o g~@sXdZddlZddlZddlmZddlmZddlmZm Z ddl m Z m Z m Z mZmZmZmZmZmZmZmZmZddlmZmZmZddlmZgd ZeZze d d Wn e!yce Z"Ynwe e d d Z"e#ed d dZ$ddZ%dddZ&ddZ'dddZ(dddZ)dddZ*e+fddZ,ddZ-e-Z.dd Z/d!d"Z0d#d$Z1dd%d&Z2d'd(Z3zdd)l m4Z5Wn e6ye3Z4Yn wd*d+Z4e3je4_Gd,d-d-e7Z8d.d/Z9d0d1Z:dd3d4Z;d5d6Zdd;d<Z?dd=d>Z@dd@dAZAddBdCZBddDdEZCdFdGdHdIZDddJdKZEdLdMZFdNdOZGdPdQZHdRdSZIdTdUZJdVdWZKdXdYZLdZd[ZMd\d]ZNd^d_ZOd`daZPdbdcZQddddeZRdfdgZSd?d dhdiZTedjkrdddkl mUZVd?d dldmZUneTZUeTjeU_dndoZWdpdqZXdrdsZYdtduZZe[eSdvZ\dwdxZ]dydzZ^d{d|Z_d}d~Z`ddZagdZbeddZcddZdddZeddZfdS)aImported 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_justseenTstrictsumprodcCs t||SN)r)xyrLO/home/ubuntu/webapp/venv/lib/python3.10/site-packages/more_itertools/recipes.pyas rNcCtt||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 niterablerLrLrMr?ds r?cCst|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 )functionstartrLrLrMr=tsr=ccsJt|trt|tdt||dEdHdStt||dEdHdS)zReturn an iterator over the last *n* items of *iterable*. >>> t = tail(3, 'ABCDEFG') >>> list(t) ['E', 'F', 'G'] rNmaxlen) isinstancerr maxleniterrrQrLrLrMr>s $r>cCs.|dur t|dddStt|||ddS)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 NrrW)rnextr )iteratorrRrLrLrMrs rcCstt||d|S)zReturns the nth item or a default value. >>> l = range(10) >>> nth(l, 3) 3 >>> nth(l, 20, "zebra") 'zebra' N)r]r )rSrRdefaultrLrLrMr&s r&cCs,t||}|D] }|D]}dSdSdS)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 )rSkeyr^firstsecondrLrLrMrs rcCrO)zcReturn the how many times the predicate is true. >>> quantify([True, False, True]) 2 )sumrT)rSpredrLrLrMr1sr1cCst|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)rrrSrLrLrMr)s r)cCsttt||S)zvReturns the sequence elements *n* times >>> list(ncycles(["a", "b"], 3)) ['a', 'b', 'a', 'b', 'a', 'b'] )r from_iterablertuplerSrRrLrLrMr%sr%cCstttj||S)zcReturns the dot product of the two iterables. >>> dotproduct([10, 10], [20, 20]) 400 )rcrToperatormul)vec1vec2rLrLrMr srcCs t|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. )rrf) listOfListsrLrLrMrs rcGs&|dur 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] N)rr)functimesargsrLrLrMr7"sr7cCs t|\}}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`. Nrr]zip)rSabrLrLrM _pairwise=s  ru)r*cCst|SrI)itertools_pairwisererLrLrMr*Qsr*cseZdZdfdd ZZS)UnequalIterablesErrorNcs*d}|dur |dj|7}t|dS)Nz Iterables have different lengthsz/: index 0 has length {}; index {} has length {})formatsuper__init__)selfdetailsmsg __class__rLrMrzXs zUnequalIterablesError.__init__rI)__name__ __module__ __qualname__rz __classcell__rLrLr~rMrwWsrwccs8t|dtiD]}|D] }|turtq |VqdS)N fillvalue)r_markerrw) iterablescombovalrLrLrM_zip_equal_generatorbsrcGsnz)t|d}t|dddD]\}}t|}||kr$t|||fdqt|WSty6t|YSw)Nr)r|)r[ enumeraterwrr TypeErrorr)r first_sizeiitsizerLrLrM _zip_equaljs    rfillcCsLt|g|}|dkrt|d|iS|dkrt|S|dkr"t|Std)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 rrrGignorez Expected fill, strict, or ignore)r\rrrr ValueError)rSrR incompleterrprLrLrMrzsrcgsDtt|}tt|ddD]}tt||}tt|EdHqdS)aJYields an item from each iterable, alternating between them. >>> 