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Edit File : random.py \| Size : 28.13 KB
"""Random variable generators. 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 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. """ from warnings import warn as _warn 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 os import urandom as _urandom from _collections_abc import Set as _Set, Sequence as _Sequence from itertools import accumulate as _accumulate, repeat as _repeat from bisect import bisect as _bisect import os as _os try: # hashlib is pretty heavy to load, try lean internal module first from _sha512 import sha512 as _sha512 except ImportError: # fallback to official implementation from hashlib import sha512 as _sha512 __all__ = ["Random","seed","random","uniform","randint","choice","sample", "randrange","shuffle","normalvariate","lognormvariate", "expovariate","vonmisesvariate","gammavariate","triangular", "gauss","betavariate","paretovariate","weibullvariate", "getstate","setstate", "getrandbits", "choices", "SystemRandom"] NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0) TWOPI = 2.0_pi LOG4 = _log(4.0) SG_MAGICCONST = 1.0 + _log(4.5) BPF = 53 # Number of bits in a float RECIP_BPF = 2-BPF # 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. import _random 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 __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 seed(self, a=None, version=2): """Initialize internal state from hashable object. 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 if version == 2 and isinstance(a, (str, bytes, bytearray)): if isinstance(a, str): a = a.encode() a += _sha512(a).digest() a = int.from_bytes(a, 'big') 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 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() ## -------------------- integer methods ------------------- def randrange(self, start, stop=None, step=1, _int=int): """Choose a random item from range(start, stop[, step]). This fixes the problem with randint() which includes the endpoint; in Python this is usually not what you want. """ # This code is a bit messy to make it fast for the # common case while still doing adequate error checking. istart = _int(start) if istart != start: raise ValueError("non-integer arg 1 for randrange()") if stop is None: if istart > 0: return self._randbelow(istart) raise ValueError("empty range for randrange()") # stop argument supplied. istop = _int(stop) if istop != stop: raise ValueError("non-integer stop for randrange()") width = istop - istart if step == 1 and width > 0: return istart + self._randbelow(width) if step == 1: raise ValueError("empty range for randrange() (%d, %d, %d)" % (istart, istop, width)) # Non-unit step argument supplied. istep = _int(step) if istep != step: raise ValueError("non-integer step for randrange()") 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("empty range for randrange()") return istart + istepself._randbelow(n) def randint(self, a, b): """Return random integer in range [a, b], including both end points. """ return self.randrange(a, b+1) def _randbelow_with_getrandbits(self, n): "Return a random int in the range [0,n). Raises ValueError if n==0." getrandbits = self.getrandbits k = n.bit_length() # don't use (n-1) here because n can be 1 r = getrandbits(k) # 0 <= r < 2*k while r >= n: r = getrandbits(k) return r def _randbelow_without_getrandbits(self, n, int=int, maxsize=1<<BPF): """Return a random int in the range [0,n). Raises ValueError if n==0. The implementation does not use getrandbits, but only random. """ random = self.random if n >= maxsize: _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 int(random() n) if n == 0: raise ValueError("Boundary cannot be zero") rem = maxsize % n limit = (maxsize - rem) / maxsize # int(limit * maxsize) % n == 0 r = random() while r >= limit: r = random() return int(rmaxsize) % n _randbelow = _randbelow_with_getrandbits ## -------------------- sequence methods ------------------- def choice(self, seq): """Choose a random element from a non-empty sequence.""" try: i = self._randbelow(len(seq)) except ValueError: raise IndexError('Cannot choose from an empty sequence') from None return seq[i] def shuffle(self, x, random=None): """Shuffle list x in place, and return None. Optional argument random is a 0-argument function returning a random float in [0.0, 1.0); if it is the default None, the standard random.random will be used. """ if random is 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] else: _int = int for i in reversed(range(1, len(x))): # pick an element in x[:i+1] with which to exchange x[i] j = _int(random() (i+1)) x[i], x[j] = x[j], x[i] def sample(self, population, k): """Chooses k unique random elements from a population sequence or set. 