Average loop length
You are encouraged to solve this task according to the task description, using any language you may know.
Let f be a uniformly-randomly chosen mapping from the numbers 1..N to the numbers 1..N (note: not necessarily a permutation of 1..N; the mapping could produce a number in more than one way or not at all). At some point, the sequence 1, f(1), f(f(1))... will contain a repetition, a number that occurring for the second time in the sequence.
- Task
Write a program or a script that estimates, for each N, the average length until the first such repetition.
Also calculate this expected length using an analytical formula, and optionally compare the simulated result with the theoretical one.
This problem comes from the end of Donald Knuth's Christmas tree lecture 2011.
Example of expected output:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.4992 1.5000 ( 0.05%) 3 1.8784 1.8889 ( 0.56%) 4 2.2316 2.2188 ( 0.58%) 5 2.4982 2.5104 ( 0.49%) 6 2.7897 2.7747 ( 0.54%) 7 3.0153 3.0181 ( 0.09%) 8 3.2429 3.2450 ( 0.07%) 9 3.4536 3.4583 ( 0.14%) 10 3.6649 3.6602 ( 0.13%) 11 3.8091 3.8524 ( 1.12%) 12 3.9986 4.0361 ( 0.93%) 13 4.2074 4.2123 ( 0.12%) 14 4.3711 4.3820 ( 0.25%) 15 4.5275 4.5458 ( 0.40%) 16 4.6755 4.7043 ( 0.61%) 17 4.8877 4.8579 ( 0.61%) 18 4.9951 5.0071 ( 0.24%) 19 5.1312 5.1522 ( 0.41%) 20 5.2699 5.2936 ( 0.45%)
Ada
<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Generic_Elementary_Functions; with Ada.Numerics.Discrete_Random; procedure Avglen is
package IIO is new Ada.Text_IO.Integer_IO (Positive); use IIO; package LFIO is new Ada.Text_IO.Float_IO (Long_Float); use LFIO; subtype FactN is Natural range 0..20; TESTS : constant Natural := 1_000_000;
function Factorial (N : FactN) return Long_Float is
Result : Long_Float := 1.0;
begin
for I in 2..N loop Result := Result * Long_Float(I); end loop;
return Result;
end Factorial;
function Analytical (N : FactN) return Long_Float is
Sum : Long_Float := 0.0;
begin
for I in 1..N loop
Sum := Sum + Factorial(N) / Factorial(N - I) / Long_Float(N)**I;
end loop;
return Sum;
end Analytical;
function Experimental (N : FactN) return Long_Float is
subtype RandInt is Natural range 1..N;
package Random is new Ada.Numerics.Discrete_Random(RandInt);
seed : Random.Generator;
Num : RandInt;
count : Natural := 0;
bits : array(RandInt'Range) of Boolean;
begin
Random.Reset(seed);
for run in 1..TESTS loop
bits := (others => false);
for I in RandInt'Range loop
Num := Random.Random(seed); exit when bits(Num);
bits(Num) := True; count := count + 1;
end loop;
end loop;
return Long_Float(count)/Long_Float(TESTS);
end Experimental;
A, E, err : Long_Float;
begin
Put_Line(" N avg calc %diff");
for I in 1..20 loop
A := Analytical(I); E := Experimental(I); err := abs(E-A)/A*100.0;
Put(I, Width=>2); Put(E ,Aft=>4, exp=>0); Put(A, Aft=>4, exp=>0);
Put(err, Fore=>3, Aft=>3, exp=>0); New_line;
end loop;
end Avglen;</lang>
- Output:
N avg calc %diff 1 1.0000 1.0000 0.000 2 1.5000 1.5000 0.003 3 1.8886 1.8889 0.015 4 2.2180 2.2188 0.033 5 2.5104 2.5104 0.000 6 2.7745 2.7747 0.006 7 3.0191 3.0181 0.033 8 3.2433 3.2450 0.052 9 3.4583 3.4583 0.001 10 3.6597 3.6602 0.015 11 3.8524 3.8524 0.001 12 4.0352 4.0361 0.022 13 4.2147 4.2123 0.055 14 4.3853 4.3820 0.075 15 4.5453 4.5458 0.011 16 4.7055 4.7043 0.027 17 4.8592 4.8579 0.028 18 5.0062 5.0071 0.017 19 5.1535 5.1522 0.025 20 5.2955 5.2936 0.035
BBC BASIC
<lang bbcbasic> @% = &2040A
MAX_N = 20
TIMES = 1000000
FOR n = 1 TO MAX_N
avg = FNtest(n, TIMES)
theory = FNanalytical(n)
diff = (avg / theory - 1) * 100
PRINT STR$(n), avg, theory, diff "%"
NEXT
END
DEF FNanalytical(n)
LOCAL i, s
FOR i = 1 TO n
s += FNfactorial(n) / n^i / FNfactorial(n-i)
NEXT
= s
DEF FNtest(n, times)
LOCAL i, b, c, x
FOR i = 1 TO times
x = 1 : b = 0
WHILE (b AND x) = 0
c += 1
b OR= x
x = 1 << (RND(n) - 1)
ENDWHILE
NEXT
= c / times
DEF FNfactorial(n)
IF n=1 OR n=0 THEN =1 ELSE = n * FNfactorial(n-1)</lang>
- Output:
1 1.0000 1.0000 0.0000% 2 1.4995 1.5000 -0.0366% 3 1.8879 1.8889 -0.0509% 4 2.2193 2.2188 0.0240% 5 2.5105 2.5104 0.0057% 6 2.7755 2.7747 0.0293% 7 3.0199 3.0181 0.0573% 8 3.2396 3.2450 -0.1664% 9 3.4562 3.4583 -0.0609% 10 3.6578 3.6602 -0.0659% 11 3.8523 3.8524 -0.0025% 12 4.0336 4.0361 -0.0602% 13 4.2139 4.2123 0.0366% 14 4.3816 4.3820 -0.0105% 15 4.5432 4.5458 -0.0570% 16 4.7108 4.7043 0.1386% 17 4.8578 4.8579 -0.0018% 18 5.0063 5.0071 -0.0144% 19 5.1564 5.1522 0.0814% 20 5.2945 5.2936 0.0166%
C
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- include <time.h>
- define MAX_N 20
- define TIMES 1000000
double factorial(int n) { double f = 1; int i; for (i = 1; i <= n; i++) f *= i; return f; }
double expected(int n) { double sum = 0; int i; for (i = 1; i <= n; i++) sum += factorial(n) / pow(n, i) / factorial(n - i); return sum; }
int randint(int n) { int r, rmax = RAND_MAX / n * n; while ((r = rand()) >= rmax); return r / (RAND_MAX / n); }
int test(int n, int times) { int i, count = 0; for (i = 0; i < times; i++) { int x = 1, bits = 0; while (!(bits & x)) { count++; bits |= x; x = 1 << randint(n); } } return count; }
int main(void) { srand(time(0)); puts(" n\tavg\texp.\tdiff\n-------------------------------");
int n; for (n = 1; n <= MAX_N; n++) { int cnt = test(n, TIMES); double avg = (double)cnt / TIMES; double theory = expected(n); double diff = (avg / theory - 1) * 100; printf("%2d %8.4f %8.4f %6.3f%%\n", n, avg, theory, diff); } return 0; }</lang>
- Output:
n avg exp. diff ------------------------------- 1 1.0000 1.0000 0.000% 2 1.4998 1.5000 -0.015% 3 1.8879 1.8889 -0.051% 4 2.2181 2.2188 -0.029% 5 2.5107 2.5104 0.012% 6 2.7741 2.7747 -0.021% 7 3.0168 3.0181 -0.044% 8 3.2455 3.2450 0.014% 9 3.4591 3.4583 0.023% 10 3.6596 3.6602 -0.017% 11 3.8519 3.8524 -0.013% 12 4.0384 4.0361 0.059% 13 4.2106 4.2123 -0.042% 14 4.3840 4.3820 0.044% 15 4.5449 4.5458 -0.020% 16 4.7058 4.7043 0.033% 17 4.8549 4.8579 -0.060% 18 5.0084 5.0071 0.026% 19 5.1479 5.1522 -0.084% 20 5.2957 5.2936 0.040%
C++
Partial translation of C using stl and std. <lang cpp>#include <random>
- include <vector>
- include <iostream>
- define MAX_N 20
- define TIMES 1000000
/**
* Used to generate a uniform random distribution */
static std::random_device rd; //Will be used to obtain a seed for the random number engine static std::mt19937 gen(rd()); //Standard mersenne_twister_engine seeded with rd() static std::uniform_int_distribution<> dis;
int randint(int n) {
int r, rmax = RAND_MAX / n * n; dis=std::uniform_int_distribution(0,rmax) ; r = dis(gen); return r / (RAND_MAX / n);
}
unsigned long factorial(size_t n) {
//Factorial using dynamic programming to memoize the values.
