itertools — Functions creating iterators for efficient looping

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Цей модуль реалізує кілька будівельних блоків iterator, натхненних конструкціями з APL, Haskell і SML. Кожен був перероблений у формі, придатній для Python.

Модуль стандартизує основний набір швидких, ефективних інструментів пам’яті, які корисні окремо або в комбінації. Разом вони утворюють «алгебру ітераторів», що дає змогу створювати спеціалізовані інструменти лаконічно та ефективно на чистому Python.

Наприклад, SML надає інструмент табуляції: tabulate(f), який створює послідовність f(0), f(1), .... Такого ж ефекту можна досягти в Python, об’єднавши map() і count() для формування map(f, count()).

Нескінченні ітератори:

Ітератор	Аргументи	Результати	приклад
count()	[start[, step]]	початок, початок+крок, початок+2*крок, …	count(10) → 10 11 12 13 14 ...
cycle()	стор	p0, p1, … plast, p0, p1, …	cycle('ABCD') → A B C D A B C D ...
repeat()	елемент [,n]	елем, елем, елем, … нескінченно або до n разів	repeat(10, 3) → 10 10 10

Ітератори, що завершуються на найкоротшій вхідній послідовності:

Ітератор	Аргументи	Результати	приклад
accumulate()	p [,func]	p0, p0+p1, p0+p1+p2, …	accumulate([1,2,3,4,5]) → 1 3 6 10 15
batched()	p, n	(p0, p1, …, p_n-1), …	batched('ABCDEFG', n=3) → ABC DEF G
chain()	p, q, …	p0, p1, … plast, q0, q1, …	chain('ABC', 'DEF') → A B C D E F
chain.from_iterable()	ітерований	p0, p1, … plast, q0, q1, …	chain.from_iterable(['ABC', 'DEF']) → A B C D E F
compress()	дані, селектори	(d[0], якщо s[0]), (d[1], якщо s[1]), …	compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
dropwhile()	predicate, seq	seq[n], seq[n+1], starting when predicate fails	dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8
filterfalse()	predicate, seq	elements of seq where predicate(elem) fails	filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8
groupby()	ітерований [, ключ]	субітератори, згруповані за значенням ключа (v)	groupby(['A','B','DEF'], len) → (1, A B) (3, DEF)
islice()	seq, [початок,] зупинка [, крок]	елементи з seq[start:stop:step]	islice('ABCDEFG', 2, None) → C D E F G
pairwise()	ітерований	(p[0], p[1]), (p[1], p[2])	pairwise('ABCDEFG') → AB BC CD DE EF FG
starmap()	функція, посл	func(*seq[0]), func(*seq[1]), …	starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
takewhile()	predicate, seq	seq[0], seq[1], until predicate fails	takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
tee()	воно, п	it1, it2, … itn розділяє один ітератор на n
zip_longest()	p, q, …	(p[0], q[0]), (p[1], q[1]), …	zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

Комбінаторні ітератори:

Ітератор	Аргументи	Результати
product()	p, q, … [repeat=1]	декартовий добуток, еквівалентний вкладеному циклу for
permutations()	p[, r]	кортежі довжини r, усі можливі впорядкування, відсутність повторюваних елементів
combinations()	п, р	кортежі довжини r, у відсортованому порядку, без повторюваних елементів
комбінації_із_заміною()	п, р	кортежі довжини r, у відсортованому порядку, з повторюваними елементами

Приклади	Результати
product('ABCD', repeat=2)	AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD
перестановки('ABCD', 2)	AB AC AD BA BC BD CA CB CD DA DB DC
combinations('ABCD', 2)	AB AC AD BC BD CD
combinations_with_replacement('ABCD', 2)	AA AB AC AD BB BC BD CC CD DD

Itertool Functions

The following functions all construct and return iterators. Some provide streams of infinite length, so they should only be accessed by functions or loops that truncate the stream.

itertools.accumulate(iterable[, function, *, initial=None])

Make an iterator that returns accumulated sums or accumulated results from other binary functions.

