absl/random/internal/generate_real_test.cc - third_party/abseil-cpp - Git at Google

 // 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/internal/generate_real.h"

 #include <cfloat>
 #include <cstddef>
 #include <cstdint>
 #include <string>

 #include "gtest/gtest.h"
 #include "absl/flags/flag.h"
 #include "absl/numeric/bits.h"

 ABSL_FLAG(int64_t, absl_random_test_trials, 50000,
           "Number of trials for the probability tests.");

 using absl::random_internal::GenerateNegativeTag;
 using absl::random_internal::GeneratePositiveTag;
 using absl::random_internal::GenerateRealFromBits;
 using absl::random_internal::GenerateSignedTag;

 namespace {

 TEST(GenerateRealTest, U64ToFloat_Positive_NoZero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GeneratePositiveTag, false>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 2.710505431e-20f);
   EXPECT_EQ(ToFloat(0x0000000000000001), 5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x8000000000000000), 0.5);
   EXPECT_EQ(ToFloat(0x8000000000000001), 0.5);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Positive_Zero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 0.0);
   EXPECT_EQ(ToFloat(0x0000000000000001), 5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x8000000000000000), 0.5);
   EXPECT_EQ(ToFloat(0x8000000000000001), 0.5);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Negative_NoZero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GenerateNegativeTag, false>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), -2.710505431e-20f);
   EXPECT_EQ(ToFloat(0x0000000000000001), -5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x8000000000000000), -0.5);
   EXPECT_EQ(ToFloat(0x8000000000000001), -0.5);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Negative_Zero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GenerateNegativeTag, true>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 0.0);
   EXPECT_EQ(ToFloat(0x0000000000000001), -5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x8000000000000000), -0.5);
   EXPECT_EQ(ToFloat(0x8000000000000001), -0.5);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Signed_NoZero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GenerateSignedTag, false>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x0000000000000001), 1.084202172e-19f);
   EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 0.9999999404f);
   EXPECT_EQ(ToFloat(0x8000000000000000), -5.421010862e-20f);
   EXPECT_EQ(ToFloat(0x8000000000000001), -1.084202172e-19f);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Signed_Zero_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GenerateSignedTag, true>(a);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 0);
   EXPECT_EQ(ToFloat(0x0000000000000001), 1.084202172e-19f);
   EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 0.9999999404f);
   EXPECT_EQ(ToFloat(0x8000000000000000), 0);
   EXPECT_EQ(ToFloat(0x8000000000000001), -1.084202172e-19f);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloat_Signed_Bias_Test) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GenerateSignedTag, true>(a, 1);
   };
   EXPECT_EQ(ToFloat(0x0000000000000000), 0);
   EXPECT_EQ(ToFloat(0x0000000000000001), 2 * 1.084202172e-19f);
   EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 2 * 0.9999999404f);
   EXPECT_EQ(ToFloat(0x8000000000000000), 0);
   EXPECT_EQ(ToFloat(0x8000000000000001), 2 * -1.084202172e-19f);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 2 * -0.9999999404f);
 }

 TEST(GenerateRealTest, U64ToFloatTest) {
   auto ToFloat = [](uint64_t a) -> float {
     return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
   };

   EXPECT_EQ(ToFloat(0x0000000000000000), 0.0f);

   EXPECT_EQ(ToFloat(0x8000000000000000), 0.5f);
   EXPECT_EQ(ToFloat(0x8000000000000001), 0.5f);
   EXPECT_EQ(ToFloat(0x800000FFFFFFFFFF), 0.5f);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);

   EXPECT_GT(ToFloat(0x0000000000000001), 0.0f);

   EXPECT_NE(ToFloat(0x7FFFFF0000000000), ToFloat(0x7FFFFEFFFFFFFFFF));

   EXPECT_LT(ToFloat(0xFFFFFFFFFFFFFFFF), 1.0f);
   int32_t two_to_24 = 1 << 24;
   EXPECT_EQ(static_cast<int32_t>(ToFloat(0xFFFFFFFFFFFFFFFF) * two_to_24),
             two_to_24 - 1);
   EXPECT_NE(static_cast<int32_t>(ToFloat(0xFFFFFFFFFFFFFFFF) * two_to_24 * 2),
             two_to_24 * 2 - 1);
   EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), ToFloat(0xFFFFFF0000000000));
   EXPECT_NE(ToFloat(0xFFFFFFFFFFFFFFFF), ToFloat(0xFFFFFEFFFFFFFFFF));
   EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), ToFloat(0x7FFFFF8000000000));
   EXPECT_NE(ToFloat(0x7FFFFFFFFFFFFFFF), ToFloat(0x7FFFFF7FFFFFFFFF));
   EXPECT_EQ(ToFloat(0x3FFFFFFFFFFFFFFF), ToFloat(0x3FFFFFC000000000));
   EXPECT_NE(ToFloat(0x3FFFFFFFFFFFFFFF), ToFloat(0x3FFFFFBFFFFFFFFF));