static std::vector<unsigned long>factorials{1,1,2};
for (;factorials.size() <= n;) factorials.push_back(factorials.back()*factorials.size()); return factorials[n]; }
long double expected(size_t n) {
long double sum = 0;
for (size_t i = 1; i <= n; i++)
sum += factorial(n) / pow(n, i) / factorial(n - i);
return sum;
}
int test(int n, int times) {
int i, count = 0;
for (i = 0; i < times; i++) {
unsigned int x = 1, bits = 0;
while (!(bits & x)) {
count++;
bits |= x;
x = static_cast<unsigned int>(1 << randint(n));
}
}
return count;
}
int main() {
puts(" n\tavg\texp.\tdiff\n-------------------------------");
int n;
for (n = 1; n <= MAX_N; n++) {
int cnt = test(n, TIMES);
long double avg = (double)cnt / TIMES;
long double theory = expected(static_cast<size_t>(n));
long double diff = (avg / theory - 1) * 100;
printf("%2d %8.4f %8.4f %6.3f%%\n", n, static_cast<double>(avg), static_cast<double>(theory), static_cast<double>(diff));
}
return 0;
} </lang>
- Output:
n avg exp. diff ------------------------------- 1 1.0000 1.0000 0.000% 2 1.4998 1.5000 -0.016% 3 1.8883 1.8889 -0.032% 4 2.2188 2.2188 0.000% 5 2.5105 2.5104 0.004% 6 2.7760 2.7747 0.047% 7 3.0180 3.0181 -0.004% 8 3.2448 3.2450 -0.007% 9 3.4580 3.4583 -0.010% 10 3.6614 3.6602 0.032% 11 3.8532 3.8524 0.022% 12 4.0349 4.0361 -0.029% 13 4.2153 4.2123 0.070% 14 4.3819 4.3820 -0.003% 15 4.5494 4.5458 0.079% 16 4.7082 4.7043 0.084% 17 4.8576 4.8579 -0.005% 18 5.0028 5.0071 -0.084% 19 5.1484 5.1522 -0.073% 20 5.2939 5.2936 0.006%
Clojure
<lang lisp>(ns cyclelengths
(:gen-class))
(defn factorial [n]
" n! " (apply *' (range 1 (inc n)))) ; Use *' (vs. *) to allow arbitrary length arithmetic
(defn pow [n i]
" n^i" (apply *' (repeat i n)))
(defn analytical [n]
" Analytical Computation "
(->>(range 1 (inc n))
(map #(/ (factorial n) (pow n %) (factorial (- n %)))) ;calc n %))
(reduce + 0)))
- Number of random times to test each n
(def TIMES 1000000)
(defn single-test-cycle-length [n]
" Single random test of cycle length "
(loop [count 0
bits 0
x 1]
(if (zero? (bit-and x bits))
(recur (inc count) (bit-or bits x) (bit-shift-left 1 (rand-int n)))
count)))
(defn avg-cycle-length [n times]
" Average results of single tests of cycle lengths "
(/
(reduce +
(for [i (range times)]
(single-test-cycle-length n)))
times))
- Show Results
(println "\tAvg\t\tExp\t\tDiff") (doseq [q (range 1 21)
:let [anal (double (analytical q))
avg (double (avg-cycle-length q TIMES))
diff (Math/abs (* 100 (- 1 (/ avg anal))))]]
(println (format "%3d\t%.4f\t%.4f\t%.2f%%" q avg anal diff)))
</lang>
- Output:
Avg Exp Diff 1 1.0000 1.0000 0.00% 2 1.4995 1.5000 0.03% 3 1.8899 1.8889 0.05% 4 2.2178 2.2188 0.04% 5 2.5118 2.5104 0.06% 6 2.7773 2.7747 0.09% 7 3.0177 3.0181 0.02% 8 3.2448 3.2450 0.01% 9 3.4587 3.4583 0.01% 10 3.6594 3.6602 0.02% 11 3.8553 3.8524 0.08% 12 4.0335 4.0361 0.06% 13 4.2113 4.2123 0.03% 14 4.3823 4.3820 0.01% 15 4.5491 4.5458 0.07% 16 4.7035 4.7043 0.02% 17 4.8580 4.8579 0.00% 18 5.0050 5.0071 0.04% 19 5.1543 5.1522 0.04% 20 5.2956 5.2936 0.04%
D
<lang d>import std.stdio, std.random, std.math, std.algorithm, std.range, std.format;
real analytical(in int n) pure nothrow @safe /*@nogc*/ {
enum aux = (int k) => reduce!q{a * b}(1.0L, iota(n - k + 1, n + 1));
return iota(1, n + 1)
.map!(k => (aux(k) * k ^^ 2) / (real(n) ^^ (k + 1)))
.sum;
}
size_t loopLength(size_t maxN)(in int size, ref Xorshift rng) {
__gshared static bool[maxN + 1] seen;
seen[0 .. size + 1] = false;
int current = 1;
size_t steps = 0;
while (!seen[current]) {
seen[current] = true;
current = uniform(1, size + 1, rng);
steps++;
}
return steps;
}
void main() {
enum maxN = 40;
enum nTrials = 300_000;
auto rng = Xorshift(unpredictableSeed);
writeln(" n average analytical (error)");
writeln("=== ========= ============ ==========");
foreach (immutable n; 1 .. maxN + 1) {
long total = 0;
foreach (immutable _; 0 .. nTrials)
total += loopLength!maxN(n, rng);
immutable average = total / real(nTrials);
immutable an = n.analytical;
immutable percentError = abs(an - average) / an * 100;
immutable errorS = format("%2.4f", percentError);
writefln("%3d %9.5f %12.5f (%7s%%)",
n, average, an, errorS);
}
}</lang>
- Output:
n average analytical (error) === ========= ============ ========== 1 1.00000 1.00000 ( 0.0000%) 2 1.50017 1.50000 ( 0.0111%) 3 1.88932 1.88889 ( 0.0226%) 4 2.21795 2.21875 ( 0.0362%) 5 2.51159 2.51040 ( 0.0474%) 6 2.77373 2.77469 ( 0.0345%) 7 3.01894 3.01814 ( 0.0264%) 8 3.24734 3.24502 ( 0.0716%) 9 3.45876 3.45832 ( 0.0127%) 10 3.66595 3.66022 ( 0.1567%) 11 3.85000 3.85237 ( 0.0616%) 12 4.03532 4.03607 ( 0.0187%) 13 4.20879 4.21235 ( 0.0843%) 14 4.37664 4.38203 ( 0.1230%) 15 4.54986 4.54581 ( 0.0892%) 16 4.70431 4.70426 ( 0.0010%) 17 4.85640 4.85787 ( 0.0302%) 18 5.01359 5.00706 ( 0.1303%) 19 5.15487 5.15220 ( 0.0519%) 20 5.29486 5.29358 ( 0.0241%) 21 5.43276 5.43150 ( 0.0231%) 22 5.56570 5.56620 ( 0.0088%) 23 5.70611 5.69788 ( 0.1443%) 24 5.82618 5.82675 ( 0.0098%) 25 5.94846 5.95298 ( 0.0759%) 26 6.07440 6.07672 ( 0.0381%) 27 6.20717 6.19811 ( 0.1461%) 28 6.31546 6.31729 ( 0.0290%) 29 6.44201 6.43437 ( 0.1187%) 30 6.54592 6.54946 ( 0.0540%) 31 6.65818 6.66265 ( 0.0671%) 32 6.77215 6.77405 ( 0.0279%) 33 6.88381 6.88372 ( 0.0013%) 34 6.99790 6.99175 ( 0.0880%) 35 7.10990 7.09820 ( 0.1648%) 36 7.20391 7.20316 ( 0.0104%) 37 7.30085 7.30667 ( 0.0796%) 38 7.40366 7.40880 ( 0.0693%) 39 7.51864 7.50959 ( 0.1204%) 40 7.60255 7.60911 ( 0.0863%)
EchoLisp
<lang scheme> (lib 'math) ;; Σ aka (sigma f(n) nfrom nto)
(define (f-count N (times 100000))
(define count 0)
(for ((i times))
;; new random f mapping from 0..N-1 to 0..N-1
;; (f n) is NOT (random N)
;; because each call (f n) must return the same value
(define f (build-vector N (lambda(i) (random N))))
(define hits (make-vector N))
(define n 0)
(while (zero? [hits n])
(++ count)
(vector+= hits n 1)
(set! n [f n])))
(// count times))
(define (f-anal N)
(Σ (lambda(i) (// (! N) (! (- N i)) (^ N i))) 1 N))
(decimals 5) (define (f-print (maxN 21)) (for ((N (in-range 1 maxN))) (define fc (f-count N)) (define fa (f-anal N)) (printf "%3d %10d %10d %10.2d %%" N fc fa (// (abs (- fa fc)) fc 0.01)))) </lang>
- Output:
(f-print) 1 1 1 0 % 2 1.49908 1.5 0.06 % 3 1.89059 1.88889 0.09 % 4 2.21709 2.21875 0.07 % 5 2.50629 2.5104 0.16 % 6 2.77027 2.77469 0.16 % 7 3.01739 3.01814 0.02 % 8 3.23934 3.24502 0.18 % 9 3.45862 3.45832 0.01 % 10 3.65959 3.66022 0.02 % 11 3.85897 3.85237 0.17 % 12 4.04188 4.03607 0.14 % 13 4.21226 4.21235 0 % 14 4.38021 4.38203 0.04 % 15 4.54158 4.54581 0.09 % 16 4.70633 4.70426 0.04 % 17 4.86109 4.85787 0.07 % 18 4.99903 5.00706 0.16 % 19 5.15873 5.1522 0.13 % 20 5.30243 5.29358 0.17 %
Elixir
<lang elixir>defmodule RC do
def factorial(0), do: 1
def factorial(n), do: Enum.reduce(1..n, 1, &(&1 * &2))
def loop_length(n), do: loop_length(n, MapSet.new)
defp loop_length(n, set) do
r = :rand.uniform(n)
if r in set, do: MapSet.size(set), else: loop_length(n, MapSet.put(set, r))
end
def task(runs) do
IO.puts " N average analytical (error) "
IO.puts "=== ========= ========== ========="
Enum.each(1..20, fn n ->
avg = Enum.reduce(1..runs, 0, fn _,sum -> sum + loop_length(n) end) / runs
analytical = Enum.reduce(1..n, 0, fn i,sum ->
sum + (factorial(n) / :math.pow(n, i) / factorial(n-i))
end)
:io.format "~3w ~9.4f ~9.4f (~6.2f%)~n", [n, avg, analytical, abs(avg/analytical - 1)*100]
end)
end
end
runs = 1_000_000 RC.task(runs)</lang>
- Output:
N average analytical (error) === ========= ========== ========= 1 1.0000 1.0000 ( 0.00%) 2 1.5001 1.5000 ( 0.00%) 3 1.8892 1.8889 ( 0.02%) 4 2.2189 2.2188 ( 0.01%) 5 2.5113 2.5104 ( 0.04%) 6 2.7749 2.7747 ( 0.01%) 7 3.0185 3.0181 ( 0.01%) 8 3.2456 3.2450 ( 0.02%) 9 3.4612 3.4583 ( 0.08%) 10 3.6573 3.6602 ( 0.08%) 11 3.8524 3.8524 ( 0.00%) 12 4.0357 4.0361 ( 0.01%) 13 4.2102 4.2123 ( 0.05%) 14 4.3813 4.3820 ( 0.02%) 15 4.5422 4.5458 ( 0.08%) 16 4.7057 4.7043 ( 0.03%) 17 4.8581 4.8579 ( 0.01%) 18 5.0045 5.0071 ( 0.05%) 19 5.1533 5.1522 ( 0.02%) 20 5.2951 5.2936 ( 0.03%)
F#
But uses the Gamma function instead of factorials.