The function defaults to addition. The function should accept two arguments, an accumulated total and a value from the iterable.

If an initial value is provided, the accumulation will start with that value and the output will have one more element than the input iterable.

Приблизно еквівалентно:

def accumulate(iterable, function=operator.add, *, initial=None):
    'Return running totals'
    # accumulate([1,2,3,4,5]) → 1 3 6 10 15
    # accumulate([1,2,3,4,5], initial=100) → 100 101 103 106 110 115
    # accumulate([1,2,3,4,5], operator.mul) → 1 2 6 24 120

    iterator = iter(iterable)
    total = initial
    if initial is None:
        try:
            total = next(iterator)
        except StopIteration:
            return

    yield total
    for element in iterator:
        total = function(total, element)
        yield total

To compute a running minimum, set function to min(). For a running maximum, set function to max(). Or for a running product, set function to operator.mul(). To build an amortization table, accumulate the interest and apply payments:

>>> data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
>>> list(accumulate(data, max))              # running maximum
[3, 4, 6, 6, 6, 9, 9, 9, 9, 9]
>>> list(accumulate(data, operator.mul))     # running product
[3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]

# Amortize a 5% loan of 1000 with 10 annual payments of 90
>>> update = lambda balance, payment: round(balance * 1.05) - payment
>>> list(accumulate(repeat(90, 10), update, initial=1_000))
[1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

Перегляньте functools.reduce() подібну функцію, яка повертає лише остаточне накопичене значення.

Added in version 3.2.

Змінено в версії 3.3: Added the optional function parameter.

Змінено в версії 3.8: Додано необов’язковий початковий параметр.

itertools.batched(iterable, n, *, strict=False)

Batch data from the iterable into tuples of length n. The last batch may be shorter than n.

If strict is true, will raise a ValueError if the final batch is shorter than n.

Loops over the input iterable and accumulates data into tuples up to size n. The input is consumed lazily, just enough to fill a batch. The result is yielded as soon as the batch is full or when the input iterable is exhausted:

>>> flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
>>> unflattened = list(batched(flattened_data, 2))
>>> unflattened
[('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

Приблизно еквівалентно:

def batched(iterable, n, *, strict=False):
    # batched('ABCDEFG', 3) → ABC DEF G
    if n < 1:
        raise ValueError('n must be at least one')
    iterator = iter(iterable)
    while batch := tuple(islice(iterator, n)):
        if strict and len(batch) != n:
            raise ValueError('batched(): incomplete batch')
        yield batch

Added in version 3.12.

Змінено в версії 3.13: Added the strict option.

itertools.chain(*iterables)

Make an iterator that returns elements from the first iterable until it is exhausted, then proceeds to the next iterable, until all of the iterables are exhausted. This combines multiple data sources into a single iterator. Roughly equivalent to:

def chain(*iterables):
    # chain('ABC', 'DEF') → A B C D E F
    for iterable in iterables:
        yield from iterable

classmethod chain.from_iterable(iterable)

Альтернативний конструктор для chain(). Отримує ланцюгові вхідні дані з одного ітерованого аргументу, який обчислюється ліниво. Приблизно еквівалентно:

def from_iterable(iterables):
    # chain.from_iterable(['ABC', 'DEF']) → A B C D E F
    for iterable in iterables:
        yield from iterable

itertools.combinations(iterable, r)

Повертає r довжину підпослідовностей елементів із вхідного iterable.

The output is a subsequence of product() keeping only entries that are subsequences of the iterable. The length of the output is given by math.comb() which computes n! / r! / (n - r)! when 0 ≤ r ≤ n or zero when r > n.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within each combination.

Приблизно еквівалентно:

def combinations(iterable, r):
    # combinations('ABCD', 2) → AB AC AD BC BD CD
    # combinations(range(4), 3) → 012 013 023 123

    pool = tuple(iterable)
    n = len(pool)
    if r > n:
        return
    indices = list(range(r))

    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != i + n - r:
                break
        else:
            return
        indices[i] += 1
        for j in range(i+1, r):
            indices[j] = indices[j-1] + 1
        yield tuple(pool[i] for i in indices)

itertools.combinations_with_replacement(iterable, r)

Повертає r довжину підпослідовностей елементів із вхідного iterable, що дозволяє повторювати окремі елементи більше одного разу.