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. To choose a sample in a range of integers, use range as an 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 isinstance(population, _Set): population = tuple(population) if not isinstance(population, _Sequence): raise TypeError("Population must be a sequence or set. For dicts, use list(d).") randbelow = self._randbelow n = len(population) 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 pool = list(population) for i in range(k): # invariant: non-selected at [0,n-i) 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: _int = int n += 0.0 # convert to float for a small speed improvement return [population[_int(random() n)] for i in _repeat(None, k)] cum_weights = list(_accumulate(weights)) 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') bisect = _bisect total = cum_weights[-1] + 0.0 # convert to float hi = n - 1 return [population[bisect(cum_weights, random() * total, 0, hi)] for i in _repeat(None, k)] ## -------------------- real-valued distributions ------------------- ## -------------------- uniform distribution ------------------- def uniform(self, a, b): "Get a random number in the range [a, b) or [a, b] depending on rounding." return a + (b-a) * self.random() ## -------------------- triangular -------------------- 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 """ 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) ## -------------------- normal distribution -------------------- def normalvariate(self, mu, sigma): """Normal distribution. mu is the mean, and sigma is the standard deviation. """ # mu = mean, sigma = 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 1: u1 = random() u2 = 1.0 - random() z = NV_MAGICCONST(u1-0.5)/u2 zz = zz/4.0 if zz <= -_log(u2): break return mu + zsigma ## -------------------- lognormal distribution -------------------- 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)) ## -------------------- exponential distribution -------------------- def expovariate(self, lambd): """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. """ # lambd: rate lambd = 1/mean # ('lambda' is a Python reserved word) # we use 1-random() instead of random() to preclude the # possibility of taking the log of zero. return -_log(1.0 - self.random())/lambd ## -------------------- von Mises distribution -------------------- def vonmisesvariate(self, mu, kappa): """Circular data distribution. mu is the mean angle, expressed in radians between 0 and 2pi, 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 2pi. """ # mu: mean angle (in radians between 0 and 2pi) # kappa: concentration parameter kappa (>= 0) # if kappa = 0 generate uniform random angle # 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 1: 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 ## -------------------- gamma distribution -------------------- 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 """ # alpha > 0, beta > 0, mean is alphabeta, variance is alphabeta*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 1: u1 = random() if not 1e-7 < u1 < .9999999: continue u2 = 1.0 - random() v = _log(u1/(1.0-u1))/ainv x = alpha_exp(v) z = u1u1u2 r = bbb+cccv-x if r + SG_MAGICCONST - 4.5z >= 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 1: u = random() b = (_e + alpha)/_e p = bu 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 ## -------------------- Gauss (faster alternative) -------------------- def gauss(self, mu, sigma): """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(2pix)sqrt(-2log(1-y)) # sin(2pix)sqrt(-2log(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 + zsigma ## -------------------- beta -------------------- ## 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. def betavariate(self, alpha, beta): """Beta distribution. Conditions on the parameters are alpha > 0 and beta > 0. Returned values range between 0 and 1. """ # 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 == 0: return 0.0 else: return y / (y + self.gammavariate(beta, 1.0)) ## -------------------- Pareto -------------------- def paretovariate(self, alpha): """Pareto distribution. alpha is the shape parameter.""" # Jain, pg. 495 u = 1.0 - self.random() return 1.0 / u * (1.0/alpha) ## -------------------- Weibull -------------------- 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) ## --------------- 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, 1.0).""" return (int.from_bytes(_urandom(7), 'big') >> 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 greater than zero') numbytes = (k + 7) // 8 # bits / 8 and rounded up x = int.from_bytes(_urandom(numbytes), 'big') return x >> (numbytes * 8 - k) # trim excess bits 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 ## -------------------- test program -------------------- def _test_generator(n, func, args): import time print(n, 'times', func.__name__) total = 0.0 sqsum = 0.0 smallest = 1e10 largest = -1e10 t0 = time.perf_counter() for i in range(n): x = func(args) total += x sqsum = sqsum + xx smallest = min(x, smallest) largest = max(x, largest) t1 = time.perf_counter() print(round(t1-t0, 3), 'sec,', end=' ') avg = total/n stddev = _sqrt(sqsum/n - avgavg) print('avg %g, stddev %g, min %g, max %g\n' % \ (avg, stddev, smallest, largest)) def _test(N=2000): _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, 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)) # 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 paretovariate = _inst.paretovariate weibullvariate = _inst.weibullvariate getstate = _inst.getstate setstate = _inst.setstate getrandbits = _inst.getrandbits if hasattr(_os, "fork"): _os.register_at_fork(after_in_child=_inst.seed) if __name__ == '__main__': _test() Back

