# Source code for random

    """Random variable generators.

        bytes
        -----
               uniform bytes (values between 0 and 255)

        integers
        --------
               uniform within range

        sequences
        ---------
               pick random element
               pick random sample
               pick weighted random sample
               generate random permutation

        distributions on the real line:
        ------------------------------
               uniform
               triangular
               normal (Gaussian)
               lognormal
               negative exponential
               gamma
               beta
               pareto
               Weibull

        distributions on the circle (angles 0 to 2pi)
        ---------------------------------------------
               circular uniform
               von Mises

        discrete distributions
        ----------------------
               binomial

    General notes on the underlying Mersenne Twister core generator:

    * The period is 2**19937-1.
    * It is one of the most extensively tested generators in existence.
    * The random() method is implemented in C, executes in a single Python step,
      and is, therefore, threadsafe.

    """

    # Translated by Guido van Rossum from C source provided by
    # Adrian Baddeley.  Adapted by Raymond Hettinger for use with
    # the Mersenne Twister  and os.urandom() core generators.

    from math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceil
    from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin
    from math import tau as TWOPI, floor as _floor, isfinite as _isfinite
    from math import lgamma as _lgamma, fabs as _fabs, log2 as _log2
    from os import urandom as _urandom
    from _collections_abc import Sequence as _Sequence
    from operator import index as _index
    from itertools import accumulate as _accumulate, repeat as _repeat
    from bisect import bisect as _bisect
    import os as _os
    import _random

    __all__ = [
        "Random",
        "SystemRandom",
        "betavariate",
        "binomialvariate",
        "choice",
        "choices",
        "expovariate",
        "gammavariate",
        "gauss",
        "getrandbits",
        "getstate",
        "lognormvariate",
        "normalvariate",
        "paretovariate",
        "randbytes",
        "randint",
        "random",
        "randrange",
        "sample",
        "seed",
        "setstate",
        "shuffle",
        "triangular",
        "uniform",
        "vonmisesvariate",
        "weibullvariate",
    ]

    NV_MAGICCONST = 4 * _exp(-0.5) / _sqrt(2.0)
    LOG4 = _log(4.0)
    SG_MAGICCONST = 1.0 + _log(4.5)
    BPF = 53        # Number of bits in a float
    RECIP_BPF = 2 ** -BPF
    _ONE = 1
    _sha512 = None

    class Random(_random.Random):
        """Random number generator base class used by bound module functions.

        Used to instantiate instances of Random to get generators that don't
        share state.

        Class Random can also be subclassed if you want to use a different basic
        generator of your own devising: in that case, override the following
        methods:  random(), seed(), getstate(), and setstate().
        Optionally, implement a getrandbits() method so that randrange()
        can cover arbitrarily large ranges.

        """

        VERSION = 3     # used by getstate/setstate

        def __init__(self, x=None):
            """Initialize an instance.

            Optional argument x controls seeding, as for Random.seed().
            """

            self.seed(x)
            self.gauss_next = None

        def seed(self, a=None, version=2):
            """Initialize internal state from a seed.

            The only supported seed types are None, int, float,
            str, bytes, and bytearray.

            None or no argument seeds from current time or from an operating
            system specific randomness source if available.

            If *a* is an int, all bits are used.

            For version 2 (the default), all of the bits are used if *a* is a str,
            bytes, or bytearray.  For version 1 (provided for reproducing random
            sequences from older versions of Python), the algorithm for str and
            bytes generates a narrower range of seeds.

            """

            if version == 1 and isinstance(a, (str, bytes)):
                a = a.decode('latin-1') if isinstance(a, bytes) else a
                x = ord(a[0]) << 7 if a else 0
                for c in map(ord, a):
                    x = ((1000003 * x) ^ c) & 0xFFFFFFFFFFFFFFFF
                x ^= len(a)
                a = -2 if x == -1 else x

            elif version == 2 and isinstance(a, (str, bytes, bytearray)):
                global _sha512
                if _sha512 is None:
                    try:
                        # hashlib is pretty heavy to load, try lean internal
                        # module first
                        from _sha2 import sha512 as _sha512
                    except ImportError:
                        # fallback to official implementation
                        from hashlib import sha512 as _sha512

                if isinstance(a, str):
                    a = a.encode()
                a = int.from_bytes(a + _sha512(a).digest())

            elif not isinstance(a, (type(None), int, float, str, bytes, bytearray)):
                raise TypeError('The only supported seed types are:\n'
                                'None, int, float, str, bytes, and bytearray.')