The output is a subsequence of product() that keeps only entries that are subsequences (with possible repeated elements) of the iterable. The number of subsequence returned is (n + r - 1)! / r! / (n - 1)! when n > 0.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. if the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, the generated combinations will also be unique.

Приблизно еквівалентно:

def combinations_with_replacement(iterable, r):
    # combinations_with_replacement('ABC', 2) → AA AB AC BB BC CC

    pool = tuple(iterable)
    n = len(pool)
    if not n and r:
        return
    indices = [0] * r

    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != n - 1:
                break
        else:
            return
        indices[i:] = [indices[i] + 1] * (r - i)
        yield tuple(pool[i] for i in indices)

Added in version 3.1.

itertools.compress(data, selectors)

Make an iterator that returns elements from data where the corresponding element in selectors is true. Stops when either the data or selectors iterables have been exhausted. Roughly equivalent to:

def compress(data, selectors):
    # compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
    return (datum for datum, selector in zip(data, selectors) if selector)

Added in version 3.1.

itertools.count(start=0, step=1)

Make an iterator that returns evenly spaced values beginning with start. Can be used with map() to generate consecutive data points or with zip() to add sequence numbers. Roughly equivalent to:

def count(start=0, step=1):
    # count(10) → 10 11 12 13 14 ...
    # count(2.5, 0.5) → 2.5 3.0 3.5 ...
    n = start
    while True:
        yield n
        n += step

When counting with floating-point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: (start + step * i for i in count()).

Змінено в версії 3.1: Додано аргумент step і дозволено нецілі аргументи.

itertools.cycle(iterable)

Make an iterator returning elements from the iterable and saving a copy of each. When the iterable is exhausted, return elements from the saved copy. Repeats indefinitely. Roughly equivalent to:

def cycle(iterable):
    # cycle('ABCD') → A B C D A B C D A B C D ...

    saved = []
    for element in iterable:
        yield element
        saved.append(element)

    while saved:
        for element in saved:
            yield element

This itertool may require significant auxiliary storage (depending on the length of the iterable).

itertools.dropwhile(predicate, iterable)

Make an iterator that drops elements from the iterable while the predicate is true and afterwards returns every element. Roughly equivalent to:

def dropwhile(predicate, iterable):
    # dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8

    iterator = iter(iterable)
    for x in iterator:
        if not predicate(x):
            yield x
            break

    for x in iterator:
        yield x

Note this does not produce any output until the predicate first becomes false, so this itertool may have a lengthy start-up time.

itertools.filterfalse(predicate, iterable)

Make an iterator that filters elements from the iterable returning only those for which the predicate returns a false value. If predicate is None, returns the items that are false. Roughly equivalent to:

def filterfalse(predicate, iterable):
    # filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8

    if predicate is None:
        predicate = bool

    for x in iterable:
        if not predicate(x):
            yield x

itertools.groupby(iterable, key=None)

Створіть ітератор, який повертає послідовні ключі та групи з iterable. Ключ — це функція, яка обчислює значення ключа для кожного елемента. Якщо не вказано або має значення None, key за замовчуванням використовується як функція ідентифікації та повертає елемент без змін. Як правило, iterable вже має бути відсортований за тією самою ключовою функцією.

Робота groupby() подібна до фільтра uniq в Unix. Він генерує розрив або нову групу щоразу, коли змінюється значення ключової функції (саме тому зазвичай необхідно відсортувати дані за допомогою тієї самої ключової функції). Така поведінка відрізняється від GROUP BY SQL, яка агрегує загальні елементи незалежно від порядку введення.

