SKIP (2, 3, 4) This equivalent to taking a random selection from ``itertools.combinations(iterable, r)``. c3.># UH nTUv M g7frWrrrrs rZr%random_combination..*'Qa')r}rkrrr)rarr`indicesrs @rZrArAs= ?D D AVE!Ha()G *'* **r\c^^[U5m[T5m[U4Sj[U555n[U4SjU55$)a;Return a random *r* length subsequence of elements in *iterable*, allowing individual elements to be repeated. >>> random_combination_with_replacement(range(3), 5) # doctest:+SKIP (0, 0, 1, 2, 2) This equivalent to taking a random selection from ``itertools.combinations_with_replacement(iterable, r)``. c3:># UHn[T5v M g7frW)rrrr`s rZr6random_combination_with_replacement..s48aYq\\8sc3.># UH nTUv M g7frWrrs rZrrrr)r}rkrr)rarrr`rs @@rZr@r@s< ?D D A45844G *'* **r\c[U5n[U5nUS:dX:a[eSn[XU- 5n[ SUS-5HnXTU- U--U-nM US:aX%- nUS:dX%:a[ e/nU(aNXQ-U-US- US- pnX%:aX%-nXTU- -U-US- pEX%:aMUR USU- 5 U(aMN[U5$)aEquivalent to ``list(combinations(iterable, r))[index]``. The subsequences of *iterable* that are of length *r* can be ordered lexicographically. :func:`nth_combination` computes the subsequence at sort position *index* directly, without computing the previous subsequences. >>> nth_combination(range(5), 3, 5) (0, 3, 4) ``ValueError`` will be raised If *r* is negative or greater than the length of *iterable*. ``IndexError`` will be raised if the given *index* is invalid. rrr)r}rkrminr IndexErrorr) rarindexrr`crrresults rZr4r4s ?D D A A15 A A1u A 1a!e_ QOq  qy   uz F %1*a!eQUaj JEA;!#QUqj  d26l# ! =r\c[U/U5$)aYield *value*, followed by the elements in *iterator*. >>> value = '0' >>> iterator = ['1', '2', '3'] >>> list(prepend(value, iterator)) ['0', '1', '2', '3'] To prepend multiple values, see :func:`itertools.chain` or :func:`value_chain`. )r )valueros rZr=r=s %( ##r\c## [U5SSS2n[U5n[S/US9U-n[U[ SUS- 55H!nUR U5 [ X5v M# g7f)u-Discrete linear convolution of two iterables. Equivalent to polynomial multiplication. For example, multiplying ``(x² -x - 20)`` by ``(x - 3)`` gives ``(x³ -4x² -17x + 60)``. >>> list(convolve([1, -1, -20], [1, -3])) [1, -4, -17, 60] Examples of popular kinds of kernels: * The kernel ``[0.25, 0.25, 0.25, 0.25]`` computes a moving average. For image data, this blurs the image and reduces noise. * The kernel ``[1/2, 0, -1/2]`` estimates the first derivative of a function evaluated at evenly spaced inputs. * The kernel ``[1, -2, 1]`` estimates the second derivative of a function evaluated at evenly spaced inputs. Convolutions are mathematically commutative; however, the inputs are evaluated differently. The signal is consumed lazily and can be infinite. The kernel is fully consumed before the calculations begin. Supports all numeric types: int, float, complex, Decimal, Fraction. References: * Article: https://betterexplained.com/articles/intuitive-convolution/ * Video by 3Blue1Brown: https://www.youtube.com/watch?v=KuXjwB4LzSA Nrrrgr)r}rkrr rr_sumprod)signalkernelr`windowrXs rZr&r&siF6]4R4 F F A A3q !A %F 66!QU+ , av&&-sA*A,c\^^^[T5m/mUUU4Sjn[TT5nU"5U4$)aA variant of :func:`takewhile` that allows complete access to the remainder of the iterator. >>> it = iter('ABCdEfGhI') >>> all_upper, remainder = before_and_after(str.isupper, it) >>> ''.join(all_upper) 'ABC' >>> ''.join(remainder) # takewhile() would lose the 'd' 'dEfGhI' Note that the first iterator must be fully consumed before the second iterator can generate valid results. c3h># TH'nT"U5(aUv MTRU5 g g7frW)r)elemr predicate transitions rZ true_iterator'before_and_after..true_iterator-s/D !!