   // For values where every bit counts, the values scale as multiples of the
   // input.
   for (int i = 0; i < 100; ++i) {
     EXPECT_EQ(i * ToFloat(0x0000000000000001), ToFloat(i));
   }

   // For each i: value generated from (1 << i).
   float exp_values[64];
   exp_values[63] = 0.5f;
   for (int i = 62; i >= 0; --i) exp_values[i] = 0.5f * exp_values[i + 1];
   constexpr uint64_t one = 1;
   for (int i = 0; i < 64; ++i) {
     EXPECT_EQ(ToFloat(one << i), exp_values[i]);
     for (int j = 1; j < FLT_MANT_DIG && i - j >= 0; ++j) {
       EXPECT_NE(exp_values[i] + exp_values[i - j], exp_values[i]);
       EXPECT_EQ(ToFloat((one << i) + (one << (i - j))),
                 exp_values[i] + exp_values[i - j]);
     }
     for (int j = FLT_MANT_DIG; i - j >= 0; ++j) {
       EXPECT_EQ(exp_values[i] + exp_values[i - j], exp_values[i]);
       EXPECT_EQ(ToFloat((one << i) + (one << (i - j))), exp_values[i]);
     }
   }
 }

 TEST(GenerateRealTest, U64ToDouble_Positive_NoZero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GeneratePositiveTag, false>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 2.710505431213761085e-20);
   EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x0000000000000002), 1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);
   EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_Positive_Zero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GeneratePositiveTag, true>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
   EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);
   EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_Negative_NoZero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateNegativeTag, false>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), -2.710505431213761085e-20);
   EXPECT_EQ(ToDouble(0x0000000000000001), -5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x0000000000000002), -1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0x8000000000000000), -0.5);
   EXPECT_EQ(ToDouble(0x8000000000000001), -0.5);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_Negative_Zero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateNegativeTag, true>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
   EXPECT_EQ(ToDouble(0x0000000000000001), -5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x0000000000000002), -1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0x8000000000000000), -0.5);
   EXPECT_EQ(ToDouble(0x8000000000000001), -0.5);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_Signed_NoZero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateSignedTag, false>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978);
   EXPECT_EQ(ToDouble(0x8000000000000000), -5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_Signed_Zero_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateSignedTag, true>(a);
   };
   EXPECT_EQ(ToDouble(0x0000000000000000), 0);
   EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978);
   EXPECT_EQ(ToDouble(0x8000000000000000), 0);
   EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
 }

 TEST(GenerateRealTest, U64ToDouble_GenerateSignedTag_Bias_Test) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateSignedTag, true>(a, -1);
   };
   EXPECT_EQ(ToDouble(0x0000000000000000), 0);
   EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19 / 2);
   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978 / 2);
   EXPECT_EQ(ToDouble(0x8000000000000000), 0);
   EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19 / 2);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978 / 2);
 }

 TEST(GenerateRealTest, U64ToDoubleTest) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GeneratePositiveTag, true>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
   EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);

   EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x7fffffffffffffef), 0.499999999999999944489);
   EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);

   // For values > 0.5, RandU64ToDouble discards up to 11 bits. (64-53).
   EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0x80000000000007FF), 0.5);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
   EXPECT_NE(ToDouble(0x7FFFFFFFFFFFF800), ToDouble(0x7FFFFFFFFFFFF7FF));

   EXPECT_LT(ToDouble(0xFFFFFFFFFFFFFFFF), 1.0);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), ToDouble(0xFFFFFFFFFFFFF800));
   EXPECT_NE(ToDouble(0xFFFFFFFFFFFFFFFF), ToDouble(0xFFFFFFFFFFFFF7FF));
   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFC00));
   EXPECT_NE(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFBFF));
   EXPECT_EQ(ToDouble(0x3FFFFFFFFFFFFFFF), ToDouble(0x3FFFFFFFFFFFFE00));
   EXPECT_NE(ToDouble(0x3FFFFFFFFFFFFFFF), ToDouble(0x3FFFFFFFFFFFFDFF));

   EXPECT_EQ(ToDouble(0x1000000000000001), 0.0625);
   EXPECT_EQ(ToDouble(0x2000000000000001), 0.125);
   EXPECT_EQ(ToDouble(0x3000000000000001), 0.1875);
   EXPECT_EQ(ToDouble(0x4000000000000001), 0.25);
   EXPECT_EQ(ToDouble(0x5000000000000001), 0.3125);
   EXPECT_EQ(ToDouble(0x6000000000000001), 0.375);
   EXPECT_EQ(ToDouble(0x7000000000000001), 0.4375);
   EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0x9000000000000001), 0.5625);
   EXPECT_EQ(ToDouble(0xa000000000000001), 0.625);
   EXPECT_EQ(ToDouble(0xb000000000000001), 0.6875);
   EXPECT_EQ(ToDouble(0xc000000000000001), 0.75);
   EXPECT_EQ(ToDouble(0xd000000000000001), 0.8125);
   EXPECT_EQ(ToDouble(0xe000000000000001), 0.875);
   EXPECT_EQ(ToDouble(0xf000000000000001), 0.9375);

