text: // 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.
  
  #ifndef ABSL_RANDOM_POISSON_DISTRIBUTION_H_
  #define ABSL_RANDOM_POISSON_DISTRIBUTION_H_
  
  #include <cassert>
  #include <cmath>
  #include <istream>
  #include <limits>
  #include <ostream>
  #include <type_traits>
  
  #include "absl/random/internal/distribution_impl.h"
  #include "absl/random/internal/fast_uniform_bits.h"
  #include "absl/random/internal/fastmath.h"
  #include "absl/random/internal/iostream_state_saver.h"
  
  namespace absl {
  inline namespace lts_2019_08_08 {
  
  // absl::poisson_distribution:
  // Generates discrete variates conforming to a Poisson distribution.
  //   p(n) = (mean^n / n!) exp(-mean)
  //
  // Depending on the parameter, the distribution selects one of the following
  // algorithms:
  // * The standard algorithm, attributed to Knuth, extended using a split method
  // for larger values
  // * The "Ratio of Uniforms as a convenient method for sampling from classical
  // discrete distributions", Stadlober, 1989.
  // http://www.sciencedirect.com/science/article/pii/0377042790903495
  //
  // NOTE: param_type.mean() is a double, which permits values larger than
  // poisson_distribution<IntType>::max(), however this should be avoided and
  // the distribution results are limited to the max() value.
  //
  // The goals of this implementation are to provide good performance while still
  // beig thread-safe: This limits the implementation to not using lgamma provided
  // by <math.h>.
  //
  template <typename IntType = int>
  class poisson_distribution {
   public:
    using result_type = IntType;
  
    class param_type {
     public:
      using distribution_type = poisson_distribution;
      explicit param_type(double mean = 1.0);
  
      double mean() const { return mean_; }
  
      friend bool operator==(const param_type& a, const param_type& b) {
        return a.mean_ == b.mean_;
      }
  
      friend bool operator!=(const param_type& a, const param_type& b) {
        return !(a == b);
      }
  
     private:
      friend class poisson_distribution;
  
      double mean_;
      double emu_;  // e ^ -mean_
      double lmu_;  // ln(mean_)
      double s_;
      double log_k_;
      int split_;
  
      static_assert(std::is_integral<IntType>::value,
                    "Class-template absl::poisson_distribution<> must be "
                    "parameterized using an integral type.");
    };
  
    poisson_distribution() : poisson_distribution(1.0) {}
  
    explicit poisson_distribution(double mean) : param_(mean) {}
  
    explicit poisson_distribution(const param_type& p) : param_(p) {}
  
    void reset() {}
  
    // generating functions
    template <typename URBG>
    result_type operator()(URBG& g) {  // NOLINT(runtime/references)
      return (*this)(g, param_);
    }
  
    template <typename URBG>
    result_type operator()(URBG& g,  // NOLINT(runtime/references)
                           const param_type& p);
  
    param_type param() const { return param_; }
    void param(const param_type& p) { param_ = p; }
  
    result_type(min)() const { return 0; }
    result_type(max)() const { return (std::numeric_limits<result_type>::max)(); }
  
    double mean() const { return param_.mean(); }
  
    friend bool operator==(const poisson_distribution& a,
                           const poisson_distribution& b) {
      return a.param_ == b.param_;
    }
    friend bool operator!=(const poisson_distribution& a,
                           const poisson_distribution& b) {
      return a.param_ != b.param_;
    }
  
   private:
    param_type param_;
    random_internal::FastUniformBits<uint64_t> fast_u64_;
  };
  
  // -----------------------------------------------------------------------------
  // Implementation details follow
  // -----------------------------------------------------------------------------
  
  template <typename IntType>
  poisson_distribution<IntType>::param_type::param_type(double mean)
      : mean_(mean), split_(0) {
    assert(mean >= 0);
    assert(mean <= (std::numeric_limits<result_type>::max)());
    // As a defensive measure, avoid large values of the mean.  The rejection
    // algorithm used does not support very large values well.  It my be worth
    // changing algorithms to better deal with these cases.
    assert(mean <= 1e10);
    if (mean_ < 10) {
      // For small lambda, use the knuth method.
      split_ = 1;
      emu_ = std::exp(-mean_);
    } else if (mean_ <= 50) {
      // Use split-knuth method.
      split_ = 1 + static_cast<int>(mean_ / 10.0);
      emu_ = std::exp(-mean_ / static_cast<double>(split_));
    } else {
      // Use ratio of uniforms method.
      constexpr double k2E = 0.7357588823428846;
      constexpr double kSA = 0.4494580810294493;
  