$' s/2)rlr )rrrremainder_iteratorrs`` @rZr$r$s5 bBJz2. ?. ..r\c[US5upn[US5 [US5 [US5 [XU5$)zReturn overlapping triplets from *iterable*. >>> list(triplewise('ABCDE')) [('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')] rNr)rarrt3s rZrOrO=s;Xq!JBBTNTNTN rr?r\c|[X5n[U5Hup4[[XCU5S5 M [ U6$rW)rrrnrr)rar`rrros rZ_sliding_window_islicer Ms8H I +  VH #T*,  ?r\c## [U5n[[X!S- 5US9nUH!nURU5 [ U5v M# g7f)Nrrg)rlrrrr})rar`rorrXs rZ_sliding_window_dequer UsCH~H 6(E*1 5F  aFmsA A cUS:a [X5$US:a [X5$US:Xa [U5$US:Xa [U5$[ SU35e)a=Return a sliding window of width *n* over *iterable*. >>> list(sliding_window(range(6), 4)) [(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5)] If *iterable* has fewer than *n* items, then nothing is yielded: >>> list(sliding_window(range(3), 4)) [] For a variant with more features, see :func:`windowed`. rzn should be at least one, not )r r r7rrr~s rZrGrG^s^ 2v$X11 Q%h22 a!! a8}9!=>>r\c [U5n[[[[ [ U5S-5S55n[ [[U5U5$)zReturn all contiguous non-empty subslices of *iterable*. >>> list(subslices('ABC')) [['A'], ['A', 'B'], ['A', 'B', 'C'], ['B'], ['B', 'C'], ['C']] This is similar to :func:`substrings`, but emits items in a different order. rr) r^rslicer rrkrcrr)raseqslicess rZrHrHws@ x.C ULs3x!|)>> roots = [5, -4, 3] # (x - 5) * (x + 4) * (x - 3) >>> polynomial_from_roots(roots) # x³ - 4 x² - 17 x + 60 [1, -4, -17, 60] Supports all numeric types: int, float, complex, Decimal, Fraction. r)r^r&)rootspolyroots rZr:r:s1 3DHTAu:./ Kr\c## [USS5nUc0[XU5n[XR5HupgXqLdXq:XdMUv M gUc [U5OUnUS- n[ [ 5 U"XS-U5=nv M!,(df  g=f7f)aYield the index of each place in *iterable* that *value* occurs, beginning with index *start* and ending before index *stop*. >>> list(iter_index('AABCADEAF', 'A')) [0, 1, 4, 7] >>> list(iter_index('AABCADEAF', 'A', 1)) # start index is inclusive [1, 4, 7] >>> list(iter_index('AABCADEAF', 'A', 1, 7)) # stop index is not inclusive [1, 4] The behavior for non-scalar *values* matches the built-in Python types. >>> list(iter_index('ABCDABCD', 'AB')) [0, 4] >>> list(iter_index([0, 1, 2, 3, 0, 1, 2, 3], [0, 1])) [] >>> list(iter_index([[0, 1], [2, 3], [0, 1], [2, 3]], [0, 1])) [0, 2] See :func:`locate` for a more general means of finding the indexes associated with particular values. rNr)getattrrrrkrr)rarrestop seq_indexrorrs rZr.r.s2'40I(40#H4JA7#35 !% s8}$ AI j !%eUD99q:" !s4B -B 'A;; B B c #H# US:aSv Sn[S5US--n[USU[U5S-S9HLn[USXU-5ShvN [[ [ X3-XU-555X#U-XU-2'X3-nMN [USU5ShvN gNON7f)zYYield the primes less than n. >>> list(sieve(30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] rr)rrr)rN) bytearrayr.rbytesrkr)r`redatars rZrFrFs 1u E V Q 'D aU1X\ :dAu!e444"'E!