module(random) """Random variable generators. integers -------- uniform within range sequences --------- pick random element pick random sample pick weighted random sample generate random permutation distributions on the real line: ------------------------------ uniform triangular normal (Gaussian) lognormal negative exponential gamma beta pareto Weibull distributions on the circle (angles 0 to 2pi) --------------------------------------------- circular uniform von Mises General notes on the underlying Mersenne Twister core generator: * The period is 2**19937-1. * It is one of the most extensively tested generators in existence. * The random() method is implemented in C, executes in a single Python step, and is, therefore, threadsafe. """ from warnings import warn as _warn from types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodType from math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceil from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin from os import urandom as _urandom from collections.abc import Set as _Set, Sequence as _Sequence from hashlib import sha512 as _sha512 import itertools as _itertools import bisect as _bisect __all__ = ["Random","seed","random","uniform","randint","choice","sample", "randrange","shuffle","normalvariate","lognormvariate", "expovariate","vonmisesvariate","gammavariate","triangular", "gauss","betavariate","paretovariate","weibullvariate", "getstate","setstate", "getrandbits", "choices", "SystemRandom"] NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0) TWOPI = 2.0*_pi LOG4 = _log(4.0) SG_MAGICCONST = 1.0 + _log(4.5) BPF = 53 # Number of bits in a float RECIP_BPF = 2**-BPF # Translated by Guido van Rossum from C source provided by # Adrian Baddeley. Adapted by Raymond Hettinger for use with # the Mersenne Twister and os.urandom() core generators. import _random class Random (_random.Random): """Random number generator base class used by bound module functions. Used to instantiate instances of Random to get generators that don't share state. Class Random can also be subclassed if you want to use a different basic generator of your own devising: in that case, override the following methods: random(), seed(), getstate(), and setstate(). Optionally, implement a getrandbits() method so that randrange() can cover arbitrarily large ranges. """ VERSION = 3 # used by getstate/setstate def __init__(self, x=None): """Initialize an instance. Optional argument x controls seeding, as for Random.seed(). """ self.seed(x) self.gauss_next = False def seed(self, a=None, version=2): """Initialize internal state from hashable object. None or no argument seeds from current time or from an operating system specific randomness source if available. If *a* is an int, all bits are used. For version 2 (the default), all of the bits are used if *a* is a str, bytes, or bytearray. For version 1 (provided for reproducing random sequences from older versions of Python), the algorithm for str and bytes generates a narrower range of seeds. """ if version == 1 and isinstance(a, (str, bytes)): a = a.decode('latin-1') if isinstance(a, bytes) else a x = ord(a[0]) << 7 if a else 0 for c in map(ord, a): x = ((1000003 * x) ^ c) & 0xFFFFFFFFFFFFFFFF x ^= len(a) a = -2 if x == -1 else x if version == 2 and isinstance(a, (str, bytes, bytearray)): if isinstance(a, str): a = a.encode() a += _sha512(a).digest() a = int.from_bytes(a, 'big') super().seed(a) self.gauss_next = False 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, step, _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) ## -------------------- sequence methods ------------------- def choice(self, seq): """Choose a random element from a non-empty sequence.""" try: i = self._randbelow(len(seq)) except ValueError: raise IndexError('Cannot choose from an empty sequence') from None return seq[i] def shuffle(self, x, random=None): """Shuffle list x in place, and return None. Optional argument random is a 0-argument function returning a random float in [0.0, 1.0); if it is the default None, the standard random.random will be used. """ if random is None: randbelow = self._randbelow for i in reversed(range(1, len(x))): # pick an element in x[:i+1] with which to exchange x[i] j = randbelow(i+1) x[i], x[j] = x[j], x[i] else: _int = int for i in reversed(range(1, len(x))): # pick an element in x[:i+1] with which to exchange x[i] j = _int(random() * (i+1)) x[i], x[j] = x[j], x[i] def sample(self, population, k): """Chooses k unique random elements from a population sequence or set. Returns a new list containing elements from the population while leaving the original population unchanged. The resulting list is in selection order so that all sub-slices will also be valid random samples. This allows raffle winners (the