      lmu_ = std::log(mean_);
      double a = mean_ + 0.5;
      s_ = kSA + std::sqrt(k2E * a);
      const double mode = std::ceil(mean_) - 1;
      log_k_ = lmu_ * mode - absl::random_internal::StirlingLogFactorial(mode);
    }
  }
  
  template <typename IntType>
  template <typename URBG>
  typename poisson_distribution<IntType>::result_type
  poisson_distribution<IntType>::operator()(
      URBG& g,  // NOLINT(runtime/references)
      const param_type& p) {
    using random_internal::PositiveValueT;
    using random_internal::RandU64ToDouble;
    using random_internal::SignedValueT;
  
    if (p.split_ != 0) {
      // Use Knuth's algorithm with range splitting to avoid floating-point
      // errors. Knuth's algorithm is: Ui is a sequence of uniform variates on
      // (0,1); return the number of variates required for product(Ui) <
      // exp(-lambda).
      //
      // The expected number of variates required for Knuth's method can be
      // computed as follows:
      // The expected value of U is 0.5, so solving for 0.5^n < exp(-lambda) gives
      // the expected number of uniform variates
      // required for a given lambda, which is:
      //  lambda = [2, 5,  9, 10, 11, 12, 13, 14, 15, 16, 17]
      //  n      = [3, 8, 13, 15, 16, 18, 19, 21, 22, 24, 25]
      //
      result_type n = 0;
      for (int split = p.split_; split > 0; --split) {
        double r = 1.0;
        do {
          r *= RandU64ToDouble<PositiveValueT, true>(fast_u64_(g));
          ++n;
        } while (r > p.emu_);
        --n;
      }
      return n;
    }
  
    // Use ratio of uniforms method.
    //
    // Let u ~ Uniform(0, 1), v ~ Uniform(-1, 1),
    //     a = lambda + 1/2,
    //     s = 1.5 - sqrt(3/e) + sqrt(2(lambda + 1/2)/e),
    //     x = s * v/u + a.
    // P(floor(x) = k | u^2 < f(floor(x))/k), where
    // f(m) = lambda^m exp(-lambda)/ m!, for 0 <= m, and f(m) = 0 otherwise,
    // and k = max(f).
    const double a = p.mean_ + 0.5;
    for (;;) {
      const double u =
          RandU64ToDouble<PositiveValueT, false>(fast_u64_(g));  // (0, 1)
      const double v =
          RandU64ToDouble<SignedValueT, false>(fast_u64_(g));  // (-1, 1)
      const double x = std::floor(p.s_ * v / u + a);
      if (x < 0) continue;  // f(negative) = 0
      const double rhs = x * p.lmu_;
      // clang-format off
      double s = (x <= 1.0) ? 0.0
               : (x == 2.0) ? 0.693147180559945
               : absl::random_internal::StirlingLogFactorial(x);
      // clang-format on
      const double lhs = 2.0 * std::log(u) + p.log_k_ + s;
      if (lhs < rhs) {
        return x > (max)() ? (max)()
                           : static_cast<result_type>(x);  // f(x)/k >= u^2
      }
    }
  }
  
  template <typename CharT, typename Traits, typename IntType>
  std::basic_ostream<CharT, Traits>& operator<<(
      std::basic_ostream<CharT, Traits>& os,  // NOLINT(runtime/references)
      const poisson_distribution<IntType>& x) {
    auto saver = random_internal::make_ostream_state_saver(os);
    os.precision(random_internal::stream_precision_helper<double>::kPrecision);
    os << x.mean();
    return os;
  }
  
  template <typename CharT, typename Traits, typename IntType>
  std::basic_istream<CharT, Traits>& operator>>(
      std::basic_istream<CharT, Traits>& is,  // NOLINT(runtime/references)
      poisson_distribution<IntType>& x) {     // NOLINT(runtime/references)
    using param_type = typename poisson_distribution<IntType>::param_type;
  
    auto saver = random_internal::make_istream_state_saver(is);
    double mean = random_internal::read_floating_point<double>(is);
    if (!is.fail()) {
      x.param(param_type(mean));
    }
    return is;
  }
  
  }  // inline namespace lts_2019_08_08
  }  // namespace absl
  
  #endif  // ABSL_RANDOM_POISSON_DISTRIBUTION_H_
  