<lang fsharp>open System
let gamma z =
let lanczosCoefficients = [76.18009172947146;-86.50532032941677;24.01409824083091;-1.231739572450155;0.1208650973866179e-2;-0.5395239384953e-5]
let rec sumCoefficients acc i coefficients =
match coefficients with
| [] -> acc
| h::t -> sumCoefficients (acc + (h/i)) (i+1.0) t
let gamma = 5.0
let x = z - 1.0
Math.Pow(x + gamma + 0.5, x + 0.5) * Math.Exp( -(x + gamma + 0.5) ) * Math.Sqrt( 2.0 * Math.PI ) * sumCoefficients 1.000000000190015 (x + 1.0) lanczosCoefficients
let factorial n = gamma ((float n) + 1.)
let expected n =
seq {for i in 1 .. n do yield (factorial n) / System.Math.Pow((float n), (float i)) / (factorial (n - i)) }
|> Seq.sum
let r = System.Random()
let trial n =
let count = ref 0
let x = ref 1
let bits = ref 0
while (!bits &&& !x) = 0 do
count := !count + 1
bits := !bits ||| !x
x := 1 <<< r.Next(n)
!count
let tested n times = (float (Seq.sum (seq { for i in 1 .. times do yield (trial n) }))) / (float times)
let results = seq {
for n in 1 .. 20 do
let avg = tested n 1000000
let theory = expected n
yield n, avg, theory
}
[<EntryPoint>] let main argv =
printfn " N average analytical (error)"
printfn "------------------------------------"
results
|> Seq.iter (fun (n, avg, theory) ->
printfn "%2i %2.6f %2.6f %+2.3f%%" n avg theory ((avg / theory - 1.) * 100.))
0
</lang>
- Output:
N average analytical (error) ------------------------------------ 1 1.000000 1.000000 +0.000% 2 1.498934 1.500000 -0.071% 3 1.889318 1.888889 +0.023% 4 2.219397 2.218750 +0.029% 5 2.510618 2.510400 +0.009% 6 2.771914 2.774691 -0.100% 7 3.014726 3.018139 -0.113% 8 3.245022 3.245018 +0.000% 9 3.457096 3.458316 -0.035% 10 3.660337 3.660216 +0.003% 11 3.849770 3.852372 -0.068% 12 4.038977 4.036074 +0.072% 13 4.213248 4.212348 +0.021% 14 4.380451 4.382029 -0.036% 15 4.541868 4.545807 -0.087% 16 4.704117 4.704258 -0.003% 17 4.858934 4.857871 +0.022% 18 5.004236 5.007063 -0.056% 19 5.154166 5.152196 +0.038% 20 5.298119 5.293585 +0.086%
Go
<lang go>package main
import (
"fmt" "math" "math/rand"
)
const nmax = 20
func main() {
fmt.Println(" N average analytical (error)")
fmt.Println("=== ========= ============ =========")
for n := 1; n <= nmax; n++ {
a := avg(n)
b := ana(n)
fmt.Printf("%3d %9.4f %12.4f (%6.2f%%)\n",
n, a, b, math.Abs(a-b)/b*100)
}
}
func avg(n int) float64 {
const tests = 1e4
sum := 0
for t := 0; t < tests; t++ {
var v [nmax]bool
for x := 0; !v[x]; x = rand.Intn(n) {
v[x] = true
sum++
}
}
return float64(sum) / tests
}
func ana(n int) float64 {
nn := float64(n)
term := 1.
sum := 1.
for i := nn - 1; i >= 1; i-- {
term *= i / nn
sum += term
}
return sum
}</lang>
- Output:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.5007 1.5000 ( 0.05%) 3 1.8959 1.8889 ( 0.37%) 4 2.2138 2.2188 ( 0.22%) 5 2.5013 2.5104 ( 0.36%) 6 2.7940 2.7747 ( 0.70%) 7 3.0197 3.0181 ( 0.05%) 8 3.2715 3.2450 ( 0.82%) 9 3.4147 3.4583 ( 1.26%) 10 3.6758 3.6602 ( 0.43%) 11 3.8672 3.8524 ( 0.38%) 12 4.0309 4.0361 ( 0.13%) 13 4.2153 4.2123 ( 0.07%) 14 4.3380 4.3820 ( 1.00%) 15 4.5030 4.5458 ( 0.94%) 16 4.7563 4.7043 ( 1.11%) 17 4.8616 4.8579 ( 0.08%) 18 4.9933 5.0071 ( 0.27%) 19 5.1534 5.1522 ( 0.02%) 20 5.3031 5.2936 ( 0.18%)
Haskell
<lang Haskell>import System.Random import qualified Data.Set as S import Text.Printf
findRep :: (Random a, Integral a, RandomGen b) => a -> b -> (a, b) findRep n gen = findRep' (S.singleton 1) 1 gen
where
findRep' seen len gen'
| S.member fx seen = (len, gen)
| otherwise = findRep' (S.insert fx seen) (len + 1) gen
where
(fx, gen) = randomR (1, n) gen'
statistical :: (Integral a, Random b, Integral b, RandomGen c, Fractional d) =>
a -> b -> c -> (d, c)
statistical samples size gen =
let (total, gen') = sar samples gen 0
in ((fromIntegral total) / (fromIntegral samples), gen')
where
sar 0 gen' acc = (acc, gen')
sar samples' gen' acc =
let (len, gen) = findRep size gen'
in sar (samples' - 1) gen (acc + len)
factorial :: (Integral a) => a -> a factorial n = foldl (*) 1 [1..n]
analytical :: (Integral a, Fractional b) => a -> b analytical n = sum [fromIntegral num /
fromIntegral (factorial (n - i)) /
fromIntegral (n ^ i) |
i <- [1..n]]
where num = factorial n
test :: (Integral a, Random b, Integral b, PrintfArg b, RandomGen c) =>
a -> [b] -> c -> IO c
test _ [] gen = return gen test samples (x:xs) gen = do
let (st, gen') = statistical samples x gen
an = analytical x
err = abs (st - an) / st * 100.0
str = printf "%3d %9.4f %12.4f (%6.2f%%)\n"
x (st :: Float) (an :: Float) (err :: Float)
putStr str
test samples xs gen'
main :: IO () main = do
putStrLn " N average analytical (error)"
putStrLn "=== ========= ============ ========="
let samples = 10000 :: Integer
range = [1..20] :: [Integer]
_ <- test samples range $ mkStdGen 0
return ()</lang>
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.4941 1.5000 ( 0.39%) 3 1.8895 1.8889 ( 0.03%) 4 2.2246 2.2188 ( 0.26%) 5 2.5158 2.5104 ( 0.21%) 6 2.7875 2.7747 ( 0.46%) 7 3.0425 3.0181 ( 0.80%) 8 3.2157 3.2450 ( 0.91%) 9 3.4534 3.4583 ( 0.14%) 10 3.6561 3.6602 ( 0.11%) 11 3.8357 3.8524 ( 0.43%) 12 4.0291 4.0361 ( 0.17%) 13 4.1819 4.2123 ( 0.73%) 14 4.3469 4.3820 ( 0.81%) 15 4.4942 4.5458 ( 1.15%) 16 4.7093 4.7043 ( 0.11%) 17 4.8288 4.8579 ( 0.60%) 18 5.0021 5.0071 ( 0.10%) 19 5.1980 5.1522 ( 0.88%) 20 5.2961 5.2936 ( 0.05%)
J
First, let's consider an exact, brute force approach.
Since J array indices start at 0, we'll work with 0..N-1 instead of 1..N, dealing with the difference at the boundaries.