   // Large powers of 2.
   int64_t two_to_53 = int64_t{1} << 53;
   EXPECT_EQ(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53),
             two_to_53 - 1);
   EXPECT_NE(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53 * 2),
             two_to_53 * 2 - 1);

   // For values where every bit counts, the values scale as multiples of the
   // input.
   for (int i = 0; i < 100; ++i) {
     EXPECT_EQ(i * ToDouble(0x0000000000000001), ToDouble(i));
   }

   // For each i: value generated from (1 << i).
   double exp_values[64];
   exp_values[63] = 0.5;
   for (int i = 62; i >= 0; --i) exp_values[i] = 0.5 * exp_values[i + 1];
   constexpr uint64_t one = 1;
   for (int i = 0; i < 64; ++i) {
     EXPECT_EQ(ToDouble(one << i), exp_values[i]);
     for (int j = 1; j < DBL_MANT_DIG && i - j >= 0; ++j) {
       EXPECT_NE(exp_values[i] + exp_values[i - j], exp_values[i]);
       EXPECT_EQ(ToDouble((one << i) + (one << (i - j))),
                 exp_values[i] + exp_values[i - j]);
     }
     for (int j = DBL_MANT_DIG; i - j >= 0; ++j) {
       EXPECT_EQ(exp_values[i] + exp_values[i - j], exp_values[i]);
       EXPECT_EQ(ToDouble((one << i) + (one << (i - j))), exp_values[i]);
     }
   }
 }

 TEST(GenerateRealTest, U64ToDoubleSignedTest) {
   auto ToDouble = [](uint64_t a) {
     return GenerateRealFromBits<double, GenerateSignedTag, false>(a);
   };

   EXPECT_EQ(ToDouble(0x0000000000000000), 5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);

   EXPECT_EQ(ToDouble(0x8000000000000000), -5.42101086242752217004e-20);
   EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);

   const double e_plus = ToDouble(0x0000000000000001);
   const double e_minus = ToDouble(0x8000000000000001);
   EXPECT_EQ(e_plus, 1.084202172485504434e-19);
   EXPECT_EQ(e_minus, -1.084202172485504434e-19);

   EXPECT_EQ(ToDouble(0x3fffffffffffffef), 0.499999999999999944489);
   EXPECT_EQ(ToDouble(0xbfffffffffffffef), -0.499999999999999944489);

   // For values > 0.5, RandU64ToDouble discards up to 10 bits. (63-53).
   EXPECT_EQ(ToDouble(0x4000000000000000), 0.5);
   EXPECT_EQ(ToDouble(0x4000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0x40000000000003FF), 0.5);

   EXPECT_EQ(ToDouble(0xC000000000000000), -0.5);
   EXPECT_EQ(ToDouble(0xC000000000000001), -0.5);
   EXPECT_EQ(ToDouble(0xC0000000000003FF), -0.5);

   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFe), 0.999999999999999888978);
   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFe), -0.999999999999999888978);

   EXPECT_NE(ToDouble(0x7FFFFFFFFFFFF800), ToDouble(0x7FFFFFFFFFFFF7FF));

   EXPECT_LT(ToDouble(0x7FFFFFFFFFFFFFFF), 1.0);
   EXPECT_GT(ToDouble(0x7FFFFFFFFFFFFFFF), 0.9999999999);

   EXPECT_GT(ToDouble(0xFFFFFFFFFFFFFFFe), -1.0);
   EXPECT_LT(ToDouble(0xFFFFFFFFFFFFFFFe), -0.999999999);

   EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFe), ToDouble(0xFFFFFFFFFFFFFC00));
   EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFC00));
   EXPECT_NE(ToDouble(0xFFFFFFFFFFFFFFFe), ToDouble(0xFFFFFFFFFFFFF3FF));
   EXPECT_NE(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFF3FF));

   EXPECT_EQ(ToDouble(0x1000000000000001), 0.125);
   EXPECT_EQ(ToDouble(0x2000000000000001), 0.25);
   EXPECT_EQ(ToDouble(0x3000000000000001), 0.375);
   EXPECT_EQ(ToDouble(0x4000000000000001), 0.5);
   EXPECT_EQ(ToDouble(0x5000000000000001), 0.625);
   EXPECT_EQ(ToDouble(0x6000000000000001), 0.75);
   EXPECT_EQ(ToDouble(0x7000000000000001), 0.875);
   EXPECT_EQ(ToDouble(0x7800000000000001), 0.9375);
   EXPECT_EQ(ToDouble(0x7c00000000000001), 0.96875);
   EXPECT_EQ(ToDouble(0x7e00000000000001), 0.984375);
   EXPECT_EQ(ToDouble(0x7f00000000000001), 0.9921875);

