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    // Copyright 2018 Developers of the Rand project.
    // Copyright 2016-2017 The Rust Project Developers.
    //
    // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
    // https://www.apache.org/licenses/LICENSE-2.0> or the MIT license
    // <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your
    // option. This file may not be copied, modified, or distributed
    // except according to those terms.

    //! The Poisson distribution.
    #![allow(deprecated)]

    use crate::Rng;
    use crate::distributions::{Distribution, Cauchy};
    use crate::distributions::utils::log_gamma;

    /// The Poisson distribution `Poisson(lambda)`.
    ///
    /// This distribution has a density function:
    /// `f(k) = lambda^k * exp(-lambda) / k!` for `k >= 0`.
    #[deprecated(since="0.7.0", note="moved to rand_distr crate")]
    #[derive(Clone, Copy, Debug)]
    pub struct Poisson {
        lambda: f64,
        // precalculated values
        exp_lambda: f64,
        log_lambda: f64,
        sqrt_2lambda: f64,
        magic_val: f64,
    }

    impl Poisson {
        /// Construct a new `Poisson` with the given shape parameter
        /// `lambda`. Panics if `lambda <= 0`.
        pub fn new(lambda: f64) -> Poisson {
            assert!(lambda > 0.0, "Poisson::new called with lambda <= 0");
            let log_lambda = lambda.ln();
            Poisson {
                lambda,
                exp_lambda: (-lambda).exp(),
                log_lambda,
                sqrt_2lambda: (2.0 * lambda).sqrt(),
                magic_val: lambda * log_lambda - log_gamma(1.0 + lambda),
            }
        }
    }

    impl Distribution<u64> for Poisson {
        fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
            // using the algorithm from Numerical Recipes in C

            // for low expected values use the Knuth method
            if self.lambda < 12.0 {
                let mut result = 0;
                let mut p = 1.0;
                while p > self.exp_lambda {
                    p *= rng.gen::<f64>();
                    result += 1;
                }
                result - 1
            }
            // high expected values - rejection method
            else {
                let mut int_result: u64;

                // we use the Cauchy distribution as the comparison distribution
                // f(x) ~ 1/(1+x^2)
                let cauchy = Cauchy::new(0.0, 1.0);

                loop {
                    let mut result;
                    let mut comp_dev;

                    loop {
                        // draw from the Cauchy distribution
                        comp_dev = rng.sample(cauchy);
                        // shift the peak of the comparison ditribution
                        result = self.sqrt_2lambda * comp_dev + self.lambda;
                        // repeat the drawing until we are in the range of possible values
                        if result >= 0.0 {
                            break;
                        }
                    }
                    // now the result is a random variable greater than 0 with Cauchy distribution
                    // the result should be an integer value
                    result = result.floor();
                    int_result = result as u64;

                    // this is the ratio of the Poisson distribution to the comparison distribution
                    // the magic value scales the distribution function to a range of approximately 0-1
                    // since it is not exact, we multiply the ratio by 0.9 to avoid ratios greater than 1
                    // this doesn't change the resulting distribution, only increases the rate of failed drawings
                    let check = 0.9 * (1.0 + comp_dev * comp_dev)
                        * (result * self.log_lambda - log_gamma(1.0 + result) - self.magic_val).exp();

                    // check with uniform random value - if below the threshold, we are within the target distribution
                    if rng.gen::<f64>() <= check {
                        break;
                    }
                }
                int_result
            }
        }
    }

    #[cfg(test)]
    mod test {
        use crate::distributions::Distribution;
        use super::Poisson;

        #[test]
        #[cfg(not(miri))] // Miri is too slow
        fn test_poisson_10() {
            let poisson = Poisson::new(10.0);
            let mut rng = crate::test::rng(123);
            let mut sum = 0;
            for _ in 0..1000 {
                sum += poisson.sample(&mut rng);
            }
            let avg = (sum as f64) / 1000.0;
            println!("Poisson average: {}", avg);
            assert!((avg - 10.0).abs() < 0.5); // not 100% certain, but probable enough
        }

        #[test]
        #[cfg(not(miri))] // Miri doesn't support transcendental functions
        fn test_poisson_15() {
            // Take the 'high expected values' path
            let poisson = Poisson::new(15.0);
            let mut rng = crate::test::rng(123);
            let mut sum = 0;
            for _ in 0..1000 {
                sum += poisson.sample(&mut rng);
            }
            let avg = (sum as f64) / 1000.0;
            println!("Poisson average: {}", avg);
            assert!((avg - 15.0).abs() < 0.5); // not 100% certain, but probable enough
        }

        #[test]
        #[should_panic]
        fn test_poisson_invalid_lambda_zero() {
            Poisson::new(0.0);
        }

        #[test]
        #[should_panic]
        fn test_poisson_invalid_lambda_neg() {
            Poisson::new(-10.0);
        }
    }

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