We can implement f as {&LIST where LIST is an arbitrary list of N numbers, each picked independently from the range 0..(N-1). We can incrementally build the described sequence using (, f@{:) - here we extend the sequence by applying f to the last element of the sequence. Since we are only concerned with the sequence up to the point of the first repeat, we can select the unique values giving us (~.@, f@{:). This routine stops changing when we reach the desired length, so we can repeatedly apply it forever. For example: <lang J> (~.@, {&0 0@{:)^:_] 0 0
(~.@, {&0 0@{:)^:_] 1
1 0</lang> Once we have the sequence, we can count how many elements are in it. <lang J> 0 0 ([: # (] ~.@, {:@] { [)^:_) 1 2</lang> Meanwhile, we can also generate all possible values of 1..N by counting out N^N values and breaking out the result as a base N list of digits. <lang J> (#.inv i.@^~)2 0 0 0 1 1 0 1 1</lang> All that's left is to count the lengths of all possible sequences for all possible distinct instances of f and average the results: <lang J> (+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)1 1
(+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)2
1.5
(+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)3
1.88889
(+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)4
2.21875
(+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)5
2.5104
(+/ % #)@,@((#.inv i.@^~) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)6
2.77469</lang> Meanwhile the analytic solution (derived by reading the Ada implementation) looks like this: <lang J> ana=: +/@(!@[ % !@- * ^) 1+i.
ana"0]1 2 3 4 5 6
1 1.5 1.88889 2.21875 2.5104 2.77469</lang> To get our simulation, we can take the exact approach and replace the part that generates all possible values for f with a random mechanism. Since the task does not specify how long to run the simulation, and to make this change easy, we'll use N*1e4 tests. <lang J> sim=: (+/ % #)@,@((]?@$~1e4,]) ([: # (] ~.@, {:@] { [)^:_)"1 0/ i.)
sim"0]1 2 3 4 5 6
1 1.5034 1.8825 2.22447 2.51298 2.76898</lang> The simulation approach is noticeably slower than the analytic approach, while being less accurate.
Finally, we can generate our desired results: <lang J> (;:'N average analytic error'),:,.each(;ana"0 ([;];-|@%[) sim"0)1+i.20 +--+-------+--------+-----------+ |N |average|analytic|error | +--+-------+--------+-----------+ | 1| 1| 1 | 0| | 2| 1.5|1.49955 | 0.0003| | 3|1.88889| 1.8928 | 0.00207059| | 4|2.21875|2.23082 | 0.00544225| | 5| 2.5104|2.52146 | 0.00440567| | 6|2.77469|2.78147 | 0.00244182| | 7|3.01814| 3.0101 | 0.00266346| | 8|3.24502|3.25931 | 0.00440506| | 9|3.45832|3.45314 | 0.00149532| |10|3.66022| 3.6708 | 0.00289172| |11|3.85237|3.84139 | 0.00285049| |12|4.03607|4.03252 |0.000881304| |13|4.21235|4.18358 | 0.00682833| |14|4.38203|4.38791 | 0.00134132| |15|4.54581|4.54443 |0.000302246| |16|4.70426|4.71351 | 0.00196721| |17|4.85787|4.85838 |0.000104089| |18|5.00706|5.00889 |0.000365752| |19| 5.1522|5.14785 |0.000843052| |20|5.29358|5.28587 | 0.00145829| +--+-------+--------+-----------+</lang> Here, error is the difference between the two values divided by the analytic value.
Java
This uses a 0-based index (0, 1, ..., n-1) as opposed to the 1-based index (1, 2, ..., n) specified in the question, because it fits better with the native structure of Java.
<lang java>import java.util.HashSet; import java.util.Random; import java.util.Set;
public class AverageLoopLength {
private static final int N = 100000;
//analytical(n) = sum_(i=1)^n (n!/(n-i)!/n**i)
private static double analytical(int n) {
double[] factorial = new double[n + 1];
double[] powers = new double[n + 1];
powers[0] = 1.0;
factorial[0] = 1.0;
for (int i = 1; i <= n; i++) {
factorial[i] = factorial[i - 1] * i;
powers[i] = powers[i - 1] * n;
}
double sum = 0;
//memoized factorial and powers
for (int i = 1; i <= n; i++) {
sum += factorial[n] / factorial[n - i] / powers[i];
}
return sum;
}
private static double average(int n) {
Random rnd = new Random();
double sum = 0.0;
for (int a = 0; a < N; a++) {
int[] random = new int[n];
for (int i = 0; i < n; i++) {
random[i] = rnd.nextInt(n);
}
Set<Integer> seen = new HashSet<>(n);
int current = 0;
int length = 0;
while (seen.add(current)) {
length++;
current = random[current];
}
sum += length;
}
return sum / N;
}
public static void main(String[] args) {
System.out.println(" N average analytical (error)");
System.out.println("=== ========= ============ =========");
for (int i = 1; i <= 20; i++) {
double avg = average(i);
double ana = analytical(i);
System.out.println(String.format("%3d %9.4f %12.4f (%6.2f%%)", i, avg, ana, ((ana - avg) / ana * 100)));
}
}
}</lang>
Julia
<lang julia># Version 5.2 analytical(n::Integer) = sum(factorial(n) / big(n) ^ i / factorial(n - i) for i = 1:n)
function test(n::Integer, times::Integer = 1000000)
c = 0
for i = range(0, times)
x, bits = 1, 0
while (bits & x) == 0
c += 1
bits |= x
x = 1 << rand(0:(n - 1))
end
end
return c / times
end
function main(n::Integer)
println(" n\tavg\texp.\tdiff\n-------------------------------")
for n in 1:n
avg = test(n)
theory = analytical(n)
diff = (avg / theory - 1) * 100
@printf(STDOUT, "%2d %8.4f %8.4f %6.3f%%\n", n, avg, theory, diff)
end
end
main(20) </lang>
- Output:
n avg exp. diff ------------------------------- 1 1.0000 1.0000 0.000% 2 1.4998 1.5000 -0.015% 3 1.8895 1.8889 0.034% 4 2.2171 2.2188 -0.075% 5 2.5082 2.5104 -0.088% 6 2.7729 2.7747 -0.063% 7 3.0171 3.0181 -0.033% 8 3.2439 3.2450 -0.034% 9 3.4578 3.4583 -0.016% 10 3.6616 3.6602 0.038% 11 3.8525 3.8524 0.004% 12 4.0353 4.0361 -0.020% 13 4.2126 4.2123 0.006% 14 4.3835 4.3820 0.034% 15 4.5428 4.5458 -0.067% 16 4.7027 4.7043 -0.033% 17 4.8560 4.8579 -0.039% 18 5.0054 5.0071 -0.033% 19 5.1492 5.1522 -0.058% 20 5.2896 5.2936 -0.076%
Kotlin
<lang scala>const val NMAX = 20 const val TESTS = 1000000 val rand = java.util.Random()
fun avg(n: Int): Double {
var sum = 0
for (t in 0 until TESTS) {
val v = BooleanArray(NMAX)
var x = 0
while (!v[x]) {
v[x] = true
sum++
x = rand.nextInt(n)
}
}
return sum.toDouble() / TESTS
}
fun ana(n: Int): Double {
val nn = n.toDouble()
var term = 1.0
var sum = 1.0
for (i in n - 1 downTo 1) {
term *= i / nn
sum += term
}
return sum
}
fun main(args: Array<String>) {
println(" N average analytical (error)")
println("=== ========= ============ =========")
for (n in 1..NMAX) {
val a = avg(n)
val b = ana(n)
println(String.format("%3d %6.4f %10.4f (%4.2f%%)", n, a, b, Math.abs(a - b) / b * 100.0))
}
}</lang> Sample output:
- Output:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 (0.00%) 2 1.5004 1.5000 (0.03%) 3 1.8890 1.8889 (0.00%) 4 2.2179 2.2188 (0.04%) 5 2.5108 2.5104 (0.02%) 6 2.7738 2.7747 (0.03%) 7 3.0178 3.0181 (0.01%) 8 3.2482 3.2450 (0.10%) 9 3.4572 3.4583 (0.03%) 10 3.6608 3.6602 (0.02%) 11 3.8545 3.8524 (0.06%) 12 4.0378 4.0361 (0.04%) 13 4.2131 4.2123 (0.02%) 14 4.3795 4.3820 (0.06%) 15 4.5481 4.5458 (0.05%) 16 4.7044 4.7043 (0.00%) 17 4.8610 4.8579 (0.06%) 18 5.0027 5.0071 (0.09%) 19 5.1498 5.1522 (0.05%) 20 5.2941 5.2936 (0.01%)
Liberty BASIC
<lang lb> MAXN = 20 TIMES = 10000'00
't0=time$("ms") FOR n = 1 TO MAXN
avg = FNtest(n, TIMES)
theory = FNanalytical(n)
diff = (avg / theory - 1) * 100
PRINT n, avg, theory, using("##.####",diff); "%"
NEXT 't1=time$("ms") 'print t1-t0; " ms" END
function FNanalytical(n)
FOR i = 1 TO n
s = s+ FNfactorial(n) / n^i / FNfactorial(n-i)
NEXT
FNanalytical = s
end function
function FNtest(n, times)
FOR i = 1 TO times
x = 1 : b = 0
WHILE (b AND x) = 0
c = c + 1
b = b OR x
x = 2^int(n*RND(1))
WEND
NEXT
FNtest = c / times
end function
function FNfactorial(n)
IF n=1 OR n=0 THEN FNfactorial=1 ELSE FNfactorial= n * FNfactorial(n-1)
end function </lang>
- Output:
1 1 1 0.0000% 2 1.4759 1.5 -1.6067% 3 1.8868 1.88888889 -0.1106% 4 2.2139 2.21875 -0.2186% 5 2.4784 2.5104 -1.2747% 6 2.7888 2.77469136 0.5085% 7 2.9846 3.0181387 -1.1112% 8 3.2645 3.24501801 0.6004% 9 3.464 3.45831574 0.1644% 10 3.6602 3.66021568 -0.0004% 11 3.8255 3.85237205 -0.6975% 12 4.019 4.03607368 -0.4230% 13 4.2033 4.21234791 -0.2148% 14 4.3985 4.38202942 0.3759% 15 4.5868 4.54580729 0.9018% 16 4.6705 4.70425825 -0.7176% 17 4.8807 4.85787082 0.4699% 18 4.9759 5.0070631 -0.6224% 19 5.1755 5.1521962 0.4523% 20 5.2792 5.29358459 -0.2717%
Mathematica / Wolfram Language
<lang mathematica>Grid@Prepend[
Table[{n, #1, #2,
Row[{Round[10000 Abs[#1 - #2]/#2]/100., "%"}]} &@
N[{Mean[Array[
Length@NestWhileList[#, 1, UnsameQ[##] &, All] - 1 &[# /.