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)rTr\ranger[r r r])r iterators num_activerLrLrMr8s  r8cCsJ|durt}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, ' ']) N)boolrrTrrinot_)rdrSt1t2pp1p2rLrLrMr+s r+cs,t|tfddttdDS)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`. c3|]}t|VqdSrI)r).0rsrLrM zpowerset..r)rPrrfrr[rerLrrMr/s$r/c cst}|j}g}|j}|du}|D]+}|r||n|}z ||vr(|||VWqty=||vr;|||VYqwdS)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) rSr`seenset seenset_addseenlist seenlist_adduse_keyelementkrLrLrMrDs( rDcCs8|durttdt|Sttttdt||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'] Nrr)rTri itemgetterr r])rSr`rLrLrMrEs rEFcCstt|||d|dS)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. )r`reverse)r`)rEsorted)rSr`rrLrLrMrC!srCccs4z|dur |V |Vq |yYdSw)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))) [] NrL)rn exceptionrarLrLrMr!4s r!cCstt|||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' )r]filter)rSr_rdrLrLrMrSsrrrcGs$dd|D|}tdd|DS)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)``. cSsg|]}t|qSrLrgrpoolrLrLrM wsz"random_product..css|]}t|VqdSrI)rrrLrLrMrxz!random_product..r)rrppoolsrLrLrMr6gsr6cCs*t|}|dur 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)``. N)rgr[r)rSrrrLrLrMr5{s r5cs8t|t}ttt||}tfdd|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|]}|VqdSrIrLrrrrLrMrrz%random_combination..)rgr[rrr)rSrrRindicesrLrrMr4s r4cs@t|ttfddt|D}tfdd|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)``. c3s|]}tVqdSrIrrrRrLrMrrz6random_combination_with_replacement..c3rrIrLrrrLrMrr)rgr[rr)rSrrrL)rRrrMr3s r3c Cst|}t|}|dks||krtd}t|||}td|dD] }|||||}q"|dkr7||7}|dks?||krAtg}|ry||||d|d}}}||krn||8}|||||d}}||ksY||d||sEt|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)rgr[rminr IndexErrorr) rSrindexrrRcrrresultrLrLrMr's, r'cCs t|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)valuer^rLrLrMr0s r0ccsbt|ddd}t|}tdg|d|}t|td|dD] }||t||Vq!dS)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. NrrrWr)rgr[rrrr_sumprod)signalkernelrRwindowrJrLrLrMrs rcs0tgfdd}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. c3s.D]}|r |Vq|dSdSrI)r)elemr predicate transitionrLrM true_iterators z'before_and_after..true_iterator)r\r)rrrremainder_iteratorrLrrMrs  rcCs:t|d\}}}t|dt|dt|dt|||S)zReturn overlapping triplets from *iterable*. >>> list(triplewise('ABCDE')) [('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')] rNrq)rSrrt3rLrLrMrBs     rBcCs6t||}t|D] \}}tt|||dq t|SrI)rrr]r rr)rSrRrrr^rLrLrM_sliding_window_islice.s rccsBt|}tt||d|d}|D] }||t|VqdS)NrrW)r\rr rrg)rSrRrrrJrLrLrM_sliding_window_deque6s  rcCsR|dkr t||S|dkrt||S|dkrt|S|dkr"t|Std|)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*rrrrhrLrLrMr:?s  r:cCs6t|}ttttt|dd}ttjt ||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) rPrslicerrr[rTrigetitemr)rSseqslicesrLrLrMr;Xs r;cCs(dg}|D] }tt|d| f}q|S)zCompute a polynomial's coefficients from its roots. >>> 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)rPr)rootspolyrootrLrLrMr-fsr-ccst|dd}|dur(t|||}t||D]\}}||us"||kr%|VqdS|dur0t|n|}|d}z |||d|}Vq8tyMYdSw)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)rSrrVstop