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"""Random variable generators.
uniform bytes (values between 0 and 255)
uniform within range
pick random element
pick random sample
pick weighted random sample
generate random permutation
distributions on the real line:
normal (Gaussian)
negative exponential
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.
# Translated by Guido van Rossum from C source provided by
# Adrian Baddeley. Adapted by Raymond Hettinger for use with
# the Mersenne Twister and os.urandom() core generators.
from 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 math import tau as TWOPI, floor as _floor, isfinite as _isfinite
from os import urandom as _urandom
from _collections_abc import Set as _Set, Sequence as _Sequence
from operator import index as _index
from itertools import accumulate as _accumulate, repeat as _repeat
from bisect import bisect as _bisect
import os as _os
import _random
# 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__ = [
NV_MAGICCONST = 4 * _exp(-0.5) / _sqrt(2.0)
LOG4 = _log(4.0)
SG_MAGICCONST = 1.0 + _log(4.5)
BPF = 53 # Number of bits in a float
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.gauss_next = None
def seed(self, a=None, version=2):
"""Initialize internal state from a seed.
The only supported seed types are None, int, float,
str, bytes, and bytearray.
None or no argument seeds from current time or from an operating
system specific randomness source if available.
If *a* is an int, all bits are used.
For version 2 (the default), all of the bits are used if *a* is a str,
bytes, or bytearray. For version 1 (provided for reproducing random
sequences from older versions of Python), the algorithm for str and
bytes generates a narrower range of seeds.
if version == 1 and isinstance(a, (str, bytes)):
a = a.decode('latin-1') if isinstance(a, bytes) else a
x = ord(a[0]) << 7 if a else 0
for c in map(ord, a):
x = ((1000003 * x) ^ c) & 0xFFFFFFFFFFFFFFFF
x ^= len(a)
a = -2 if x == -1 else x
elif version == 2 and isinstance(a, (str, bytes, bytearray)):
if isinstance(a, str):
a = a.encode()
a += _sha512(a).digest()
a = int.from_bytes(a, 'big')
elif not isinstance(a, (type(None), int, float, str, bytes, bytearray)):
_warn('Seeding based on hashing is deprecated\n'
'since Python 3.9 and will be removed in a subsequent '
'version. The only \n'
'supported seed types are: None, '
'int, float, str, bytes, and bytearray.',
DeprecationWarning, 2)
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
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.
internalstate = tuple(x % (2 ** 32) for x in internalstate)
except ValueError as e:
raise TypeError from e
raise ValueError("state with version %s passed to "
"Random.setstate() of version %s" %
(version, self.VERSION))
## -------------------------------------------------------
## ---- Methods below this point do not need to be overridden or extended
## ---- when subclassing for the purpose of using a different core generator.
## -------------------- pickle support -------------------
# Issue 17489: Since __reduce__ was defined to fix #759889 this is no
# longer called; we leave it here because it has been here since random was
# rewritten back in 2001 and why risk breaking something.
def __getstate__(self): # for pickle
return self.getstate()
def __setstate__(self, state): # for pickle
def __reduce__(self):
return self.__class__, (), self.getstate()
## ---- internal support method for evenly distributed integers ----
def __init_subclass__(cls, /, **kwargs):
"""Control how subclasses generate random integers.
The algorithm a subclass can use depends on the random() and/or
getrandbits() implementation available to it and determines
whether it can generate random integers from arbitrarily large
for c in cls.__mro__:
if '_randbelow' in c.__dict__:
# just inherit it
if 'getrandbits' in c.__dict__:
cls._randbelow = cls._randbelow_with_getrandbits
if 'random' in c.__dict__:
cls._randbelow = cls._randbelow_without_getrandbits
def _randbelow_with_getrandbits(self, n):
"Return a random int in the range [0,n). Returns 0 if n==0."
