o U h@sdZddlZddlmZddlmZddlmZddlm Z m Z ddl m Z m Z mZmZmZmZmZmZmZmZmZmZmZddlmZmZmZmZdd lmZm Z m!Z!m"Z"dd lm#Z#m$Z$m%Z%dd l&m'Z'gd Z(e)Z*ze+d dWn e,ye+Z-Ynwe e+d dZ-zddlm.Z/Wn e0yddZ/YnwddZ1dddZ2ddZ3dddZ4dddZ5dddZ6e7fddZ8d d!Z9e9Z:d"d#Z;d$d%Zd*d+Z?zdd,l m@ZAWn e0ye?Z@Yn wd-d.Z@e?je@_Gd/d0d0eBZCd1d2ZDd3d4ZEdd6d7ZFd8d9ZGd:d;ZHdd?ZJdd@dAZKddCdDZLddEdFZMddGdHZNdIdJdKdLZOddMdNZPdOdPZQdQdRZRdSdTZSdUdVZTdWdXZUdYdZZVd[d\ZWd]d^ZXd_d`ZYdadbZZdcddZ[dedfZ\ddgdhZ]didjZ^dBddkdlZ_e'dmkrddnl m`ZadBddodpZ`ne_Z`e_je`_dqdrZbdsdtZcdudvZddwdxZeefe^dyZgdzd{Zhd|d}Zid~dZjddZkddZlgdZme ddZnddZoepj#ZqddZrddZsddZtdS)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)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)sumprodcCs t||SN)r&)xyrXj/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/more_itertools/recipes.pyis rZcCtt||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] )listrniterablerXrXrYrKls rKcCst|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 )functionstartrXrXrYrI|srIcCs4t|trt|tdt||dStt||dS)zReturn an iterator over the last *n* items of *iterable*. >>> t = tail(3, 'ABCDEFG') >>> list(t) ['E', 'F', 'G'] rNmaxlen) isinstancerrmaxleniterrr]rXrXrYrJs rJcCs.|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 Nrrc)rnextr)iteratorr^rXrXrYr$s r$cCstt||d|S)zReturns the nth item or a default value. >>> l = range(10) >>> nth(l, 3) 3 >>> nth(l, 20, "zebra") 'zebra' N)rir)r_r^defaultrXrXrYr2s r2cCs,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)r_keyrjfirstsecondrXrXrYr!s r!cCr[)zcReturn the how many times the predicate is true. >>> quantify([True, False, True]) 2 )sumr`)r_predrXrXrYr=sr=cCst|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)r rr_rXrXrYr5s r5cCsttt||S)zvReturns the sequence elements *n* times >>> list(ncycles(["a", "b"], 3)) ['a', 'b', 'a', 'b', 'a', 'b'] )r from_iterablertupler_r^rXrXrYr1 sr1cCsttt||S)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. )ror`r)vec1vec2rXrXrYr&s r&cCs 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. )r rr) listOfListsrXrXrYr)!s r)cGs&|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)functimesargsrXrXrYrC-srCcCs 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`. Nrrizip)r_abrXrXrY _pairwiseHs  r)r6cCst|SrU)itertools_pairwiserqrXrXrYr6\sr6cseZdZdfdd ZZS)UnequalIterablesErrorNcs*d}|dur |dj|7}t|dS)Nz Iterables have different lengthsz/: index 0 has length {}; index {} has length {})formatsuper__init__)selfdetailsmsg __class__rXrYrcs zUnequalIterablesError.__init__rU)__name__ __module__ __qualname__r __classcell__rXrXrrYrbsrccs8t|dtiD]}|D] }|turtq |VqdS)N fillvalue)r_markerr) iterablescombovalrXrXrY_zip_equal_generatormsrcGsnz)t|d}t|dddD]\}}t|}||kr$t|||fdqt|WSty6t|YSw)Nr)r)rg enumeraterr| TypeErrorr)r first_sizeiitsizerXrXrY _zip_equalus    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. >>> iterator = grouper('ABCDEFG', 3, incomplete='strict') >>> list(iterator) # doctest: +IGNORE_EXCEPTION_DETAIL Traceback (most recent call last): ... UnequalIterablesError rrrSignorez Expected fill, strict, or ignore)rhrrr| ValueError)r_r^ incompleter iteratorsrXrXrYr*sr*cgsDtt|}tt|ddD]}tt||}tt|EdHqdS)aGVisit