Ng~XdZddlZddlZddlmZddlmZddlmZm Z ddl m Z m Z m Z mZmZmZmZmZmZmZmZmZddlmZmZmZddlmZgd ZeZ e d e e d Z!e#ed d Z$dZ%dPdZ&dZ'dQdZ(dQdZ)dQdZ*e+fdZ,dZ-e-Z.dZ/dZ0dZ1dQdZ2dZ3 ddl m4Z5dZ4e3je4_Gdde7Z8dZ9d Z:dRd!Z;d"ZdQd%Z?dQd&Z@dSd(ZAdQd)ZBdTd*ZCd+d,d-ZDdQd.ZEd/ZFd0ZGd1ZHd2ZId3ZJd4ZKd5ZLd6ZMd7ZNd8ZOd9ZPd:ZQdUd;ZRd<ZSd'd d=ZTed>k\r dd?l mUZVd'd d@ZUneTZUeTjeU_dAZWdBZXdCZYdDZZe[eSdEZ\dFZ]dGZ^dHZ_dIZ`dJZagdKZbedLZcdMZddNZedOZfy#e"$re Z!Y1wxYw#e6$re3Z4YwxYw)VaImported 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)deque)Sized) lru_cachepartial) chain combinationscompresscountcyclegroupbyisliceproductrepeatstarmaptee zip_longest) randrangesamplechoice) hexversion)0 all_equalbatchedbefore_and_afterconsumeconvolve dotproduct first_truefactorflattengrouperis_prime iter_except iter_indexloopsmatmulncyclesnthnth_combinationpadnonepad_nonepairwise partitionpolynomial_evalpolynomial_from_rootspolynomial_derivativepowersetprependquantifyreshape#random_combination_with_replacementrandom_combinationrandom_permutationrandom_product repeatfunc roundrobinsievesliding_window subslicessum_of_squarestabulatetailtaketotient transpose triplewiseuniqueunique_everseenunique_justseenTstrictsumprodct||SN)r)xys Q/var/www/djangounited/venv/lib/python3.12/site-packages/more_itertools/recipes.pyrOas Aq1Ac,tt||S)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] )listr niterables rNr@r@ds x# $$rPc,t|t|S)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 rNr>r>ts xu &&rPc #Kt|tr,t|tdt ||z dEd{yt t ||Ed{y7$7w)zReturn an iterator over the last *n* items of *iterable*. >>> t = tail(3, 'ABCDEFG') >>> list(t) ['E', 'F', 'G'] rNmaxlen) isinstancerr maxleniterrrSs rNr?r?sS(E"(C3x=1+<$=tDDDhq1222 E2s!7A"AA"A A" A"cR|t|dytt|||dy)aXAdvance *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 Nrr[)rnextr )iteratorrTs rNrrs)@ y hq! VHa #T*rPc0tt||d|S)zReturns the nth item or a default value. >>> l = range(10) >>> nth(l, 3) 3 >>> nth(l, 20, "zebra") 'zebra' N)rbr )rUrTdefaults rNr'r's xD)7 33rPc>t||}|D] }|D]}yyy)a 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 )rUkeyrcfirstseconds rNrrs9$x%H F  rPc,tt||S)zcReturn the how many times the predicate is true. >>> quantify([True, False, True]) 2 )sumrW)rUpreds rNr2r2s s4" ##rPc,t|tdS)aReturns the sequence of elements and then returns ``None`` indefinitely. >>> take(5, pad_none(range(3))) [0, 1, 2, None, None] Useful for emulating the behavior of the built-in :func:`map` function. See also :func:`padded`. N)rrrUs rNr*r*s 6$< ((rPcRtjtt||S)zvReturns the sequence elements *n* times >>> list(ncycles(["a", "b"], 3)) ['a', 'b', 'a', 'b', 'a', 'b'] )r from_iterablertuplerUrTs rNr&r&s    veHoq9 ::rPcJtttj||S)zcReturns the dot product of the two iterables. >>> dotproduct([10, 10], [20, 20]) 400 )rkrWoperatormul)vec1vec2s rNrr s s8<<t, --rPc,tj|S)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. )rrp) listOfListss rNrrs   { ++rPc\|t|t|St|t||S)aGCall *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 rNr8r8"s., }tVD\** 4e, --rPcNt|\}}t|dt||S)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`. Nrrbzip)rUabs rN _pairwiser=s& x=DAqDM q!9rP)r+ct|SrK)itertools_pairwiserns rNr+r+Qs !