auto merge of #10223 : huonw/rust/gamma, r=cmr

Implements the [Gamma distribution](https://en.wikipedia.org/wiki/Gamma_distribution), using the algorithm described by Marsaglia & Tsang 2000[1]. I added tests checking that the mean and variance of this implementation is as expected for a range of values of the parameters in 5d87c00a0f (they pass locally, but obviously won't even build on Travis until this is merged).

Also, moves `std::rand::distributions` to a subfolder, and performs a minor clean-up of the benchmarking (makes the number of iterations shared by the whole `std::rand` subtree).

[1]: George Marsaglia and Wai Wan Tsang. 2000. "A Simple Method for Generating Gamma Variables" *ACM Trans. Math. Softw.* 26, 3 (September 2000), 363-372. DOI:[10.1145/358407.358414](http://doi.acm.org/10.1145/358407.358414).

src/libstd/rand/distributions/gamma.rs

@@ -0,0 +1,216 @@
+// Copyright 2013 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+//! The Gamma distribution.
+
+use rand::Rng;
+use super::{IndependentSample, Sample, StandardNormal, Exp};
+use num;
+
+/// The Gamma distribution `Gamma(shape, scale)` distribution.
+///
+/// The density function of this distribution is
+///
+/// ```
+/// f(x) =  x^(k - 1) * exp(-x / θ) / (Γ(k) * θ^k)
+/// ```
+///
+/// where `Γ` is the Gamma function, `k` is the shape and `θ` is the
+/// scale and both `k` and `θ` are strictly positive.
+///
+/// The algorithm used is that described by Marsaglia & Tsang 2000[1],
+/// falling back to directly sampling from an Exponential for `shape
+/// == 1`, and using the boosting technique described in [1] for
+/// `shape < 1`.
+///
+/// # Example
+///
+/// ```rust
+/// use std::rand;
+/// use std::rand::distributions::{IndependentSample, Gamma};
+///
+/// fn main() {
+///     let gamma = Gamma::new(2.0, 5.0);
+///     let v = gamma.ind_sample(rand::task_rng());
+///     println!("{} is from a Gamma(2, 5) distribution", v);
+/// }
+/// ```
+///
+/// [1]: George Marsaglia and Wai Wan Tsang. 2000. "A Simple Method
+/// for Generating Gamma Variables" *ACM Trans. Math. Softw.* 26, 3
+/// (September 2000),
+/// 363-372. DOI:[10.1145/358407.358414](http://doi.acm.org/10.1145/358407.358414)
+pub enum Gamma {
+    priv Large(GammaLargeShape),
+    priv One(Exp),
+    priv Small(GammaSmallShape)
+}
+
+// These two helpers could be made public, but saving the
+// match-on-Gamma-enum branch from using them directly (e.g. if one
+// knows that the shape is always > 1) doesn't appear to be much
+// faster.
+
+/// Gamma distribution where the shape parameter is less than 1.
+///
+/// Note, samples from this require a compulsory floating-point `pow`
+/// call, which makes it significantly slower than sampling from a
+/// gamma distribution where the shape parameter is greater than or
+/// equal to 1.
+///
+/// See `Gamma` for sampling from a Gamma distribution with general
+/// shape parameters.
+struct GammaSmallShape {
+    inv_shape: f64,
+    large_shape: GammaLargeShape
+}
+
+/// Gamma distribution where the shape parameter is larger than 1.
+///
+/// See `Gamma` for sampling from a Gamma distribution with general
+/// shape parameters.
+struct GammaLargeShape {
+    shape: f64,
+    scale: f64,
+    c: f64,
+    d: f64
+}
+
+impl Gamma {
+    /// Construct an object representing the `Gamma(shape, scale)`
+    /// distribution.
+    ///
+    /// Fails if `shape <= 0` or `scale <= 0`.
+    pub fn new(shape: f64, scale: f64) -> Gamma {
+        assert!(shape > 0.0, "Gamma::new called with shape <= 0");
+        assert!(scale > 0.0, "Gamma::new called with scale <= 0");