   // 0x8000000000000000 ~= 0
   EXPECT_EQ(ToDouble(0x9000000000000001), -0.125);
   EXPECT_EQ(ToDouble(0xa000000000000001), -0.25);
   EXPECT_EQ(ToDouble(0xb000000000000001), -0.375);
   EXPECT_EQ(ToDouble(0xc000000000000001), -0.5);
   EXPECT_EQ(ToDouble(0xd000000000000001), -0.625);
   EXPECT_EQ(ToDouble(0xe000000000000001), -0.75);
   EXPECT_EQ(ToDouble(0xf000000000000001), -0.875);

   // Large powers of 2.
   int64_t two_to_53 = int64_t{1} << 53;
   EXPECT_EQ(static_cast<int64_t>(ToDouble(0x7FFFFFFFFFFFFFFF) * two_to_53),
             two_to_53 - 1);
   EXPECT_EQ(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53),
             -(two_to_53 - 1));

   EXPECT_NE(static_cast<int64_t>(ToDouble(0x7FFFFFFFFFFFFFFF) * two_to_53 * 2),
             two_to_53 * 2 - 1);

   // For values where every bit counts, the values scale as multiples of the
   // input.
   for (int i = 1; i < 100; ++i) {
     EXPECT_EQ(i * e_plus, ToDouble(i)) << i;
     EXPECT_EQ(i * e_minus, ToDouble(0x8000000000000000 | i)) << i;
   }
 }

 TEST(GenerateRealTest, ExhaustiveFloat) {
   auto ToFloat = [](uint64_t a) {
     return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
   };

   // Rely on RandU64ToFloat generating values from greatest to least when
   // supplied with uint64_t values from greatest (0xfff...) to least (0x0).
   // Thus, this algorithm stores the previous value, and if the new value is at
   // greater than or equal to the previous value, then there is a collision in
   // the generation algorithm.
   //
   // Use the computation below to convert the random value into a result:
   //   double res = a() * (1.0f - sample) + b() * sample;
   float last_f = 1.0, last_g = 2.0;
   uint64_t f_collisions = 0, g_collisions = 0;
   uint64_t f_unique = 0, g_unique = 0;
   uint64_t total = 0;
   auto count = [&](const float r) {
     total++;
     // `f` is mapped to the range [0, 1) (default)
     const float f = 0.0f * (1.0f - r) + 1.0f * r;
     if (f >= last_f) {
       f_collisions++;
     } else {
       f_unique++;
       last_f = f;
     }
     // `g` is mapped to the range [1, 2)
     const float g = 1.0f * (1.0f - r) + 2.0f * r;
     if (g >= last_g) {
       g_collisions++;
     } else {
       g_unique++;
       last_g = g;
     }
   };

   size_t limit = absl::GetFlag(FLAGS_absl_random_test_trials);

   // Generate all uint64_t which have unique floating point values.
   // Counting down from 0xFFFFFFFFFFFFFFFFu ... 0x0u
   uint64_t x = ~uint64_t(0);
   for (; x != 0 && limit > 0;) {
     constexpr int kDig = (64 - FLT_MANT_DIG);
     // Set a decrement value & the next point at which to change
     // the decrement value. By default these are 1, 0.
     uint64_t dec = 1;
     uint64_t chk = 0;

     // Adjust decrement and check value based on how many leading 0
     // bits are set in the current value.
     const int clz = absl::countl_zero(x);
     if (clz < kDig) {
       dec <<= (kDig - clz);
       chk = (~uint64_t(0)) >> (clz + 1);
     }
     for (; x > chk && limit > 0; x -= dec) {
       count(ToFloat(x));
       --limit;
     }
   }

   static_assert(FLT_MANT_DIG == 24,
                 "The float type is expected to have a 24 bit mantissa.");

   if (limit != 0) {
     // There are between 2^28 and 2^29 unique values in the range [0, 1).  For
     // the low values of x, there are 2^24 -1 unique values.  Once x > 2^24,
     // there are 40 * 2^24 unique values. Thus:
     // (2 + 4 + 8 ... + 2^23) + 40 * 2^23
     EXPECT_LT(1 << 28, f_unique);
     EXPECT_EQ((1 << 24) + 40 * (1 << 23) - 1, f_unique);
     EXPECT_EQ(total, f_unique);
     EXPECT_EQ(0, f_collisions);

     // Expect at least 2^23 unique values for the range [1, 2)
     EXPECT_LE(1 << 23, g_unique);
     EXPECT_EQ(total - g_unique, g_collisions);
   }
 }

 }  // namespace
	// 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/internal/generate_real.h"

	#include <cfloat>
	#include <cstddef>
	#include <cstdint>
	#include <string>

	#include "gtest/gtest.h"
	#include "absl/flags/flag.h"
	#include "absl/numeric/bits.h"