            super().seed(a)
            self.gauss_next = None

        def getstate(self):
            """Return internal state; can be passed to setstate() later."""
            return self.VERSION, super().getstate(), self.gauss_next

        def setstate(self, state):
            """Restore internal state from object returned by getstate()."""
            version = state[0]
            if version == 3:
                version, internalstate, self.gauss_next = state
                super().setstate(internalstate)
            elif version == 2:
                version, internalstate, self.gauss_next = state
                # In version 2, the state was saved as signed ints, which causes
                #   inconsistencies between 32/64-bit systems. The state is
                #   really unsigned 32-bit ints, so we convert negative ints from
                #   version 2 to positive longs for version 3.
                try:
                    internalstate = tuple(x % (2 ** 32) for x in internalstate)
                except ValueError as e:
                    raise TypeError from e
                super().setstate(internalstate)
            else:
                raise ValueError("state with version %s passed to "
                                 "Random.setstate() of version %s" %
                                 (version, self.VERSION))

        ## -------------------------------------------------------
        ## ---- Methods below this point do not need to be overridden or extended
        ## ---- when subclassing for the purpose of using a different core generator.

        ## -------------------- pickle support  -------------------

        # Issue 17489: Since __reduce__ was defined to fix #759889 this is no
        # longer called; we leave it here because it has been here since random was
        # rewritten back in 2001 and why risk breaking something.
        def __getstate__(self):  # for pickle
            return self.getstate()

        def __setstate__(self, state):  # for pickle
            self.setstate(state)

        def __reduce__(self):
            return self.__class__, (), self.getstate()

        ## ---- internal support method for evenly distributed integers ----

        def __init_subclass__(cls, /, **kwargs):
            """Control how subclasses generate random integers.

            The algorithm a subclass can use depends on the random() and/or
            getrandbits() implementation available to it and determines
            whether it can generate random integers from arbitrarily large
            ranges.
            """

            for c in cls.__mro__:
                if '_randbelow' in c.__dict__:
                    # just inherit it
                    break
                if 'getrandbits' in c.__dict__:
                    cls._randbelow = cls._randbelow_with_getrandbits
                    break
                if 'random' in c.__dict__:
                    cls._randbelow = cls._randbelow_without_getrandbits
                    break

        def _randbelow_with_getrandbits(self, n):
            "Return a random int in the range [0,n).  Defined for n > 0."

            getrandbits = self.getrandbits
            k = n.bit_length()
            r = getrandbits(k)  # 0 <= r < 2**k
            while r >= n:
                r = getrandbits(k)
            return r

        def _randbelow_without_getrandbits(self, n, maxsize=1<<BPF):
            """Return a random int in the range [0,n).  Defined for n > 0.

            The implementation does not use getrandbits, but only random.
            """

            random = self.random
            if n >= maxsize:
                from warnings import warn
                warn("Underlying random() generator does not supply \n"
                     "enough bits to choose from a population range this large.\n"
                     "To remove the range limitation, add a getrandbits() method.")
                return _floor(random() * n)
            rem = maxsize % n
            limit = (maxsize - rem) / maxsize   # int(limit * maxsize) % n == 0
            r = random()
            while r >= limit:
                r = random()
            return _floor(r * maxsize) % n

        _randbelow = _randbelow_with_getrandbits

        ## --------------------------------------------------------
        ## ---- Methods below this point generate custom distributions
        ## ---- based on the methods defined above.  They do not
        ## ---- directly touch the underlying generator and only
        ## ---- access randomness through the methods:  random(),
        ## ---- getrandbits(), or _randbelow().