(++rPc eZdZdfd ZxZS)UnequalIterablesErrorcPd}||dj|z }t| |y)Nz Iterables have different lengthsz/: index 0 has length {}; index {} has length {})formatsuper__init__)selfdetailsmsg __class__s rNrzUnequalIterablesError.__init__Xs;0   MEMM C rPrK)__name__ __module__ __qualname__r __classcell__)rs@rNrrWs rPrc#nKt|dtiD]}|D]}|tus t|!yw)N fillvalue)r_markerr) iterablescombovals rN_zip_equal_generatorrbsEi;7; .Cg~+-- . s 55c t|d}t|dddD]$\}}t|}||k7st|||ft|S#t$rt |cYSwxYw)Nr)r)r_ enumeraterr TypeErrorr)r first_sizeiitsizes rN _zip_equalrjs /1& y}a0 KEArr7Dz!+ZD4IJJ K I /#I../s3AAA%$A%ct|g|z}|dk(r t|d|iS|dk(rt|S|dk(rt|St d)aGroup elements from *iterable* into fixed-length groups of length *n*. >>> list(grouper('ABCDEF', 3)) [('A', 'B', 'C'), ('D', 'E', 'F')] The keyword arguments *incomplete* and *fillvalue* control what happens for iterables whose length is not a multiple of *n*. When *incomplete* is `'fill'`, the last group will contain instances of *fillvalue*. >>> list(grouper('ABCDEFG', 3, incomplete='fill', fillvalue='x')) [('A', 'B', 'C'), ('D', 'E', 'F'), ('G', 'x', 'x')] When *incomplete* is `'ignore'`, the last group will not be emitted. >>> list(grouper('ABCDEFG', 3, incomplete='ignore', fillvalue='x')) [('A', 'B', 'C'), ('D', 'E', 'F')] When *incomplete* is `'strict'`, a subclass of `ValueError` will be raised. >>> it = grouper('ABCDEFG', 3, incomplete='strict') >>> list(it) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... UnequalIterablesError fillrrHignorez Expected fill, strict, or ignore)r`rrr ValueError)rUrT incompleterr}s rNr r zs^: N a DVD6I66X4  XDz;<>> list(roundrobin('ABC', 'D', 'EF')) ['A', 'D', 'E', 'B', 'F', 'C'] This function produces the same output as :func:`interleave_longest`, but may perform better for some inputs (in particular when the number of iterables is small). rN)rWr`ranger_r r rb)r iterators num_actives rNr9r9sTD)$IC NAr2( &J78 tY'''('sAAAAc|t}t|d\}}}tt||\}}t|ttj |t||fS)a 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, ' ']) )boolrrWr rtnot_)rlrUt1t2pp1p2s rNr,r,sX( |Ha IBA T1 FB RX]]B/ 0(2r2B CCrPct|tjfdtt dzDS)a1Yields 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`. c36K|]}t|ywrK)r).0rss rN zpowerset..sMa|Aq1Mr)rRrrprr_)rUrs @rNr0r0s2" XA   M5Q!;LM MMrPc#Kt}|j}g}|j}|du}|D]}|r||n|} ||vr |||!y#t$r||vr |||Y>wxYww)a Yield unique elements, preserving order. >>> list(unique_everseen('AAAABBBCCDAABBB')) ['A', 'B', 'C', 'D'] >>> list(unique_everseen('ABBCcAD', str.lower)) ['A', 'B', 'C', 'D'] Sequences with a mix of hashable and unhashable items can be used. The function will be slower (i.e., `O(n^2)`) for unhashable items. Remember that ``list`` objects are unhashable - you can use the *key* parameter to transform the list to a tuple (which is hashable) to avoid a slowdown. >>> iterable = ([1, 2], [2, 3], [1, 2]) >>> list(unique_everseen(iterable)) # Slow [[1, 2], [2, 3]] >>> list(unique_everseen(iterable, key=tuple)) # Faster [[1, 2], [2, 3]] Similarly, you may want to convert unhashable ``set`` objects with ``key=frozenset``. For ``dict`` objects, ``key=lambda x: frozenset(x.items())`` can be used. N)setaddappendr) rUrgseenset seenset_addseenlist seenlist_adduse_keyelementks rNrErEs6eG++KH??LoG #CL A    