Creation of LTS branch "lts_2019_08_08"20190808

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GitOrigin-RevId: 9ee91d3e430fb33a4590486573792eb0fa146c2d
Change-Id: Ia06e548bc106cc9d136f6c65714be6645317aced

1 files changed, 614 insertions, 0 deletions

diff --git a/absl/random/beta_distribution_test.cc b/absl/random/beta_distribution_test.cc
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+// 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/beta_distribution.h"
+
+#include <algorithm>
+#include <cstddef>
+#include <cstdint>
+#include <iterator>
+#include <random>
+#include <sstream>
+#include <string>
+#include <unordered_map>
+#include <vector>
+
+#include "gmock/gmock.h"
+#include "gtest/gtest.h"
+#include "absl/base/internal/raw_logging.h"
+#include "absl/random/internal/chi_square.h"
+#include "absl/random/internal/distribution_test_util.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"
+
+namespace {
+
+template <typename IntType>
+class BetaDistributionInterfaceTest : public ::testing::Test {};
+
+using RealTypes = ::testing::Types<float, double, long double>;
+TYPED_TEST_CASE(BetaDistributionInterfaceTest, RealTypes);
+
+TYPED_TEST(BetaDistributionInterfaceTest, SerializeTest) {
+  // The threshold for whether std::exp(1/a) is finite.
+  const TypeParam kSmallA =
+      1.0f / std::log((std::numeric_limits<TypeParam>::max)());
+  // The threshold for whether a * std::log(a) is finite.
+  const TypeParam kLargeA =
+      std::exp(std::log((std::numeric_limits<TypeParam>::max)()) -
+               std::log(std::log((std::numeric_limits<TypeParam>::max)())));
+  const TypeParam kLargeAPPC = std::exp(
+      std::log((std::numeric_limits<TypeParam>::max)()) -
+      std::log(std::log((std::numeric_limits<TypeParam>::max)())) - 10.0f);
+  using param_type = typename absl::beta_distribution<TypeParam>::param_type;
+
+  constexpr int kCount = 1000;
+  absl::InsecureBitGen gen;
+  const TypeParam kValues[] = {
+      TypeParam(1e-20), TypeParam(1e-12), TypeParam(1e-8), TypeParam(1e-4),
+      TypeParam(1e-3), TypeParam(0.1), TypeParam(0.25),
+      std::nextafter(TypeParam(0.5), TypeParam(0)),  // 0.5 - epsilon
+      std::nextafter(TypeParam(0.5), TypeParam(1)),  // 0.5 + epsilon
+      TypeParam(0.5), TypeParam(1.0),                //
+      std::nextafter(TypeParam(1), TypeParam(0)),    // 1 - epsilon
+      std::nextafter(TypeParam(1), TypeParam(2)),    // 1 + epsilon
+      TypeParam(12.5), TypeParam(1e2), TypeParam(1e8), TypeParam(1e12),
+      TypeParam(1e20),                        //
+      kSmallA,                                //
+      std::nextafter(kSmallA, TypeParam(0)),  //
+      std::nextafter(kSmallA, TypeParam(1)),  //
+      kLargeA,                                //
+      std::nextafter(kLargeA, TypeParam(0)),  //
+      std::nextafter(kLargeA, std::numeric_limits<TypeParam>::max()),
+      kLargeAPPC,  //
+      std::nextafter(kLargeAPPC, TypeParam(0)),
+      std::nextafter(kLargeAPPC, std::numeric_limits<TypeParam>::max()),
+      // Boundary cases.
+      std::numeric_limits<TypeParam>::max(),
+      std::numeric_limits<TypeParam>::epsilon(),
+      std::nextafter(std::numeric_limits<TypeParam>::min(),
+                     TypeParam(1)),                  // min + epsilon
+      std::numeric_limits<TypeParam>::min(),         // smallest normal
+      std::numeric_limits<TypeParam>::denorm_min(),  // smallest denorm
+      std::numeric_limits<TypeParam>::min() / 2,     // denorm
+      std::nextafter(std::numeric_limits<TypeParam>::min(),
+                     TypeParam(0)),  // denorm_max
+  };
+  for (TypeParam alpha : kValues) {
+    for (TypeParam beta : kValues) {
+      ABSL_INTERNAL_LOG(
+          INFO, absl::StrFormat("Smoke test for Beta(%f, %f)", alpha, beta));
+
+      param_type param(alpha, beta);
+      absl::beta_distribution<TypeParam> before(alpha, beta);
+      EXPECT_EQ(before.alpha(), param.alpha());
+      EXPECT_EQ(before.beta(), param.beta());
+
+      {
+        absl::beta_distribution<TypeParam> via_param(param);
+        EXPECT_EQ(via_param, before);
+        EXPECT_EQ(via_param.param(), before.param());
+      }
+
+      // Smoke test.
