  void operator()(OpKernelContext* ctx, const CPUDevice& d, const T* rate_flat,
                  int num_rate, int num_samples,
                  const random::PhiloxRandom& rng, U* samples_flat) {
    // Two different algorithms are employed, depending on the size of
    // rate.
    // If rate < 10, we use an algorithm attributed to Knuth:
    // Seminumerical Algorithms. Art of Computer Programming, Volume 2.
    //
    // This algorithm runs in O(rate) time, and will require O(rate)
    // uniform variates.
    //
    // If rate >= 10 we use a transformation-rejection algorithm from
    // pairs of uniform random variables due to Hormann.
    // http://www.sciencedirect.com/science/article/pii/0167668793909974
    //
    // The algorithm has an acceptance rate of ~89% for the smallest rate
    // (~10),
    // and higher accept rates for higher rate, so runtime is
    // O(NumRate * NumSamples * k) with k ~ 1 / 0.89.
    //
    // We partition work first across rates then across
    // samples-per-rate to
    // avoid a couple flops which can be done on a per-rate basis.

    typedef random::UniformDistribution<random::PhiloxRandom, CT> Uniform;

    auto DoWork = [num_samples, num_rate, &rng, samples_flat, rate_flat](
                      int start_output, int limit_output) {
      // Capturing "rng" by value would only make a copy for the _shared_
      // lambda.  Since we want to let each worker have its own copy, we pass
      // "rng" by reference and explicitly do a copy assignment.

      Uniform uniform;
      typename Uniform::ResultType uniform_result;
      for (int64 output_idx = start_output; output_idx < limit_output;
           /* output_idx incremented within inner loop below */) {
        const int64 rate_idx = output_idx / num_samples;

        // Several calculations can be done on a per-rate basis.
        const CT rate = CT(rate_flat[rate_idx]);

        auto samples_rate_output = samples_flat + rate_idx;

        if (rate < CT(10)) {
          // Knuth's algorithm for generating Poisson random variates.
          // Given a Poisson process, the time between events is exponentially
          // distributed. If we have a Poisson process with rate lambda, then,
          // the time between events is distributed Exp(lambda). If X ~
          // Uniform(0, 1), then Y ~ Exp(lambda), where Y = -log(X) / lambda.
          // Thus to simulate a Poisson draw, we can draw X_i ~ Exp(lambda),
          // and N ~ Poisson(lambda), where N is the least number such that
          // \sum_i^N X_i > 1.
          const CT exp_neg_rate = Eigen::numext::exp(-rate);

          // Compute the rest of the samples for the current rate value.
          for (int64 sample_idx = output_idx % num_samples;
               sample_idx < num_samples && output_idx < limit_output;
               sample_idx++, output_idx++) {
            random::PhiloxRandom gen = rng;
            gen.Skip(kReservedSamplesPerOutput * output_idx);
            int16 uniform_remaining = 0;

            CT prod = 1;
            CT x = 0;

            // Keep trying until we surpass e^(-rate). This will take
            // expected time proportional to rate.
            while (true) {
              UNIFORM(u);
              prod = prod * u;
              if (prod <= exp_neg_rate &&
                  x <= CT(Eigen::NumTraits<U>::highest())) {
                samples_rate_output[sample_idx * num_rate] = U(x);
                break;
              }
              x += 1;
            }
          }
          continue;
        }
        // Transformed rejection due to Hormann.
        //
        // Given a CDF F(x), and G(x), a dominating distribution chosen such
        // that it is close to the inverse CDF F^-1(x), compute the following
        // steps:
        //
        // 1) Generate U and V, two independent random variates. Set U = U - 0.5
        // (this step isn't strictly necessary, but is done to make some
        // calculations symmetric and convenient. Henceforth, G is defined on
        // [-0.5, 0.5]).
        //
        // 2) If V <= alpha * F'(G(U)) * G'(U), return floor(G(U)), else return
        // to step 1. alpha is the acceptance probability of the rejection
        // algorithm.
        //
        // For more details on transformed rejection, see:
        // http://citeseer.ist.psu.edu/viewdoc/citations;jsessionid=1BEB35946CC807879F55D42512E5490C?doi=10.1.1.48.3054.
        //
        // The dominating distribution in this case:
        //
        // G(u) = (2 * a / (2 - |u|) + b) * u + c

        using Eigen::numext::log;
        const CT log_rate = log(rate);