Повернена група сама є ітератором, який ділиться основним ітератором із groupby(). Оскільки джерело є спільним, коли об’єкт groupby() розширено, попередня група більше не відображається. Отже, якщо ці дані знадобляться пізніше, їх слід зберегти як список:

groups = []
uniquekeys = []
data = sorted(data, key=keyfunc)
for k, g in groupby(data, keyfunc):
    groups.append(list(g))      # Store group iterator as a list
    uniquekeys.append(k)

groupby() приблизно еквівалентно:

def groupby(iterable, key=None):
    # [k for k, g in groupby('AAAABBBCCDAABBB')] → A B C D A B
    # [list(g) for k, g in groupby('AAAABBBCCD')] → AAAA BBB CC D

    keyfunc = (lambda x: x) if key is None else key
    iterator = iter(iterable)
    exhausted = False

    def _grouper(target_key):
        nonlocal curr_value, curr_key, exhausted
        yield curr_value
        for curr_value in iterator:
            curr_key = keyfunc(curr_value)
            if curr_key != target_key:
                return
            yield curr_value
        exhausted = True

    try:
        curr_value = next(iterator)
    except StopIteration:
        return
    curr_key = keyfunc(curr_value)

    while not exhausted:
        target_key = curr_key
        curr_group = _grouper(target_key)
        yield curr_key, curr_group
        if curr_key == target_key:
            for _ in curr_group:
                pass

itertools.islice(iterable, stop)

itertools.islice(iterable, start, stop[, step])

Make an iterator that returns selected elements from the iterable. Works like sequence slicing but does not support negative values for start, stop, or step.

If start is zero or None, iteration starts at zero. Otherwise, elements from the iterable are skipped until start is reached.

If stop is None, iteration continues until the input is exhausted, if at all. Otherwise, it stops at the specified position.

If step is None, the step defaults to one. Elements are returned consecutively unless step is set higher than one which results in items being skipped.

Приблизно еквівалентно:

def islice(iterable, *args):
    # islice('ABCDEFG', 2) → A B
    # islice('ABCDEFG', 2, 4) → C D
    # islice('ABCDEFG', 2, None) → C D E F G
    # islice('ABCDEFG', 0, None, 2) → A C E G

    s = slice(*args)
    start = 0 if s.start is None else s.start
    stop = s.stop
    step = 1 if s.step is None else s.step
    if start < 0 or (stop is not None and stop < 0) or step <= 0:
        raise ValueError

    indices = count() if stop is None else range(max(start, stop))
    next_i = start
    for i, element in zip(indices, iterable):
        if i == next_i:
            yield element
            next_i += step

If the input is an iterator, then fully consuming the islice advances the input iterator by max(start, stop) steps regardless of the step value.

itertools.pairwise(iterable)

Повертає послідовні пари, що перекриваються, взяті з вхідних даних iterable.

Кількість 2-кортежів у вихідному ітераторі буде на один менше, ніж кількість входів. Він буде порожнім, якщо вхідний ітератор має менше двох значень.

Приблизно еквівалентно:

def pairwise(iterable):
    # pairwise('ABCDEFG') → AB BC CD DE EF FG

    iterator = iter(iterable)
    a = next(iterator, None)

    for b in iterator:
        yield a, b
        a = b

Added in version 3.10.

itertools.permutations(iterable, r=None)

Return successive r length permutations of elements from the iterable.

Якщо r не вказано або має значення None, тоді r за замовчуванням відповідає довжині iterable і генеруються всі можливі перестановки повної довжини.

The output is a subsequence of product() where entries with repeated elements have been filtered out. The length of the output is given by math.perm() which computes n! / (n - r)! when 0 ≤ r ≤ n or zero when r > n.

The permutation tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within a permutation.

Приблизно еквівалентно:

def permutations(iterable, r=None):
    # permutations('ABCD', 2) → AB AC AD BA BC BD CA CB CD DA DB DC
    # permutations(range(3)) → 012 021 102 120 201 210

    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r > n:
        return

    indices = list(range(n))
    cycles = list(range(n, n-r, -1))
    yield tuple(pool[i] for i in indices[:r])

    while n:
        for i in reversed(range(r)):
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            return

itertools.product(*iterables, repeat=1)

Cartesian product of the input iterables.

