summaryrefslogtreecommitdiff
path: root/absl/random/poisson_distribution_test.cc
blob: 54755960e108a0a5e642f1b6aaf4f6a594585f22 (plain)
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
// Copyright 2017 The Abseil Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#include "absl/random/poisson_distribution.h"

#include <algorithm>
#include <cstddef>
#include <cstdint>
#include <iterator>
#include <random>
#include <sstream>
#include <string>
#include <vector>

#include "gmock/gmock.h"
#include "gtest/gtest.h"
#include "absl/base/macros.h"
#include "absl/container/flat_hash_map.h"
#include "absl/log/log.h"
#include "absl/random/internal/chi_square.h"
#include "absl/random/internal/distribution_test_util.h"
#include "absl/random/internal/pcg_engine.h"
#include "absl/random/internal/sequence_urbg.h"
#include "absl/random/random.h"
#include "absl/strings/str_cat.h"
#include "absl/strings/str_format.h"
#include "absl/strings/str_replace.h"
#include "absl/strings/strip.h"

// Notes about generating poisson variates:
//
// It is unlikely that any implementation of std::poisson_distribution
// will be stable over time and across library implementations. For instance
// the three different poisson variate generators listed below all differ:
//
// https://github.com/ampl/gsl/tree/master/randist/poisson.c
// * GSL uses a gamma + binomial + knuth method to compute poisson variates.
//
// https://github.com/gcc-mirror/gcc/blob/master/libstdc%2B%2B-v3/include/bits/random.tcc
// * GCC uses the Devroye rejection algorithm, based on
// Devroye, L. Non-Uniform Random Variates Generation. Springer-Verlag,
// New York, 1986, Ch. X, Sects. 3.3 & 3.4 (+ Errata!), ~p.511
//   http://www.nrbook.com/devroye/
//
// https://github.com/llvm-mirror/libcxx/blob/master/include/random
// * CLANG uses a different rejection method, which appears to include a
// normal-distribution approximation and an exponential distribution to
// compute the threshold, including a similar factorial approximation to this
// one, but it is unclear where the algorithm comes from, exactly.
//

namespace {

using absl::random_internal::kChiSquared;

// The PoissonDistributionInterfaceTest provides a basic test that
// absl::poisson_distribution conforms to the interface and serialization
// requirements imposed by [rand.req.dist] for the common integer types.

template <typename IntType>
class PoissonDistributionInterfaceTest : public ::testing::Test {};

using IntTypes = ::testing::Types<int, int8_t, int16_t, int32_t, int64_t,
                                  uint8_t, uint16_t, uint32_t, uint64_t>;
TYPED_TEST_SUITE(PoissonDistributionInterfaceTest, IntTypes);

TYPED_TEST(PoissonDistributionInterfaceTest, SerializeTest) {
  using param_type = typename absl::poisson_distribution<TypeParam>::param_type;
  const double kMax =
      std::min(1e10 /* assertion limit */,
               static_cast<double>(std::numeric_limits<TypeParam>::max()));

  const double kParams[] = {
      // Cases around 1.
      1,                         //
      std::nextafter(1.0, 0.0),  // 1 - epsilon
      std::nextafter(1.0, 2.0),  // 1 + epsilon
      // Arbitrary values.
      1e-8, 1e-4,
      0.0000005,  // ~7.2e-7
      0.2,        // ~0.2x
      0.5,        // 0.72
      2,          // ~2.8
      20,         // 3x ~9.6
      100, 1e4, 1e8, 1.5e9, 1e20,
      // Boundary cases.
      std::numeric_limits<double>::max(),
      std::numeric_limits<double>::epsilon(),
      std::nextafter(std::numeric_limits<double>::min(),
                     1.0),                        // min + epsilon
      std::numeric_limits<double>::min(),         // smallest normal
      std::numeric_limits<double>::denorm_min(),  // smallest denorm
      std::numeric_limits<double>::min() / 2,     // denorm
      std::nextafter(std::numeric_limits<double>::min(),
                     0.0),  // denorm_max
  };


  constexpr int kCount = 1000;
  absl::InsecureBitGen gen;
  for (const double m : kParams) {
    const double mean = std::min(kMax, m);
    const param_type param(mean);

    // Validate parameters.
    absl::poisson_distribution<TypeParam> before(mean);
    EXPECT_EQ(before.mean(), param.mean());

    {
      absl::poisson_distribution<TypeParam> via_param(param);
      EXPECT_EQ(via_param, before);
      EXPECT_EQ(via_param.param(), before.param());
    }

