-
-
Notifications
You must be signed in to change notification settings - Fork 34.2k
Expand file tree
/
Copy pathrandom.py
More file actions
1078 lines (855 loc) · 36.4 KB
/
random.py
File metadata and controls
1078 lines (855 loc) · 36.4 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
"""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
discrete distributions
----------------------
binomial
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 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 math import lgamma as _lgamma, fabs as _fabs, log2 as _log2
from os import urandom as _urandom
from _collections_abc import 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
__all__ = [
"Random",
"SystemRandom",
"betavariate",
"binomialvariate",
"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
_sha512 = None
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)):
global _sha512
if _sha512 is None:
try:
# hashlib is pretty heavy to load, try lean internal
# module first
from _sha2 import sha512 as _sha512
except ImportError:
# fallback to official implementation
from hashlib import sha512 as _sha512
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:\n'
'None, 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."
k = n.bit_length()
r = self.getrandbits(k) # 0 <= r < 2**k
while r >= n:
r = self.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:
from warnings import warn
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.
istart = _index(start)
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.
istop = _index(stop)
width = istop - istart
istep = _index(step)
# Fast path.
if istep == 1:
if width > 0:
return istart + self._randbelow(width)
raise ValueError(f"empty range in randrange({start}, {stop})")
# 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(f"empty range in randrange({start}, {stop}, {step})")
return istart + istep * self._randbelow(n)
def randint(self, a, b):
"""Return random integer in range [a, b], including both end points.
"""
a = _index(a)
b = _index(b)
if b < a:
raise ValueError(f"empty range in randint({a}, {b})")
return a + self._randbelow(b - a + 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 cum_counts else 0
if not isinstance(total, int):
raise TypeError('Counts must be integers')
if total < 0:
raise ValueError('Counts must be non-negative')
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.
The mean (expected value) and variance of the random variable are:
E[X] = (a + b) / 2
Var[X] = (b - a) ** 2 / 12
"""
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
The mean (expected value) and variance of the random variable are:
E[X] = (low + high + mode) / 3
Var[X] = (low**2 + high**2 + mode**2 - low*high - low*mode - high*mode) / 18
"""
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=1.0):
"""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.
The mean (expected value) and variance of the random variable are:
E[X] = 1 / lambd
Var[X] = 1 / lambd ** 2
"""
# 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
The mean (expected value) and variance of the random variable are:
E[X] = alpha * beta
Var[X] = 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.
The mean (expected value) and variance of the random variable are:
E[X] = alpha / (alpha + beta)
Var[X] = alpha * beta / ((alpha + beta)**2 * (alpha + beta + 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)
## -------------------- discrete distributions ---------------------
def binomialvariate(self, n=1, p=0.5):
"""Binomial random variable.
Gives the number of successes for *n* independent trials
with the probability of success in each trial being *p*:
sum(random() < p for i in range(n))
Returns an integer in the range:
0 <= X <= n
The integer is chosen with the probability:
P(X == k) = math.comb(n, k) * p ** k * (1 - p) ** (n - k)
The mean (expected value) and variance of the random variable are:
E[X] = n * p
Var[X] = n * p * (1 - p)
"""
# Error check inputs and handle edge cases
if n < 0:
raise ValueError("n must be non-negative")
if p <= 0.0 or p >= 1.0:
if p == 0.0:
return 0
if p == 1.0:
return n
raise ValueError("p must be in the range 0.0 <= p <= 1.0")
random = self.random
# Fast path for a common case
if n == 1:
return _index(random() < p)
# Exploit symmetry to establish: p <= 0.5
if p > 0.5:
return n - self.binomialvariate(n, 1.0 - p)
if n * p < 10.0:
# BG: Geometric method by Devroye with running time of O(np).
# https://dl.acm.org/doi/pdf/10.1145/42372.42381
x = y = 0
c = _log2(1.0 - p)
if not c:
return x
while True:
y += _floor(_log2(random()) / c) + 1
if y > n:
return x
x += 1
# BTRS: Transformed rejection with squeeze method by Wolfgang Hörmann
# https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.8407&rep=rep1&type=pdf
assert n*p >= 10.0 and p <= 0.5
setup_complete = False
spq = _sqrt(n * p * (1.0 - p)) # Standard deviation of the distribution
b = 1.15 + 2.53 * spq
a = -0.0873 + 0.0248 * b + 0.01 * p
c = n * p + 0.5
vr = 0.92 - 4.2 / b
while True:
u = random()
u -= 0.5
us = 0.5 - _fabs(u)
k = _floor((2.0 * a / us + b) * u + c)
if k < 0 or k > n:
continue
# The early-out "squeeze" test substantially reduces
# the number of acceptance condition evaluations.
v = random()
if us >= 0.07 and v <= vr:
return k
# Acceptance-rejection test.
# Note, the original paper erroneously omits the call to log(v)
# when comparing to the log of the rescaled binomial distribution.
if not setup_complete:
alpha = (2.83 + 5.1 / b) * spq
lpq = _log(p / (1.0 - p))
m = _floor((n + 1) * p) # Mode of the distribution
h = _lgamma(m + 1) + _lgamma(n - m + 1)
setup_complete = True # Only needs to be done once
v *= alpha / (a / (us * us) + b)
if _log(v) <= h - _lgamma(k + 1) - _lgamma(n - k + 1) + (k - m) * lpq:
return k
## ------------------------------------------------------------------
## --------------- 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
binomialvariate = _inst.binomialvariate
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__}{args!r}')
print('avg %g, stddev %g, min %g, max %g\n' % (xbar, sigma, low, high))
def _test(N=10_000):
_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, binomialvariate, (15, 0.60))
_test_generator(N, binomialvariate, (100, 0.75))
_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))
## ------------------------------------------------------