Приблизно еквівалентно вкладеним циклам for у виразі генератора. Наприклад, product(A, B) повертає те саме, що ((x,y) для x в A для y у B).

Вкладені цикли обертаються як одометр із крайнім правим елементом, що просувається на кожній ітерації. Цей шаблон створює лексикографічне впорядкування, так що якщо ітеровані вхідні елементи відсортовані, кортежі продукту видаються в відсортованому порядку.

Щоб обчислити добуток iterable із самим собою, вкажіть кількість повторень за допомогою необов’язкового аргументу repeat. Наприклад, product(A, repeat=4) означає те саме, що product(A, A, A, A).

Ця функція приблизно еквівалентна наступному коду, за винятком того, що фактична реалізація не накопичує проміжні результати в пам’яті:

def product(*iterables, repeat=1):
    # product('ABCD', 'xy') → Ax Ay Bx By Cx Cy Dx Dy
    # product(range(2), repeat=3) → 000 001 010 011 100 101 110 111

    if repeat < 0:
        raise ValueError('repeat argument cannot be negative')
    pools = [tuple(pool) for pool in iterables] * repeat

    result = [[]]
    for pool in pools:
        result = [x+[y] for x in result for y in pool]

    for prod in result:
        yield tuple(prod)

Перед запуском product() він повністю споживає вхідні ітерації, зберігаючи пули значень у пам’яті для генерації продуктів. Відповідно, це корисно лише з обмеженими вхідними даними.

itertools.repeat(object[, times])

Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified.

Приблизно еквівалентно:

def repeat(object, times=None):
    # repeat(10, 3) → 10 10 10
    if times is None:
        while True:
            yield object
    else:
        for i in range(times):
            yield object

A common use for repeat is to supply a stream of constant values to map or zip:

>>> list(map(pow, range(10), repeat(2)))
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

itertools.starmap(function, iterable)

Make an iterator that computes the function using arguments obtained from the iterable. Used instead of map() when argument parameters have already been «pre-zipped» into tuples.

The difference between map() and starmap() parallels the distinction between function(a,b) and function(*c). Roughly equivalent to:

def starmap(function, iterable):
    # starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
    for args in iterable:
        yield function(*args)

itertools.takewhile(predicate, iterable)

Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:

def takewhile(predicate, iterable):
    # takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
    for x in iterable:
        if not predicate(x):
            break
        yield x

Note, the element that first fails the predicate condition is consumed from the input iterator and there is no way to access it. This could be an issue if an application wants to further consume the input iterator after takewhile has been run to exhaustion. To work around this problem, consider using more-itertools before_and_after() instead.

itertools.tee(iterable, n=2)

Повертає n незалежних ітераторів з одного ітератора.

Приблизно еквівалентно:

def tee(iterable, n=2):
    if n < 0:
        raise ValueError
    if n == 0:
        return ()
    iterator = _tee(iterable)
    result = [iterator]
    for _ in range(n - 1):
        result.append(_tee(iterator))
    return tuple(result)

class _tee:

    def __init__(self, iterable):
        it = iter(iterable)
        if isinstance(it, _tee):
            self.iterator = it.iterator
            self.link = it.link
        else:
            self.iterator = it
            self.link = [None, None]

    def __iter__(self):
        return self

    def __next__(self):
        link = self.link
        if link[1] is None:
            link[0] = next(self.iterator)
            link[1] = [None, None]
        value, self.link = link
        return value

When the input iterable is already a tee iterator object, all members of the return tuple are constructed as if they had been produced by the upstream tee() call. This «flattening step» allows nested tee() calls to share the same underlying data chain and to have a single update step rather than a chain of calls.

The flattening property makes tee iterators efficiently peekable:

def lookahead(tee_iterator):
     "Return the next value without moving the input forward"
     [forked_iterator] = tee(tee_iterator, 1)
     return next(forked_iterator)

>>> iterator = iter('abcdef')
>>> [iterator] = tee(iterator, 1)   # Make the input peekable
>>> next(iterator)                  # Move the iterator forward
'a'
>>> lookahead(iterator)             # Check next value
'b'
>>> next(iterator)                  # Continue moving forward
'b'

tee iterators are not threadsafe. A RuntimeError may be raised when simultaneously using iterators returned by the same tee() call, even if the original iterable is threadsafe.