Edit File : random.py | Size : 28.13 KB

"""Random variable generators.

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

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.

"""

from warnings import warn as _warn
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 os import urandom as _urandom
from _collections_abc import Set as _Set, Sequence as _Sequence
from itertools import accumulate as _accumulate, repeat as _repeat
from bisect import bisect as _bisect
import os as _os

try:
    # hashlib is pretty heavy to load, try lean internal module first
    from _sha512 import sha512 as _sha512
except ImportError:
    # fallback to official implementation
    from hashlib import sha512 as _sha512

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

NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)
TWOPI = 2.0*_pi
LOG4 = _log(4.0)
SG_MAGICCONST = 1.0 + _log(4.5)
BPF = 53        # Number of bits in a float
RECIP_BPF = 2**-BPF

# 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.

import _random

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

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

if version == 2 and isinstance(a, (str, bytes, bytearray)):
            if isinstance(a, str):
                a = a.encode()
            a += _sha512(a).digest()
            a = int.from_bytes(a, 'big')

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 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()

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

def randrange(self, start, stop=None, step=1, _int=int):
        """Choose a random item from range(start, stop[, step]).

This fixes the problem with randint() which includes the
        endpoint; in Python this is usually not what you want.

"""

# This code is a bit messy to make it fast for the
        # common case while still doing adequate error checking.
        istart = _int(start)
        if istart != start:
            raise ValueError("non-integer arg 1 for randrange()")
        if stop is None:
            if istart > 0:
                return self._randbelow(istart)
            raise ValueError("empty range for randrange()")

# stop argument supplied.
        istop = _int(stop)
        if istop != stop:
            raise ValueError("non-integer stop for randrange()")
        width = istop - istart
        if step == 1 and width > 0:
            return istart + self._randbelow(width)
        if step == 1:
            raise ValueError("empty range for randrange() (%d, %d, %d)" % (istart, istop, width))

# Non-unit step argument supplied.
        istep = _int(step)
        if istep != step:
            raise ValueError("non-integer step for randrange()")
        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("empty range for randrange()")

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)

def _randbelow_with_getrandbits(self, n):
        "Return a random int in the range [0,n).  Raises ValueError if n==0."

getrandbits = self.getrandbits
        k = n.bit_length()  # don't use (n-1) here because n can be 1
        r = getrandbits(k)          # 0 <= r < 2**k
        while r >= n:
            r = getrandbits(k)
        return r

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

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

random = self.random
        if n >= maxsize:
            _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 int(random() * n)
        if n == 0:
            raise ValueError("Boundary cannot be zero")
        rem = maxsize % n
        limit = (maxsize - rem) / maxsize   # int(limit * maxsize) % n == 0
        r = random()
        while r >= limit:
            r = random()
        return int(r*maxsize) % n

_randbelow = _randbelow_with_getrandbits

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

def choice(self, seq):
        """Choose a random element from a non-empty sequence."""
        try:
            i = self._randbelow(len(seq))
        except ValueError:
            raise IndexError('Cannot choose from an empty sequence') from None
        return seq[i]

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

Optional argument random is a 0-argument function returning a
        random float in [0.0, 1.0); if it is the default None, the
        standard random.random will be used.

