diff --git a/benches/misc.rs b/benches/misc.rs
index 55a9805fcb6..4e9cbda37ae 100644
--- a/benches/misc.rs
+++ b/benches/misc.rs
@@ -63,9 +63,8 @@ fn misc_gen_ratio_var(b: &mut Bencher) {
 #[bench]
 fn misc_bernoulli_const(b: &mut Bencher) {
     let mut rng = StdRng::from_rng(&mut thread_rng()).unwrap();
-    let d = rand::distributions::Bernoulli::new(0.18);
     b.iter(|| {
-        // Can be evaluated at compile time.
+        let d = rand::distributions::Bernoulli::new(0.18);
         let mut accum = true;
         for _ in 0..::RAND_BENCH_N {
             accum ^= rng.sample(d);
diff --git a/src/distributions/bernoulli.rs b/src/distributions/bernoulli.rs
index 8bf70c82558..76bf924730d 100644
--- a/src/distributions/bernoulli.rs
+++ b/src/distributions/bernoulli.rs
@@ -37,6 +37,31 @@ pub struct Bernoulli {
     p_int: u64,
 }
 
+// To sample from the Bernoulli distribution we use a method that compares a
+// random `u64` value `v < (p * 2^64)`.
+//
+// If `p == 1.0`, the integer `v` to compare against can not represented as a
+// `u64`. We manually set it to `u64::MAX` instead (2^64 - 1 instead of 2^64).
+// Note that  value of `p < 1.0` can never result in `u64::MAX`, because an
+// `f64` only has 53 bits of precision, and the next largest value of `p` will
+// result in `2^64 - 2048`.
+//
+// Also there is a 100% theoretical concern: if someone consistenly wants to
+// generate `true` using the Bernoulli distribution (i.e. by using a probability
+// of `1.0`), just using `u64::MAX` is not enough. On average it would return
+// false once every 2^64 iterations. Some people apparently care about this
+// case.
+//
+// That is why we special-case `u64::MAX` to always return `true`, without using
+// the RNG, and pay the performance price for all uses that *are* reasonable.
+// Luckily, if `new()` and `sample` are close, the compiler can optimize out the
+// extra check.
+const ALWAYS_TRUE: u64 = ::core::u64::MAX;
+
+// This is just `2.0.powi(64)`, but written this way because it is not available
+// in `no_std` mode.
+const SCALE: f64 = 2.0 * (1u64 << 63) as f64;
+
 impl Bernoulli {
     /// Construct a new `Bernoulli` with the given probability of success `p`.
     ///
@@ -54,18 +79,11 @@ impl Bernoulli {
     /// 2<sup>-64</sup> in `[0, 1]` can be represented as a `f64`.)
     #[inline]
     pub fn new(p: f64) -> Bernoulli {
-        assert!((p >= 0.0) & (p <= 1.0), "Bernoulli::new not called with 0 <= p <= 0");
-        // Technically, this should be 2^64 or `u64::MAX + 1` because we compare
-        // using `<` when sampling. However, `u64::MAX` rounds to an `f64`
-        // larger than `u64::MAX` anyway.
-        const MAX_P_INT: f64 = ::core::u64::MAX as f64;
-        let p_int = if p < 1.0 {
-            (p * MAX_P_INT) as u64
-        } else {
-            // Avoid overflow: `MAX_P_INT` cannot be represented as u64.
-            ::core::u64::MAX
-        };
-        Bernoulli { p_int }
+        if p < 0.0 || p >= 1.0 {
+            if p == 1.0 { return Bernoulli { p_int: ALWAYS_TRUE } }
+            panic!("Bernoulli::new not called with 0.0 <= p <= 1.0");
+        }
+        Bernoulli { p_int: (p * SCALE) as u64 }
     }
 
     /// Construct a new `Bernoulli` with the probability of success of
@@ -85,7 +103,6 @@ impl Bernoulli {
         if numerator == denominator {
             return Bernoulli { p_int: ::core::u64::MAX }
         }
-        const SCALE: f64 = 2.0 * (1u64 << 63) as f64;
         let p_int = ((numerator as f64 / denominator as f64) * SCALE) as u64;
         Bernoulli { p_int }
     }
@@ -95,11 +112,9 @@ impl Distribution<bool> for Bernoulli {
     #[inline]
     fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> bool {
         // Make sure to always return true for p = 1.0.
-        if self.p_int == ::core::u64::MAX {
-            return true;
-        }
-        let r: u64 = rng.gen();
-        r < self.p_int
+        if self.p_int == ALWAYS_TRUE { return true; }
+        let v: u64 = rng.gen();
+        v < self.p_int
     }
 }