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"""Random variable generators. bytes ----- uniform bytes (values between 0 and 255) 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. """ # 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 try: # hashlib is pretty heavy to load, try lean internal module first from _sha512 import sha512 as _sha512 except ImportError: # fallback to official implementation from hashlib import sha512 as _sha512 __all__ = [ "Random", "SystemRandom", "betavariate", "choice", "choices", "expovariate", "gammavariate", "gauss", "getrandbits", "getstate", "lognormvariate", "normalvariate", "paretovariate", "randbytes", "randint", "random", "randrange", "sample", "seed", "setstate", "shuffle", "triangular", "uniform", "vonmisesvariate", "weibullvariate", ] 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 RECIP_BPF = 2 ** -BPF _ONE = 1 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 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 = int.from_bytes(a + _sha512(a).digest()) elif not isinstance(a, (type(None), int, float, str, bytes, bytearray)): raise TypeError('The only supported seed types are: None,\n' 'int, float, str, bytes, and bytearray.') 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 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 self.setstate(state) 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 ranges. """ for c in cls.__mro__: if '_randbelow' in c.__dict__: # just inherit it break if 'getrandbits' in c.__dict__: cls._randbelow = cls._randbelow_with_getrandbits break if 'random' in c.__dict__: cls._randbelow = cls._randbelow_without_getrandbits break def _randbelow_with_getrandbits(self, n): "Return a random int in the range [0,n). Defined for n > 0." getrandbits = self.getrandbits k = n.bit_length() # don't use (n-1) here because n can be 1 r = getrandbits(k) # 0 <= r < 2**k while r >= n: r = getrandbits(k) return r def _randbelow_without_getrandbits(self, n, maxsize=1<<BPF): """Return a random int in the range [0,n). Defined for 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) 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=_ONE): """Choose a random item from range(stop) or range(start, stop[, step]). Roughly equivalent to ``choice(range(start, stop, step))`` but supports arbitrarily large ranges and is optimized for common cases. """ # This code is a bit messy to make it fast for the # common case while still doing adequate error checking. try: istart = _index(start) except TypeError: istart = int(start) if istart != start: _warn('randrange() will raise TypeError in the future', DeprecationWarning, 2) raise ValueError("non-integer arg 1 for randrange()") _warn('non-integer arguments to randrange() have been deprecated ' 'since Python 3.10 and will be removed in a subsequent ' 'version', DeprecationWarning, 2) if stop is None: # We don't check for "step != 1" because it hasn't been # type checked and converted to an integer yet. if step is not _ONE: raise TypeError('Missing a non-None stop argument') if istart > 0: return self._randbelow(istart) raise ValueError("empty range for randrange()") # stop argument supplied. try: istop = _index(stop) except TypeError: istop = int(stop) if istop != stop: _warn('randrange() will raise TypeError in the future', DeprecationWarning, 2) raise ValueError("non-integer stop for randrange()") _warn('non-integer arguments to randrange() have been deprecated ' 'since Python 3.10 and will be removed in a subsequent ' 'version', DeprecationWarning, 2) width = istop - istart try: istep = _index(step) except TypeError: istep = int(step) if istep != step: _warn('randrange() will raise TypeError in the future', DeprecationWarning, 2) raise ValueError("non-integer step for randrange()") _warn('non-integer arguments to randrange() have been deprecated ' 'since Python 3.10 and will be removed in a subsequent ' 'version', DeprecationWarning, 2) # Fast path. if istep == 1: if width > 0: return istart + self._randbelow(width) 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 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) ## -------------------- sequence methods ------------------- def choice(self, seq): """Choose a random element from a non-empty sequence.""" # As an accommodation for NumPy, we don't use "if not seq" # because bool(numpy.array()) raises a ValueError. if not len(seq): raise IndexError('Cannot choose from an empty sequence') return seq[self._randbelow(len(seq))] def shuffle(self, x): """Shuffle list x in place, and return 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] def sample(self, population, k, *, counts=None): """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. 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): 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 = self.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 else: selected = set() selected_add = selected.add for i in range(k): j = randbelow(n) while j in selected: j = randbelow(n) selected_add(j) result[i] = population[j] return result def choices(self, population, weights=None, *, cum_weights=None, k=1): """Return a k sized list of population elements chosen with replacement. If the relative weights or cumulative weights are not specified, the selections are made with equal probability. """ random = self.random n = len(population) if cum_weights is None: if weights is None: floor = _floor n += 0.0 # convert to float for a small speed improvement return [population[floor(random() * n)] for i in _repeat(None, k)] try: cum_weights = list(_accumulate(weights)) except TypeError: if not isinstance(weights, int): raise k = weights raise TypeError( f'The number of choices must be a keyword argument: {k=}' ) from None 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. 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) def normalvariate(self, mu=0.0, sigma=1.0): """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): break return mu + z * sigma def gauss(self, mu=0.0, sigma=1.0): """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): 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 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: continue u2 = 1.0 - random() v = _log(u1 / (1.0 - u1)) / ainv x = alpha * _exp(v) z = u1 * u1 * u2 r = bbb + ccc * v - x if r + SG_MAGICCONST - 4.5 * z >= 0.0 or r >= _log(z): return x * beta elif alpha == 1.0: # expovariate(1/beta) return -_log(1.0 - random()) * beta else: # alpha is between 0 and 1 (exclusive) # Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle while True: 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 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 ## 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. # 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 <= X < 1.0.""" return (int.from_bytes(_urandom(7)) >> 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)) 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 _repeat(None, 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"): _os.register_at_fork(after_in_child=_inst.seed) if __name__ == '__main__': _test()