	ABSL_FLAG(int64_t, absl_random_test_trials, 50000,
	"Number of trials for the probability tests.");

	using absl::random_internal::GenerateNegativeTag;
	using absl::random_internal::GeneratePositiveTag;
	using absl::random_internal::GenerateRealFromBits;
	using absl::random_internal::GenerateSignedTag;

	namespace {

	TEST(GenerateRealTest, U64ToFloat_Positive_NoZero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GeneratePositiveTag, false>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 2.710505431e-20f);
	EXPECT_EQ(ToFloat(0x0000000000000001), 5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x8000000000000000), 0.5);
	EXPECT_EQ(ToFloat(0x8000000000000001), 0.5);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Positive_Zero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 0.0);
	EXPECT_EQ(ToFloat(0x0000000000000001), 5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x8000000000000000), 0.5);
	EXPECT_EQ(ToFloat(0x8000000000000001), 0.5);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Negative_NoZero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GenerateNegativeTag, false>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), -2.710505431e-20f);
	EXPECT_EQ(ToFloat(0x0000000000000001), -5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x8000000000000000), -0.5);
	EXPECT_EQ(ToFloat(0x8000000000000001), -0.5);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Negative_Zero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GenerateNegativeTag, true>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 0.0);
	EXPECT_EQ(ToFloat(0x0000000000000001), -5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x8000000000000000), -0.5);
	EXPECT_EQ(ToFloat(0x8000000000000001), -0.5);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Signed_NoZero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GenerateSignedTag, false>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x0000000000000001), 1.084202172e-19f);
	EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 0.9999999404f);
	EXPECT_EQ(ToFloat(0x8000000000000000), -5.421010862e-20f);
	EXPECT_EQ(ToFloat(0x8000000000000001), -1.084202172e-19f);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Signed_Zero_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GenerateSignedTag, true>(a);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 0);
	EXPECT_EQ(ToFloat(0x0000000000000001), 1.084202172e-19f);
	EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 0.9999999404f);
	EXPECT_EQ(ToFloat(0x8000000000000000), 0);
	EXPECT_EQ(ToFloat(0x8000000000000001), -1.084202172e-19f);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), -0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloat_Signed_Bias_Test) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GenerateSignedTag, true>(a, 1);
	};
	EXPECT_EQ(ToFloat(0x0000000000000000), 0);
	EXPECT_EQ(ToFloat(0x0000000000000001), 2 * 1.084202172e-19f);
	EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), 2 * 0.9999999404f);
	EXPECT_EQ(ToFloat(0x8000000000000000), 0);
	EXPECT_EQ(ToFloat(0x8000000000000001), 2 * -1.084202172e-19f);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 2 * -0.9999999404f);
	}

	TEST(GenerateRealTest, U64ToFloatTest) {
	auto ToFloat = [](uint64_t a) -> float {
	return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
	};

	EXPECT_EQ(ToFloat(0x0000000000000000), 0.0f);

	EXPECT_EQ(ToFloat(0x8000000000000000), 0.5f);
	EXPECT_EQ(ToFloat(0x8000000000000001), 0.5f);
	EXPECT_EQ(ToFloat(0x800000FFFFFFFFFF), 0.5f);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), 0.9999999404f);

	EXPECT_GT(ToFloat(0x0000000000000001), 0.0f);

	EXPECT_NE(ToFloat(0x7FFFFF0000000000), ToFloat(0x7FFFFEFFFFFFFFFF));

	EXPECT_LT(ToFloat(0xFFFFFFFFFFFFFFFF), 1.0f);
	int32_t two_to_24 = 1 << 24;
	EXPECT_EQ(static_cast<int32_t>(ToFloat(0xFFFFFFFFFFFFFFFF) * two_to_24),
	two_to_24 - 1);
	EXPECT_NE(static_cast<int32_t>(ToFloat(0xFFFFFFFFFFFFFFFF) * two_to_24 * 2),
	two_to_24 * 2 - 1);
	EXPECT_EQ(ToFloat(0xFFFFFFFFFFFFFFFF), ToFloat(0xFFFFFF0000000000));
	EXPECT_NE(ToFloat(0xFFFFFFFFFFFFFFFF), ToFloat(0xFFFFFEFFFFFFFFFF));
	EXPECT_EQ(ToFloat(0x7FFFFFFFFFFFFFFF), ToFloat(0x7FFFFF8000000000));
	EXPECT_NE(ToFloat(0x7FFFFFFFFFFFFFFF), ToFloat(0x7FFFFF7FFFFFFFFF));
	EXPECT_EQ(ToFloat(0x3FFFFFFFFFFFFFFF), ToFloat(0x3FFFFFC000000000));
	EXPECT_NE(ToFloat(0x3FFFFFFFFFFFFFFF), ToFloat(0x3FFFFFBFFFFFFFFF));

	// For values where every bit counts, the values scale as multiples of the
	// input.
	for (int i = 0; i < 100; ++i) {
	EXPECT_EQ(i * ToFloat(0x0000000000000001), ToFloat(i));
	}