+
+        match shape {
+            1.0        => One(Exp::new(1.0 / scale)),
+            0.0 .. 1.0 => Small(GammaSmallShape::new_raw(shape, scale)),
+            _          => Large(GammaLargeShape::new_raw(shape, scale))
+        }
+    }
+}
+
+impl GammaSmallShape {
+    fn new_raw(shape: f64, scale: f64) -> GammaSmallShape {
+        GammaSmallShape {
+            inv_shape: 1. / shape,
+            large_shape: GammaLargeShape::new_raw(shape + 1.0, scale)
+        }
+    }
+}
+
+impl GammaLargeShape {
+    fn new_raw(shape: f64, scale: f64) -> GammaLargeShape {
+        let d = shape - 1. / 3.;
+        GammaLargeShape {
+            shape: shape,
+            scale: scale,
+            c: 1. / num::sqrt(9. * d),
+            d: d
+        }
+    }
+}
+
+impl Sample<f64> for Gamma {
+    fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
+}
+impl Sample<f64> for GammaSmallShape {
+    fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
+}
+impl Sample<f64> for GammaLargeShape {
+    fn sample<R: Rng>(&mut self, rng: &mut R) -> f64 { self.ind_sample(rng) }
+}
+
+impl IndependentSample<f64> for Gamma {
+    fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
+        match *self {
+            Small(ref g) => g.ind_sample(rng),
+            One(ref g) => g.ind_sample(rng),
+            Large(ref g) => g.ind_sample(rng),
+        }
+    }
+}
+impl IndependentSample<f64> for GammaSmallShape {
+    fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
+        // Need (0, 1) here.
+        let mut u = rng.gen::<f64>();
+        while u == 0. {
+            u = rng.gen();
+        }
+
+        self.large_shape.ind_sample(rng) * num::pow(u, self.inv_shape)
+    }
+}
+impl IndependentSample<f64> for GammaLargeShape {
+    fn ind_sample<R: Rng>(&self, rng: &mut R) -> f64 {
+        loop {
+            let x = *rng.gen::<StandardNormal>();
+            let v_cbrt = 1.0 + self.c * x;
+            if v_cbrt <= 0.0 { // a^3 <= 0 iff a <= 0
+                continue
+            }
+
+            let v = v_cbrt * v_cbrt * v_cbrt;
+            // Need (0, 1) here, not [0, 1). This would be faster if
+            // we were generating an f64 in (0, 1) directly.
+            let mut u = rng.gen::<f64>();
+            while u == 0.0 {
+                u = rng.gen();
+            }
+
+            let x_sqr = x * x;
+            if u < 1.0 - 0.0331 * x_sqr * x_sqr ||
+                num::ln(u) < 0.5 * x_sqr + self.d * (1.0 - v + num::ln(v)) {
+                return self.d * v * self.scale
+            }
+        }
+    }
+}
+
+#[cfg(test)]
+mod bench {
+    use super::*;
+    use mem::size_of;
+    use rand::distributions::IndependentSample;
+    use rand::{StdRng, RAND_BENCH_N};
+    use extra::test::BenchHarness;
+    use iter::range;
+    use option::{Some, None};
+
+
+    #[bench]
+    fn bench_gamma_large_shape(bh: &mut BenchHarness) {
+        let gamma = Gamma::new(10., 1.0);
+        let mut rng = StdRng::new();
+
+        do bh.iter {
+            for _ in range(0, RAND_BENCH_N) {
+                gamma.ind_sample(&mut rng);
+            }
+        }
+        bh.bytes = size_of::<f64>() as u64 * RAND_BENCH_N;
+    }
+
+    #[bench]
+    fn bench_gamma_small_shape(bh: &mut BenchHarness) {
+        let gamma = Gamma::new(0.1, 1.0);
+        let mut rng = StdRng::new();
+
+        do bh.iter {
+            for _ in range(0, RAND_BENCH_N) {
+                gamma.ind_sample(&mut rng);
+            }
+        }
+        bh.bytes = size_of::<f64>() as u64 * RAND_BENCH_N;
+    }
+}