+      for (int i = 0; i < kCount; ++i) {
+        auto sample = before(gen);
+        EXPECT_TRUE(std::isfinite(sample));
+        EXPECT_GE(sample, before.min());
+        EXPECT_LE(sample, before.max());
+      }
+
+      // Validate stream serialization.
+      std::stringstream ss;
+      ss << before;
+      absl::beta_distribution<TypeParam> after(3.8f, 1.43f);
+      EXPECT_NE(before.alpha(), after.alpha());
+      EXPECT_NE(before.beta(), after.beta());
+      EXPECT_NE(before.param(), after.param());
+      EXPECT_NE(before, after);
+
+      ss >> after;
+
+#if defined(__powerpc64__) || defined(__PPC64__) || defined(__powerpc__) || \
+    defined(__ppc__) || defined(__PPC__)
+      if (std::is_same<TypeParam, long double>::value) {
+        // Roundtripping floating point values requires sufficient precision
+        // to reconstruct the exact value. It turns out that long double
+        // has some errors doing this on ppc.
+        if (alpha <= std::numeric_limits<double>::max() &&
+            alpha >= std::numeric_limits<double>::lowest()) {
+          EXPECT_EQ(static_cast<double>(before.alpha()),
+                    static_cast<double>(after.alpha()))
+              << ss.str();
+        }
+        if (beta <= std::numeric_limits<double>::max() &&
+            beta >= std::numeric_limits<double>::lowest()) {
+          EXPECT_EQ(static_cast<double>(before.beta()),
+                    static_cast<double>(after.beta()))
+              << ss.str();
+        }
+        continue;
+      }
+#endif
+
+      EXPECT_EQ(before.alpha(), after.alpha());
+      EXPECT_EQ(before.beta(), after.beta());
+      EXPECT_EQ(before, after)           //
+          << ss.str() << " "             //
+          << (ss.good() ? "good " : "")  //
+          << (ss.bad() ? "bad " : "")    //
+          << (ss.eof() ? "eof " : "")    //
+          << (ss.fail() ? "fail " : "");
+    }
+  }
+}
+
+TYPED_TEST(BetaDistributionInterfaceTest, DegenerateCases) {
+  // Extreme cases when the params are abnormal.
+  absl::InsecureBitGen gen;
+  constexpr int kCount = 1000;
+  const TypeParam kSmallValues[] = {
+      std::numeric_limits<TypeParam>::min(),
+      std::numeric_limits<TypeParam>::denorm_min(),
+      std::nextafter(std::numeric_limits<TypeParam>::min(),
+                     TypeParam(0)),  // denorm_max
+      std::numeric_limits<TypeParam>::epsilon(),
+  };
+  const TypeParam kLargeValues[] = {
+      std::numeric_limits<TypeParam>::max() * static_cast<TypeParam>(0.9999),
+      std::numeric_limits<TypeParam>::max() - 1,
+      std::numeric_limits<TypeParam>::max(),
+  };
+  {
+    // Small alpha and beta.
+    // Useful WolframAlpha plots:
+    //   * plot InverseBetaRegularized[x, 0.0001, 0.0001] from 0.495 to 0.505
+    //   * Beta[1.0, 0.0000001, 0.0000001]
+    //   * Beta[0.9999, 0.0000001, 0.0000001]
+    for (TypeParam alpha : kSmallValues) {
+      for (TypeParam beta : kSmallValues) {
+        int zeros = 0;
+        int ones = 0;
+        absl::beta_distribution<TypeParam> d(alpha, beta);
+        for (int i = 0; i < kCount; ++i) {
+          TypeParam x = d(gen);
+          if (x == 0.0) {
+            zeros++;
+          } else if (x == 1.0) {
+            ones++;
+          }
+        }
+        EXPECT_EQ(ones + zeros, kCount);
+        if (alpha == beta) {
+          EXPECT_NE(ones, 0);
+          EXPECT_NE(zeros, 0);
+        }
+      }
+    }
+  }
+  {
+    // Small alpha, large beta.
