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```"""Random variable generators.

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

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

distributions on the real line:
------------------------------
uniform
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.
* Without a direct way to compute N steps forward, the semantics of
on the large period to avoid overlapping sequences.
* The random() method is implemented in C, executes in a single Python step,

"""

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 binascii import hexlify as _hexlify

__all__ = ["Random","seed","random","uniform","randint","choice","sample",
"randrange","shuffle","normalvariate","lognormvariate",
"expovariate","vonmisesvariate","gammavariate",
"gauss","betavariate","paretovariate","weibullvariate",
"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
# 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.  Especially useful for multi-threaded programs, creating
a different instance of Random for each thread, and using the jumpahead()
method to ensure that the generated sequences seen by each thread don't
overlap.

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(), setstate() and jumpahead().
Optionally, implement a getrandombits() method so that randrange()
can cover arbitrarily large ranges.

"""

VERSION = 2     # 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):
"""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 not None or an int or long, hash(a) is used instead.
"""

if a is None:
try:
a = long(_hexlify(_urandom(16)), 16)
except NotImplementedError:
import time
a = long(time.time() * 256) # use fractional seconds

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

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

def setstate(self, state):
"""Restore internal state from object returned by getstate()."""
version = state[0]
if version == 2:
version, internalstate, self.gauss_next = state
super(Random, self).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  -------------------

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, default=None,
maxwidth=1L<<BPF):
"""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.
Do not supply the 'int', 'default', and 'maxwidth' arguments.
"""

# 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 default:
if istart > 0:
if istart >= maxwidth:
return self._randbelow(istart)
return int(self.random() * 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:
# Note that
#     int(istart + self.random()*width)
# instead would be incorrect.  For example, consider istart
# = -2 and istop = 0.  Then the guts would be in
# -2.0 to 0.0 exclusive on both ends (ignoring that random()
# might return 0.0), and because int() truncates toward 0, the
# final result would be -1 or 0 (instead of -2 or -1).
#     istart + int(self.random()*width)
# would also be incorrect, for a subtler reason:  the RHS
# can return a long, and then randrange() would also return
# a long, but we're supposed to return an int (for backward
# compatibility).

if width >= maxwidth:
return int(istart + self._randbelow(width))
return int(istart + int(self.random()*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()"

if n >= maxwidth:
return istart + istep*self._randbelow(n)
return istart + istep*int(self.random() * 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, _log=_log, int=int, _maxwidth=1L<<BPF,
_Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):
"""Return a random int in the range [0,n)

Handles the case where n has more bits than returned
by a single call to the underlying generator.
"""

try:
getrandbits = self.getrandbits
except AttributeError:
pass
else:
# Only call self.getrandbits if the original random() builtin method
# has not been overridden or if a new getrandbits() was supplied.
# This assures that the two methods correspond.
if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:
k = int(1.00001 + _log(n-1, 2.0))   # 2**k > n-1 > 2**(k-2)
r = getrandbits(k)
while r >= n:
r = getrandbits(k)
return r
if n >= _maxwidth:
_warn("Underlying random() generator does not supply \n"
"enough bits to choose from a population range this large")
return int(self.random() * n)

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

def choice(self, seq):
"""Choose a random element from a non-empty sequence."""
return seq[int(self.random() * len(seq))]  # raises IndexError if seq is empty

def shuffle(self, x, random=None, int=int):
"""x, random=random.random -> shuffle list x in place; return None.

Optional arg random is a 0-argument function returning a random
float in [0.0, 1.0); by default, the standard random.random.
"""

if random is None:
random = self.random
for i in reversed(xrange(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.

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 xrange as an argument.
This is especially fast and space efficient for sampling from a
large population:   sample(xrange(10000000), 60)
"""

# XXX Although the documentation says `population` is "a sequence",
# XXX attempts are made to cater to any iterable with a __len__
# XXX method.  This has had mixed success.  Examples from both
# XXX sides:  sets work fine, and should become officially supported;
# XXX dicts are much harder, and have failed in various subtle
# XXX ways across attempts.  Support for mapping types should probably
# XXX be dropped (and users should pass mapping.keys() or .values()
# XXX explicitly).

# 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.

n = len(population)
if not 0 <= k <= n:
raise ValueError, "sample larger than population"
random = self.random
_int = int
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 or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
pool = list(population)
for i in xrange(k):         # invariant:  non-selected at [0,n-i)
j = _int(random() * (n-i))
result[i] = pool[j]
pool[j] = pool[n-i-1]   # move non-selected item into vacancy
else:
try:
selected = set()
for i in xrange(k):
j = _int(random() * n)
while j in selected:
j = _int(random() * n)
result[i] = population[j]
except (TypeError, KeyError):   # handle (at least) sets
if isinstance(population, list):
raise
return self.sample(tuple(population), k)
return result

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

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

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

## -------------------- 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.  (The parameter would be
called "lambda", but that is a reserved word in Python.)  Returned
values range from 0 to positive infinity.

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

random = self.random
u = random()
while u <= 1e-7:
u = random()
return -_log(u)/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()

a = 1.0 + _sqrt(1.0 + 4.0 * kappa * kappa)
b = (a - _sqrt(2.0 * a))/(2.0 * kappa)
r = (1.0 + b * b)/(2.0 * b)

while 1:
u1 = random()

z = _cos(_pi * u1)
f = (1.0 + r * z)/(r + z)
c = kappa * (r - f)

u2 = random()

if u2 < c * (2.0 - c) or u2 <= c * _exp(1.0 - c):
break

u3 = random()
if u3 > 0.5:
theta = (mu % TWOPI) + _acos(f)
else:
theta = (mu % TWOPI) - _acos(f)

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.

