"""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 types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodType
	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 hashlib import sha512 as _sha512
	import itertools as _itertools
	import bisect as _bisect
	import os as _os
	__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 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(self, n, int=int, maxsize=1<<BPF, type=type,
	Method=_MethodType, BuiltinMethod=_BuiltinMethodType):
	"Return a random int in the range [0,n). Raises ValueError if n==0."
	random = self.random
	getrandbits = self.getrandbits
	# Only call self.getrandbits if the original random() builtin method
	# has not been overridden or if a new getrandbits() was supplied.
	if type(random) is BuiltinMethod or type(getrandbits) is Method:
	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
	# There's an overridden random() method but no new getrandbits() method,
	# so we can only use random() from here.
	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
	## -------------------- 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.
	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
	if cum_weights is None:
	if weights is None:
	_int = int
	total = len(population)
	return [population[_int(random() * total)] for i in range(k)]
	cum_weights = list(_itertools.accumulate(weights))
	elif weights is not None:
	raise TypeError('Cannot specify both weights and cumulative weights')
	if len(cum_weights) != len(population):
	raise ValueError('The number of weights does not match the population')
	bisect = _bisect.bisect
	total = cum_weights[-1]
	hi = len(cum_weights) - 1
	return [population[bisect(cum_weights, random() * total, 0, hi)]
	for i in range(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)
	u = random()
	while u <= 1e-7:
	u = random()
	return -_log(u) * 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')
	if k != int(k):
	raise TypeError('number of bits should be an integer')
	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()