Цей інструмент itertool може потребувати значного допоміжного сховища (залежно від того, скільки тимчасових даних потрібно зберегти). Загалом, якщо один ітератор використовує більшість або всі дані перед запуском іншого ітератора, швидше використовувати list() замість tee().

itertools.zip_longest(*iterables, fillvalue=None)

Make an iterator that aggregates elements from each of the iterables.

If the iterables are of uneven length, missing values are filled-in with fillvalue. If not specified, fillvalue defaults to None.

Iteration continues until the longest iterable is exhausted.

Приблизно еквівалентно:

def zip_longest(*iterables, fillvalue=None):
    # zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

    iterators = list(map(iter, iterables))
    num_active = len(iterators)
    if not num_active:
        return

    while True:
        values = []
        for i, iterator in enumerate(iterators):
            try:
                value = next(iterator)
            except StopIteration:
                num_active -= 1
                if not num_active:
                    return
                iterators[i] = repeat(fillvalue)
                value = fillvalue
            values.append(value)
        yield tuple(values)

If one of the iterables is potentially infinite, then the zip_longest() function should be wrapped with something that limits the number of calls (for example islice() or takewhile()).

Рецепти Itertools

У цьому розділі наведено рецепти для створення розширеного набору інструментів з використанням існуючих itertools як будівельних блоків.

The primary purpose of the itertools recipes is educational. The recipes show various ways of thinking about individual tools — for example, that chain.from_iterable is related to the concept of flattening. The recipes also give ideas about ways that the tools can be combined — for example, how starmap() and repeat() can work together. The recipes also show patterns for using itertools with the operator and collections modules as well as with the built-in itertools such as map(), filter(), reversed(), and enumerate().

A secondary purpose of the recipes is to serve as an incubator. The accumulate(), compress(), and pairwise() itertools started out as recipes. Currently, the sliding_window(), iter_index(), and sieve() recipes are being tested to see whether they prove their worth.

Substantially all of these recipes and many, many others can be installed from the more-itertools project found on the Python Package Index:

python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying toolset. Superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style. High speed is retained by preferring «vectorized» building blocks over the use of for-loops and generators which incur interpreter overhead.

from collections import deque
from contextlib import suppress
from functools import reduce
from math import sumprod, isqrt
from operator import itemgetter, getitem, mul, neg

def take(n, iterable):
    "Return first n items of the iterable as a list."
    return list(islice(iterable, n))

def prepend(value, iterable):
    "Prepend a single value in front of an iterable."
    # prepend(1, [2, 3, 4]) → 1 2 3 4
    return chain([value], iterable)

def tabulate(function, start=0):
    "Return function(0), function(1), ..."
    return map(function, count(start))

def repeatfunc(function, times=None, *args):
    "Repeat calls to a function with specified arguments."
    if times is None:
        return starmap(function, repeat(args))
    return starmap(function, repeat(args, times))

def flatten(list_of_lists):
    "Flatten one level of nesting."
    return chain.from_iterable(list_of_lists)

def ncycles(iterable, n):
    "Returns the sequence elements n times."
    return chain.from_iterable(repeat(tuple(iterable), n))

def loops(n):
    "Loop n times. Like range(n) but without creating integers."
    # for _ in loops(100): ...
    return repeat(None, n)

def tail(n, iterable):
    "Return an iterator over the last n items."
    # tail(3, 'ABCDEFG') → E F G
    return iter(deque(iterable, maxlen=n))

def consume(iterator, n=None):
    "Advance the iterator n-steps ahead. If n is None, consume entirely."
    # Use functions that consume iterators at C speed.
    if n is None:
        deque(iterator, maxlen=0)
    else:
        next(islice(iterator, n, n), None)

def nth(iterable, n, default=None):
    "Returns the nth item or a default value."
    return next(islice(iterable, n, None), default)