Q  s(:A/A A/A,)A/+A,,A/c |(ttjdt|Sttttjdt||S)zYields elements in order, ignoring serial duplicates >>> list(unique_justseen('AAAABBBCCDAABBB')) ['A', 'B', 'C', 'D', 'A', 'B'] >>> list(unique_justseen('ABBCcAD', str.lower)) ['A', 'B', 'C', 'A', 'D'] rr)rWrt itemgetterr rb)rUrgs rNrFrFsL {8&&q)78+<== tS,,Q/31GH IIrPFc4tt||||S)aYields unique elements in sorted order. >>> list(unique([[1, 2], [3, 4], [1, 2]])) [[1, 2], [3, 4]] *key* and *reverse* are passed to :func:`sorted`. >>> list(unique('ABBcCAD', str.casefold)) ['A', 'B', 'c', 'D'] >>> list(unique('ABBcCAD', str.casefold, reverse=True)) ['D', 'c', 'B', 'A'] The elements in *iterable* need not be hashable, but they must be comparable for sorting to work. )rgreverse)rg)rFsorted)rUrgrs rNrDrD!s 6(WE3 OOrPc#NK | | | #|$rYywxYww)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))) [] N)r{ exceptionrhs rNr"r"4s8,  'M&L    s%"%"%c.tt|||S)a 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' )rbfilter)rUrerls rNrrSs" tX& 00rPrrch|Dcgc] }t|c}|z}td|DScc}w)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)``. c32K|]}t|ywrK)r)rpools rNrz!random_product..xs0$0s)rq)rr}rpoolss rNr7r7gs3 &* *TU4[ *V 3E 0%0 00 +s/c`t|}| t|n|}tt||S)abReturn 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)``. )rqr_r)rUrrs rNr6r6{s- ?DYD AA a !!rPct|t}ttt ||}tfd|DS)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(K|] }| ywrKrrrrs rNrz%random_combination..*Qa*)rqr_rrr)rUrrTindicesrs @rNr5r5s= ?D D AVE!Ha()G *'* **rPct|ttfdt|D}tfd|DS)aSReturn 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)``. c34K|]}tywrK)r)rrrTs rNrz6random_combination_with_replacement..s4aYq\4sc3(K|] }| ywrKrrs rNrz6random_combination_with_replacement..rr)rqr_rr)rUrrrTrs @@rNr4r4s< ?D D A45844G *'* **rPct|}t|}|dks||kDrtd}t|||z }t d|dzD]}|||z |zz|z}|dkr||z }|dks||k\rt g}|rL||z|z|dz |dz }}}||k\r||z}|||z z|z|dz }}||k\r|j |d|z |rLt|S)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)rqr_rminr IndexErrorr) rUrindexrrTcrrresults rNr(r(s ?D D A A1q5 A Aq1u A 1a!e_! QOq ! qy    uz F a%1*a!eQUa1qj QJEA;!#QUqAqj  d26l#  =rPct|g|S)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)valuercs rNr1r1s %( ##rPc#Kt|ddd}t|}tdg||z}t|t d|dz D]!}|j |t ||#yw)aBConvolve the iterable *signal* with the iterable *kernel*. >>> signal = (1, 2, 3, 4, 5) >>> kernel = [3, 2, 1] >>> list(convolve(signal, kernel)) [3, 8, 14, 20, 26, 14, 5] Note: the input arguments are not interchangeable, as the *kernel* is immediately consumed and stored. Nrrr[r)rqr_rrrr_sumprod)signalkernelrTwindowrLs rNrrsp6]4R4 F F A A3q !A %F 66!QU+ ,' avv&&'sA,A.cZtgfd}t}||fS)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. c3ZKD]!}|r|j|yywrK)r)elemr predicate transitions rN true_iteratorz'before_and_after..true_iterators3 D !!