+    // Useful WolframAlpha plots:
+    //   * plot InverseBetaRegularized[x, 0.0001, 10000] from 0.995 to 1
+    //   * Beta[0, 0.0000001, 1000000]
+    //   * Beta[0.001, 0.0000001, 1000000]
+    //   * Beta[1, 0.0000001, 1000000]
+    for (TypeParam alpha : kSmallValues) {
+      for (TypeParam beta : kLargeValues) {
+        absl::beta_distribution<TypeParam> d(alpha, beta);
+        for (int i = 0; i < kCount; ++i) {
+          EXPECT_EQ(d(gen), 0.0);
+        }
+      }
+    }
+  }
+  {
+    // Large alpha, small beta.
+    // Useful WolframAlpha plots:
+    //   * plot InverseBetaRegularized[x, 10000, 0.0001] from 0 to 0.001
+    //   * Beta[0.99, 1000000, 0.0000001]
+    //   * Beta[1, 1000000, 0.0000001]
+    for (TypeParam alpha : kLargeValues) {
+      for (TypeParam beta : kSmallValues) {
+        absl::beta_distribution<TypeParam> d(alpha, beta);
+        for (int i = 0; i < kCount; ++i) {
+          EXPECT_EQ(d(gen), 1.0);
+        }
+      }
+    }
+  }
+  {
+    // Large alpha and beta.
+    absl::beta_distribution<TypeParam> d(std::numeric_limits<TypeParam>::max(),
+                                         std::numeric_limits<TypeParam>::max());
+    for (int i = 0; i < kCount; ++i) {
+      EXPECT_EQ(d(gen), 0.5);
+    }
+  }
+  {
+    // Large alpha and beta but unequal.
+    absl::beta_distribution<TypeParam> d(
+        std::numeric_limits<TypeParam>::max(),
+        std::numeric_limits<TypeParam>::max() * 0.9999);
+    for (int i = 0; i < kCount; ++i) {
+      TypeParam x = d(gen);
+      EXPECT_NE(x, 0.5f);
+      EXPECT_FLOAT_EQ(x, 0.500025f);
+    }
+  }
+}
+
+class BetaDistributionModel {
+ public:
+  explicit BetaDistributionModel(::testing::tuple<double, double> p)
+      : alpha_(::testing::get<0>(p)), beta_(::testing::get<1>(p)) {}
+
+  double Mean() const { return alpha_ / (alpha_ + beta_); }
+
+  double Variance() const {
+    return alpha_ * beta_ / (alpha_ + beta_ + 1) / (alpha_ + beta_) /
+           (alpha_ + beta_);
+  }
+
+  double Kurtosis() const {
+    return 3 + 6 *
+                   ((alpha_ - beta_) * (alpha_ - beta_) * (alpha_ + beta_ + 1) -
+                    alpha_ * beta_ * (2 + alpha_ + beta_)) /
+                   alpha_ / beta_ / (alpha_ + beta_ + 2) / (alpha_ + beta_ + 3);
+  }
+
+ protected:
+  const double alpha_;
+  const double beta_;
+};
+
+class BetaDistributionTest
+    : public ::testing::TestWithParam<::testing::tuple<double, double>>,
+      public BetaDistributionModel {
+ public:
+  BetaDistributionTest() : BetaDistributionModel(GetParam()) {}
+
+ protected:
+  template <class D>
+  bool SingleZTestOnMeanAndVariance(double p, size_t samples);
+
+  template <class D>
+  bool SingleChiSquaredTest(double p, size_t samples, size_t buckets);
+
+  absl::InsecureBitGen rng_;
+};
+
+template <class D>
+bool BetaDistributionTest::SingleZTestOnMeanAndVariance(double p,
+                                                        size_t samples) {
+  D dis(alpha_, beta_);
+
+  std::vector<double> data;
+  data.reserve(samples);
+  for (size_t i = 0; i < samples; i++) {
+    const double variate = dis(rng_);
+    EXPECT_FALSE(std::isnan(variate));
+    // Note that equality is allowed on both sides.
+    EXPECT_GE(variate, 0.0);
+    EXPECT_LE(variate, 1.0);
+    data.push_back(variate);
+  }
+
+  // We validate that the sample mean and sample variance are indeed from a
+  // Beta distribution with the given shape parameters.
+  const auto m = absl::random_internal::ComputeDistributionMoments(data);
+
+  // The variance of the sample mean is variance / n.
+  const double mean_stddev = std::sqrt(Variance() / static_cast<double>(m.n));
+
+  // The variance of the sample variance is (approximately):
+  //   (kurtosis - 1) * variance^2 / n
+  const double variance_stddev = std::sqrt(
+      (Kurtosis() - 1) * Variance() * Variance() / static_cast<double>(m.n));
+  // z score for the sample variance.