def quantify(iterable, predicate=bool):
    "Given a predicate that returns True or False, count the True results."
    return sum(map(predicate, iterable))

def first_true(iterable, default=False, predicate=None):
    "Returns the first true value or the *default* if there is no true value."
    # first_true([a,b,c], x) → a or b or c or x
    # first_true([a,b], x, f) → a if f(a) else b if f(b) else x
    return next(filter(predicate, iterable), default)

def all_equal(iterable, key=None):
    "Returns True if all the elements are equal to each other."
    # all_equal('4٤௪౪໔', key=int) → True
    return len(take(2, groupby(iterable, key))) <= 1

def unique_justseen(iterable, key=None):
    "Yield unique elements, preserving order. Remember only the element just seen."
    # unique_justseen('AAAABBBCCDAABBB') → A B C D A B
    # unique_justseen('ABBcCAD', str.casefold) → A B c A D
    if key is None:
        return map(itemgetter(0), groupby(iterable))
    return map(next, map(itemgetter(1), groupby(iterable, key)))

def unique_everseen(iterable, key=None):
    "Yield unique elements, preserving order. Remember all elements ever seen."
    # unique_everseen('AAAABBBCCDAABBB') → A B C D
    # unique_everseen('ABBcCAD', str.casefold) → A B c D
    seen = set()
    if key is None:
        for element in filterfalse(seen.__contains__, iterable):
            seen.add(element)
            yield element
    else:
        for element in iterable:
            k = key(element)
            if k not in seen:
                seen.add(k)
                yield element

def unique(iterable, key=None, reverse=False):
   "Yield unique elements in sorted order. Supports unhashable inputs."
   # unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4]
   sequenced = sorted(iterable, key=key, reverse=reverse)
   return unique_justseen(sequenced, key=key)

def sliding_window(iterable, n):
    "Collect data into overlapping fixed-length chunks or blocks."
    # sliding_window('ABCDEFG', 4) → ABCD BCDE CDEF DEFG
    iterator = iter(iterable)
    window = deque(islice(iterator, n - 1), maxlen=n)
    for x in iterator:
        window.append(x)
        yield tuple(window)

def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
    "Collect data into non-overlapping fixed-length chunks or blocks."
    # grouper('ABCDEFG', 3, fillvalue='x') → ABC DEF Gxx
    # grouper('ABCDEFG', 3, incomplete='strict') → ABC DEF ValueError
    # grouper('ABCDEFG', 3, incomplete='ignore') → ABC DEF
    iterators = [iter(iterable)] * n
    match incomplete:
        case 'fill':
            return zip_longest(*iterators, fillvalue=fillvalue)
        case 'strict':
            return zip(*iterators, strict=True)
        case 'ignore':
            return zip(*iterators)
        case _:
            raise ValueError('Expected fill, strict, or ignore')

def roundrobin(*iterables):
    "Visit input iterables in a cycle until each is exhausted."
    # roundrobin('ABC', 'D', 'EF') → A D E B F C
    # Algorithm credited to George Sakkis
    iterators = map(iter, iterables)
    for num_active in range(len(iterables), 0, -1):
        iterators = cycle(islice(iterators, num_active))
        yield from map(next, iterators)

def subslices(seq):
    "Return all contiguous non-empty subslices of a sequence."
    # subslices('ABCD') → A AB ABC ABCD B BC BCD C CD D
    slices = starmap(slice, combinations(range(len(seq) + 1), 2))
    return map(getitem, repeat(seq), slices)

def iter_index(iterable, value, start=0, stop=None):
    "Return indices where a value occurs in a sequence or iterable."
    # iter_index('AABCADEAF', 'A') → 0 1 4 7
    seq_index = getattr(iterable, 'index', None)
    if seq_index is None:
        iterator = islice(iterable, start, stop)
        for i, element in enumerate(iterator, start):
            if element is value or element == value:
                yield i
    else:
        stop = len(iterable) if stop is None else stop
        i = start
        with suppress(ValueError):
            while True:
                yield (i := seq_index(value, i, stop))
                i += 1

def iter_except(function, exception, first=None):
    "Convert a call-until-exception interface to an iterator interface."
    # iter_except(d.popitem, KeyError) → non-blocking dictionary iterator
    with suppress(exception):
        if first is not None:
            yield first()
        while True:
            yield function()