$'  s(+)r`r)rrrremainder_iteratorrs`` @rNrrs5 bBJz2. ?. ..rPct|d\}}}t|dt|dt|dt|||S)zReturn overlapping triplets from *iterable*. >>> list(triplewise('ABCDE')) [('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')] rNr)rUrrt3s rNrCrCs?Xq!JBBTNTNTN r2r?rPc~t||}t|D]\}}tt|||dt |SrK)rrrbr r)rUrTrrrcs rN_sliding_window_islicer.sCHa I ++ 8 VHa #T*+  ?rPc#Kt|}tt||dz |}|D] }|j|t |"yw)Nrr[)r`rr rrq)rUrTrrrLs rN_sliding_window_dequer6sK hB 6"a!e$Q /F  aFmsA Ac|dkDr t||S|dkDr t||S|dk(r t|S|dk(r t|St d|)aYReturn a sliding window of width *n* over *iterable*. >>> list(sliding_window(range(6), 4)) [(0, 1, 2, 3), (1, 2, 3, 4), (2, 3, 4, 5)] If *iterable* has fewer than *n* items, then nothing is yielded: >>> list(sliding_window(range(3), 4)) [] For a variant with more features, see :func:`windowed`. rzn should be at least one, not )rrr+rrrrs rNr;r;?sb 2v$Xq11 Q%h22 a!! a8}9!=>>rPc t|}tttt t |dzd}t tjt||S)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) rRrslicerrr_rWrtgetitemr)rUseqslicess rNr<r<XsF x.C ULs3x!|)>> roots = [5, -4, 3] # (x - 5) * (x + 4) * (x - 3) >>> polynomial_from_roots(roots) # x^3 - 4 * x^2 - 17 * x + 60 [1, -4, -17, 60] r)rRr)rootspolyroots rNr.r.fs6 3D0HTAu:./0 KrPc#Kt|dd}|0t|||}t||D]\}}||us||k(s|y| t|n|}|dz } |||dz|x}#t$rYywxYww)aYield the index of each place in *iterable* that *value* occurs, beginning with index *start* and ending before index *stop*. >>> list(iter_index('AABCADEAF', 'A')) [0, 1, 4, 7] >>> list(iter_index('AABCADEAF', 'A', 1)) # start index is inclusive [1, 4, 7] >>> list(iter_index('AABCADEAF', 'A', 1, 7)) # stop index is not inclusive [1, 4] The behavior for non-scalar *values* matches the built-in Python types. >>> list(iter_index('ABCDABCD', 'AB')) [0, 4] >>> list(iter_index([0, 1, 2, 3, 0, 1, 2, 3], [0, 1])) [] >>> list(iter_index([[0, 1], [2, 3], [0, 1], [2, 3]], [0, 1])) [0, 2] See :func:`locate` for a more general means of finding the indexes associated with particular values. rNr)getattrr rr_r)rUrrYstop seq_indexrrrs rNr#r#ss2'40I HeT *#B. JAw%7e#3  !% s8}$ AI %eQUD99q:   s(8A9A9A** A63A95A66A9c #jK|dkDrdd}td|dzz}tj|dz}t|d||D]Q}t|d|||zEd{t t t ||z|||z|||z|||z<||z}St|d|Ed{y7R7w)zdYield the primes less than n. >>> list(sieve(30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] rr)rrrN) bytearraymathisqrtr#bytesr_r)rTrYdatalimitrs rNr:r:s  1u E V Q 'D JJqMA E a .dAua!e444"'E!a%AE,B(C"DQUQQ A$5))) 5*s%AB3B/A B3)B1*B31B3c#K|dkr tdt|}tt||x}r8|rt ||k7r td|tt||x}r7yyw)aBatch data into tuples of length *n*. If the number of items in *iterable* is not divisible by *n*: * The last batch will be shorter if *strict* is ``False``. * :exc:`ValueError` will be raised if *strict* is ``True``. >>> list(batched('ABCDEFG', 3)) [('A', 'B', 'C'), ('D', 'E', 'F'), ('G',)] On Python 3.13 and above, this is an alias for :func:`itertools.batched`. rzn must be at least onezbatched(): incomplete batchN)rr`rqr r_)rUrTrHrbatchs rN_batchedrsp 1u122 hBA' '% ' c%jAo:; ; A' '% 's A)A.,A.i )rct|||S)NrG)itertools_batched)rUrTrHs rNrrs 1V<>> list(transpose([(1, 2, 3), (11, 22, 33)])) [(1, 11), (2, 22), (3, 33)] The caller should ensure that the dimensions of the input are compatible. If the input is empty, no output will be produced. ) _zip_strictrs rNrBrBs  rPc@ttj||S)zReshape the 2-D input *matrix* to have a column count given by *cols*. >>> matrix = [(0, 1), (2, 3), (4, 5)] >>> cols = 3 >>> list(reshape(matrix, cols)) [(0, 1, 2), (3, 4, 5)] )rrrp)matrixcolss rNr3r3s 5&&v. 55rPc xt|d}tttt |t ||S)zMultiply two matrices. >>> list(matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)])) [(49, 80), (41, 60)] The caller should ensure that the dimensions of the input matrices are compatible with each other. r)r_rrrrrB)m1m2rTs rNr%r%s0 BqE A 78WR2%?