if not n:
return 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, maxsize=1<<BPF):
"""Return a random int in the range [0,n). Returns 0 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 _floor(random() * n)
if n == 0:
return 0
rem = maxsize % n
limit = (maxsize - rem) / maxsize # int(limit * maxsize) % n == 0
r = random()
while r >= limit:
r = random()
return _floor(r * maxsize) % n
_randbelow = _randbelow_with_getrandbits
## --------------------------------------------------------
## ---- Methods below this point generate custom distributions
## ---- based on the methods defined above. They do not
## ---- directly touch the underlying generator and only
## ---- access randomness through the methods: random(),
## ---- getrandbits(), or _randbelow().
## -------------------- bytes methods ---------------------
def randbytes(self, n):
"""Generate n random bytes."""
return self.getrandbits(n * 8).to_bytes(n, 'little')
## -------------------- integer methods -------------------
def randrange(self, start, stop=None, step=1):
"""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 = _index(start)
except TypeError:
if int(start) == start:
istart = int(start)
_warn('Float arguments to randrange() have been deprecated\n'
'since Python 3.10 and will be removed in a subsequent '
DeprecationWarning, 2)
_warn('randrange() will raise TypeError in the future',
DeprecationWarning, 2)
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 = _index(stop)
except TypeError:
if int(stop) == stop:
istop = int(stop)
_warn('Float arguments to randrange() have been deprecated\n'
'since Python 3.10 and will be removed in a subsequent '
DeprecationWarning, 2)
_warn('randrange() will raise TypeError in the future',
DeprecationWarning, 2)
raise ValueError("non-integer stop for randrange()")
istep = _index(step)
except TypeError:
if int(step) == step:
istep = int(step)
_warn('Float arguments to randrange() have been deprecated\n'
'since Python 3.10 and will be removed in a subsequent '
DeprecationWarning, 2)
_warn('randrange() will raise TypeError in the future',
DeprecationWarning, 2)
raise ValueError("non-integer step for randrange()")
width = istop - istart
if istep == 1 and width > 0:
return istart + self._randbelow(width)
if istep == 1:
raise ValueError("empty range for randrange() (%d, %d, %d)" % (istart, istop, width))
# Non-unit step argument supplied.
if istep > 0:
n = (width + istep - 1) // istep
elif istep < 0:
n = (width + istep + 1) // istep
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)
## -------------------- sequence methods -------------------
def choice(self, seq):
"""Choose a random element from a non-empty sequence."""
# raises IndexError if seq is empty
return seq[self._randbelow(len(seq))]
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]
_warn('The *random* parameter to shuffle() has been deprecated\n'
'since Python 3.9 and will be removed in a subsequent '
DeprecationWarning, 2)
floor = _floor
for i in reversed(range(1, len(x))):
# pick an element in x[:i+1] with which to exchange x[i]
j = floor(random() * (i + 1))
x[i], x[j] = x[j], x[i]
def sample(self, population, k, *, counts=None):
"""Chooses k unique random elements from a population sequence 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.
Repeated elements can be specified one at a time or with the optional
counts parameter. For example:
sample(['red', 'blue'], counts=[4, 2], k=5)
is equivalent to:
sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k=5)
To choose a sample from a range of integers, use range() for the
population argument. This is especially fast and space efficient
for sampling from a large population:
sample(range(10000000), 60)
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
# The number of calls to _randbelow() is kept at or near k, the
# theoretical minimum. This is important because running time
# is dominated by _randbelow() and because it extracts the
# least entropy from the underlying random number generators.
# Memory requirements are kept to the smaller of a k-length
# set or an n-length list.
# There are other sampling algorithms that do not require
# auxiliary memory, but they were rejected because they made
# too many calls to _randbelow(), making them slower and
# causing them to eat more entropy than necessary.
if not isinstance(population, _Sequence):
if isinstance(population, _Set):
_warn('Sampling from a set deprecated\n'
'since Python 3.9 and will be removed in a subsequent version.',
DeprecationWarning, 2)
population = tuple(population)
raise TypeError("Population must be a sequence. For dicts or sets, use sorted(d).")