The following recipes have a more mathematical flavor:

def powerset(iterable):
    "Subsequences of the iterable from shortest to longest."
    # powerset([1,2,3]) → () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

def sum_of_squares(iterable):
    "Add up the squares of the input values."
    # sum_of_squares([10, 20, 30]) → 1400
    return sumprod(*tee(iterable))

def reshape(matrix, columns):
    "Reshape a 2-D matrix to have a given number of columns."
    # reshape([(0, 1), (2, 3), (4, 5)], 3) →  (0, 1, 2), (3, 4, 5)
    return batched(chain.from_iterable(matrix), columns, strict=True)

def transpose(matrix):
    "Swap the rows and columns of a 2-D matrix."
    # transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33)
    return zip(*matrix, strict=True)

def matmul(m1, m2):
    "Multiply two matrices."
    # matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) → (49, 80), (41, 60)
    n = len(m2[0])
    return batched(starmap(sumprod, product(m1, transpose(m2))), n)

def convolve(signal, kernel):
    """Discrete linear convolution of two iterables.
    Equivalent to polynomial multiplication.

    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.

    Article:  https://betterexplained.com/articles/intuitive-convolution/
    Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
    """
    # convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60
    # convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)
    # convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate
    # convolve(data, [1, -2, 1]) → 2nd derivative estimate
    kernel = tuple(kernel)[::-1]
    n = len(kernel)
    padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
    windowed_signal = sliding_window(padded_signal, n)
    return map(sumprod, repeat(kernel), windowed_signal)

def polynomial_from_roots(roots):
    """Compute a polynomial's coefficients from its roots.

       (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
    """
    # polynomial_from_roots([5, -4, 3]) → [1, -4, -17, 60]
    factors = zip(repeat(1), map(neg, roots))
    return list(reduce(convolve, factors, [1]))

def polynomial_eval(coefficients, x):
    """Evaluate a polynomial at a specific value.

    Computes with better numeric stability than Horner's method.
    """
    # Evaluate x³ -4x² -17x + 60 at x = 5
    # polynomial_eval([1, -4, -17, 60], x=5) → 0
    n = len(coefficients)
    if not n:
        return type(x)(0)
    powers = map(pow, repeat(x), reversed(range(n)))
    return sumprod(coefficients, powers)

def polynomial_derivative(coefficients):
    """Compute the first derivative of a polynomial.

       f(x)  =  x³ -4x² -17x + 60
       f'(x) = 3x² -8x  -17
    """
    # polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]
    n = len(coefficients)
    powers = reversed(range(1, n))
    return list(map(mul, coefficients, powers))

def sieve(n):
    "Primes less than n."
    # sieve(30) → 2 3 5 7 11 13 17 19 23 29
    if n > 2:
        yield 2
    data = bytearray((0, 1)) * (n // 2)
    for p in iter_index(data, 1, start=3, stop=isqrt(n) + 1):
        data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
    yield from iter_index(data, 1, start=3)

def factor(n):
    "Prime factors of n."
    # factor(99) → 3 3 11
    # factor(1_000_000_000_000_007) → 47 59 360620266859
    # factor(1_000_000_000_000_403) → 1000000000000403
    for prime in sieve(isqrt(n) + 1):
        while not n % prime:
            yield prime
            n //= prime
            if n == 1:
                return
    if n > 1:
        yield n

def is_prime(n):
    "Return True if n is prime."
    # is_prime(1_000_000_000_000_403) → True
    return n > 1 and next(factor(n)) == n

def totient(n):
    "Count of natural numbers up to n that are coprime to n."
    # https://mathworld.wolfram.com/TotientFunction.html
    # totient(12) → 4 because len([1, 5, 7, 11]) == 4
    for prime in set(factor(n)):
        n -= n // prime
    return n