	// For each i: value generated from (1 << i).
	float exp_values[64];
	exp_values[63] = 0.5f;
	for (int i = 62; i >= 0; --i) exp_values[i] = 0.5f * exp_values[i + 1];
	constexpr uint64_t one = 1;
	for (int i = 0; i < 64; ++i) {
	EXPECT_EQ(ToFloat(one << i), exp_values[i]);
	for (int j = 1; j < FLT_MANT_DIG && i - j >= 0; ++j) {
	EXPECT_NE(exp_values[i] + exp_values[i - j], exp_values[i]);
	EXPECT_EQ(ToFloat((one << i) + (one << (i - j))),
	exp_values[i] + exp_values[i - j]);
	}
	for (int j = FLT_MANT_DIG; i - j >= 0; ++j) {
	EXPECT_EQ(exp_values[i] + exp_values[i - j], exp_values[i]);
	EXPECT_EQ(ToFloat((one << i) + (one << (i - j))), exp_values[i]);
	}
	}
	}

	TEST(GenerateRealTest, U64ToDouble_Positive_NoZero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GeneratePositiveTag, false>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 2.710505431213761085e-20);
	EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x0000000000000002), 1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);
	EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_Positive_Zero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GeneratePositiveTag, true>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
	EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);
	EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_Negative_NoZero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateNegativeTag, false>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), -2.710505431213761085e-20);
	EXPECT_EQ(ToDouble(0x0000000000000001), -5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x0000000000000002), -1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0x8000000000000000), -0.5);
	EXPECT_EQ(ToDouble(0x8000000000000001), -0.5);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_Negative_Zero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateNegativeTag, true>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
	EXPECT_EQ(ToDouble(0x0000000000000001), -5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x0000000000000002), -1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0x8000000000000000), -0.5);
	EXPECT_EQ(ToDouble(0x8000000000000001), -0.5);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_Signed_NoZero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateSignedTag, false>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978);
	EXPECT_EQ(ToDouble(0x8000000000000000), -5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_Signed_Zero_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateSignedTag, true>(a);
	};
	EXPECT_EQ(ToDouble(0x0000000000000000), 0);
	EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978);
	EXPECT_EQ(ToDouble(0x8000000000000000), 0);
	EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978);
	}

	TEST(GenerateRealTest, U64ToDouble_GenerateSignedTag_Bias_Test) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateSignedTag, true>(a, -1);
	};
	EXPECT_EQ(ToDouble(0x0000000000000000), 0);
	EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19 / 2);
	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), 0.999999999999999888978 / 2);
	EXPECT_EQ(ToDouble(0x8000000000000000), 0);
	EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19 / 2);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), -0.999999999999999888978 / 2);
	}

	TEST(GenerateRealTest, U64ToDoubleTest) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GeneratePositiveTag, true>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);
	EXPECT_EQ(ToDouble(0x0000000000000000), 0.0);

	EXPECT_EQ(ToDouble(0x0000000000000001), 5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x7fffffffffffffef), 0.499999999999999944489);
	EXPECT_EQ(ToDouble(0x8000000000000000), 0.5);

	// For values > 0.5, RandU64ToDouble discards up to 11 bits. (64-53).
	EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0x80000000000007FF), 0.5);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), 0.999999999999999888978);
	EXPECT_NE(ToDouble(0x7FFFFFFFFFFFF800), ToDouble(0x7FFFFFFFFFFFF7FF));

	EXPECT_LT(ToDouble(0xFFFFFFFFFFFFFFFF), 1.0);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFF), ToDouble(0xFFFFFFFFFFFFF800));
	EXPECT_NE(ToDouble(0xFFFFFFFFFFFFFFFF), ToDouble(0xFFFFFFFFFFFFF7FF));
	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFC00));
	EXPECT_NE(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFBFF));
	EXPECT_EQ(ToDouble(0x3FFFFFFFFFFFFFFF), ToDouble(0x3FFFFFFFFFFFFE00));
	EXPECT_NE(ToDouble(0x3FFFFFFFFFFFFFFF), ToDouble(0x3FFFFFFFFFFFFDFF));

	EXPECT_EQ(ToDouble(0x1000000000000001), 0.0625);
	EXPECT_EQ(ToDouble(0x2000000000000001), 0.125);
	EXPECT_EQ(ToDouble(0x3000000000000001), 0.1875);
	EXPECT_EQ(ToDouble(0x4000000000000001), 0.25);
	EXPECT_EQ(ToDouble(0x5000000000000001), 0.3125);
	EXPECT_EQ(ToDouble(0x6000000000000001), 0.375);
	EXPECT_EQ(ToDouble(0x7000000000000001), 0.4375);
	EXPECT_EQ(ToDouble(0x8000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0x9000000000000001), 0.5625);
	EXPECT_EQ(ToDouble(0xa000000000000001), 0.625);
	EXPECT_EQ(ToDouble(0xb000000000000001), 0.6875);
	EXPECT_EQ(ToDouble(0xc000000000000001), 0.75);
	EXPECT_EQ(ToDouble(0xd000000000000001), 0.8125);
	EXPECT_EQ(ToDouble(0xe000000000000001), 0.875);
	EXPECT_EQ(ToDouble(0xf000000000000001), 0.9375);