MapIndexed[#21 -> #1 &,
RandomInteger[{1, n}, n]] &] &, 10000]],
Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]/n^(n - 1)}, 5], {n, 1,
20}], {"N", "average", "analytical", "error"}]</lang>
- Output:
N average analytical error 1 1.0000 1.0000 0.% 2 1.5017 1.5000 0.11% 3 1.8910 1.8889 0.11% 4 2.2334 2.2188 0.66% 5 2.5090 2.5104 0.06% 6 2.8092 2.7747 1.24% 7 3.0468 3.0181 0.95% 8 3.2253 3.2450 0.61% 9 3.4695 3.4583 0.32% 10 3.6661 3.6602 0.16% 11 3.8662 3.8524 0.36% 12 4.0393 4.0361 0.08% 13 4.2232 4.2123 0.26% 14 4.3496 4.3820 0.74% 15 4.5706 4.5458 0.55% 16 4.6963 4.7043 0.17% 17 4.8548 4.8579 0.06% 18 5.0671 5.0071 1.2% 19 5.1702 5.1522 0.35% 20 5.2264 5.2936 1.27%
Nim
<lang nim>import random, math, strfmt randomize()
const
maxN = 20 times = 1_000_000
proc factorial(n: int): float =
result = 1 for i in 1 .. n: result *= i.float
proc expected(n: int): float =
for i in 1 .. n: result += factorial(n) / pow(n.float, i.float) / factorial(n - i)
proc test(n, times: int): int =
for i in 1 .. times:
var
x = 1
bits = 0
while (bits and x) == 0:
inc result
bits = bits or x
x = 1 shl random(n)
echo " n\tavg\texp.\tdiff" echo "-------------------------------" for n in 1 .. maxN:
let cnt = test(n, times)
let avg = cnt.float / times
let theory = expected(n)
let diff = (avg / theory - 1) * 100
printlnfmt "{:2} {:8.4f} {:8.4f} {:6.3f}%", n, avg, theory, diff</lang>
- Output:
n avg exp. diff ------------------------------- 1 1.0000 1.0000 0% 2 1.5001 1.5000 0.008% 3 1.8884 1.8889 -0.025% 4 2.2187 2.2187 -0.000% 5 2.5098 2.5104 -0.025% 6 2.7752 2.7747 0.017% 7 3.0175 3.0181 -0.020% 8 3.2411 3.2450 -0.120% 9 3.4565 3.4583 -0.054% 10 3.6599 3.6602 -0.010% 11 3.8555 3.8524 0.081% 12 4.0381 4.0361 0.051% 13 4.2124 4.2123 0.000% 14 4.3813 4.3820 -0.017% 15 4.5471 4.5458 0.027% 16 4.7009 4.7043 -0.072% 17 4.8589 4.8579 0.021% 18 5.0054 5.0071 -0.034% 19 5.1554 5.1522 0.061% 20 5.2915 5.2936 -0.040%
Oberon-2
<lang oberon2> MODULE AvgLoopLen; (* Oxford Oberon-2 *) IMPORT Random, Out;
PROCEDURE Fac(n: INTEGER; f: REAL): REAL; BEGIN IF n = 0 THEN RETURN f ELSE RETURN Fac(n - 1,n*f) END END Fac;
PROCEDURE Power(n,i: INTEGER): REAL; VAR p: REAL; BEGIN p := 1.0; WHILE i > 0 DO p := p * n; DEC(i) END; RETURN p END Power;
PROCEDURE Abs(x: REAL): REAL; BEGIN IF x < 0 THEN RETURN -x ELSE RETURN x END END Abs;
PROCEDURE Analytical(n: INTEGER): REAL; VAR i: INTEGER; res: REAL; BEGIN res := 0.0; FOR i := 1 TO n DO res := res + (Fac(n,1.0) / Power(n,i) / Fac(n - i,1.0)); END; RETURN res END Analytical;
PROCEDURE Averages(n: INTEGER): REAL; CONST times = 100000; VAR rnds: SET; r,count,i: INTEGER; BEGIN count := 0; i := 0; WHILE i < times DO rnds := {}; LOOP r := Random.Roll(n); IF r IN rnds THEN EXIT ELSE INCL(rnds,r); INC(count) END END; INC(i) END;
RETURN count / times END Averages;
VAR i: INTEGER; av,an,df: REAL; BEGIN Random.Randomize; Out.String(" Averages Analytical Diff% ");Out.Ln; FOR i := 1 TO 20 DO Out.Int(i,3); Out.String(": "); av := Averages(i);an := Analytical(i);df := Abs(av - an) / an * 100.0; Out.Fixed(av,10,4);Out.Fixed(an,11,4);Out.Fixed(df,10,4);Out.Ln END END AvgLoopLen. </lang>
- Output:
Averages Analytical Diff% 1: 1.0000 1.0000 0.0000 2: 1.5015 1.5000 0.0993 3: 1.8868 1.8889 0.1085 4: 2.2187 2.2188 0.0005 5: 2.5119 2.5104 0.0578 6: 2.7785 2.7747 0.1366 7: 3.0184 3.0181 0.0090 8: 3.2435 3.2450 0.0471 9: 3.4585 3.4583 0.0056 10: 3.6549 3.6602 0.1463 11: 3.8559 3.8524 0.0918 12: 4.0452 4.0361 0.2264 13: 4.2097 4.2123 0.0628 14: 4.3740 4.3820 0.1830 15: 4.5583 4.5458 0.2739 16: 4.7001 4.7043 0.0882 17: 4.8654 4.8579 0.1556 18: 5.0157 5.0071 0.1731 19: 5.1515 5.1522 0.0135 20: 5.2930 5.2936 0.0105
PARI/GP
<lang parigp>expected(n)=sum(i=1,n,n!/(n-i)!/n^i,0.); test(n, times)={
my(ct); for(i=1,times, my(x=1,bits); while(!bitand(bits,x),ct++; bits=bitor(bits,x); x = 1<<random(n)) ); ct
}; TIMES=1000000; {for(n=1,20,
my(cnt=test(n, TIMES),avg=cnt/TIMES,ex=expected(n),diff=(avg/ex-1)*100.); print(n"\t"avg*1."\t"ex*1."\t"diff);
)}</lang>
- Output:
1 1.0000 1.0000 0.E-7 2 1.4998 1.5000 -0.012933 3 1.8891 1.8889 0.013559 4 2.2198 2.2188 0.047369 5 2.5095 2.5104 -0.034616 6 2.7744 2.7747 -0.010248 7 3.0177 3.0181 -0.012945 8 3.2467 3.2450 0.050600 9 3.4611 3.4583 0.080278 10 3.6595 3.6602 -0.018651 11 3.8541 3.8524 0.044880 12 4.0428 4.0361 0.16690 13 4.2116 4.2123 -0.017921 14 4.3825 4.3820 0.011150 15 4.5467 4.5458 0.020562 16 4.7087 4.7043 0.095058 17 4.8573 4.8579 -0.011997 18 5.0080 5.0071 0.018312 19 5.1530 5.1522 0.015970 20 5.2970 5.2936 0.065143
Perl
<lang perl>use List::Util qw(sum reduce);
sub find_loop {
my($n) = @_;
my($r,@seen);
while () { $seen[$r] = $seen[($r = int(1+rand $n))] ? return sum @seen : 1 }
}
print " N empiric theoric (error)\n"; print "=== ========= ============ =========\n";
my $MAX = 20; my $TRIALS = 1000;
for my $n (1 .. $MAX) {
my $empiric = ( sum map { find_loop($n) } 1..$TRIALS ) / $TRIALS;
my $theoric = sum map { (reduce { $a*$b } $_**2, ($n-$_+1)..$n ) / $n ** ($_+1) } 1..$n;
printf "%3d %9.4f %12.4f (%5.2f%%)\n",
$n, $empiric, $theoric, 100 * ($empiric - $theoric) / $theoric;
}</lang>
- Output:
N empiric theoric (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.4950 1.5000 (-0.33%) 3 1.9190 1.8889 ( 1.59%) 4 2.2400 2.2188 ( 0.96%) 5 2.5120 2.5104 ( 0.06%) 6 2.7500 2.7747 (-0.89%) 7 3.0360 3.0181 ( 0.59%) 8 3.2600 3.2450 ( 0.46%) 9 3.4440 3.4583 (-0.41%) 10 3.6670 3.6602 ( 0.19%) 11 3.8340 3.8524 (-0.48%) 12 4.0450 4.0361 ( 0.22%) 13 4.2160 4.2123 ( 0.09%) 14 4.4420 4.3820 ( 1.37%) 15 4.5600 4.5458 ( 0.31%) 16 4.7940 4.7043 ( 1.91%) 17 4.7830 4.8579 (-1.54%) 18 4.9140 5.0071 (-1.86%) 19 5.2490 5.1522 ( 1.88%) 20 5.2930 5.2936 (-0.01%)
Perl 6
<lang perl6>constant MAX_N = 20; constant TRIALS = 100;
for 1 .. MAX_N -> $N {
my $empiric = TRIALS R/ [+] find-loop(random-mapping($N)).elems xx TRIALS;
my $theoric = [+]
map -> $k { $N ** ($k + 1) R/ [*] flat $k**2, $N - $k + 1 .. $N }, 1 .. $N;
FIRST say " N empiric theoric (error)";
FIRST say "=== ========= ============ =========";
printf "%3d %9.4f %12.4f (%4.2f%%)\n",
$N, $empiric,
$theoric, 100 * abs($theoric - $empiric) / $theoric;
}
sub random-mapping { hash .list Z=> .roll given ^$^size } sub find-loop { 0, | %^mapping{*} ...^ { (%){$_}++ } }</lang>
- Example:
N empiric theoric (error) === ========= ============ ========= 1 1.0000 1.0000 (0.00%) 2 1.5600 1.5000 (4.00%) 3 1.7800 1.8889 (5.76%) 4 2.1800 2.2188 (1.75%) 5 2.6200 2.5104 (4.37%) 6 2.8300 2.7747 (1.99%) 7 3.1200 3.0181 (3.37%) 8 3.1400 3.2450 (3.24%) 9 3.4500 3.4583 (0.24%) 10 3.6700 3.6602 (0.27%) 11 3.8300 3.8524 (0.58%) 12 4.3600 4.0361 (8.03%) 13 3.9000 4.2123 (7.42%) 14 4.4900 4.3820 (2.46%) 15 4.9500 4.5458 (8.89%) 16 4.9800 4.7043 (5.86%) 17 4.9100 4.8579 (1.07%) 18 4.9700 5.0071 (0.74%) 19 5.1000 5.1522 (1.01%) 20 5.2300 5.2936 (1.20%)
Phix
<lang Phix>constant MAX = 20,
ITER = 1000000
function expected(integer n) atom sum = 0
for i=1 to n do
sum += factorial(n) / power(n,i) / factorial(n-i)
end for