+  const double z_variance = (m.variance - Variance()) / variance_stddev;
+
+  const double max_err = absl::random_internal::MaxErrorTolerance(p);
+  const double z_mean = absl::random_internal::ZScore(Mean(), m);
+  const bool pass =
+      absl::random_internal::Near("z", z_mean, 0.0, max_err) &&
+      absl::random_internal::Near("z_variance", z_variance, 0.0, max_err);
+  if (!pass) {
+    ABSL_INTERNAL_LOG(
+        INFO,
+        absl::StrFormat(
+            "Beta(%f, %f), "
+            "mean: sample %f, expect %f, which is %f stddevs away, "
+            "variance: sample %f, expect %f, which is %f stddevs away.",
+            alpha_, beta_, m.mean, Mean(),
+            std::abs(m.mean - Mean()) / mean_stddev, m.variance, Variance(),
+            std::abs(m.variance - Variance()) / variance_stddev));
+  }
+  return pass;
+}
+
+template <class D>
+bool BetaDistributionTest::SingleChiSquaredTest(double p, size_t samples,
+                                                size_t buckets) {
+  constexpr double kErr = 1e-7;
+  std::vector<double> cutoffs, expected;
+  const double bucket_width = 1.0 / static_cast<double>(buckets);
+  int i = 1;
+  int unmerged_buckets = 0;
+  for (; i < buckets; ++i) {
+    const double p = bucket_width * static_cast<double>(i);
+    const double boundary =
+        absl::random_internal::BetaIncompleteInv(alpha_, beta_, p);
+    // The intention is to add `boundary` to the list of `cutoffs`. It becomes
+    // problematic, however, when the boundary values are not monotone, due to
+    // numerical issues when computing the inverse regularized incomplete
+    // Beta function. In these cases, we merge that bucket with its previous
+    // neighbor and merge their expected counts.
+    if ((cutoffs.empty() && boundary < kErr) ||
+        (!cutoffs.empty() && boundary <= cutoffs.back())) {
+      unmerged_buckets++;
+      continue;
+    }
+    if (boundary >= 1.0 - 1e-10) {
+      break;
+    }
+    cutoffs.push_back(boundary);
+    expected.push_back(static_cast<double>(1 + unmerged_buckets) *
+                       bucket_width * static_cast<double>(samples));
+    unmerged_buckets = 0;
+  }
+  cutoffs.push_back(std::numeric_limits<double>::infinity());
+  // Merge all remaining buckets.
+  expected.push_back(static_cast<double>(buckets - i + 1) * bucket_width *
+                     static_cast<double>(samples));
+  // Make sure that we don't merge all the buckets, making this test
+  // meaningless.
+  EXPECT_GE(cutoffs.size(), 3) << alpha_ << ", " << beta_;
+
+  D dis(alpha_, beta_);
+
+  std::vector<int32_t> counts(cutoffs.size(), 0);
+  for (int i = 0; i < samples; i++) {
+    const double x = dis(rng_);
+    auto it = std::upper_bound(cutoffs.begin(), cutoffs.end(), x);
+    counts[std::distance(cutoffs.begin(), it)]++;
+  }
+
+  // Null-hypothesis is that the distribution is beta distributed with the
+  // provided alpha, beta params (not estimated from the data).