	// Large powers of 2.
	int64_t two_to_53 = int64_t{1} << 53;
	EXPECT_EQ(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53),
	two_to_53 - 1);
	EXPECT_NE(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53 * 2),
	two_to_53 * 2 - 1);

	// For values where every bit counts, the values scale as multiples of the
	// input.
	for (int i = 0; i < 100; ++i) {
	EXPECT_EQ(i * ToDouble(0x0000000000000001), ToDouble(i));
	}

	// For each i: value generated from (1 << i).
	double exp_values[64];
	exp_values[63] = 0.5;
	for (int i = 62; i >= 0; --i) exp_values[i] = 0.5 * exp_values[i + 1];
	constexpr uint64_t one = 1;
	for (int i = 0; i < 64; ++i) {
	EXPECT_EQ(ToDouble(one << i), exp_values[i]);
	for (int j = 1; j < DBL_MANT_DIG && i - j >= 0; ++j) {
	EXPECT_NE(exp_values[i] + exp_values[i - j], exp_values[i]);
	EXPECT_EQ(ToDouble((one << i) + (one << (i - j))),
	exp_values[i] + exp_values[i - j]);
	}
	for (int j = DBL_MANT_DIG; i - j >= 0; ++j) {
	EXPECT_EQ(exp_values[i] + exp_values[i - j], exp_values[i]);
	EXPECT_EQ(ToDouble((one << i) + (one << (i - j))), exp_values[i]);
	}
	}
	}

	TEST(GenerateRealTest, U64ToDoubleSignedTest) {
	auto ToDouble = [](uint64_t a) {
	return GenerateRealFromBits<double, GenerateSignedTag, false>(a);
	};

	EXPECT_EQ(ToDouble(0x0000000000000000), 5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x0000000000000001), 1.084202172485504434e-19);

	EXPECT_EQ(ToDouble(0x8000000000000000), -5.42101086242752217004e-20);
	EXPECT_EQ(ToDouble(0x8000000000000001), -1.084202172485504434e-19);

	const double e_plus = ToDouble(0x0000000000000001);
	const double e_minus = ToDouble(0x8000000000000001);
	EXPECT_EQ(e_plus, 1.084202172485504434e-19);
	EXPECT_EQ(e_minus, -1.084202172485504434e-19);

	EXPECT_EQ(ToDouble(0x3fffffffffffffef), 0.499999999999999944489);
	EXPECT_EQ(ToDouble(0xbfffffffffffffef), -0.499999999999999944489);

	// For values > 0.5, RandU64ToDouble discards up to 10 bits. (63-53).
	EXPECT_EQ(ToDouble(0x4000000000000000), 0.5);
	EXPECT_EQ(ToDouble(0x4000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0x40000000000003FF), 0.5);

	EXPECT_EQ(ToDouble(0xC000000000000000), -0.5);
	EXPECT_EQ(ToDouble(0xC000000000000001), -0.5);
	EXPECT_EQ(ToDouble(0xC0000000000003FF), -0.5);

	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFe), 0.999999999999999888978);
	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFe), -0.999999999999999888978);

	EXPECT_NE(ToDouble(0x7FFFFFFFFFFFF800), ToDouble(0x7FFFFFFFFFFFF7FF));

	EXPECT_LT(ToDouble(0x7FFFFFFFFFFFFFFF), 1.0);
	EXPECT_GT(ToDouble(0x7FFFFFFFFFFFFFFF), 0.9999999999);

	EXPECT_GT(ToDouble(0xFFFFFFFFFFFFFFFe), -1.0);
	EXPECT_LT(ToDouble(0xFFFFFFFFFFFFFFFe), -0.999999999);

	EXPECT_EQ(ToDouble(0xFFFFFFFFFFFFFFFe), ToDouble(0xFFFFFFFFFFFFFC00));
	EXPECT_EQ(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFFC00));
	EXPECT_NE(ToDouble(0xFFFFFFFFFFFFFFFe), ToDouble(0xFFFFFFFFFFFFF3FF));
	EXPECT_NE(ToDouble(0x7FFFFFFFFFFFFFFF), ToDouble(0x7FFFFFFFFFFFF3FF));

	EXPECT_EQ(ToDouble(0x1000000000000001), 0.125);
	EXPECT_EQ(ToDouble(0x2000000000000001), 0.25);
	EXPECT_EQ(ToDouble(0x3000000000000001), 0.375);
	EXPECT_EQ(ToDouble(0x4000000000000001), 0.5);
	EXPECT_EQ(ToDouble(0x5000000000000001), 0.625);
	EXPECT_EQ(ToDouble(0x6000000000000001), 0.75);
	EXPECT_EQ(ToDouble(0x7000000000000001), 0.875);
	EXPECT_EQ(ToDouble(0x7800000000000001), 0.9375);
	EXPECT_EQ(ToDouble(0x7c00000000000001), 0.96875);
	EXPECT_EQ(ToDouble(0x7e00000000000001), 0.984375);
	EXPECT_EQ(ToDouble(0x7f00000000000001), 0.9921875);

