About the subtitle

The subtitle for this blog may be misleading. Actually it reads "When the pupil is ready to learn, a teacher will appear."
I picked this Zen proverb because in early 2008 I was looking for a language with the following requirements: strong metaprogrammability, strong concurrency support, strong functional bias and bonus points for running on the JVM. I had prepared myself to not find the rare bird when I met Clojure (and it was nearly love at first sight).
So in that perspective I am the pupil and Clojure (the language, Rich Hickey and the whole community) the teacher.

Clojure as fast as Java!

I wrote a prototypal implementation of IPersistentSet in Clojure written with new new and, surprisingly, on its first benchmark it's already on par with Java-based implementations.

(dotimes [i 10]
  (let [n (int (Math/pow 2 (+ 10 i)))
        _ (println "n=" n)   
        a (do 
            (print "clojure\t")
            (time (doall (let [s (reduce conj empty-set (range 0 n 2))] (map #(get s %) (range n))))))
        b (do 
            (print "java\t")
            (time (doall (let [s (reduce conj #{} (range 0 n 2))] (map #(get s %) (range n))))))]
    (println (= a b))))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

n= 1024
clojure "Elapsed time: 1.006064 msecs"
java "Elapsed time: 0.736197 msecs"
true
n= 2048
clojure "Elapsed time: 1.813009 msecs"
java "Elapsed time: 1.328102 msecs"
true
n= 4096
clojure "Elapsed time: 3.590191 msecs"
java "Elapsed time: 2.608153 msecs"
true
n= 8192
clojure "Elapsed time: 7.046566 msecs"
java "Elapsed time: 5.807302 msecs"
true
n= 16384
clojure "Elapsed time: 16.015862 msecs"
java "Elapsed time: 10.284897 msecs"
true
n= 32768
clojure "Elapsed time: 29.803928 msecs"
java "Elapsed time: 23.850378 msecs"
true
n= 65536
clojure "Elapsed time: 68.79778 msecs"
java "Elapsed time: 63.947582 msecs"
true
n= 131072
clojure "Elapsed time: 132.082499 msecs"
java "Elapsed time: 113.433411 msecs"
true
n= 262144
clojure "Elapsed time: 292.149631 msecs"
java "Elapsed time: 265.39197 msecs"
true
n= 524288
clojure "Elapsed time: 595.265321 msecs"
java "Elapsed time: 698.711009 msecs"
true

To tell the truth, this benchmark is a bit unfair for Java: no dummy map and slightly different algorithms (half-full bitmap nodes become array-nodes, no leaf-node etc.).
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What *warn-on-reflection* doesn't tell you about arrays

user=> (time (let [a (make-array Object 100)] (dotimes [i 10000000] (aset a (rem i 100) nil))))
Reflection warning, NO_SOURCE_PATH:840 - call to aset can't be resolved.
"Elapsed time: 136063.015272 msecs"

With aget/aset, the index must be hinted to be an int:

user=> (time (let [a (make-array Object 100)] (dotimes [i 10000000] (aset a (int (rem i 100)) nil))))
"Elapsed time: 1064.546402 msecs"

Wow, more than 100x faster (reflection is bad) but despite the compiler doesn't complain one can help it to choose a faster path:

user=> (time (let [a #^"[Ljava.lang.Object;" (make-array Object 100)] (dotimes [i 10000000] (aset a (int (rem i 100)) nil))))
"Elapsed time: 247.446882 msecs"

On the whole we get a 500x speed-up with only two type hints.

Everybody loves the Sieve of Eratosthenes

If I judge by traffic logs for this blog, many newcomers want to compute prime numbers in Clojure.

A recent thread on the mailing list prompted me to test an idea I had for a while: to use a map to implement the Sieve of Eratosthenes.