return sum
end function
function test(integer n) integer count = 0, x, bits
for i=1 to ITER do
x = 1
bits = 0
while not and_bits(bits,x) do
count += 1
bits = or_bits(bits,x)
x = power(2,rand(n)-1)
end while
end for
return count/ITER
end function
atom av, ex
puts(1," n avg. exp. (error%)\n");
puts(1,"== ====== ====== ========\n");
for n=1 to MAX do
av = test(n)
ex = expected(n)
printf(1,"%2d %8.4f %8.4f (%5.3f%%)\n", {n,av,ex,abs(1-av/ex)*100})
end for</lang>
- Output:
n avg. exp. (error%) == ====== ====== ======== 1 1.0000 1.0000 (0.000%) 2 1.5003 1.5000 (0.018%) 3 1.8880 1.8889 (0.046%) 4 2.2176 2.2188 (0.052%) 5 2.5104 2.5104 (0.001%) 6 2.7734 2.7747 (0.046%) 7 3.0198 3.0181 (0.055%) 8 3.2464 3.2450 (0.042%) 9 3.4562 3.4583 (0.062%) 10 3.6618 3.6602 (0.043%) 11 3.8511 3.8524 (0.033%) 12 4.0357 4.0361 (0.009%) 13 4.2158 4.2123 (0.083%) 14 4.3843 4.3820 (0.052%) 15 4.5410 4.5458 (0.105%) 16 4.7084 4.7043 (0.087%) 17 4.8603 4.8579 (0.049%) 18 5.0044 5.0071 (0.052%) 19 5.1516 5.1522 (0.011%) 20 5.2955 5.2936 (0.037%)
PicoLisp
<lang PicoLisp>(scl 4) (seed (in "/dev/urandom" (rd 8)))
(de fact (N)
(if (=0 N) 1 (apply * (range 1 N))) )
(de analytical (N)
(sum
'((I)
(/
(* (fact N) 1.0)
(** N I)
(fact (- N I)) ) )
(range 1 N) ) )
(de testing (N)
(let (C 0 N (dec N) X 0 B 0 I 1000000)
(do I
(zero B)
(one X)
(while (=0 (& B X))
(inc 'C)
(setq
B (| B X)
X (** 2 (rand 0 N)) ) ) )
(*/ C 1.0 I) ) )
(let F (2 8 8 6)
(tab F "N" "Avg" "Exp" "Diff")
(for I 20
(let (A (testing I) B (analytical I))
(tab F
I
(round A 4)
(round B 4)
(round
(*
(abs (- (*/ A 1.0 B) 1.0))
100 )
2 ) ) ) ) )
(bye)</lang>
PowerShell
<lang PowerShell> function Get-AnalyticalLoopAverage ( [int]$N )
{
# Expected loop average = sum from i = 1 to N of N! / (N-i)! / N^(N-i+1)
# Equivalently, Expected loop average = sum from i = 1 to N of F(i)
# where F(N) = 1, and F(i) = F(i+1)*i/N
$LoopAverage = $Fi = 1
If ( $N -eq 1 ) { return $LoopAverage }
ForEach ( $i in ($N-1)..1 )
{
$Fi *= $i / $N
$LoopAverage += $Fi
}
return $LoopAverage
}
function Get-ExperimentalLoopAverage ( [int]$N, [int]$Tests = 100000 )
{
If ( $N -eq 1 ) { return 1 }
# Using 0 through N-1 instead of 1 through N for speed and simplicity
$NMO = $N - 1
# Create array to hold mapping function
$F = New-Object int[] ( $N )
$Count = 0
$Random = New-Object System.Random
ForEach ( $Test in 1..$Tests )
{
# Map each number to a random number
ForEach ( $i in 0..$NMO )
{
$F[$i] = $Random.Next( $N )
}
# For each number...
ForEach ( $i in 0..$NMO )
{
# Add the number to the list
$List = @()
$Count++
$List += $X = $i
# If loop does not yet exist in list...
While ( $F[$X] -notin $List )
{
# Go to the next mapped number and add it to the list
$Count++
$List += $X = $F[$X]
}
}
}
$LoopAvereage = $Count / $N / $Tests
return $LoopAvereage
}
</lang> Note: The use of the [pscustomobject] type accelerator to simplify making the test result table look pretty requires PowerShell 3.0. <lang PowerShell>
- Display results for N = 1 through 20
ForEach ( $N in 1..20 )
{
$AnalyticalAverage = Get-AnalyticalLoopAverage $N
$ExperimentalAverage = Get-ExperimentalLoopAverage $N
[pscustomobject] @{
N = $N.ToString().PadLeft( 2, ' ' )
Analytical = $AnalyticalAverage.ToString( '0.00000000' )
Experimental = $ExperimentalAverage.ToString( '0.00000000' )
'Error (%)' = ( [math]::Abs( $AnalyticalAverage - $ExperimentalAverage ) / $AnalyticalAverage * 100 ).ToString( '0.00000000' )
}
}
</lang>
- Output:
N Analytical Experimental Error (%) - ---------- ------------ --------- 1 1.00000000 1.00000000 0.00000000 2 1.50000000 1.49985500 0.00966667 3 1.88888889 1.88713000 0.09311765 4 2.21875000 2.22103500 0.10298592 5 2.51040000 2.51069200 0.01163161 6 2.77469136 2.77264833 0.07363070 7 3.01813870 3.01547143 0.08837474 8 3.24501801 3.25003875 0.15472163 9 3.45831574 3.45067667 0.22089013 10 3.66021568 3.65659000 0.09905646 11 3.85237205 3.85669273 0.11215626 12 4.03607368 4.03813500 0.05107253 13 4.21234791 4.20946231 0.06850349 14 4.38202942 4.38458786 0.05838465 15 4.54580729 4.54466400 0.02515032 16 4.70425825 4.70146375 0.05940356 17 4.85787082 4.86807647 0.21008483 18 5.00706310 5.01939278 0.24624572 19 5.15219620 5.15179263 0.00783296 20 5.29358459 5.29214950 0.02710991
Python
<lang python>from __future__ import division # Only necessary for Python 2.X from math import factorial from random import randrange
MAX_N = 20 TIMES = 1000000
def analytical(n): return sum(factorial(n) / pow(n, i) / factorial(n -i) for i in range(1, n+1))
def test(n, times):
count = 0
for i in range(times):
x, bits = 1, 0
while not (bits & x):
count += 1
bits |= x
x = 1 << randrange(n)
return count / times
if __name__ == '__main__':
print(" n\tavg\texp.\tdiff\n-------------------------------")
for n in range(1, MAX_N+1):
avg = test(n, TIMES)
theory = analytical(n)
diff = (avg / theory - 1) * 100
print("%2d %8.4f %8.4f %6.3f%%" % (n, avg, theory, diff))</lang>
- Output:
n avg exp. diff ------------------------------- 1 1.0000 1.0000 0.000% 2 1.5006 1.5000 0.037% 3 1.8887 1.8889 -0.012% 4 2.2190 2.2188 0.011% 5 2.5101 2.5104 -0.012% 6 2.7750 2.7747 0.012% 7 3.0158 3.0181 -0.076% 8 3.2447 3.2450 -0.009% 9 3.4586 3.4583 0.009% 10 3.6598 3.6602 -0.010% 11 3.8510 3.8524 -0.036% 12 4.0368 4.0361 0.017% 13 4.2099 4.2123 -0.058% 14 4.3784 4.3820 -0.083% 15 4.5484 4.5458 0.058% 16 4.7045 4.7043 0.006% 17 4.8611 4.8579 0.067% 18 5.0074 5.0071 0.007% 19 5.1534 5.1522 0.024% 20 5.2927 5.2936 -0.017%
Racket
<lang racket>
- lang racket
(require (only-in math factorial))
(define (analytical n)
(for/sum ([i (in-range 1 (add1 n))]) (/ (factorial n) (expt n i) (factorial (- n i)))))
(define (test n times)
(define (count-times seen times)
(define x (random n))
(if (memq x seen) times (count-times (cons x seen) (add1 times))))
(/ (for/fold ([count 0]) ([i times]) (count-times '() count))
times))
(define (test-table max-n times)
(displayln " n avg theory error\n------------------------")
(for ([i (in-range 1 (add1 max-n))])
(define average (test i times))
(define theory (analytical i))
(define difference (* (abs (sub1 (/ average theory))) 100))
(displayln (~a (~a i #:width 2 #:align 'right)
" " (real->decimal-string average 4)
" " (real->decimal-string theory 4)
" " (real->decimal-string difference 4)
"%"))))
(test-table 20 10000) </lang>
- Output:
n avg theory error ------------------------ 1 1.0000 1.0000 0.0000% 2 1.5082 1.5000 0.5467% 3 1.8966 1.8889 0.4082% 4 2.2251 2.2188 0.2862% 5 2.5138 2.5104 0.1354% 6 2.7582 2.7747 0.5943% 7 3.0253 3.0181 0.2373% 8 3.2293 3.2450 0.4844% 9 3.4602 3.4583 0.0545% 10 3.6831 3.6602 0.6252% 11 3.8459 3.8524 0.1680% 12 4.0348 4.0361 0.0316% 13 4.1896 4.2123 0.5400% 14 4.3555 4.3820 0.6054% 15 4.5678 4.5458 0.4838% 16 4.6950 4.7043 0.1968% 17 4.8524 4.8579 0.1126% 18 5.0224 5.0071 0.3063% 19 5.1017 5.1522 0.9801% 20 5.3316 5.2936 0.7181%
REXX
This REXX program automatically adjusts the precision (decimal digits) to be used based on the size of the
factorial (product) for RUNS.