input iterables in a cycle until each is exhausted. >>> 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)r`rhrangergr rri)rr num_activerXrXrYrDs  rDcCsH|durt}t|d\}}}tt||\}}t|tt|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)boolrr`r r)rpr_t1t2pp1p2rXrXrYr7s r7cs,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|VqdSrU)r ).0rsrXrY zpowerset..r)r\r rrrrgrqrXrrYr;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) r_rlseenset seenset_addseenlist seenlist_adduse_keyelementkrXrXrYrPs( rPcCs4|dur ttdt|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)r`rrri)r_rlrXrXrYrQs rQFcCst|||d}t||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. )rlreverse)rl)sortedrQ)r_rlr sequencedrXrXrYrO,s rOccs<t||dur|V |Vq1swYdS)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)rx exceptionrmrXrXrYr,@s 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' )rifilter)r_rkrprXrXrYr']sr'rrcGs$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|qSrXrsrpoolrXrXrY sz"random_product..css|]}t|VqdSrU)rrrXrXrYrz!random_product..r)rrzpoolsrXrXrYrBqsrBcCs*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)rsrgr)r_rrrXrXrYrAs rAcs8t|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|]}|VqdSrUrXrrrrXrYrrz%random_combination..)rsrgrrr)r_rr^indicesrXrrYr@s r@cs@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|]}tVqdSrU)rrr^rXrYrrz6random_combination_with_replacement..c3rrUrXrrrXrYrr)rsrgrr)r_rrrX)r^rrYr?s r?c 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)rsrgrminr IndexErrorr) r_rindexrr^crrresultrXrXrYr3s, r3cCs 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 )valuerjrXrXrYr<s r<ccsbt|ddd}t|}tdg|d|}t|td|dD] }||t||Vq!dS)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 Nrrrcr)rsrgrr rr_sumprod)signalkernelr^windowrVrXrXrYr%s# r%cs0tgfdd}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|dSdSrU)r)elemr predicate transitionrXrY true_iterator-s z'before_and_after..true_iterator)rhr )rrrremainder_iteratorrXrrYr#s  r#cCs:t|d\}}}t|dt|dt|dt|||S)zReturn overlapping triplets from *iterable*. >>> list(triplewise('ABCDE')) [('A', 'B', 'C'), ('B', 'C', 'D'), ('C', 'D', 'E')] rNr{)r_rrt3rXrXrYrN=s     rNcCs6t||}t|D] \}}tt|||dq t|SrU)rrrirr|)r_r^rrrjrXrXrY_sliding_window_isliceMs rccsBt|}tt||d|d}|D] }||t|VqdS)Nrrc)rhrrrrs)r_r^rjrrVrXrXrY_sliding_window_dequeUs  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 )rrr6r|rrtrXrXrYrF^s  rFcCs4t|}ttttt|dd}ttt||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) r\rslicer rrgr`rr)r_seqslicesrXrXrYrGws rGcCs(dg}|D] }tt|d| f}q|S)uCompute a polynomial's coefficients from its roots. >>> 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%)rootspolyrootrXrXrYr9s r9ccst|dd}|dur(t|||}t||D]\}}||us"||kr%|VqdS|dur0t|n|}|d}tt |||d|}Vq<1sKwYdS)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)getattrrrrgrr)r_rrbstop seq_indexrjrrrXrXrYr-s    r-ccs|dkrdVd}td|d}t|d|t|ddD])}t|d|||EdHttt|||||||||||<||}qt|d|EdHdS)zeYield the primes less than n. >>> list(sieve(30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] rr)rrr)rN) bytearrayr-rbytesrgr)r^rbdatarrXrXrYrEs . rEccsd|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)rrhrsrrg)r_r^rSrjbatchrXrXrY_batcheds ri )r"cCst|||dS)NrR)itertools_batched)r_r^rSrXrXrYr"sr"cCst|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_strictrrXrXrYrMs rMcCstt||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)] )r"r rr)matrixcolsrXrXrYr>sr>cCs&t|d}tttt|t||S)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)rgr"rrrrM)m1m2r^rXrXrYr/ s r/cCstd|D]7}d}}d}|dkr4||||}||||}||||}t|||}|dks||kr<|Sqtd)Nrrzprime or under 5)rrr)r^r~rVrWdrXrXrY_factor_pollardsrccs|dkrdStD]}||s|V||}||rq g}|dkr$|gng}|D]}|dks2t|r8||q(t|}||||f7}q(t|EdHdS)aYield the prime factors of n. >>> list(factor(360)) [2, 2, 2, 3, 3, 5] Finds small factors with trial division. Larger factors are either verified as prime with ``is_prime`` or split into smaller factors with Pollard's rho algorithm. rNri)_primes_below_211r+rrr)r^primeprimestodofactrXrXrYr(.s"  r(cCs>t|}|dkrt|dSttt|tt|}t||S)adEvaluate a polynomial at a specific value. Computes with better numeric stability than Horner's method. Evaluate ``x^3 - 4 * x^2 - 17 * x + 60`` at ``x = 2.5``: >>> coefficients = [1, -4, -17, 60] >>> x = 2.5 >>> polynomial_eval(coefficients, x) 8.125 Supports all numeric types: int, float, complex, Decimal, Fraction. r)rgtyper`powrreversedrr) coefficientsrVr^powersrXrXrYr8Os   r8cCs tt|S)zReturn the sum of the squares of the input values. >>> sum_of_squares([10, 20, 30]) 1400 Supports all numeric types: int, float, complex, Decimal, Fraction. )rrrrXrXrYrHds rHcCs&t|}ttd|}ttt||S)uXCompute the first derivative of a polynomial. Evaluate the derivative of ``x³ - 4 x² - 17 x + 60``: >>> coefficients = [1, -4, -17, 60] >>> derivative_coefficients = polynomial_derivative(coefficients) >>> derivative_coefficients [3, -8, -17] Supports all numeric types: int, float, complex, Decimal, Fraction. r)rgrrr\r`r)rr^rrXrXrYr:os r:cCs"tt|D]}|||8}q|S)uReturn the count of natural numbers up to *n* that are coprime with *n*. Euler's totient function φ(n) gives the number of totatives. Totative are integers k in the range 1 ≤ k ≤ n such that gcd(n, k) = 1. >>> n = 9 >>> totient(n) 6 >>> totatives = [x for x in range(1, n) if gcd(n, x) == 1] >>> totatives [1, 2, 4, 5, 7, 8] >>> len(totatives) 6 Reference: https://en.wikipedia.org/wiki/Euler%27s_totient_function )rr()r^rrXrXrYrLsrL))i)r)i)I)ltT7)r=)lay)r iS_)l;n>)rrr )lp )rrr rr r)l)riEi$inii=ik)l%!Hn fW) rrr rr rr r%)cCsH|d|Ad}||?}d|>||kr|d@r|dks J||fS)z#Return s, d such that 2**s * d == nrr) bit_length)r^rrrXrXrY _shift_to_odds$rcCs|dkr|d@rd|kr|ksJJt|d\}}t|||}|dks.||dkr0dSt|dD]}|||}||dkrGdSq6dS)NrrTF)rrr)r^baserrrV_rXrXrY_strong_probable_primes,   rcsdkrdvSd@r dr dr dr dr ds"d StD] \}}|kr.n q$fd d td D}tfd d |DS)aReturn ``True`` if *n* is prime and ``False`` otherwise. 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. r >rrr rr rrrr rr rFc3s|] }tddVqdS)rrN)_private_randrangerrrXrYrszis_prime..@c3rrU)r)rrrrXrYrr)_perfect_testsrall)r^limitbasesrXrrYr+s!0 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 NrrrXrXrYr.s r.cGstttt||S)uNumber 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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