%E,B(C"DUQQ ;$5))) 5*s%A B"BA B"B B" B"c## US:a [S5e[U5n[[X155=n(aCU(a[ U5U:wa [S5eUv [[X155=n(aMBgg7f)arBatch data into tuples of length *n*. If the number of items in *iterable* is not divisible by *n*: * The last batch will be shorter if *strict* is ``False``. * :exc:`ValueError` will be raised if *strict* is ``True``. >>> list(batched('ABCDEFG', 3)) [('A', 'B', 'C'), ('D', 'E', 'F'), ('G',)] On Python 3.13 and above, this is an alias for :func:`itertools.batched`. rzn must be at least onezbatched(): incomplete batchN)rrlr}rrk)rar`rTrobatchs rZ_batchedr#sp 1u122H~H,- -% - c%jAo:; ; ,- -% - -s A8A><A>i )r#c[XUS9$)NrS)itertools_batched)rar`rTs rZr#r#s V<>> list(transpose([(1, 2, 3), (11, 22, 33)])) [(1, 11), (2, 22), (3, 33)] The caller should ensure that the dimensions of the input are compatible. If the input is empty, no output will be produced. ) _zip_strictrs rZrNrNs  r\cB[[R"U5U5$)zReshape the 2-D input *matrix* to have a column count given by *cols*. >>> matrix = [(0, 1), (2, 3), (4, 5)] >>> cols = 3 >>> list(reshape(matrix, cols)) [(0, 1, 2), (3, 4, 5)] )r#r r|)matrixcolss rZr?r?s 5&&v. 55r\c x[US5n[[[[ U[ U555U5$)a Multiply two matrices. >>> list(matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)])) [(49, 80), (41, 60)] The caller should ensure that the dimensions of the input matrices are compatible with each other. Supports all numeric types: int, float, complex, Decimal, Fraction. r)rkr#rrrrN)m1m2r`s rZr0r0 s0 BqE A 78WR2%?@! DDr\c[SU5HKnS=p#SnUS:Xa4X"-U-U-nX3-U-U-nX3-U-U-n[X#- U5nUS:XaM4X@:wdMIUs $ [S5e)Nrrzprime or under 5)rrr)r`rrXrYds rZ_factor_pollardr1s1a[  1faAaAaAAE1 A 1f 6H ' ((r\c#># US:ag[H!nX-(aMUv X-nX-(dMM# /nUS:aU/O/nUH?nUS:d[U5(aURU5 M,[U5nX4X-4- nMA [ U5ShvN gN7f)zYield the prime factors of n. >>> list(factor(360)) [2, 2, 2, 3, 3, 5] Finds small factors with trial division. Larger factors are either verified as prime with ``is_prime`` or split into smaller factors with Pollard's rho algorithm. rNri)_primes_below_211r,rr1r)r`primeprimestodofacts rZr)r).s 1u#))K KA))# Fa%A3RD  v:! MM! "1%D 19% %D  f~sBBA$BBBc [U5nUS:Xa[U5"S5$[[[ U5[ [ U555n[X5$)aDEvaluate a polynomial at a specific value. Computes with better numeric stability than Horner's method. Evaluate ``x^3 - 4 * x^2 - 17 * x + 60`` at ``x = 2.5``: >>> coefficients = [1, -4, -17, 60] >>> x = 2.5 >>> polynomial_eval(coefficients, x) 8.125 Supports all numeric types: int, float, complex, Decimal, Fraction. r)rktypercpowrreversedrr) coefficientsrXr`powerss rZr9r9OsI LAAvAwqz fQi%(!3 4F L ))r\c$[[U56$)zReturn the sum of the squares of the input values. >>> sum_of_squares([10, 20, 30]) 1400 Supports all numeric types: int, float, complex, Decimal, Fraction. )rrr(s rZrIrIds SW r\ct[U5n[[SU55n[[ [ X55$)u<Compute the first derivative of a polynomial. Evaluate the derivative of ``x³ - 4 x² - 17 x + 60``: >>> coefficients = [1, -4, -17, 60] >>> derivative_coefficients = polynomial_derivative(coefficients) >>> derivative_coefficients [3, -8, -17] Supports all numeric types: int, float, complex, Decimal, Fraction. r)rkr<rr^rcr)r=r`r>s rZr;r;os0 LA eAqk "F C. //r\cJ[[U55H nXU--nM U$)uReturn the count of natural numbers up to *n* that are coprime with *n*. Euler's totient function φ(n) gives the number of totatives. Totative are integers k in the range 1 ≤ k ≤ n such that gcd(n, k) = 1. >>> n = 9 >>> totient(n) 6 >>> totatives = [x for x in range(1, n) if gcd(n, x) == 1] >>> totatives [1, 2, 4, 5, 7, 