        // Constants used to define the dominating distribution. Names taken
        // from Hormann's paper. Constants were chosen to define the tightest
        // G(u) for the inverse Poisson CDF.
        const CT b = CT(0.931) + CT(2.53) * Eigen::numext::sqrt(rate);
        const CT a = CT(-0.059) + CT(0.02483) * b;

        // This is the inverse acceptance rate. At a minimum (when rate = 10),
        // this corresponds to ~75% acceptance. As the rate becomes larger, this
        // approaches ~89%.
        const CT inv_alpha = CT(1.1239) + CT(1.1328) / (b - CT(3.4));

        // Compute the rest of the samples for the current rate value.
        for (int64 sample_idx = output_idx % num_samples;
             sample_idx < num_samples && output_idx < limit_output;
             sample_idx++, output_idx++) {
          random::PhiloxRandom gen = rng;
          gen.Skip(kReservedSamplesPerOutput * output_idx);
          int16 uniform_remaining = 0;

          while (true) {
            UNIFORM(u);
            u -= CT(0.5);
            UNIFORM(v);

            CT u_shifted = CT(0.5) - Eigen::numext::abs(u);
            CT k = Eigen::numext::floor((CT(2) * a / u_shifted + b) * u + rate +
                                        CT(0.43));

            if (k > CT(Eigen::NumTraits<U>::highest())) {
              // retry in case of overflow.
              continue;
            }

            // When alpha * f(G(U)) * G'(U) is close to 1, it is possible to
            // find a rectangle (-u_r, u_r) x (0, v_r) under the curve, such
            // that if v <= v_r and |u| <= u_r, then we can accept.
            // Here v_r = 0.9227 - 3.6224 / (b - 2) and u_r = 0.43.
            if (u_shifted >= CT(0.07) &&
                v <= CT(0.9277) - CT(3.6224) / (b - CT(2))) {
              samples_rate_output[sample_idx * num_rate] = U(k);
              break;
            }

            if (k < 0 || (u_shifted < CT(0.013) && v > u_shifted)) {
              continue;
            }

            // The expression below is equivalent to the computation of step 2)
            // in transformed rejection (v <= alpha * F'(G(u)) * G'(u)).
            CT s = log(v * inv_alpha / (a / (u_shifted * u_shifted) + b));
            CT t = -rate + k * log_rate - Eigen::numext::lgamma(k + 1);
            if (s <= t) {
              samples_rate_output[sample_idx * num_rate] = U(k);
              break;
            }
          }
        }
      }
    };

    // This will depend on rate.
    // For rate < 10, on average, O(rate) calls to uniform are
    // needed, with that
    // many multiplies. ~10 uniform calls on average with ~25 cost op calls.
    //
    // Very roughly, for rate >= 10, the single call to log + call to
    // lgamma
    // occur for ~60 percent of samples.
    // 2 x 100 (64-bit cycles per log) * 0.62 = ~124
    // Additionally, there are ~10 other ops (+, *, /, ...) at 3-6 cycles each:
    // 40 * .62  = ~25.
    //
    // Finally, there are several other ops that are done every loop along with
    // 2 uniform generations along with 5 other ops at 3-6 cycles each.
    // ~15 / .89 = ~16
    //
    // In total this should be ~165 + 2 * Uniform::kElementCost.
    // We assume that half the tensor has rate < 10, so on average 6
    // uniform's
    // will be needed. We will upper bound the other op cost by the one for
    // rate > 10.
    static const int kElementCost = 165 + 6 * Uniform::kElementCost +
                                    6 * random::PhiloxRandom::kElementCost;
    auto worker_threads = *(ctx->device()->tensorflow_cpu_worker_threads());
    Shard(worker_threads.num_threads, worker_threads.workers,
          num_rate * num_samples, kElementCost, DoWork);
  }
  void operator()(OpKernelContext* ctx, const CPUDevice& d, const T* rate_flat,
                  int num_rate, int num_samples,
                  const random::PhiloxRandom& rng, U* samples_flat) {
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