        ## -------------------- bytes methods ---------------------

        def randbytes(self, n):
            """Generate n random bytes."""
            return self.getrandbits(n * 8).to_bytes(n, 'little')

        ## -------------------- integer methods  -------------------

        def randrange(self, start, stop=None, step=_ONE):
            """Choose a random item from range(stop) or range(start, stop[, step]).

            Roughly equivalent to ``choice(range(start, stop, step))`` but
            supports arbitrarily large ranges and is optimized for common cases.

            """

            # This code is a bit messy to make it fast for the
            # common case while still doing adequate error checking.
            istart = _index(start)
            if stop is None:
                # We don't check for "step != 1" because it hasn't been
                # type checked and converted to an integer yet.
                if step is not _ONE:
                    raise TypeError("Missing a non-None stop argument")
                if istart > 0:
                    return self._randbelow(istart)
                raise ValueError("empty range for randrange()")

            # Stop argument supplied.
            istop = _index(stop)
            width = istop - istart
            istep = _index(step)
            # Fast path.
            if istep == 1:
                if width > 0:
                    return istart + self._randbelow(width)
                raise ValueError(f"empty range in randrange({start}, {stop})")

            # Non-unit step argument supplied.
            if istep > 0:
                n = (width + istep - 1) // istep
            elif istep < 0:
                n = (width + istep + 1) // istep
            else:
                raise ValueError("zero step for randrange()")
            if n <= 0:
                raise ValueError(f"empty range in randrange({start}, {stop}, {step})")
            return istart + istep * self._randbelow(n)

        def randint(self, a, b):
            """Return random integer in range [a, b], including both end points.
            """

            return self.randrange(a, b+1)

        ## -------------------- sequence methods  -------------------

        def choice(self, seq):
            """Choose a random element from a non-empty sequence."""

            # As an accommodation for NumPy, we don't use "if not seq"
            # because bool(numpy.array()) raises a ValueError.
            if not len(seq):
                raise IndexError('Cannot choose from an empty sequence')
            return seq[self._randbelow(len(seq))]

        def shuffle(self, x):
            """Shuffle list x in place, and return None."""

            randbelow = self._randbelow
            for i in reversed(range(1, len(x))):
                # pick an element in x[:i+1] with which to exchange x[i]
                j = randbelow(i + 1)
                x[i], x[j] = x[j], x[i]

        def sample(self, population, k, *, counts=None):
            """Chooses k unique random elements from a population sequence.

            Returns a new list containing elements from the population while
            leaving the original population unchanged.  The resulting list is
            in selection order so that all sub-slices will also be valid random
            samples.  This allows raffle winners (the sample) to be partitioned
            into grand prize and second place winners (the subslices).

            Members of the population need not be hashable or unique.  If the
            population contains repeats, then each occurrence is a possible
            selection in the sample.

            Repeated elements can be specified one at a time or with the optional
            counts parameter.  For example:

                sample(['red', 'blue'], counts=[4, 2], k=5)

            is equivalent to:

                sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k=5)

            To choose a sample from a range of integers, use range() for the
            population argument.  This is especially fast and space efficient
            for sampling from a large population:

                sample(range(10000000), 60)

            """

            # Sampling without replacement entails tracking either potential
            # selections (the pool) in a list or previous selections in a set.

            # When the number of selections is small compared to the
            # population, then tracking selections is efficient, requiring
            # only a small set and an occasional reselection.  For
            # a larger number of selections, the pool tracking method is
            # preferred since the list takes less space than the
            # set and it doesn't suffer from frequent reselections.

            # The number of calls to _randbelow() is kept at or near k, the
            # theoretical minimum.  This is important because running time
            # is dominated by _randbelow() and because it extracts the
            # least entropy from the underlying random number generators.