seq_indexrrrrLrLrMr"ss$   r"ccs|dkrdVd}td|d}t|d}t|d||D])}t|d|||EdHttt|||||||||||<||}q t|d|EdHdS)zdYield the primes less than n. >>> list(sieve(30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] rr)rrrN) bytearraymathisqrtr"bytesr[r)rRrVdatalimitrrLrLrMr9s. r9ccsd|dkr tdt|}tt||}r0|r"t||kr"td|Vtt||}sdSdS)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\rgr r[)rSrRrGrbatchrLrLrM_batcheds ri )rcCst|||dS)NrF)itertools_batched)rSrRrGrLrLrMrsrcCst|S)a Swap the rows and columns of the input matrix. >>> 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_strictrrLrLrMrAs rAcCstt||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)] )rrrf)matrixcolsrLrLrMr2sr2cCs&t|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[rrrr rA)m1m2rRrLrLrMr$s r$cCstj}td|dD]7}d}}d}|dkr9||||}||||}||||}||||}|dks||krA|Sq td)Nrrzprime or under 5)rgcdrr)rRrrtrJrKdrLrLrM_factor_pollardsrccs|dkrdStD]}||s|V||}||rq g}|dkr$|gng}|D]}|dks2t|r8||q(t|}||||f7}q(t|EdHdS)aYield the prime factors of n. >>> list(factor(360)) [2, 2, 2, 3, 3, 5] Finds small factors with trial division. Larger factors are either verified as prime with ``is_prime`` or split into smaller factors with Pollard's rho algorithm. rNri)_primes_below_211r rrr)rRprimeprimestodofactrLrLrMrs"  rcCs:t|}|dkr |dSttt|tt|}t||S)zEvaluate a polynomial at a specific value. Example: evaluating x^3 - 4 * x^2 - 17 * x + 60 at x = 2.5: >>> coefficients = [1, -4, -17, 60] >>> x = 2.5 >>> polynomial_eval(coefficients, x) 8.125 r)r[rTpowrreversedrr) coefficientsrJrRpowersrLrLrMr,(s   r,cCs tt|S)zfReturn the sum of the squares of the input values. >>> sum_of_squares([10, 20, 30]) 1400 )rrrrLrLrMr<9s r<cCs(t|}ttd|}tttj||S)aCompute the first derivative of a polynomial. Example: evaluating the derivative of x^3 - 4 * x^2 - 17 * x + 60 >>> coefficients = [1, -4, -17, 60] >>> derivative_coefficients = polynomial_derivative(coefficients) >>> derivative_coefficients [3, -8, -17] r)r[rrrPrTrirj)rrRrrLrLrMr.Bs r.cCs"tt|D]}|||8}q|S)zReturn the count of natural numbers up to *n* that are coprime with *n*. >>> totient(9) 6 >>> totient(12) 4 )rr)rRrrLrLrMr@Qsr@))i)r)i)I)ltT7)r=)lay)r iS_)l;n>)rrr )lp )rrrrrr)l)riEi$inii=ik)l%!Hn fW) rrrrrrrr%)cCsH|d|Ad}||?}d|>||kr|d@r|dks J||fS)z#Return s, d such that 2**s * d == nrr) bit_length)rRrrrLrLrM _shift_to_oddns$rcCs|dkr|d@rd|kr|ksJJt|d\}}t|||}|dks.||dkr0dSt|dD]}|||}||dkrGdSq6dS)NrrTF)rrr)rRbaserrrJ_rLrLrM_strong_probable_primews,   rcsdkrdvSd@r dr dr dr dr ds"d StD] \}}|kr.n q$fd d td D}tfd d|DS)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>rrrrrrrrrrrrFcsg|] }tddqS)rrrrrrLrMrszis_prime..@c3rrI)r)rrrrLrMrrzis_prime..)_perfect_testsrall)rRrbasesrLrrMr s0 r cCs td|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 NrrrLrLrMr#s r#)rrI)rN)NF)NN)rN)g__doc__rri collectionsrcollections.abcr functoolsrr itertoolsrrrr r r r r rrrrrandomrrrsysr__all__objectrrrrrrrr?r=r>rr&rrr1r)r(r%rrr7rur*rv ImportErrorrrwrrrr8r+r/rDrErCr!rr6r5r4r3r'r0rrrBrrr:r;r-r"r9rrrrAr2r$rrgrrr,r<r.r@rrrr r#rLrLrLrMs   8 3     (       (  -    *!  +     !