"""

# 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)
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)

# 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()))

return mu + z*sigma

## -------------------- beta --------------------
## See
## http://sourceforge.net/bugs/?func=detailbug&bug_id=130030&group_id=5470
## 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.)
if y == 0:
return 0.0
else:
return y / (y + self.gammavariate(beta, 1.))

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

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

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

## -------------------- Wichmann-Hill -------------------

class WichmannHill(Random):

VERSION = 1     # used by getstate/setstate

def seed(self, a=None):
"""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 not None or an int or long, hash(a) is used instead.

If a is an int or long, a is used directly.  Distinct values between
0 and 27814431486575L inclusive are guaranteed to yield distinct
internal states (this guarantee is specific to the default
Wichmann-Hill generator).
"""

if a is None:
try:
a = long(_hexlify(_urandom(16)), 16)
except NotImplementedError:
import time
a = long(time.time() * 256) # use fractional seconds

if not isinstance(a, (int, long)):
a = hash(a)

a, x = divmod(a, 30268)
a, y = divmod(a, 30306)
a, z = divmod(a, 30322)
self._seed = int(x)+1, int(y)+1, int(z)+1

self.gauss_next = None

def random(self):
"""Get the next random number in the range [0.0, 1.0)."""

# Wichman-Hill random number generator.
#
# Wichmann, B. A. & Hill, I. D. (1982)
# Algorithm AS 183:
# An efficient and portable pseudo-random number generator
# Applied Statistics 31 (1982) 188-190
#
#        Correction to Algorithm AS 183
#        Applied Statistics 33 (1984) 123
#
#        McLeod, A. I. (1985)
#        A remark on Algorithm AS 183
#        Applied Statistics 34 (1985),198-200

# BEGIN CRITICAL SECTION
x, y, z = self._seed
x = (171 * x) % 30269
y = (172 * y) % 30307
z = (170 * z) % 30323
self._seed = x, y, z
# END CRITICAL SECTION

# Note:  on a platform using IEEE-754 double arithmetic, this can
# never return 0.0 (asserted by Tim; proof too long for a comment).
return (x/30269.0 + y/30307.0 + z/30323.0) % 1.0

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

def setstate(self, state):
"""Restore internal state from object returned by getstate()."""
version = state[0]
if version == 1:
version, self._seed, self.gauss_next = state
else:
raise ValueError("state with version %s passed to "
"Random.setstate() of version %s" %
(version, self.VERSION))

"""Act as if n calls to random() were made, but quickly.

n is an int, greater than or equal to 0.

Example use:  If you have 2 threads and know that each will
consume no more than a million random numbers, create two Random
objects r1 and r2, then do
r2.setstate(r1.getstate())
Then r1 and r2 will use guaranteed-disjoint segments of the full
period.
"""

if not n >= 0:
raise ValueError("n must be >= 0")
x, y, z = self._seed
x = int(x * pow(171, n, 30269)) % 30269
y = int(y * pow(172, n, 30307)) % 30307
z = int(z * pow(170, n, 30323)) % 30323
self._seed = x, y, z

def __whseed(self, x=0, y=0, z=0):
"""Set the Wichmann-Hill seed from (x, y, z).

These must be integers in the range [0, 256).
"""

if not type(x) == type(y) == type(z) == int:
raise TypeError('seeds must be integers')
if not (0 <= x < 256 and 0 <= y < 256 and 0 <= z < 256):
raise ValueError('seeds must be in range(0, 256)')
if 0 == x == y == z:
# Initialize from current time
import time
t = long(time.time() * 256)
t = int((t&0xffffff) ^ (t>>24))
t, x = divmod(t, 256)
t, y = divmod(t, 256)
t, z = divmod(t, 256)
# Zero is a poor seed, so substitute 1
self._seed = (x or 1, y or 1, z or 1)

self.gauss_next = None

def whseed(self, a=None):
"""Seed from hashable object's hash code.

None or no argument seeds from current time.  It is not guaranteed
that objects with distinct hash codes lead to distinct internal
states.

This is obsolete, provided for compatibility with the seed routine
used prior to Python 2.1.  Use the .seed() method instead.
"""

if a is None:
self.__whseed()
return
a = hash(a)
a, x = divmod(a, 256)
a, y = divmod(a, 256)
a, z = divmod(a, 256)
x = (x + a) % 256 or 1
y = (y + a) % 256 or 1
z = (z + a) % 256 or 1
self.__whseed(x, y, z)

## --------------- 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 (long(_hexlify(_urandom(7)), 16) >> 3) * RECIP_BPF

def getrandbits(self, k):
"""getrandbits(k) -> x.  Generates a long 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')
bytes = (k + 7) // 8                    # bits / 8 and rounded up
x = long(_hexlify(_urandom(bytes)), 16)
return x >> (bytes * 8 - k)             # trim excess bits

def _stub(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.time()
for i in range(n):
x = func(*args)
total += x
sqsum = sqsum + x*x
smallest = min(x, smallest)
largest = max(x, largest)
t1 = time.time()
print round(t1-t0, 3), 'sec,',
avg = total/n
stddev = _sqrt(sqsum/n - avg*avg)
print 'avg %g, stddev %g, min %g, max %g' % \
(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))

# 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
randint = _inst.randint
choice = _inst.choice
randrange = _inst.randrange
sample = _inst.sample
shuffle = _inst.shuffle
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