            # Memory requirements are kept to the smaller of a k-length
            # set or an n-length list.

            # There are other sampling algorithms that do not require
            # auxiliary memory, but they were rejected because they made
            # too many calls to _randbelow(), making them slower and
            # causing them to eat more entropy than necessary.

            if not isinstance(population, _Sequence):
                raise TypeError("Population must be a sequence.  "
                                "For dicts or sets, use sorted(d).")
            n = len(population)
            if counts is not None:
                cum_counts = list(_accumulate(counts))
                if len(cum_counts) != n:
                    raise ValueError('The number of counts does not match the population')
                total = cum_counts.pop() if cum_counts else 0
                if not isinstance(total, int):
                    raise TypeError('Counts must be integers')
                if total < 0:
                    raise ValueError('Counts must be non-negative')
                selections = self.sample(range(total), k=k)
                bisect = _bisect
                return [population[bisect(cum_counts, s)] for s in selections]
            randbelow = self._randbelow
            if not 0 <= k <= n:
                raise ValueError("Sample larger than population or is negative")
            result = [None] * k
            setsize = 21        # size of a small set minus size of an empty list
            if k > 5:
                setsize += 4 ** _ceil(_log(k * 3, 4))  # table size for big sets
            if n <= setsize:
                # An n-length list is smaller than a k-length set.
                # Invariant:  non-selected at pool[0 : n-i]
                pool = list(population)
                for i in range(k):
                    j = randbelow(n - i)
                    result[i] = pool[j]
                    pool[j] = pool[n - i - 1]  # move non-selected item into vacancy
            else:
                selected = set()
                selected_add = selected.add
                for i in range(k):
                    j = randbelow(n)
                    while j in selected:
                        j = randbelow(n)
                    selected_add(j)
                    result[i] = population[j]
            return result

        def choices(self, population, weights=None, *, cum_weights=None, k=1):
            """Return a k sized list of population elements chosen with replacement.

            If the relative weights or cumulative weights are not specified,
            the selections are made with equal probability.

            """
            random = self.random
            n = len(population)
            if cum_weights is None:
                if weights is None:
                    floor = _floor
                    n += 0.0    # convert to float for a small speed improvement
                    return [population[floor(random() * n)] for i in _repeat(None, k)]
                try:
                    cum_weights = list(_accumulate(weights))
                except TypeError:
                    if not isinstance(weights, int):
                        raise
                    k = weights
                    raise TypeError(
                        f'The number of choices must be a keyword argument: {k=}'
                    ) from None
            elif weights is not None:
                raise TypeError('Cannot specify both weights and cumulative weights')
            if len(cum_weights) != n:
                raise ValueError('The number of weights does not match the population')
            total = cum_weights[-1] + 0.0   # convert to float
            if total <= 0.0:
                raise ValueError('Total of weights must be greater than zero')
            if not _isfinite(total):
                raise ValueError('Total of weights must be finite')
            bisect = _bisect
            hi = n - 1
            return [population[bisect(cum_weights, random() * total, 0, hi)]
                    for i in _repeat(None, k)]

        ## -------------------- real-valued distributions  -------------------

        def uniform(self, a, b):
            """Get a random number in the range [a, b) or [a, b] depending on rounding.

            The mean (expected value) and variance of the random variable are:

                E[X] = (a + b) / 2
                Var[X] = (b - a) ** 2 / 12

            """
            return a + (b - a) * self.random()

        def triangular(self, low=0.0, high=1.0, mode=None):
            """Triangular distribution.

            Continuous distribution bounded by given lower and upper limits,
            and having a given mode value in-between.

            http://en.wikipedia.org/wiki/Triangular_distribution

            The mean (expected value) and variance of the random variable are:

                E[X] = (low + high + mode) / 3
                Var[X] = (low**2 + high**2 + mode**2 - low*high - low*mode - high*mode) / 18

            """
            u = self.random()
            try:
                c = 0.5 if mode is None else (mode - low) / (high - low)
            except ZeroDivisionError:
                return low
            if u > c:
                u = 1.0 - u
                c = 1.0 - c
                low, high = high, low
            return low + (high - low) * _sqrt(u * c)

        def normalvariate(self, mu=0.0, sigma=1.0):
            """Normal distribution.

            mu is the mean, and sigma is the standard deviation.

            """
            # Uses Kinderman and Monahan method. Reference: Kinderman,
            # A.J. and Monahan, J.F., "Computer generation of random
            # variables using the ratio of uniform deviates", ACM Trans
            # Math Software, 3, (1977), pp257-260.

            random = self.random
            while True:
                u1 = random()
                u2 = 1.0 - random()
                z = NV_MAGICCONST * (u1 - 0.5) / u2
                zz = z * z / 4.0
                if zz <= -_log(u2):
                    break
            return mu + z * sigma

        def gauss(self, mu=0.0, sigma=1.0):
            """Gaussian distribution.

            mu is the mean, and sigma is the standard deviation.  This is
            slightly faster than the normalvariate() function.