	// 0x8000000000000000 ~= 0
	EXPECT_EQ(ToDouble(0x9000000000000001), -0.125);
	EXPECT_EQ(ToDouble(0xa000000000000001), -0.25);
	EXPECT_EQ(ToDouble(0xb000000000000001), -0.375);
	EXPECT_EQ(ToDouble(0xc000000000000001), -0.5);
	EXPECT_EQ(ToDouble(0xd000000000000001), -0.625);
	EXPECT_EQ(ToDouble(0xe000000000000001), -0.75);
	EXPECT_EQ(ToDouble(0xf000000000000001), -0.875);

	// Large powers of 2.
	int64_t two_to_53 = int64_t{1} << 53;
	EXPECT_EQ(static_cast<int64_t>(ToDouble(0x7FFFFFFFFFFFFFFF) * two_to_53),
	two_to_53 - 1);
	EXPECT_EQ(static_cast<int64_t>(ToDouble(0xFFFFFFFFFFFFFFFF) * two_to_53),
	-(two_to_53 - 1));

	EXPECT_NE(static_cast<int64_t>(ToDouble(0x7FFFFFFFFFFFFFFF) * two_to_53 * 2),
	two_to_53 * 2 - 1);

	// For values where every bit counts, the values scale as multiples of the
	// input.
	for (int i = 1; i < 100; ++i) {
	EXPECT_EQ(i * e_plus, ToDouble(i)) << i;
	EXPECT_EQ(i * e_minus, ToDouble(0x8000000000000000 \| i)) << i;
	}
	}

	TEST(GenerateRealTest, ExhaustiveFloat) {
	auto ToFloat = [](uint64_t a) {
	return GenerateRealFromBits<float, GeneratePositiveTag, true>(a);
	};

	// Rely on RandU64ToFloat generating values from greatest to least when
	// supplied with uint64_t values from greatest (0xfff...) to least (0x0).
	// Thus, this algorithm stores the previous value, and if the new value is at
	// greater than or equal to the previous value, then there is a collision in
	// the generation algorithm.
	//
	// Use the computation below to convert the random value into a result:
	// double res = a() * (1.0f - sample) + b() * sample;
	float last_f = 1.0, last_g = 2.0;
	uint64_t f_collisions = 0, g_collisions = 0;
	uint64_t f_unique = 0, g_unique = 0;
	uint64_t total = 0;
	auto count = [&](const float r) {
	total++;
	// `f` is mapped to the range [0, 1) (default)
	const float f = 0.0f * (1.0f - r) + 1.0f * r;
	if (f >= last_f) {
	f_collisions++;
	} else {
	f_unique++;
	last_f = f;
	}
	// `g` is mapped to the range [1, 2)
	const float g = 1.0f * (1.0f - r) + 2.0f * r;
	if (g >= last_g) {
	g_collisions++;
	} else {
	g_unique++;
	last_g = g;
	}
	};

	size_t limit = absl::GetFlag(FLAGS_absl_random_test_trials);

	// Generate all uint64_t which have unique floating point values.
	// Counting down from 0xFFFFFFFFFFFFFFFFu ... 0x0u
	uint64_t x = ~uint64_t(0);
	for (; x != 0 && limit > 0;) {
	constexpr int kDig = (64 - FLT_MANT_DIG);
	// Set a decrement value & the next point at which to change
	// the decrement value. By default these are 1, 0.
	uint64_t dec = 1;
	uint64_t chk = 0;

	// Adjust decrement and check value based on how many leading 0
	// bits are set in the current value.
	const int clz = absl::countl_zero(x);
	if (clz < kDig) {
	dec <<= (kDig - clz);
	chk = (~uint64_t(0)) >> (clz + 1);
	}
	for (; x > chk && limit > 0; x -= dec) {
	count(ToFloat(x));
	--limit;
	}
	}

	static_assert(FLT_MANT_DIG == 24,
	"The float type is expected to have a 24 bit mantissa.");

	if (limit != 0) {
	// There are between 2^28 and 2^29 unique values in the range [0, 1). For
	// the low values of x, there are 2^24 -1 unique values. Once x > 2^24,
	// there are 40 * 2^24 unique values. Thus:
	// (2 + 4 + 8 ... + 2^23) + 40 * 2^23
	EXPECT_LT(1 << 28, f_unique);
	EXPECT_EQ((1 << 24) + 40 * (1 << 23) - 1, f_unique);
	EXPECT_EQ(total, f_unique);
	EXPECT_EQ(0, f_collisions);

	// Expect at least 2^23 unique values for the range [1, 2)
	EXPECT_LE(1 << 23, g_unique);
	EXPECT_EQ(total - g_unique, g_collisions);
	}
	}

	} // namespace