    // Smoke test.
    auto sample_min = before.max();
    auto sample_max = before.min();
    for (int i = 0; i < kCount; i++) {
      auto sample = before(gen);
      EXPECT_GE(sample, before.min());
      EXPECT_LE(sample, before.max());
      if (sample > sample_max) sample_max = sample;
      if (sample < sample_min) sample_min = sample;
    }

    LOG(INFO) << "Range {" << param.mean() << "}: " << sample_min << ", "
              << sample_max;

    // Validate stream serialization.
    std::stringstream ss;
    ss << before;

    absl::poisson_distribution<TypeParam> after(3.8);

    EXPECT_NE(before.mean(), after.mean());
    EXPECT_NE(before.param(), after.param());
    EXPECT_NE(before, after);

    ss >> after;

    EXPECT_EQ(before.mean(), after.mean())  //
        << ss.str() << " "                  //
        << (ss.good() ? "good " : "")       //
        << (ss.bad() ? "bad " : "")         //
        << (ss.eof() ? "eof " : "")         //
        << (ss.fail() ? "fail " : "");
  }
}

// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm

class PoissonModel {
 public:
  explicit PoissonModel(double mean) : mean_(mean) {}

  double mean() const { return mean_; }
  double variance() const { return mean_; }
  double stddev() const { return std::sqrt(variance()); }
  double skew() const { return 1.0 / mean_; }
  double kurtosis() const { return 3.0 + 1.0 / mean_; }

  // InitCDF() initializes the CDF for the distribution parameters.
  void InitCDF();

  // The InverseCDF, or the Percent-point function returns x, P(x) < v.
  struct CDF {
    size_t index;
    double pmf;
    double cdf;
  };
  CDF InverseCDF(double p) {
    CDF target{0, 0, p};
    auto it = std::upper_bound(
        std::begin(cdf_), std::end(cdf_), target,
        [](const CDF& a, const CDF& b) { return a.cdf < b.cdf; });
    return *it;
  }

  void LogCDF() {
    LOG(INFO) << "CDF (mean = " << mean_ << ")";
    for (const auto c : cdf_) {
      LOG(INFO) << c.index << ": pmf=" << c.pmf << " cdf=" << c.cdf;
    }
  }

 private:
  const double mean_;

  std::vector<CDF> cdf_;
};

// The goal is to compute an InverseCDF function, or percent point function for
// the poisson distribution, and use that to partition our output into equal
// range buckets.  However there is no closed form solution for the inverse cdf
// for poisson distributions (the closest is the incomplete gamma function).
// Instead, `InitCDF` iteratively computes the PMF and the CDF. This enables
// searching for the bucket points.
void PoissonModel::InitCDF() {
  if (!cdf_.empty()) {
    // State already initialized.
    return;
  }
  ABSL_ASSERT(mean_ < 201.0);

  const size_t max_i = 50 * stddev() + mean();
  const double e_neg_mean = std::exp(-mean());
  ABSL_ASSERT(e_neg_mean > 0);

  double d = 1;
  double last_result = e_neg_mean;
  double cumulative = e_neg_mean;
  if (e_neg_mean > 1e-10) {
    cdf_.push_back({0, e_neg_mean, cumulative});
  }
  for (size_t i = 1; i < max_i; i++) {
    d *= (mean() / i);
    double result = e_neg_mean * d;
    cumulative += result;
    if (result < 1e-10 && result < last_result && cumulative > 0.999999) {
      break;
    }
    if (result > 1e-7) {
      cdf_.push_back({i, result, cumulative});
    }
    last_result = result;
  }
  ABSL_ASSERT(!cdf_.empty());
}

// PoissonDistributionZTest implements a z-test for the poisson distribution.

struct ZParam {
  double mean;
  double p_fail;   // Z-Test probability of failure.
  int trials;      // Z-Test trials.
  size_t samples;  // Z-Test samples.
};

class PoissonDistributionZTest : public testing::TestWithParam<ZParam>,
                                 public PoissonModel {
 public:
  PoissonDistributionZTest() : PoissonModel(GetParam().mean) {}

  // ZTestImpl provides a basic z-squared test of the mean vs. expected
  // mean for data generated by the poisson distribution.
  template <typename D>
  bool SingleZTest(const double p, const size_t samples);