            Not thread-safe without a lock around calls.

            """
            # When x and y are two variables from [0, 1), uniformly
            # distributed, then
            #
            #    cos(2*pi*x)*sqrt(-2*log(1-y))
            #    sin(2*pi*x)*sqrt(-2*log(1-y))
            #
            # are two *independent* variables with normal distribution
            # (mu = 0, sigma = 1).
            # (Lambert Meertens)
            # (corrected version; bug discovered by Mike Miller, fixed by LM)

            # Multithreading note: When two threads call this function
            # simultaneously, it is possible that they will receive the
            # same return value.  The window is very small though.  To
            # avoid this, you have to use a lock around all calls.  (I
            # didn't want to slow this down in the serial case by using a
            # lock here.)

            random = self.random
            z = self.gauss_next
            self.gauss_next = None
            if z is None:
                x2pi = random() * TWOPI
                g2rad = _sqrt(-2.0 * _log(1.0 - random()))
                z = _cos(x2pi) * g2rad
                self.gauss_next = _sin(x2pi) * g2rad

            return mu + z * sigma

        def lognormvariate(self, mu, sigma):
            """Log normal distribution.

            If you take the natural logarithm of this distribution, you'll get a
            normal distribution with mean mu and standard deviation sigma.
            mu can have any value, and sigma must be greater than zero.

            """
            return _exp(self.normalvariate(mu, sigma))

        def expovariate(self, lambd=1.0):
            """Exponential distribution.

            lambd is 1.0 divided by the desired mean.  It should be
            nonzero.  (The parameter would be called "lambda", but that is
            a reserved word in Python.)  Returned values range from 0 to
            positive infinity if lambd is positive, and from negative
            infinity to 0 if lambd is negative.

            The mean (expected value) and variance of the random variable are:

                E[X] = 1 / lambd
                Var[X] = 1 / lambd ** 2

            """
            # we use 1-random() instead of random() to preclude the
            # possibility of taking the log of zero.

            return -_log(1.0 - self.random()) / lambd

        def vonmisesvariate(self, mu, kappa):
            """Circular data distribution.

            mu is the mean angle, expressed in radians between 0 and 2*pi, and
            kappa is the concentration parameter, which must be greater than or
            equal to zero.  If kappa is equal to zero, this distribution reduces
            to a uniform random angle over the range 0 to 2*pi.

            """
            # Based upon an algorithm published in: Fisher, N.I.,
            # "Statistical Analysis of Circular Data", Cambridge
            # University Press, 1993.

            # Thanks to Magnus Kessler for a correction to the
            # implementation of step 4.

            random = self.random
            if kappa <= 1e-6:
                return TWOPI * random()

            s = 0.5 / kappa
            r = s + _sqrt(1.0 + s * s)

            while True:
                u1 = random()
                z = _cos(_pi * u1)

                d = z / (r + z)
                u2 = random()
                if u2 < 1.0 - d * d or u2 <= (1.0 - d) * _exp(d):
                    break

            q = 1.0 / r
            f = (q + z) / (1.0 + q * z)
            u3 = random()
            if u3 > 0.5:
                theta = (mu + _acos(f)) % TWOPI
            else:
                theta = (mu - _acos(f)) % TWOPI

            return theta

        def gammavariate(self, alpha, beta):
            """Gamma distribution.  Not the gamma function!

            Conditions on the parameters are alpha > 0 and beta > 0.