The first implementation was this one:

(defn primes [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n factor)]
                    (assoc sieve m
                      (conj (sieve m) factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factors (sieve candidate)]
                       (reduce #(enqueue %1 candidate %2)
                         (dissoc sieve candidate)
                         factors)
                       (enqueue sieve candidate candidate)))]
    (apply concat (vals (reduce next-sieve {} (range 2 max))))))

where the sieve is a map from the next non-prime numbers to their factors. It's so naive that even numbers are tested for primality but it doesn't perform that bad: on my box, it takes 3s to compute all primes below 1,000,000 (it's roughly as fast as clojure.contrib.lazy-seqs/primes when the seq isn't yet memoized).

I wasn't happy with the way I handled the case where a non-prime was already checked off (ie was a key of the map): I was conjing onto the list of prime factors for this number. If instead I tried to check off n+p (where n is the already known non-prime and p the current prime) or n+2p or n+3p... until I found a yet unknown non-prime I wouldn't need to maintain a list of factors, nor to conj or reduce over it. And I was hoping that less allocations would yield better perfs.

Here is the second iteration:

(defn primes2 [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n factor)]
                    (if (sieve m)
                      (recur sieve m factor)
                      (assoc sieve m factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factor (sieve candidate)]
                       (-> sieve
                         (dissoc candidate)
                         (enqueue candidate factor))
                       (enqueue sieve candidate candidate)))]
    (vals (reduce next-sieve {} (range 2 max)))))

and it computes all the primes below 1,000,000 in 1.8s instead of 3s but it still tests even numbers for primality.

primes3 is primes2 modified to only test odd numbers:

(defn primes3 [max]
  (let [enqueue (fn [sieve n factor]
                  (let [m (+ n (+ factor factor))]
                    (if (sieve m)
                      (recur sieve m factor)
                      (assoc sieve m factor))))
        next-sieve (fn [sieve candidate]
                     (if-let [factor (sieve candidate)]
                       (-> sieve
                         (dissoc candidate)
                         (enqueue candidate factor))
                       (enqueue sieve candidate candidate)))]
    (cons 2 (vals (reduce next-sieve {} (range 3 max 2))))))

and it computes the same list of primes in 1.5s.

Out of curiosity, I wrote a lazy version of primes3:

(defn lazy-primes3 []
  (letfn [(enqueue [sieve n step]
            (let [m (+ n step)]
              (if (sieve m)
                (recur sieve m step)
                (assoc sieve m step))))
          (next-sieve [sieve candidate]
            (if-let [step (sieve candidate)]
              (-> sieve
                (dissoc candidate)
                (enqueue candidate step))
              (enqueue sieve candidate (+ candidate candidate))))
          (next-primes [sieve candidate]
            (if (sieve candidate)
              (recur (next-sieve sieve candidate) (+ candidate 2))
              (cons candidate 
                (lazy-seq (next-primes (next-sieve sieve candidate) 
                            (+ candidate 2))))))]
    (cons 2 (lazy-seq (next-primes {} 3)))))

and, surprisingly (better locality?), it computes the primes below 1,000,000 in 1s.

Linear Interpolation and sorted-map

Sorted collections (with subseq and rsubseq) can help when working with a partition of disjoint intervals, eg when you need to interpolate.

(defn interpolator 
 "Takes a coll of 2D points (vectors) and returns 
  their linear interpolation function."
 [points]
  (let [m (into (sorted-map) points)]
    (fn [x]
      (let [[[x1 y1]] (rsubseq m <= x)
            [[x2 y2]] (subseq m > x)]
        (if x2 
          (+ y1 (* (- x x1) (/ (- y2 y1) (- x2 x1))))
          y1)))))

;; => (map (interpolator [[0 0] [1 1] [3 2] [4 3]]) (range 0 9/2 1/2))
;; (0 1/2 1 5/4 3/2 7/4 2 5/2 3)

Moustache: syntax file and tests

Tests and a hairy syntax file.

Enlive selectors: documented!

Selectors syntax