Also note that the ! (factorial function) uses memoization for optimization. <lang rexx>/*REXX program computes the average loop length mapping a random field 1···N ───► 1···N */ parse arg runs tests seed . /*obtain optional arguments from the CL*/ if runs == | runs =="," then runs = 40 /*Not specified? Then use the default.*/ if tests == | tests =="," then tests= 1000000 /* " " " " " " */ if datatype(seed, 'W') then call random ,, seed /*Is integer? For RAND repeatability.*/ !.=0; !.0=1 /*used for factorial (!) memoization.*/ numeric digits 100000 /*be able to calculate 25k! if need be.*/ numeric digits max(9, length( !(runs) ) ) /*set the NUMERIC DIGITS for !(runs). */ say right( runs, 24) 'runs' /*display number of runs we're using.*/ say right( tests, 24) 'tests' /* " " " tests " " */ say right( digits(), 24) 'digits' /* " " " digits " " */ say say " N average exact % error " /* ◄─── title, header ►────────┐ */ hdr=" ═══ ═════════ ═════════ ═════════"; pad=left(,3) /* ◄────────┘ */ say hdr
do #=1 for runs; av=fmtD( exact(#) ) /*use four digits past decimal point. */
xa=fmtD( exper(#) ) /* " " " " " " */
say right(#,9) pad xa pad av pad fmtD( abs(xa-av) * 100 / av) /*show values.*/
end /*#*/
say hdr /*display the final header (some bars).*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure expose !.; parse arg z; if !.z\==0 then return !.z
!=1; do j=2 for z -1; !=!*j; !.j=!; end; /*compute factorial*/ return !
/*──────────────────────────────────────────────────────────────────────────────────────*/ exact: parse arg x; s=0; do j=1 for x; s=s + !(x) / !(x-j) / x**j; end; return s /*──────────────────────────────────────────────────────────────────────────────────────*/ exper: parse arg n; k=0; do tests; $.=0 /*do it TESTS times.*/
do n; r=random(1, n); if $.r then leave
$.r=1; k=k + 1 /*bump the counter. */
end /*n*/
end /*tests*/
return k/tests
/*──────────────────────────────────────────────────────────────────────────────────────*/ fmtD: parse arg y,d; d=word(d 4, 1); y=format(y, , d); parse var y w '.' f
if f=0 then return w || left(, d +1); return y</lang>
- output when using the default inputs:
40 runs
1000000 tests
48 digits
N average exact % error
═══ ═════════ ═════════ ═════════
1 1 1 0
2 1.4964 1.5000 0.2400
3 1.8876 1.8889 0.0688
4 2.2222 2.2188 0.1532
5 2.5104 2.5104 0
6 2.7758 2.7747 0.0396
7 3.0194 3.0181 0.0431
8 3.2608 3.2450 0.4869
9 3.4565 3.4583 0.0520
10 3.6583 3.6602 0.0519
11 3.8513 3.8524 0.0286
12 4.0401 4.0361 0.0991
13 4.2133 4.2123 0.0237
14 4.3835 4.3820 0.0342
15 4.5445 4.5458 0.0286
16 4.6672 4.7043 0.7886
17 4.8575 4.8579 0.0082
18 5.0105 5.0071 0.0679
19 5.1517 5.1522 0.0097
20 5.2903 5.2936 0.0623
21 5.4328 5.4315 0.0239
22 5.5674 5.5662 0.0216
23 5.6990 5.6979 0.0193
24 5.8353 5.8268 0.1459
25 5.9536 5.9530 0.0101
26 6.0801 6.0767 0.0560
27 6.1997 6.1981 0.0258
28 6.3197 6.3173 0.0380
29 6.4328 6.4344 0.0249
30 6.5485 6.5495 0.0153
31 6.6615 6.6627 0.0180
32 6.7102 6.7740 0.9418
33 6.8826 6.8837 0.0160
34 6.9878 6.9917 0.0558
35 7.0996 7.0982 0.0197
36 7.2054 7.2032 0.0305
37 7.3073 7.3067 0.0082
38 7.4089 7.4088 0.0013
39 7.5052 7.5096 0.0586
40 7.6151 7.6091 0.0789
═══ ═════════ ═════════ ═════════
Ruby
Ruby does not have a factorial method, not even in it's math library. <lang ruby>class Integer
def factorial self == 0 ? 1 : (1..self).inject(:*) end
end
def rand_until_rep(n)
rands = {}
loop do
r = rand(1..n)
return rands.size if rands[r]
rands[r] = true
end
end
runs = 1_000_000
puts " N average exp. diff ",
"=== ======== ======== ==========="
(1..20).each do |n|
sum_of_runs = runs.times.inject(0){|sum, _| sum += rand_until_rep(n)}
avg = sum_of_runs / runs.to_f
analytical = (1..n).inject(0){|sum, i| sum += (n.factorial / (n**i).to_f / (n-i).factorial)}
puts "%3d %8.4f %8.4f (%8.4f%%)" % [n, avg, analytical, (avg/analytical - 1)*100]
end</lang>
- Output:
N average exp. diff === ======== ======== =========== 1 1.0000 1.0000 ( 0.0000%) 2 1.4999 1.5000 ( -0.0054%) 3 1.8886 1.8889 ( -0.0158%) 4 2.2181 2.2188 ( -0.0293%) 5 2.5107 2.5104 ( 0.0110%) 6 2.7717 2.7747 ( -0.1074%) 7 3.0167 3.0181 ( -0.0484%) 8 3.2442 3.2450 ( -0.0257%) 9 3.4597 3.4583 ( 0.0394%) 10 3.6572 3.6602 ( -0.0821%) 11 3.8502 3.8524 ( -0.0562%) 12 4.0357 4.0361 ( -0.0084%) 13 4.2139 4.2123 ( 0.0360%) 14 4.3805 4.3820 ( -0.0360%) 15 4.5481 4.5458 ( 0.0505%) 16 4.7030 4.7043 ( -0.0265%) 17 4.8582 4.8579 ( 0.0075%) 18 5.0078 5.0071 ( 0.0151%) 19 5.1568 5.1522 ( 0.0893%) 20 5.2885 5.2936 ( -0.0961%)
Rust
<lang rust>extern crate rand;
use rand::{ThreadRng, thread_rng}; use rand::distributions::{IndependentSample, Range}; use std::collections::HashSet; use std::env; use std::process;
fn help() {
println!("usage: average_loop_length <max_N> <trials>");
}
fn main() {
let args: Vec<String> = env::args().collect(); let mut max_n: u32 = 20; let mut trials: u32 = 1000;
match args.len() {
1 => {}
3 => {
max_n = args[1].parse::<u32>().unwrap();
trials = args[2].parse::<u32>().unwrap();
}
_ => {
help();
process::exit(0);
}
}
let mut rng = thread_rng();
println!(" N average analytical (error)");
println!("=== ========= ============ =========");
for n in 1..(max_n + 1) {
let the_analytical = analytical(n);
let the_empirical = empirical(n, trials, &mut rng);
println!(" {:>2} {:3.4} {:3.4} ( {:>+1.2}%)",
n,
the_empirical,
the_analytical,
100f64 * (the_empirical / the_analytical - 1f64));
}
}
fn factorial(n: u32) -> f64 {
(1..n + 1).fold(1f64, |p, n| p * n as f64)
}
fn analytical(n: u32) -> f64 {
let sum: f64 = (1..(n + 1))
.map(|i| factorial(n) / (n as f64).powi(i as i32) / factorial(n - i))
.fold(0f64, |a, v| a + v);
sum
}
fn empirical(n: u32, trials: u32, rng: &mut ThreadRng) -> f64 {
let sum: f64 = (0..trials)
.map(|_t| {
let mut item = 1u32;
let mut seen = HashSet::new();
let range = Range::new(1u32, n + 1);
for step in 0..n {
if seen.contains(&item) {
return step as f64;
}
seen.insert(item);
item = range.ind_sample(rng);
}
n as f64
})
.fold(0f64, |a, v| a + v);
sum / trials as f64
}
</lang>
- Output:
Using default arguments:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( +0.00%) 2 1.4992 1.5000 ( -0.05%) 3 1.8881 1.8889 ( -0.04%) 4 2.2177 2.2188 ( -0.05%) 5 2.5107 2.5104 ( +0.01%) 6 2.7752 2.7747 ( +0.02%) 7 3.0172 3.0181 ( -0.03%) 8 3.2452 3.2450 ( +0.01%) 9 3.4628 3.4583 ( +0.13%) 10 3.6606 3.6602 ( +0.01%) 11 3.8515 3.8524 ( -0.02%) 12 4.0348 4.0361 ( -0.03%) 13 4.2105 4.2123 ( -0.04%) 14 4.3835 4.3820 ( +0.03%) 15 4.5477 4.5458 ( +0.04%) 16 4.7042 4.7043 ( -0.00%) 17 4.8580 4.8579 ( +0.00%) 18 5.0076 5.0071 ( +0.01%) 19 5.1554 5.1522 ( +0.06%) 20 5.2911 5.2936 ( -0.05%)
Scala