            The probability distribution function is:

                        x ** (alpha - 1) * math.exp(-x / beta)
              pdf(x) =  --------------------------------------
                          math.gamma(alpha) * beta ** alpha

            The mean (expected value) and variance of the random variable are:

                E[X] = alpha * beta
                Var[X] = alpha * beta ** 2

            """

            # Warning: a few older sources define the gamma distribution in terms
            # of alpha > -1.0
            if alpha <= 0.0 or beta <= 0.0:
                raise ValueError('gammavariate: alpha and beta must be > 0.0')

            random = self.random
            if alpha > 1.0:

                # Uses R.C.H. Cheng, "The generation of Gamma
                # variables with non-integral shape parameters",
                # Applied Statistics, (1977), 26, No. 1, p71-74

                ainv = _sqrt(2.0 * alpha - 1.0)
                bbb = alpha - LOG4
                ccc = alpha + ainv

                while True:
                    u1 = random()
                    if not 1e-7 < u1 < 0.9999999:
                        continue
                    u2 = 1.0 - random()
                    v = _log(u1 / (1.0 - u1)) / ainv
                    x = alpha * _exp(v)
                    z = u1 * u1 * u2
                    r = bbb + ccc * v - x
                    if r + SG_MAGICCONST - 4.5 * z >= 0.0 or r >= _log(z):
                        return x * beta

            elif alpha == 1.0:
                # expovariate(1/beta)
                return -_log(1.0 - random()) * beta

            else:
                # alpha is between 0 and 1 (exclusive)
                # Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
                while True:
                    u = random()
                    b = (_e + alpha) / _e
                    p = b * u
                    if p <= 1.0:
                        x = p ** (1.0 / alpha)
                    else:
                        x = -_log((b - p) / alpha)
                    u1 = random()
                    if p > 1.0:
                        if u1 <= x ** (alpha - 1.0):
                            break
                    elif u1 <= _exp(-x):
                        break
                return x * beta

        def betavariate(self, alpha, beta):
            """Beta distribution.

            Conditions on the parameters are alpha > 0 and beta > 0.
            Returned values range between 0 and 1.

            The mean (expected value) and variance of the random variable are:

                E[X] = alpha / (alpha + beta)
                Var[X] = alpha * beta / ((alpha + beta)**2 * (alpha + beta + 1))

            """
            ## See
            ## http://mail.python.org/pipermail/python-bugs-list/2001-January/003752.html
            ## for Ivan Frohne's insightful analysis of why the original implementation:
            ##
            ##    def betavariate(self, alpha, beta):
            ##        # Discrete Event Simulation in C, pp 87-88.
            ##
            ##        y = self.expovariate(alpha)
            ##        z = self.expovariate(1.0/beta)
            ##        return z/(y+z)
            ##
            ## was dead wrong, and how it probably got that way.

            # This version due to Janne Sinkkonen, and matches all the std
            # texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
            y = self.gammavariate(alpha, 1.0)
            if y:
                return y / (y + self.gammavariate(beta, 1.0))
            return 0.0

        def paretovariate(self, alpha):
            """Pareto distribution.  alpha is the shape parameter."""
            # Jain, pg. 495

            u = 1.0 - self.random()
            return u ** (-1.0 / alpha)

        def weibullvariate(self, alpha, beta):
            """Weibull distribution.

            alpha is the scale parameter and beta is the shape parameter.

            """
            # Jain, pg. 499; bug fix courtesy Bill Arms

            u = 1.0 - self.random()
            return alpha * (-_log(u)) ** (1.0 / beta)

        ## -------------------- discrete  distributions  ---------------------

        def binomialvariate(self, n=1, p=0.5):
            """Binomial random variable.

            Gives the number of successes for *n* independent trials
            with the probability of success in each trial being *p*:

                sum(random() < p for i in range(n))

            Returns an integer in the range:   0 <= X <= n

            The mean (expected value) and variance of the random variable are:

                E[X] = n * p
                Var[x] = n * p * (1 - p)

            """
            # Error check inputs and handle edge cases
            if n < 0:
                raise ValueError("n must be non-negative")
            if p <= 0.0 or p >= 1.0:
                if p == 0.0:
                    return 0
                if p == 1.0:
                    return n
                raise ValueError("p must be in the range 0.0 <= p <= 1.0")

            random = self.random

            # Fast path for a common case
            if n == 1:
                return _index(random() < p)

            # Exploit symmetry to establish:  p <= 0.5
            if p > 0.5:
                return n - self.binomialvariate(n, 1.0 - p)

            if n * p < 10.0:
                # BG: Geometric method by Devroye with running time of O(np).
                # https://dl.acm.org/doi/pdf/10.1145/42372.42381
                x = y = 0
                c = _log2(1.0 - p)
                if not c:
                    return x
                while True:
                    y += _floor(_log2(random()) / c) + 1
                    if y > n:
                        return x
                    x += 1