sample) to be partitioned into grand prize and second place winners (the subslices). Members of the population need not be hashable or unique. If the population contains repeats, then each occurrence is a possible selection in the sample. To choose a sample in a range of integers, use range as an argument. This is especially fast and space efficient for sampling from a large population: sample(range(10000000), 60) """ # Sampling without replacement entails tracking either potential # selections (the pool) in a list or previous selections in a set. # When the number of selections is small compared to the # population, then tracking selections is efficient, requiring # only a small set and an occasional reselection. For # a larger number of selections, the pool tracking method is # preferred since the list takes less space than the # set and it doesn't suffer from frequent reselections. if isinstance(population, _Set): population = tuple(population) if not isinstance(population, _Sequence): raise TypeError("Population must be a sequence or set. For dicts, use list(d).") randbelow = self._randbelow n = len(population) if not 0 <= k <= n: raise ValueError("Sample larger than population or is negative") result = [None] * k setsize = 21 # size of a small set minus size of an empty list if k > 5: setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets if n <= setsize: # An n-length list is smaller than a k-length set pool = list(population) for i in range(k): # invariant: non-selected at [0,n-i) j = randbelow(n-i) result[i] = pool[j] pool[j] = pool[n-i-1] # move non-selected item into vacancy else: selected = set() selected_add = selected.add for i in range(k): j = randbelow(n) while j in selected: j = randbelow(n) selected_add(j) result[i] = population[j] return result def choices(self, population, weights=None, *, cum_weights=None, k=1): """Return a k sized list of population elements chosen with replacement. If the relative weights or cumulative weights are not specified, the selections are made with equal probability. """ random = self.random if cum_weights is None: if weights is None: _int = int total = len(population) return [population[_int(random() * total)] for i in range(k)] cum_weights = list(_itertools.accumulate(weights)) elif weights is not None: raise TypeError('Cannot specify both weights and cumulative weights') if len(cum_weights) != len(population): raise ValueError('The number of weights does not match the population') bisect = _bisect.bisect total = cum_weights[-1] return [population[bisect(cum_weights, random() * total)] for i in range(k)] ## -------------------- real-valued distributions ------------------- ## -------------------- uniform distribution ------------------- ## --------------- 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_generator0(n, func): 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() 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)) x = _random.Random() rand = x.random def _test(N=2000): _test_generator(N, randint, (10, 20)) _test_generator(N, random, ()) _test_generator0(N, random) _test_generator0(N, rand) _test_generator(N, normalvariate, (0.0, 1.0)) _test_generator(N, lognormvariate, (0.0, 1.0)) _test_generator(N, vonmisesvariate, (0.0, 1.0)) _test_generator(N, gammavariate, (0.01, 1.0)) _test_generator(N, gammavariate, (0.1, 1.0)) _test_generator(N, gammavariate, (0.1, 2.0)) _test_generator(N, gammavariate, (0.5, 1.0)) _test_generator(N, gammavariate, (0.9, 1.0)) _test_generator(N, gammavariate, (1.0, 1.0)) _test_generator(N, gammavariate, (2.0, 1.0)) _test_generator(N, gammavariate, (20.0, 1.0)) _test_generator(N, gammavariate, (200.0, 1.0)) _test_generator(N, gauss, (0.0, 1.0)) _test_generator(N, betavariate, (3.0, 3.0)) _test_generator(N, triangular, (0.0, 1.0, 1.0/3.0)) # Create one instance, seeded from current time, and export its methods # as module-level functions. The functions share state across all uses #(both in the user's code and in the Python libraries), but that's fine # for most programs and is easier for the casual user than making them # instantiate their own Random() instance. _inst = Random() seed = _inst.seed random = _inst.random uniform = _inst.uniform triangular = _inst.triangular randint = _inst.randint choice = _inst.choice randrange = _inst.randrange sample = _inst.sample shuffle = _inst.shuffle choices = _inst.choices normalvariate = _inst.normalvariate lognormvariate = _inst.lognormvariate expovariate = _inst.expovariate vonmisesvariate = _inst.vonmisesvariate gammavariate = _inst.gammavariate gauss = _inst.gauss betavariate = _inst.betavariate paretovariate = _inst.paretovariate weibullvariate = _inst.weibullvariate getstate = _inst.getstate setstate = _inst.setstate getrandbits = _inst.getrandbits