    typedef random::UniformDistribution<random::PhiloxRandom, CT> Uniform;

    auto DoWork = [num_samples, num_rate, &rng, samples_flat, rate_flat](
                      int start_output, int limit_output) {
       
       
       

      Uniform uniform;
      typename Uniform::ResultType uniform_result;
      for (int64 output_idx = start_output; output_idx < limit_output;
            ) {
        const int64 rate_idx = output_idx / num_samples;

         
        const CT rate = CT(rate_flat[rate_idx]);

        auto samples_rate_output = samples_flat + rate_idx;

        if (rate < CT(10)) {
           
           
           
           
           
           
           
           
          const CT exp_neg_rate = Eigen::numext::exp(-rate);

           
          for (int64 sample_idx = output_idx % num_samples;
               sample_idx < num_samples && output_idx < limit_output;
               sample_idx++, output_idx++) {
            random::PhiloxRandom gen = rng;
            gen.Skip(kReservedSamplesPerOutput * output_idx);
            int16 uniform_remaining = 0;

            CT prod = 1;
            CT x = 0;

             
             
            while (true) {
              UNIFORM(u);
              prod = prod * u;
              if (prod <= exp_neg_rate &&
                  x <= CT(Eigen::NumTraits<U>::highest())) {
                samples_rate_output[sample_idx * num_rate] = U(x);
                break;
              }
              x += 1;
            }
          }
          continue;
        }
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         

        using Eigen::numext::log;
        const CT log_rate = log(rate);

         
         
         
        const CT b = CT(0.931) + CT(2.53) * Eigen::numext::sqrt(rate);
        const CT a = CT(-0.059) + CT(0.02483) * b;

         
         
         
        const CT inv_alpha = CT(1.1239) + CT(1.1328) / (b - CT(3.4));

         
        for (int64 sample_idx = output_idx % num_samples;
             sample_idx < num_samples && output_idx < limit_output;
             sample_idx++, output_idx++) {
          random::PhiloxRandom gen = rng;
          gen.Skip(kReservedSamplesPerOutput * output_idx);
          int16 uniform_remaining = 0;

          while (true) {
            UNIFORM(u);
            u -= CT(0.5);
            UNIFORM(v);

            CT u_shifted = CT(0.5) - Eigen::numext::abs(u);
            CT k = Eigen::numext::floor((CT(2) * a / u_shifted + b) * u + rate +
                                        CT(0.43));

            if (k > CT(Eigen::NumTraits<U>::highest())) {
               
              continue;
            }

             
             
             
             
            if (u_shifted >= CT(0.07) &&
                v <= CT(0.9277) - CT(3.6224) / (b - CT(2))) {
              samples_rate_output[sample_idx * num_rate] = U(k);
              break;
            }

            if (k < 0 || (u_shifted < CT(0.013) && v > u_shifted)) {
              continue;
            }

             
             
            CT s = log(v * inv_alpha / (a / (u_shifted * u_shifted) + b));
            CT t = -rate + k * log_rate - Eigen::numext::lgamma(k + 1);
            if (s <= t) {
              samples_rate_output[sample_idx * num_rate] = U(k);
              break;
            }
          }
        }
      }
    };

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
    static const int kElementCost = 165 + 6 * Uniform::kElementCost +
                                    6 * random::PhiloxRandom::kElementCost;
    auto worker_threads = *(ctx->device()->tensorflow_cpu_worker_threads());
    Shard(worker_threads.num_threads, worker_threads.workers,
          num_rate * num_samples, kElementCost, DoWork);
  }