"""

if random is 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]
        else:
            _int = int
            for i in reversed(range(1, len(x))):
                # pick an element in x[:i+1] with which to exchange x[i]
                j = _int(random() * (i+1))
                x[i], x[j] = x[j], x[i]

def sample(self, population, k):
        """Chooses k unique random elements from a population sequence or set.

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.

To choose a sample in a range of integers, use range as an 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 isinstance(population, _Set):
            population = tuple(population)
        if not isinstance(population, _Sequence):
            raise TypeError("Population must be a sequence or set.  For dicts, use list(d).")
        randbelow = self._randbelow
        n = len(population)
        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
            pool = list(population)
            for i in range(k):         # invariant:  non-selected at [0,n-i)
                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:
                _int = int
                n += 0.0    # convert to float for a small speed improvement
                return [population[_int(random() * n)] for i in _repeat(None, k)]
            cum_weights = list(_accumulate(weights))
        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')
        bisect = _bisect
        total = cum_weights[-1] + 0.0   # convert to float
        hi = n - 1
        return [population[bisect(cum_weights, random() * total, 0, hi)]
                for i in _repeat(None, k)]

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

## -------------------- uniform distribution -------------------

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

## -------------------- triangular --------------------

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

"""
        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)

## -------------------- normal distribution --------------------

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

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

"""
        # mu = mean, sigma = 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 1:
            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

## -------------------- lognormal distribution --------------------

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))

## -------------------- exponential distribution --------------------

def expovariate(self, lambd):
        """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.

"""
        # lambd: rate lambd = 1/mean
        # ('lambda' is a Python reserved word)

# we use 1-random() instead of random() to preclude the
        # possibility of taking the log of zero.
        return -_log(1.0 - self.random())/lambd

## -------------------- von Mises distribution --------------------

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.

"""
        # mu:    mean angle (in radians between 0 and 2*pi)
        # kappa: concentration parameter kappa (>= 0)
        # if kappa = 0 generate uniform random angle

# 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 1:
            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

## -------------------- gamma distribution --------------------

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

"""

# alpha > 0, beta > 0, mean is alpha*beta, variance is 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 1:
                u1 = random()
                if not 1e-7 < u1 < .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 1:
                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

## -------------------- Gauss (faster alternative) --------------------

def gauss(self, mu, sigma):
        """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

## -------------------- beta --------------------
## 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.

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

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

"""

# 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 == 0:
            return 0.0
        else:
            return y / (y + self.gammavariate(beta, 1.0))

## -------------------- Pareto --------------------

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

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

## -------------------- Weibull --------------------

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)

## --------------- 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, 1.0)."""
        return (int.from_bytes(_urandom(7), 'big') >> 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 greater than zero')
        numbytes = (k + 7) // 8                       # bits / 8 and rounded up
        x = int.from_bytes(_urandom(numbytes), 'big')
        return x >> (numbytes * 8 - k)                # trim excess bits

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

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

def _test_generator(n, func, args):
    import time
    print(n, 'times', func.__name__)
    total = 0.0
    sqsum = 0.0
    smallest = 1e10
    largest = -1e10
    t0 = time.perf_counter()
    for i in range(n):
        x = func(*args)
        total += x
        sqsum = sqsum + x*x
        smallest = min(x, smallest)
        largest = max(x, largest)
    t1 = time.perf_counter()
    print(round(t1-t0, 3), 'sec,', end=' ')
    avg = total/n
    stddev = _sqrt(sqsum/n - avg*avg)
    print('avg %g, stddev %g, min %g, max %g\n' % \
              (avg, stddev, smallest, largest))

def _test(N=2000):
    _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, 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))

# 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
paretovariate = _inst.paretovariate
weibullvariate = _inst.weibullvariate
getstate = _inst.getstate
setstate = _inst.setstate
getrandbits = _inst.getrandbits

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

if __name__ == '__main__':
    _test()

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