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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
//
//
// 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 <cfloat>
#include <cstddef>
#include <cstdint>
#include <iterator>
#include <random>
#include <sstream>
#include <string>
#include <type_traits>
#include <unordered_map>
#include <vector>
#include "gmock/gmock.h"
#include "gtest/gtest.h"
#include "absl/log/log.h"
#include "absl/numeric/internal/representation.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"
namespace {
template <typename IntType>
class BetaDistributionInterfaceTest : public ::testing::Test {};
constexpr bool ShouldExerciseLongDoubleTests() {
// long double arithmetic is not supported well by either GCC or Clang on
// most platforms specifically not when implemented in terms of double-double;
// So a conservative choice here is to disable long-double tests pretty much
// everywhere except on x64 but only if long double is not implemented as
// double-double.
#if defined(__i686__) && defined(__x86_64__)
return !absl::numeric_internal::IsDoubleDouble();
#else
return false;
#endif
}
using RealTypes = std::conditional<ShouldExerciseLongDoubleTests(),
::testing::Types<float, double, long double>,
::testing::Types<float, double>>::type;
TYPED_TEST_SUITE(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)())));
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()),
// 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) {
LOG(INFO) << absl::StreamFormat("Smoke test for Beta(%a, %a)", 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;
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) {
// 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);
// Extreme cases when the params are abnormal.
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(rng);
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(rng), 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(rng), 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(rng), 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(rng);
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) {
LOG(INFO) << "Beta(" << alpha_ << ", " << beta_ << "), mean: sample "
<< m.mean << ", expect " << Mean() << ", which is "
<< std::abs(m.mean - Mean()) / mean_stddev
<< " stddevs away, variance: sample " << m.variance << ", expect "
<< Variance() << ", which is "
<< std::abs(m.variance - Variance()) / variance_stddev
<< " stddevs away.";
}
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 (size_t i = 0; i < cutoffs.size(); i++) {
LOG(INFO) << "cutoff[" << i << "] = " << cutoffs[i] << ", actual count "
<< counts[i] << ", expected " << static_cast<int>(expected[i]);
}
LOG(INFO) << "Beta(" << alpha_ << ", " << beta_ << ") "
<< absl::random_internal::kChiSquared << " " << chi_square
<< ", p = "
<< 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_SUITE_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_SUITE_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) {
#if (defined(__i386__) || defined(_M_IX86)) && FLT_EVAL_METHOD != 0
// We're using an x87-compatible FPU, and intermediate operations are
// performed with 80-bit floats. This produces slightly different results from
// what we expect below.
GTEST_SKIP()
<< "Skipping the test because we detected x87 floating-point semantics";
#endif
{
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