            # BTRS: Transformed rejection with squeeze method by Wolfgang Hörmann
            # https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.8407&rep=rep1&type=pdf
            assert n*p >= 10.0 and p <= 0.5
            setup_complete = False

            spq = _sqrt(n * p * (1.0 - p))  # Standard deviation of the distribution
            b = 1.15 + 2.53 * spq
            a = -0.0873 + 0.0248 * b + 0.01 * p
            c = n * p + 0.5
            vr = 0.92 - 4.2 / b

            while True:

                u = random()
                u -= 0.5
                us = 0.5 - _fabs(u)
                k = _floor((2.0 * a / us + b) * u + c)
                if k < 0 or k > n:
                    continue

                # The early-out "squeeze" test substantially reduces
                # the number of acceptance condition evaluations.
                v = random()
                if us >= 0.07 and v <= vr:
                    return k

                # Acceptance-rejection test.
                # Note, the original paper erroneously omits the call to log(v)
                # when comparing to the log of the rescaled binomial distribution.
                if not setup_complete:
                    alpha = (2.83 + 5.1 / b) * spq
                    lpq = _log(p / (1.0 - p))
                    m = _floor((n + 1) * p)         # Mode of the distribution
                    h = _lgamma(m + 1) + _lgamma(n - m + 1)
                    setup_complete = True           # Only needs to be done once
                v *= alpha / (a / (us * us) + b)
                if _log(v) <= h - _lgamma(k + 1) - _lgamma(n - k + 1) + (k - m) * lpq:
                    return k

    ## ------------------------------------------------------------------
    ## --------------- Operating System Random Source  ------------------

    class SystemRandom(Random):
        """Alternate random number generator using sources provided
        by the operating system (such as /dev/urandom on Unix or
        CryptGenRandom on Windows).

         Not available on all systems (see os.urandom() for details).

        """

        def random(self):
            """Get the next random number in the range 0.0 <= X < 1.0."""
            return (int.from_bytes(_urandom(7)) >> 3) * RECIP_BPF

        def getrandbits(self, k):
            """getrandbits(k) -> x.  Generates an int with k random bits."""
            if k < 0:
                raise ValueError('number of bits must be non-negative')
            numbytes = (k + 7) // 8                       # bits / 8 and rounded up
            x = int.from_bytes(_urandom(numbytes))
            return x >> (numbytes * 8 - k)                # trim excess bits

        def randbytes(self, n):
            """Generate n random bytes."""
            # os.urandom(n) fails with ValueError for n < 0
            # and returns an empty bytes string for n == 0.
            return _urandom(n)

        def seed(self, *args, **kwds):
            "Stub method.  Not used for a system random number generator."
            return None

        def _notimplemented(self, *args, **kwds):
            "Method should not be called for a system random number generator."
            raise NotImplementedError('System entropy source does not have state.')
        getstate = setstate = _notimplemented

    # ----------------------------------------------------------------------
    # Create one instance, seeded from current time, and export its methods
    # as module-level functions.  The functions share state across all uses
    # (both in the user's code and in the Python libraries), but that's fine
    # for most programs and is easier for the casual user than making them
    # instantiate their own Random() instance.

    _inst = Random()
    seed = _inst.seed
    random = _inst.random
    uniform = _inst.uniform
    triangular = _inst.triangular
    randint = _inst.randint
    choice = _inst.choice
    randrange = _inst.randrange
    sample = _inst.sample
    shuffle = _inst.shuffle
    choices = _inst.choices
    normalvariate = _inst.normalvariate
    lognormvariate = _inst.lognormvariate
    expovariate = _inst.expovariate
    vonmisesvariate = _inst.vonmisesvariate
    gammavariate = _inst.gammavariate
    gauss = _inst.gauss
    betavariate = _inst.betavariate
    binomialvariate = _inst.binomialvariate
    paretovariate = _inst.paretovariate
    weibullvariate = _inst.weibullvariate
    getstate = _inst.getstate
    setstate = _inst.setstate
    getrandbits = _inst.getrandbits
    randbytes = _inst.randbytes