n = len(population)
if counts is not None:
cum_counts = list(_accumulate(counts))
if len(cum_counts) != n:
raise ValueError('The number of counts does not match the population')
total = cum_counts.pop()
if not isinstance(total, int):
raise TypeError('Counts must be integers')
if total <= 0:
raise ValueError('Total of counts must be greater than zero')
selections = sample(range(total), k=k)
bisect = _bisect
return [population[bisect(cum_counts, s)] for s in selections]
randbelow = self._randbelow
if not 0 <= k <= n:
raise ValueError("Sample larger than population or is negative")
result = [None] * k
setsize = 21 # size of a small set minus size of an empty list
if k > 5:
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize:
# An n-length list is smaller than a k-length set.
# Invariant: non-selected at pool[0 : n-i]
pool = list(population)
for i in range(k):
j = randbelow(n - i)
result[i] = pool[j]
pool[j] = pool[n - i - 1] # move non-selected item into vacancy
selected = set()
selected_add = selected.add
for i in range(k):
j = randbelow(n)
while j in selected:
j = randbelow(n)
result[i] = population[j]
return result
def choices(self, population, weights=None, *, cum_weights=None, k=1):
"""Return a k sized list of population elements chosen with replacement.
If the relative weights or cumulative weights are not specified,
the selections are made with equal probability.
random = self.random
n = len(population)
if cum_weights is None:
if weights is None:
floor = _floor
n += 0.0 # convert to float for a small speed improvement
return [population[floor(random() * n)] for i in _repeat(None, k)]
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')
total = cum_weights[-1] + 0.0 # convert to float
if total <= 0.0:
raise ValueError('Total of weights must be greater than zero')
if not _isfinite(total):
raise ValueError('Total of weights must be finite')
bisect = _bisect
hi = n - 1
return [population[bisect(cum_weights, random() * total, 0, hi)]
for i in _repeat(None, k)]
## -------------------- real-valued distributions -------------------
def uniform(self, a, b):
"Get a random number in the range [a, b) or [a, b] depending on rounding."
return a + (b - a) * self.random()
def triangular(self, low=0.0, high=1.0, mode=None):
"""Triangular distribution.
Continuous distribution bounded by given lower and upper limits,
and having a given mode value in-between.
u = self.random()
c = 0.5 if mode is None else (mode - low) / (high - low)
except ZeroDivisionError:
return low
if u > c:
u = 1.0 - u
c = 1.0 - c
low, high = high, low
return low + (high - low) * _sqrt(u * c)
def normalvariate(self, mu, sigma):
"""Normal distribution.
mu is the mean, and sigma is the standard deviation.
# Uses Kinderman and Monahan method. Reference: Kinderman,
# A.J. and Monahan, J.F., "Computer generation of random
# variables using the ratio of uniform deviates", ACM Trans
# Math Software, 3, (1977), pp257-260.
random = self.random
while True:
u1 = random()
u2 = 1.0 - random()
z = NV_MAGICCONST * (u1 - 0.5) / u2
zz = z * z / 4.0
if zz <= -_log(u2):
return mu + z * sigma
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
def lognormvariate(self, mu, sigma):
"""Log normal distribution.
If you take the natural logarithm of this distribution, you'll get a
normal distribution with mean mu and standard deviation sigma.
mu can have any value, and sigma must be greater than zero.
return _exp(self.normalvariate(mu, sigma))
def expovariate(self, lambd):
"""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
def vonmisesvariate(self, mu, kappa):
"""Circular data distribution.
mu is the mean angle, expressed in radians between 0 and 2*pi, and
kappa is the concentration parameter, which must be greater than or
equal to zero. If kappa is equal to zero, this distribution reduces
to a uniform random angle over the range 0 to 2*pi.
# Based upon an algorithm published in: Fisher, N.I.,
# "Statistical Analysis of Circular Data", Cambridge
# University Press, 1993.
# Thanks to Magnus Kessler for a correction to the
# implementation of step 4.
random = self.random
if kappa <= 1e-6:
return TWOPI * random()
s = 0.5 / kappa
r = s + _sqrt(1.0 + s * s)
while True:
u1 = random()
z = _cos(_pi * u1)
d = z / (r + z)
u2 = random()
if u2 < 1.0 - d * d or u2 <= (1.0 - d) * _exp(d):
q = 1.0 / r
f = (q + z) / (1.0 + q * z)
u3 = random()
if u3 > 0.5:
theta = (mu + _acos(f)) % TWOPI
theta = (mu - _acos(f)) % TWOPI
return theta
def gammavariate(self, alpha, beta):
"""Gamma distribution. Not the gamma function!