src/libstd/rand/distributions/mod.rs

@@ -27,8 +27,10 @@ use rand::{Rng,Rand};
 use clone::Clone;
 
 pub use self::range::Range;
+pub use self::gamma::Gamma;
 
 pub mod range;
+pub mod gamma;
 
 /// Types that can be used to create a random instance of `Support`.
 pub trait Sample<Support> {
@@ -554,25 +556,23 @@ mod tests {
 #[cfg(test)]
 mod bench {
     use extra::test::BenchHarness;
-    use rand::*;
+    use rand::{XorShiftRng, RAND_BENCH_N};
     use super::*;
     use iter::range;
     use option::{Some, None};
     use mem::size_of;
 
-    static N: u64 = 100;
-
     #[bench]
     fn rand_normal(bh: &mut BenchHarness) {
         let mut rng = XorShiftRng::new();
         let mut normal = Normal::new(-2.71828, 3.14159);
 
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 normal.sample(&mut rng);
             }
         }
-        bh.bytes = size_of::<f64>() as u64 * N;
+        bh.bytes = size_of::<f64>() as u64 * RAND_BENCH_N;
     }
     #[bench]
     fn rand_exp(bh: &mut BenchHarness) {
@@ -580,10 +580,10 @@ mod bench {
         let mut exp = Exp::new(2.71828 * 3.14159);
 
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 exp.sample(&mut rng);
             }
         }
-        bh.bytes = size_of::<f64>() as u64 * N;
+        bh.bytes = size_of::<f64>() as u64 * RAND_BENCH_N;
     }
 }

src/libstd/rand/mod.rs

@@ -833,58 +833,58 @@ mod test {
     }
 }
 
+static RAND_BENCH_N: u64 = 100;
+
 #[cfg(test)]
 mod bench {
     use extra::test::BenchHarness;
-    use rand::*;
+    use rand::{XorShiftRng, StdRng, IsaacRng, Isaac64Rng, Rng, RAND_BENCH_N};
     use mem::size_of;
     use iter::range;
     use option::{Some, None};
 
-    static N: u64 = 100;
-
     #[bench]
     fn rand_xorshift(bh: &mut BenchHarness) {
         let mut rng = XorShiftRng::new();
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 rng.gen::<uint>();
             }
         }
-        bh.bytes = size_of::<uint>() as u64 * N;
+        bh.bytes = size_of::<uint>() as u64 * RAND_BENCH_N;
     }
 
     #[bench]
     fn rand_isaac(bh: &mut BenchHarness) {
         let mut rng = IsaacRng::new();
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 rng.gen::<uint>();
             }
         }
-        bh.bytes = size_of::<uint>() as u64 * N;
+        bh.bytes = size_of::<uint>() as u64 * RAND_BENCH_N;
     }
 
     #[bench]
     fn rand_isaac64(bh: &mut BenchHarness) {
         let mut rng = Isaac64Rng::new();
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 rng.gen::<uint>();
             }
         }
-        bh.bytes = size_of::<uint>() as u64 * N;
+        bh.bytes = size_of::<uint>() as u64 * RAND_BENCH_N;
     }
 
     #[bench]
     fn rand_std(bh: &mut BenchHarness) {
         let mut rng = StdRng::new();
         do bh.iter {
-            for _ in range(0, N) {
+            for _ in range(0, RAND_BENCH_N) {
                 rng.gen::<uint>();
             }
         }
-        bh.bytes = size_of::<uint>() as u64 * N;
+        bh.bytes = size_of::<uint>() as u64 * RAND_BENCH_N;
     }
 
     #[bench]