    ## ------------------------------------------------------
    ## ----------------- test program -----------------------

    def _test_generator(n, func, args):
        from statistics import stdev, fmean as mean
        from time import perf_counter

        t0 = perf_counter()
        data = [func(*args) for i in _repeat(None, n)]
        t1 = perf_counter()

        xbar = mean(data)
        sigma = stdev(data, xbar)
        low = min(data)
        high = max(data)

        print(f'{t1 - t0:.3f} sec, {n} times {func.__name__}{args!r}')
        print('avg %g, stddev %g, min %g, max %g\n' % (xbar, sigma, low, high))

    def _test(N=10_000):
        _test_generator(N, random, ())
        _test_generator(N, normalvariate, (0.0, 1.0))
        _test_generator(N, lognormvariate, (0.0, 1.0))
        _test_generator(N, vonmisesvariate, (0.0, 1.0))
        _test_generator(N, binomialvariate, (15, 0.60))
        _test_generator(N, binomialvariate, (100, 0.75))
        _test_generator(N, gammavariate, (0.01, 1.0))
        _test_generator(N, gammavariate, (0.1, 1.0))
        _test_generator(N, gammavariate, (0.1, 2.0))
        _test_generator(N, gammavariate, (0.5, 1.0))
        _test_generator(N, gammavariate, (0.9, 1.0))
        _test_generator(N, gammavariate, (1.0, 1.0))
        _test_generator(N, gammavariate, (2.0, 1.0))
        _test_generator(N, gammavariate, (20.0, 1.0))
        _test_generator(N, gammavariate, (200.0, 1.0))
        _test_generator(N, gauss, (0.0, 1.0))
        _test_generator(N, betavariate, (3.0, 3.0))
        _test_generator(N, triangular, (0.0, 1.0, 1.0 / 3.0))

    ## ------------------------------------------------------
    ## ------------------ fork support  ---------------------

    if hasattr(_os, "fork"):
        _os.register_at_fork(after_in_child=_inst.seed)

    # ------------------------------------------------------
    # -------------- command-line interface ----------------

    def _parse_args(arg_list: list[str] | None):
        import argparse
        parser = argparse.ArgumentParser(
            formatter_class=argparse.RawTextHelpFormatter)
        group = parser.add_mutually_exclusive_group()
        group.add_argument(
            "-c", "--choice", nargs="+",
            help="print a random choice")
        group.add_argument(
            "-i", "--integer", type=int, metavar="N",
            help="print a random integer between 1 and N inclusive")
        group.add_argument(
            "-f", "--float", type=float, metavar="N",
            help="print a random floating-point number between 0 and N inclusive")
        group.add_argument(
            "--test", type=int, const=10_000, nargs="?",
            help=argparse.SUPPRESS)
        parser.add_argument("input", nargs="*",
                            help="""\
    if no options given, output depends on the input
        string or multiple: same as --choice
        integer: same as --integer
        float: same as --float""")
        args = parser.parse_args(arg_list)
        return args, parser.format_help()

    def main(arg_list: list[str] | None = None) -> int | str:
        args, help_text = _parse_args(arg_list)

        # Explicit arguments
        if args.choice:
            return choice(args.choice)

        if args.integer is not None:
            return randint(1, args.integer)

        if args.float is not None:
            return uniform(0, args.float)

        if args.test:
            _test(args.test)
            return ""

        # No explicit argument, select based on input
        if len(args.input) == 1:
            val = args.input[0]
            try:
                # Is it an integer?
                val = int(val)
                return randint(1, val)
            except ValueError:
                try:
                    # Is it a float?
                    val = float(val)
                    return uniform(0, val)
                except ValueError:
                    # Split in case of space-separated string: "a b c"
                    return choice(val.split())

        if len(args.input) >= 2:
            return choice(args.input)

        return help_text

    if __name__ == '__main__':
        print(main())