  // We use a fixed bit generator for distribution accuracy tests.  This allows
  // these tests to be deterministic, while still testing the qualify of the
  // implementation.
  absl::random_internal::pcg64_2018_engine rng_{0x2B7E151628AED2A6};
};

template <typename D>
bool PoissonDistributionZTest::SingleZTest(const double p,
                                           const size_t samples) {
  D dis(mean());

  absl::flat_hash_map<int32_t, int> buckets;
  std::vector<double> data;
  data.reserve(samples);
  for (int j = 0; j < samples; j++) {
    const auto x = dis(rng_);
    buckets[x]++;
    data.push_back(x);
  }

  // The null-hypothesis is that the distribution is a poisson distribution with
  // the provided mean (not estimated from the data).
  const auto m = absl::random_internal::ComputeDistributionMoments(data);
  const double max_err = absl::random_internal::MaxErrorTolerance(p);
  const double z = absl::random_internal::ZScore(mean(), m);
  const bool pass = absl::random_internal::Near("z", z, 0.0, max_err);

  if (!pass) {
    // clang-format off
    LOG(INFO)
        << "p=" << p << " max_err=" << max_err << "\n"
           " mean=" << m.mean << " vs. " << mean() << "\n"
           " stddev=" << std::sqrt(m.variance) << " vs. " << stddev() << "\n"
           " skewness=" << m.skewness << " vs. " << skew() << "\n"
           " kurtosis=" << m.kurtosis << " vs. " << kurtosis() << "\n"
           " z=" << z;
    // clang-format on
  }
  return pass;
}

TEST_P(PoissonDistributionZTest, AbslPoissonDistribution) {
  const auto& param = GetParam();
  const int expected_failures =
      std::max(1, static_cast<int>(std::ceil(param.trials * param.p_fail)));
  const double p = absl::random_internal::RequiredSuccessProbability(
      param.p_fail, param.trials);

  int failures = 0;
  for (int i = 0; i < param.trials; i++) {
    failures +=
        SingleZTest<absl::poisson_distribution<int32_t>>(p, param.samples) ? 0
                                                                           : 1;
  }
  EXPECT_LE(failures, expected_failures);
}

std::vector<ZParam> GetZParams() {
  // These values have been adjusted from the "exact" computed values to reduce
  // failure rates.
  //
  // It turns out that the actual values are not as close to the expected values
  // as would be ideal.
  return std::vector<ZParam>({
      // Knuth method.
      ZParam{0.5, 0.01, 100, 1000},
      ZParam{1.0, 0.01, 100, 1000},
      ZParam{10.0, 0.01, 100, 5000},
      // Split-knuth method.
      ZParam{20.0, 0.01, 100, 10000},
      ZParam{50.0, 0.01, 100, 10000},
      // Ratio of gaussians method.
      ZParam{51.0, 0.01, 100, 10000},
      ZParam{200.0, 0.05, 10, 100000},
      ZParam{100000.0, 0.05, 10, 1000000},
  });
}

std::string ZParamName(const ::testing::TestParamInfo<ZParam>& info) {
  const auto& p = info.param;
  std::string name = absl::StrCat("mean_", absl::SixDigits(p.mean));
  return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
}

INSTANTIATE_TEST_SUITE_P(All, PoissonDistributionZTest,
                         ::testing::ValuesIn(GetZParams()), ZParamName);

// The PoissonDistributionChiSquaredTest class provides a basic test framework
// for variates generated by a conforming poisson_distribution.
class PoissonDistributionChiSquaredTest : public testing::TestWithParam<double>,
                                          public PoissonModel {
 public:
  PoissonDistributionChiSquaredTest() : PoissonModel(GetParam()) {}

  // The ChiSquaredTestImpl provides a chi-squared goodness of fit test for data
  // generated by the poisson distribution.
  template <typename D>
  double ChiSquaredTestImpl();

 private:
  void InitChiSquaredTest(const double buckets);

  std::vector<size_t> cutoffs_;
  std::vector<double> expected_;

  // We use a fixed bit generator for distribution accuracy tests.  This allows
  // these tests to be deterministic, while still testing the qualify of the
  // implementation.
  absl::random_internal::pcg64_2018_engine rng_{0x2B7E151628AED2A6};
};

void PoissonDistributionChiSquaredTest::InitChiSquaredTest(
    const double buckets) {
  if (!cutoffs_.empty() && !expected_.empty()) {
    return;
  }
  InitCDF();