@! DDrPctj}td|dz D]I}dx}}d}|dk(r3||z|z|z}||z|z|z}||z|z|z}|||z |}|dk(r3||k7sG|cStd)Nrrzprime or under 5)rgcdrr)rTr"rrLrMds rN_factor_pollardr$s ((C 1a!e_  A 1fQaAQaAQaAAE1 A 1f 6H  ' ((rPc#K|dkrytD]}||zr |||z}||zsg}|dkDr|gng}|D]9}|dks t|r|j|%t|}||||zfz };t |Ed{y7w)aYield 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!rr$r)rTprimeprimestodofacts rNrrs 1u#e)K %KAe) Fa%A3RD & v:! MM! "1%D T19% %D & f~sB B AB BB c t|}|dk(r|dzSttt|t t |}t ||S)zEvaluate a polynomial at a specific value. Example: evaluating x^3 - 4 * x^2 - 17 * x + 60 at x = 2.5: >>> coefficients = [1, -4, -17, 60] >>> x = 2.5 >>> polynomial_eval(coefficients, x) 8.125 r)r_rWpowrreversedrr) coefficientsrLrTpowerss rNr-r-(sF LAAv1u fQi%(!3 4F L& ))rPc$tt|S)zfReturn the sum of the squares of the input values. >>> sum_of_squares([10, 20, 30]) 1400 )rrrs rNr=r=9s SW rPct|}ttd|}tt t j ||S)aCompute the first derivative of a polynomial. Example: evaluating the derivative of x^3 - 4 * x^2 - 17 * x + 60 >>> coefficients = [1, -4, -17, 60] >>> derivative_coefficients = polynomial_derivative(coefficients) >>> derivative_coefficients [3, -8, -17] r)r_r.rrRrWrtru)r/rTr0s rNr/r/Bs6 LA eAqk "F HLL,7 88rPcHtt|D] }|||zz} |S)zReturn the count of natural numbers up to *n* that are coprime with *n*. >>> totient(9) 6 >>> totient(12) 4 )rr)rTr(s rNrArAQs-VAY Q%Z HrP))i)r)i)I)ltT7)r=)lay)r iS_)l;n>)rrr6 )lp )rrr:r6r;r8)l)riEi$inii=ik)l%!Hn fW) rrr:r6r;r8r9r4%)ct|dz |z jdz }||z }d|z|z|k(r |dzr|dk\sJ||fS)z#Return s, d such that 2**s * d == nrr) bit_length)rTrr#s rN _shift_to_oddrCnsS a%1  "Q&A QA Fa<1 Q161 1 a4KrPc|dkDr|dzrd|cxkr|ksJJt|dz \}}t|||}|dk(s||dz k(ryt|dz D]}||z|z}||dz k(syy)NrrTF)rCr-r)rTbaserr#rL_s rN_strong_probable_primerGws EAAMM2 2M2 2 Q DAq D!QAAva!e 1q5\ EAI A: rPcdkrdvSdzrdzrdzrdzr dzrdzsy tD] \}}|ks n&td Dcgc]}td dz }}tfd |DScc}w) aReturn ``True`` if *n* is prime and ``False`` otherwise. >>> is_prime(37) True >>> is_prime(3 * 13) False >>> is_prime(18_446_744_073_709_551_557) True This function uses the Miller-Rabin primality test, which can return false positives for very large inputs. For values of *n* below 10**24 there are no false positives. For larger values, there is less than a 1 in 2**128 false positive rate. Multiple tests can further reduce the chance of a false positive. r<>rrr:r6r;r8rrr:r6r;r8F@rc36K|]}t|ywrK)rG)rrErTs rNrzis_prime..sA4%a.Ar)_perfect_testsrrall)rTrbasesrs` rNr!r!s  2v((( Ea!eA!a%AFq2v&9 u u9 9/4Bi81a!e$88 A5A AA9s A8ctd|S)zReturns an iterable with *n* elements for efficient looping. Like ``range(n)`` but doesn't create integers. >>> i = 0 >>> for _ in loops(5): ... i += 1 >>> i 5 Nr)rTs rNr$r$s $?rP)rrK)rN)NF)NN)rN)g__doc__rrt collectionsrcollections.abcr functoolsrr itertoolsrrr r r r r rrrrrrandomrrrsysr__all__objectrrrrr rr@r>r?rr'rrr2r*r)r&rrr8rr+r ImportErrorrrrrr r9r,r0rErFrDr"rr7r6r5r4r(r1rrrCrrr;r<r.r#r:rrrrBr3r%r$rqr'rr-r=r/rArKrCrGr!r$rrPrNrYsN !(    -,1 f (,t#d+K 4$A B % '$3$%+P 44!$ ) ;. ,.6  )8 ,!((HJ / %=P($D8N**Z JP& >1("#1("$ + +"'T $',/B  ?2 6 (V*$%*(6',=G&&GO 6 E ) %*%B*" 9     "B8 S"KdHs$ F&FFFF)(F)