+  const int dof = cutoffs.size() - 1;
+
+  const double chi_square = absl::random_internal::ChiSquare(
+      counts.begin(), counts.end(), expected.begin(), expected.end());
+  const bool pass =
+      (absl::random_internal::ChiSquarePValue(chi_square, dof) >= p);
+  if (!pass) {
+    for (int i = 0; i < cutoffs.size(); i++) {
+      ABSL_INTERNAL_LOG(
+          INFO, absl::StrFormat("cutoff[%d] = %f, actual count %d, expected %d",
+                                i, cutoffs[i], counts[i],
+                                static_cast<int>(expected[i])));
+    }
+
+    ABSL_INTERNAL_LOG(
+        INFO, absl::StrFormat(
+                  "Beta(%f, %f) %s %f, p = %f", alpha_, beta_,
+                  absl::random_internal::kChiSquared, chi_square,
+                  absl::random_internal::ChiSquarePValue(chi_square, dof)));
+  }
+  return pass;
+}
+
+TEST_P(BetaDistributionTest, TestSampleStatistics) {
+  static constexpr int kRuns = 20;
+  static constexpr double kPFail = 0.02;
+  const double p =
+      absl::random_internal::RequiredSuccessProbability(kPFail, kRuns);
+  static constexpr int kSampleCount = 10000;
+  static constexpr int kBucketCount = 100;
+  int failed = 0;
+  for (int i = 0; i < kRuns; ++i) {
+    if (!SingleZTestOnMeanAndVariance<absl::beta_distribution<double>>(
+            p, kSampleCount)) {
+      failed++;
+    }
+    if (!SingleChiSquaredTest<absl::beta_distribution<double>>(
+            0.005, kSampleCount, kBucketCount)) {
+      failed++;
+    }
+  }
+  // Set so that the test is not flaky at --runs_per_test=10000
+  EXPECT_LE(failed, 5);
+}
+
+std::string ParamName(
+    const ::testing::TestParamInfo<::testing::tuple<double, double>>& info) {
+  std::string name = absl::StrCat("alpha_", ::testing::get<0>(info.param),
+                                  "__beta_", ::testing::get<1>(info.param));
+  return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
+}
+
+INSTANTIATE_TEST_CASE_P(
+    TestSampleStatisticsCombinations, BetaDistributionTest,
+    ::testing::Combine(::testing::Values(0.1, 0.2, 0.9, 1.1, 2.5, 10.0, 123.4),
+                       ::testing::Values(0.1, 0.2, 0.9, 1.1, 2.5, 10.0, 123.4)),
+    ParamName);
+
+INSTANTIATE_TEST_CASE_P(
+    TestSampleStatistics_SelectedPairs, BetaDistributionTest,
+    ::testing::Values(std::make_pair(0.5, 1000), std::make_pair(1000, 0.5),
+                      std::make_pair(900, 1000), std::make_pair(10000, 20000),
+                      std::make_pair(4e5, 2e7), std::make_pair(1e7, 1e5)),
+    ParamName);
+
+// NOTE: absl::beta_distribution is not guaranteed to be stable.
+TEST(BetaDistributionTest, StabilityTest) {
+  // absl::beta_distribution stability relies on the stability of
+  // absl::random_interna::RandU64ToDouble, std::exp, std::log, std::pow,
+  // and std::sqrt.
+  //
+  // This test also depends on the stability of std::frexp.
+  using testing::ElementsAre;
+  absl::random_internal::sequence_urbg urbg({
+      0xffff00000000e6c8ull, 0xffff0000000006c8ull, 0x800003766295CFA9ull,
+      0x11C819684E734A41ull, 0x832603766295CFA9ull, 0x7fbe76c8b4395800ull,
+      0xB3472DCA7B14A94Aull, 0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull,
+      0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull, 0x00035C904C70A239ull,
+      0x00009E0BCBAADE14ull, 0x0000000000622CA7ull, 0x4864f22c059bf29eull,
+      0x247856d8b862665cull, 0xe46e86e9a1337e10ull, 0xd8c8541f3519b133ull,
+      0xffe75b52c567b9e4ull, 0xfffff732e5709c5bull, 0xff1f7f0b983532acull,
+      0x1ec2e8986d2362caull, 0xC332DDEFBE6C5AA5ull, 0x6558218568AB9702ull,
+      0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull, 0xECDD4775619F1510ull,
+      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,
+  });
+
+  // Convert the real-valued result into a unit64 where we compare
+  // 5 (float) or 10 (double) decimal digits plus the base-2 exponent.