Conditions on the parameters are alpha > 0 and beta > 0.
The probability distribution function is:
x ** (alpha - 1) * math.exp(-x / beta)
pdf(x) = --------------------------------------
math.gamma(alpha) * beta ** alpha
# 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 True:
u1 = random()
if not 1e-7 < u1 < 0.9999999:
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
# alpha is between 0 and 1 (exclusive)
# Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
while True:
u = random()
b = (_e + alpha) / _e
p = b * u
if p <= 1.0:
x = p ** (1.0 / alpha)
x = -_log((b - p) / alpha)
u1 = random()
if p > 1.0:
if u1 <= x ** (alpha - 1.0):
elif u1 <= _exp(-x):
return x * beta
def betavariate(self, alpha, beta):
"""Beta distribution.
Conditions on the parameters are alpha > 0 and beta > 0.
Returned values range between 0 and 1.
## See
## for Ivan Frohne's insightful analysis of why the original implementation:
## def betavariate(self, alpha, beta):
## # Discrete Event Simulation in C, pp 87-88.
## y = self.expovariate(alpha)
## z = self.expovariate(1.0/beta)
## return z/(y+z)
## was dead wrong, and how it probably got that way.
# This version due to Janne Sinkkonen, and matches all the std
# texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
y = self.gammavariate(alpha, 1.0)
if y:
return y / (y + self.gammavariate(beta, 1.0))
return 0.0
def paretovariate(self, alpha):
"""Pareto distribution. alpha is the shape parameter."""
# Jain, pg. 495
u = 1.0 - self.random()
return u ** (-1.0 / alpha)
def weibullvariate(self, alpha, beta):
"""Weibull distribution.
alpha is the scale parameter and beta is the shape parameter.
# Jain, pg. 499; bug fix courtesy Bill Arms
u = 1.0 - self.random()
return alpha * (-_log(u)) ** (1.0 / beta)
## ------------------------------------------------------------------
## --------------- 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 non-negative')
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 randbytes(self, n):
"""Generate n random bytes."""
# os.urandom(n) fails with ValueError for n < 0
# and returns an empty bytes string for n == 0.
return _urandom(n)
def seed(self, *args, **kwds):
"Stub method. Not used for a system random number generator."
return None
def _notimplemented(self, *args, **kwds):
"Method should not be called for a system random number generator."
raise NotImplementedError('System entropy source does not have state.')
getstate = setstate = _notimplemented
# ----------------------------------------------------------------------
# Create one instance, seeded from current time, and export its methods
# as module-level functions. The functions share state across all uses
# (both in the user's code and in the Python libraries), but that's fine
# for most programs and is easier for the casual user than making them
# instantiate their own Random() instance.
_inst = Random()
seed = _inst.seed
random = _inst.random
uniform = _inst.uniform
triangular = _inst.triangular
randint = _inst.randint
choice = _inst.choice
randrange = _inst.randrange
sample = _inst.sample
shuffle = _inst.shuffle
choices = _inst.choices
normalvariate = _inst.normalvariate
lognormvariate = _inst.lognormvariate
expovariate = _inst.expovariate
vonmisesvariate = _inst.vonmisesvariate
gammavariate = _inst.gammavariate
gauss = _inst.gauss
betavariate = _inst.betavariate
paretovariate = _inst.paretovariate
weibullvariate = _inst.weibullvariate
getstate = _inst.getstate
setstate = _inst.setstate
getrandbits = _inst.getrandbits
randbytes = _inst.randbytes
## ------------------------------------------------------
## ----------------- test program -----------------------
def _test_generator(n, func, args):
from statistics import stdev, fmean as mean
from time import perf_counter
t0 = perf_counter()
data = [func(*args) for i in range(n)]
t1 = perf_counter()
xbar = mean(data)
sigma = stdev(data, xbar)
low = min(data)
high = max(data)
print(f'{t1 - t0:.3f} sec, {n} times {func.__name__}')
print('avg %g, stddev %g, min %g, max %g\n' % (xbar, sigma, low, high))
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))
## ------------------------------------------------------
## ------------------ fork support ---------------------
if hasattr(_os, "fork"):
if __name__ == '__main__':