cpython/Lib/random.py at 3.5 · python/cpython · GitHub

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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

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

__all__ = ["Random","seed","random","uniform","randint","choice","sample",

"randrange","shuffle","normalvariate","lognormvariate",

"expovariate","vonmisesvariate","gammavariate","triangular",

"gauss","betavariate","paretovariate","weibullvariate",

"getstate","setstate", "getrandbits",

"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 a is None:

try:

# Seed with enough bytes to span the 19937 bit

# state space for the Mersenne Twister

a = int.from_bytes(_urandom(2500), 'big')

except NotImplementedError:

import time

a = int(time.time() * 256) # use fractional seconds

if version == 1 and isinstance(a, (str, bytes)):

x = ord(a[0]) << 7 if a else 0

for c in a:

x = ((1000003 * x) ^ ord(c)) & 0xFFFFFFFFFFFFFFFF

x ^= len(a)

a = -2 if x == -1 else x

if version == 2:

if 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)

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')

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")

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

## -------------------- 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) * (u * c) ** 0.5

## -------------------- 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)

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.)

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 / 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.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,', 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

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 __name__ == '__main__':

_test()

cpython/Lib/random.py at 3.5 · python/cpython · GitHub

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