+  auto float_to_u64 = [](float d) {
+    int exp = 0;
+    auto f = std::frexp(d, &exp);
+    return (static_cast<uint64_t>(1e5 * f) * 10000) + std::abs(exp);
+  };
+  auto double_to_u64 = [](double d) {
+    int exp = 0;
+    auto f = std::frexp(d, &exp);
+    return (static_cast<uint64_t>(1e10 * f) * 10000) + std::abs(exp);
+  };
+
+  std::vector<uint64_t> output(20);
+  {
+    // Algorithm Joehnk (float)
+    absl::beta_distribution<float> dist(0.1f, 0.2f);
+    std::generate(std::begin(output), std::end(output),
+                  [&] { return float_to_u64(dist(urbg)); });
+    EXPECT_EQ(44, urbg.invocations());
+    EXPECT_THAT(output,  //
+                testing::ElementsAre(
+                    998340000, 619030004, 500000001, 999990000, 996280000,
+                    500000001, 844740004, 847210001, 999970000, 872320000,
+                    585480007, 933280000, 869080042, 647670031, 528240004,
+                    969980004, 626050008, 915930002, 833440033, 878040015));
+  }
+
+  urbg.reset();
+  {
+    // Algorithm Joehnk (double)
+    absl::beta_distribution<double> dist(0.1, 0.2);
+    std::generate(std::begin(output), std::end(output),
+                  [&] { return double_to_u64(dist(urbg)); });
+    EXPECT_EQ(44, urbg.invocations());
+    EXPECT_THAT(
+        output,  //
+        testing::ElementsAre(
+            99834713000000, 61903356870004, 50000000000001, 99999721170000,
+            99628374770000, 99999999990000, 84474397860004, 84721276240001,
+            99997407490000, 87232528120000, 58548364780007, 93328932910000,
+            86908237770042, 64767917930031, 52824581970004, 96998544140004,
+            62605946270008, 91593604380002, 83345031740033, 87804397230015));
+  }
+
+  urbg.reset();
+  {
+    // Algorithm Cheng 1
+    absl::beta_distribution<double> dist(0.9, 2.0);
+    std::generate(std::begin(output), std::end(output),
+                  [&] { return double_to_u64(dist(urbg)); });
+    EXPECT_EQ(62, urbg.invocations());
+    EXPECT_THAT(
+        output,  //
+        testing::ElementsAre(
+            62069004780001, 64433204450001, 53607416560000, 89644295430008,
+            61434586310019, 55172615890002, 62187161490000, 56433684810003,
+            80454622050005, 86418558710003, 92920514700001, 64645184680001,
+            58549183380000, 84881283650005, 71078728590002, 69949694970000,
+            73157461710001, 68592191300001, 70747623900000, 78584696930005));
+  }
+
+  urbg.reset();
+  {
+    // Algorithm Cheng 2
+    absl::beta_distribution<double> dist(1.5, 2.5);
+    std::generate(std::begin(output), std::end(output),
+                  [&] { return double_to_u64(dist(urbg)); });
+    EXPECT_EQ(54, urbg.invocations());
+    EXPECT_THAT(
+        output,  //
+        testing::ElementsAre(
+            75000029250001, 76751482860001, 53264575220000, 69193133650005,
+            78028324470013, 91573587560002, 59167523770000, 60658618560002,
+            80075870540000, 94141320460004, 63196592770003, 78883906300002,
+            96797992590001, 76907587800001, 56645167560000, 65408302280003,
+            53401156320001, 64731238570000, 83065573750001, 79788333820001));
+  }
+}
+
+// This is an implementation-specific test. If any part of the implementation
+// changes, then it is likely that this test will change as well.  Also, if
+// dependencies of the distribution change, such as RandU64ToDouble, then this
+// is also likely to change.
+TEST(BetaDistributionTest, AlgorithmBounds) {
+  {
+    absl::random_internal::sequence_urbg urbg(
+        {0x7fbe76c8b4395800ull, 0x8000000000000000ull});
+    // u=0.499, v=0.5
+    absl::beta_distribution<double> dist(1e-4, 1e-4);
+    double a = dist(urbg);
+    EXPECT_EQ(a, 2.0202860861567108529e-09);
+    EXPECT_EQ(2, urbg.invocations());
+  }
+
+  // Test that both the float & double algorithms appropriately reject the
+  // initial draw.
+  {
+    // 1/alpha = 1/beta = 2.
+    absl::beta_distribution<float> dist(0.5, 0.5);
+
+    // first two outputs are close to 1.0 - epsilon,
+    // thus:  (u ^ 2 + v ^ 2) > 1.0
+    absl::random_internal::sequence_urbg urbg(
+        {0xffff00000006e6c8ull, 0xffff00000007c7c8ull, 0x800003766295CFA9ull,
+         0x11C819684E734A41ull});
+    {
+      double y = absl::beta_distribution<double>(0.5, 0.5)(urbg);
+      EXPECT_EQ(4, urbg.invocations());
+      EXPECT_EQ(y, 0.9810668952633862) << y;
+    }
+
+    // ...and:  log(u) * a ~= log(v) * b ~= -0.02
+    // thus z ~= -0.02 + log(1 + e(~0))
+    //        ~= -0.02 + 0.69
+    // thus z > 0
+    urbg.reset();
+    {
+      float x = absl::beta_distribution<float>(0.5, 0.5)(urbg);
+      EXPECT_EQ(4, urbg.invocations());
+      EXPECT_NEAR(0.98106688261032104, x, 0.0000005) << x << "f";
+    }
+  }
+}
+
+}  // namespace