8] >>> len(totatives) 6 Reference: https://en.wikipedia.org/wiki/Euler%27s_totient_function )rr))r`r5s rZrMrMs&&VAY %Z Hr\))i)r)i)I)ltT7)r=)lay)r iS_)l;n>)rrrD )lp )rrrHrDrIrF)l)riEi$inii=ik)l%!Hn fW) rrrHrDrIrFrGrB%)c~US- U- R5S- nX- nSU-U-U:XaUS-(aUS:deX4$)z#Return s, d such that 2**s * d == nrr) bit_length)r`rr0s rZ _shift_to_oddrQsO a%1  "Q&A A Fa<1 Q161 1 4Kr\cUS:aUS-(aSUs=::aU:de e[US- 5up#[XU5nUS:XdX@S- :Xag[US- 5HnXD-U-nX@S- :XdM g g)NrrTF)rQr;r)r`baserr0rX_s rZ_strong_probable_primerUs EAAMM2 2M2 2 Q DA DQAAv!e 1q5\ EAI A: r\c^TS:aTS;$TS-(a2TS-(a(TS-(aTS-(aTS-(a TS-(dg [H upTU:dM O U4S j[S 55n[U4S jU55$) amReturn ``True`` if *n* is prime and ``False`` otherwise. Basic examples: >>> is_prime(37) True >>> is_prime(3 * 13) False >>> is_prime(18_446_744_073_709_551_557) True Find the next prime over one billion: >>> next(filter(is_prime, count(10**9))) 1000000007 Generate random primes up to 200 bits and up to 60 decimal digits: >>> from random import seed, randrange, getrandbits >>> seed(18675309) >>> next(filter(is_prime, map(getrandbits, repeat(200)))) 893303929355758292373272075469392561129886005037663238028407 >>> next(filter(is_prime, map(randrange, repeat(10**60)))) 269638077304026462407872868003560484232362454342414618963649 This function is exact for values of *n* below 10**24. For larger inputs, the probabilistic Miller-Rabin primality test has a less than 1 in 2**128 chance of a false positive. rJ>rrrHrDrIrFrrrHrDrIrFFc3B># UHn[STS- 5v M g7f)rrN)_private_randrangers rZris_prime..s Ay!#Aq1u--ys@c3<># UHn[TU5v M g7frW)rU)rrSr`s rZrrYsA54%a..5r)_perfect_testsrall)r`limitbasess` rZr,r,syB 2v((( Ea!eA!a%AFq2v&  u9 'BuRyA A5A AAr\c[SU5$)zReturns an iterable with *n* elements for efficient looping. Like ``range(n)`` but doesn't create integers. >>> i = 0 >>> for _ in loops(5): ... i += 1 >>> i 5 Nr)r`s rZr/r/s $?r\cH[[[[U5U55$)ulNumber of distinct arrangements of a multiset. The expression ``multinomial(3, 4, 2)`` has several equivalent interpretations: * In the expansion of ``(a + b + c)⁹``, the coefficient of the ``a³b⁴c²`` term is 1260. * There are 1260 distinct ways to arrange 9 balls consisting of 3 reds, 4 greens, and 2 blues. * There are 1260 unique ways to place 9 distinct objects into three bins with sizes 3, 4, and 2. The :func:`multinomial` function computes the length of :func:`distinct_permutations`. For example, there are 83,160 distinct anagrams of the word "abracadabra": >>> from more_itertools import distinct_permutations, ilen >>> ilen(distinct_permutations('abracadabra')) 83160 This can be computed directly from the letter counts, 5a 2b 2r 1c 1d: >>> from collections import Counter >>> list(Counter('abracadabra').values()) [5, 2, 2, 1, 1] >>> multinomial(5, 2, 1, 1, 2) 83160 A binomial coefficient is a special case of multinomial where there are only two categories. For example, the number of ways to arrange 12 balls with 5 reds and 7 blues is ``multinomial(5, 7)`` or ``math.comb(12, 5)``. When the multiplicities are all just 1, :func:`multinomial` is a special case of ``math.factorial`` so that ``multinomial(1, 1, 1, 1, 1, 1, 1) == math.factorial(7)``. 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