  // The code below finds cuttoffs that yield approximately equally-sized
  // buckets to the extent that it is possible. However for poisson
  // distributions this is particularly challenging for small mean parameters.
  // Track the expected proportion of items in each bucket.
  double last_cdf = 0;
  const double inc = 1.0 / buckets;
  for (double p = inc; p <= 1.0; p += inc) {
    auto result = InverseCDF(p);
    if (!cutoffs_.empty() && cutoffs_.back() == result.index) {
      continue;
    }
    double d = result.cdf - last_cdf;
    cutoffs_.push_back(result.index);
    expected_.push_back(d);
    last_cdf = result.cdf;
  }
  cutoffs_.push_back(std::numeric_limits<size_t>::max());
  expected_.push_back(std::max(0.0, 1.0 - last_cdf));
}

template <typename D>
double PoissonDistributionChiSquaredTest::ChiSquaredTestImpl() {
  const int kSamples = 2000;
  const int kBuckets = 50;

  // The poisson CDF fails for large mean values, since e^-mean exceeds the
  // machine precision. For these cases, using a normal approximation would be
  // appropriate.
  ABSL_ASSERT(mean() <= 200);
  InitChiSquaredTest(kBuckets);

  D dis(mean());

  std::vector<int32_t> counts(cutoffs_.size(), 0);
  for (int j = 0; j < kSamples; j++) {
    const size_t x = dis(rng_);
    auto it = std::lower_bound(std::begin(cutoffs_), std::end(cutoffs_), x);
    counts[std::distance(cutoffs_.begin(), it)]++;
  }

  // Normalize the counts.
  std::vector<int32_t> e(expected_.size(), 0);
  for (int i = 0; i < e.size(); i++) {
    e[i] = kSamples * expected_[i];
  }

  // The null-hypothesis is that the distribution is a poisson distribution with
  // the provided mean (not estimated from the data).
  const int dof = static_cast<int>(counts.size()) - 1;

  // The threshold for logging is 1-in-50.
  const double threshold = absl::random_internal::ChiSquareValue(dof, 0.98);

  const double chi_square = absl::random_internal::ChiSquare(
      std::begin(counts), std::end(counts), std::begin(e), std::end(e));

  const double p = absl::random_internal::ChiSquarePValue(chi_square, dof);

  // Log if the chi_squared value is above the threshold.
  if (chi_square > threshold) {
    LogCDF();

    LOG(INFO) << "VALUES  buckets=" << counts.size()
              << "  samples=" << kSamples;
    for (size_t i = 0; i < counts.size(); i++) {
      LOG(INFO) << cutoffs_[i] << ": " << counts[i] << " vs. E=" << e[i];
    }

    LOG(INFO) << kChiSquared << "(data, dof=" << dof << ") = " << chi_square
              << " (" << p << ")\n"
              << " vs.\n"
              << kChiSquared << " @ 0.98 = " << threshold;
  }
  return p;
}

TEST_P(PoissonDistributionChiSquaredTest, AbslPoissonDistribution) {
  const int kTrials = 20;

  // Large values are not yet supported -- this requires estimating the cdf
  // using the normal distribution instead of the poisson in this case.
  ASSERT_LE(mean(), 200.0);
  if (mean() > 200.0) {
    return;
  }

  int failures = 0;
  for (int i = 0; i < kTrials; i++) {
    double p_value = ChiSquaredTestImpl<absl::poisson_distribution<int32_t>>();
    if (p_value < 0.005) {
      failures++;
    }
  }
  // There is a 0.10% chance of producing at least one failure, so raise the
  // failure threshold high enough to allow for a flake rate < 10,000.
  EXPECT_LE(failures, 4);
}

INSTANTIATE_TEST_SUITE_P(All, PoissonDistributionChiSquaredTest,
                         ::testing::Values(0.5, 1.0, 2.0, 10.0, 50.0, 51.0,
                                           200.0));

// NOTE: absl::poisson_distribution is not guaranteed to be stable.
TEST(PoissonDistributionTest, StabilityTest) {
  using testing::ElementsAre;
  // absl::poisson_distribution stability relies on stability of
  // std::exp, std::log, std::sqrt, std::ceil, std::floor, and
  // absl::FastUniformBits, absl::StirlingLogFactorial, absl::RandU64ToDouble.
  absl::random_internal::sequence_urbg urbg({
      0x035b0dc7e0a18acfull, 0x06cebe0d2653682eull, 0x0061e9b23861596bull,
      0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull, 0xC332DDEFBE6C5AA5ull,
      0x6558218568AB9702ull, 0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull,
      0xECDD4775619F1510ull, 0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull,
      0xB5735C904C70A239ull, 0xD59E9E0BCBAADE14ull, 0xEECC86BC60622CA7ull,
      0x4864f22c059bf29eull, 0x247856d8b862665cull, 0xe46e86e9a1337e10ull,
      0xd8c8541f3519b133ull, 0xe75b5162c567b9e4ull, 0xf732e5ded7009c5bull,
      0xb170b98353121eacull, 0x1ec2e8986d2362caull, 0x814c8e35fe9a961aull,
      0x0c3cd59c9b638a02ull, 0xcb3bb6478a07715cull, 0x1224e62c978bbc7full,
      0x671ef2cb04e81f6eull, 0x3c1cbd811eaf1808ull, 0x1bbc23cfa8fac721ull,
      0xa4c2cda65e596a51ull, 0xb77216fad37adf91ull, 0x836d794457c08849ull,
      0xe083df03475f49d7ull, 0xbc9feb512e6b0d6cull, 0xb12d74fdd718c8c5ull,
      0x12ff09653bfbe4caull, 0x8dd03a105bc4ee7eull, 0x5738341045ba0d85ull,
      0xf3fd722dc65ad09eull, 0xfa14fd21ea2a5705ull, 0xffe6ea4d6edb0c73ull,
      0xD07E9EFE2BF11FB4ull, 0x95DBDA4DAE909198ull, 0xEAAD8E716B93D5A0ull,
      0xD08ED1D0AFC725E0ull, 0x8E3C5B2F8E7594B7ull, 0x8FF6E2FBF2122B64ull,
      0x8888B812900DF01Cull, 0x4FAD5EA0688FC31Cull, 0xD1CFF191B3A8C1ADull,
      0x2F2F2218BE0E1777ull, 0xEA752DFE8B021FA1ull, 0xE5A0CC0FB56F74E8ull,
      0x18ACF3D6CE89E299ull, 0xB4A84FE0FD13E0B7ull, 0x7CC43B81D2ADA8D9ull,
      0x165FA26680957705ull, 0x93CC7314211A1477ull, 0xE6AD206577B5FA86ull,
      0xC75442F5FB9D35CFull, 0xEBCDAF0C7B3E89A0ull, 0xD6411BD3AE1E7E49ull,
      0x00250E2D2071B35Eull, 0x226800BB57B8E0AFull, 0x2464369BF009B91Eull,
      0x5563911D59DFA6AAull, 0x78C14389D95A537Full, 0x207D5BA202E5B9C5ull,
      0x832603766295CFA9ull, 0x11C819684E734A41ull, 0xB3472DCA7B14A94Aull,
  });

  std::vector<int> output(10);

  // Method 1.
  {
    absl::poisson_distribution<int> dist(5);
    std::generate(std::begin(output), std::end(output),
                  [&] { return dist(urbg); });
  }
  EXPECT_THAT(output,  // mean = 4.2
              ElementsAre(1, 0, 0, 4, 2, 10, 3, 3, 7, 12));

  // Method 2.
  {
    urbg.reset();
    absl::poisson_distribution<int> dist(25);
    std::generate(std::begin(output), std::end(output),
                  [&] { return dist(urbg); });
  }
  EXPECT_THAT(output,  // mean = 19.8
              ElementsAre(9, 35, 18, 10, 35, 18, 10, 35, 18, 10));

  // Method 3.
  {
    urbg.reset();
    absl::poisson_distribution<int> dist(121);
    std::generate(std::begin(output), std::end(output),
                  [&] { return dist(urbg); });
  }
  EXPECT_THAT(output,  // mean = 124.1
              ElementsAre(161, 122, 129, 124, 112, 112, 117, 120, 130, 114));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_1) {
  // This tests small values of the Knuth method.
  // The underlying uniform distribution will generate exactly 0.5.
  absl::random_internal::sequence_urbg urbg({0x8000000000000001ull});
  absl::poisson_distribution<int> dist(5);
  EXPECT_EQ(7, dist(urbg));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_2) {
  // This tests larger values of the Knuth method.
  // The underlying uniform distribution will generate exactly 0.5.
  absl::random_internal::sequence_urbg urbg({0x8000000000000001ull});
  absl::poisson_distribution<int> dist(25);
  EXPECT_EQ(36, dist(urbg));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_3) {
  // This variant uses the ratio of uniforms method.
  absl::random_internal::sequence_urbg urbg(
      {0x7fffffffffffffffull, 0x8000000000000000ull});

  absl::poisson_distribution<int> dist(121);
  EXPECT_EQ(121, dist(urbg));
}

}  // namespace