<lang Scala> import scala.util.Random
object AverageLoopLength extends App {
val factorial: Stream[Double] = 1 #:: factorial.zip(Stream.from(1)).map(n => n._2 * factorial(n._2 - 1))
def expected(n: Int) = (for (i <- 1 to n) yield factorial(n) / Math.pow(n, i) / factorial(n - i)).sum
def trial(n: Int):Double = {
var count = 0
var x = 1
var bits = 0
while ((bits & x) == 0) {
count = count + 1
bits = bits | x
x = 1 << Random.nextInt(n)
}
count
}
def tested(n: Int, times: Int) = (for (i <- 1 to times) yield trial(n)).sum / times
val results = for (n <- 1 to 20;
avg = tested(n, 1000000);
theory = expected(n)
) yield (n, avg, theory, (avg / theory - 1) * 100)
println("n avg exp diff")
println("------------------------------------")
results foreach { n => {
println(f"${n._1}%2d ${n._2}%2.6f ${n._3}%2.6f ${n._4}%2.3f%%")
}
}
} </lang>
- Output:
n avg exp diff ------------------------------------ 1 1.000000 1.000000 0.000% 2 1.499894 1.500000 -0.007% 3 1.887826 1.888889 -0.056% 4 2.217514 2.218750 -0.056% 5 2.510049 2.510400 -0.014% 6 2.773658 2.774691 -0.037% 7 3.016585 3.018139 -0.051% 8 3.246865 3.245018 0.057% 9 3.458683 3.458316 0.011% 10 3.660361 3.660216 0.004% 11 3.852663 3.852372 0.008% 12 4.036970 4.036074 0.022% 13 4.213653 4.212348 0.031% 14 4.385226 4.382029 0.073% 15 4.545667 4.545807 -0.003% 16 4.705559 4.704258 0.028% 17 4.854056 4.857871 -0.079% 18 5.007146 5.007063 0.002% 19 5.148767 5.152196 -0.067% 20 5.292875 5.293585 -0.013%
Scheme
<lang scheme> (import (scheme base)
(scheme write)
(srfi 1 lists)
(only (srfi 13 strings) string-pad-right)
(srfi 27 random-bits))
(define (analytical-function n)
(define (factorial n)
(fold * 1 (iota n 1)))
;
(fold (lambda (i sum)
(+ sum
(/ (factorial n) (expt n i) (factorial (- n i)))))
0
(iota n 1)))
(define (simulation n runs)
(define (single-simulation)
(random-source-randomize! default-random-source)
(let ((vec (make-vector n #f)))
(let loop ((count 0)
(num (random-integer n)))
(if (vector-ref vec num)
count
(begin (vector-set! vec num #t)
(loop (+ 1 count)
(random-integer n)))))))
;;
(let loop ((total 0)
(run runs))
(if (zero? run)
(/ total runs)
(loop (+ total (single-simulation))
(- run 1)))))
(display " N average formula (error) \n") (display "=== ========= ========= =========\n") (for-each
(lambda (n)
(let ((simulation (inexact (simulation n 10000)))
(formula (inexact (analytical-function n))))
(display
(string-append
" "
(string-pad-right (number->string n) 3)
" "
(string-pad-right (number->string simulation) 6)
" "
(string-pad-right (number->string formula) 6)
" ("
(string-pad-right
(number->string (* 100 (/ (- simulation formula) formula)))
5)
"%)"))
(newline)))
(iota 20 1))
</lang>
- Output:
N average formula (error) === ========= ========= ========= 1 1.0 1.0 (0.0 %) 2 1.5018 1.5 (0.120%) 3 1.8863 1.8888 (-0.13%) 4 2.2154 2.2187 (-0.15%) 5 2.5082 2.5104 (-0.08%) 6 2.7613 2.7746 (-0.48%) 7 3.036 3.0181 (0.591%) 8 3.2656 3.2450 (0.634%) 9 3.455 3.4583 (-0.09%) 10 3.682 3.6602 (0.595%) 11 3.8233 3.8523 (-0.75%) 12 4.0409 4.0360 (0.119%) 13 4.2471 4.2123 (0.825%) 14 4.3577 4.3820 (-0.55%) 15 4.5351 4.5458 (-0.23%) 16 4.7181 4.7042 (0.294%) 17 4.8877 4.8578 (0.614%) 18 5.0239 5.0070 (0.336%) 19 5.1216 5.1521 (-0.59%) 20 5.2717 5.2935 (-0.41%)
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "float.s7i";
const integer: TESTS is 1000000;
const func float: factorial (in integer: number) is func
result
var float: factorial is 1.0;
local
var integer: i is 0;
begin
for i range 2 to number do
factorial *:= flt(i);
end for;
end func;
const func float: analytical (in integer: number) is func
result
var float: sum is 0.0;
local
var integer: i is 0;
begin
for i range 1 to number do
sum +:= factorial(number) / factorial(number - i) / flt(number)**i;
end for;
end func;
const func float: experimental (in integer: number) is func
result
var float: experimental is 0.0;
local
var integer: run is 0;
var set of integer: seen is EMPTY_SET;
var integer: current is 1;
var integer: count is 0;
begin
for run range 1 to TESTS do
current := 1;
seen := EMPTY_SET;
while current not in seen do
incr(count);
incl(seen, current);
current := rand(1, number);
end while;
end for;
experimental := flt(count) / flt(TESTS);
end func;
const proc: main is func
local
var integer: number is 0;
var float: analytical is 0.0;
var float: experimental is 0.0;
var float: err is 0.0;
begin
writeln(" N avg calc %diff");
for number range 1 to 20 do
analytical := analytical(number);
experimental := experimental(number);
err := abs(experimental - analytical) / analytical * 100.0;
writeln(number lpad 2 <& experimental digits 4 lpad 7 <&
analytical digits 4 lpad 7 <& err digits 3 lpad 7);
end for;
end func;</lang>
- Output:
N avg calc %diff 1 1.0000 1.0000 0.000 2 1.4999 1.5000 0.005 3 1.8891 1.8889 0.010 4 2.2196 2.2188 0.040 5 2.5073 2.5104 0.122 6 2.7744 2.7747 0.010 7 3.0186 3.0181 0.015 8 3.2463 3.2450 0.040 9 3.4592 3.4583 0.027 10 3.6597 3.6602 0.013 11 3.8549 3.8524 0.066 12 4.0374 4.0361 0.033 13 4.2115 4.2123 0.019 14 4.3835 4.3820 0.033 15 4.5474 4.5458 0.035 16 4.7017 4.7043 0.055 17 4.8558 4.8579 0.043 18 5.0096 5.0071 0.051 19 5.1522 5.1522 0.000 20 5.2907 5.2936 0.054
Tcl
<lang tcl># Generate a list of the numbers increasing from $a to $b proc range {a b} {
for {set result {}} {$a <= $b} {incr a} {lappend result $a}
return $result
}
- Computing the expected value analytically
proc tcl::mathfunc::factorial n {
::tcl::mathop::* {*}[range 2 $n]
} proc Analytical {n} {
set sum 0.0
foreach x [range 1 $n] {
set sum [expr {$sum + factorial($n) / factorial($n-$x) / double($n)**$x}]
} return $sum
}
- Determining an approximation to the value experimentally
proc Experimental {n numTests} {
set count 0
set u0 [lrepeat $n 1]
foreach run [range 1 $numTests] {
set unseen $u0 for {set i 0} {[lindex $unseen $i]} {incr count} { lset unseen $i 0 set i [expr {int(rand()*$n)}] }
}
return [expr {$count / double($numTests)}]
}
- Tabulate the results in exactly the original format
puts " N average analytical (error)" puts "=== ========= ============ =========" foreach n [range 1 20] {
set a [Analytical $n]
set e [Experimental $n 100000]
puts [format "%3d %9.4f %12.4f (%6.2f%%)" $n $e $a [expr {abs($e-$a)/$a*100.0}]]
}</lang>
- Output:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.5003 1.5000 ( 0.02%) 3 1.8881 1.8889 ( 0.04%) 4 2.2228 2.2188 ( 0.18%) 5 2.5109 2.5104 ( 0.02%) 6 2.7804 2.7747 ( 0.20%) 7 3.0223 3.0181 ( 0.14%) 8 3.2456 3.2450 ( 0.02%) 9 3.4598 3.4583 ( 0.04%) 10 3.6590 3.6602 ( 0.03%) 11 3.8527 3.8524 ( 0.01%) 12 4.0390 4.0361 ( 0.07%) 13 4.2156 4.2123 ( 0.08%) 14 4.3821 4.3820 ( 0.00%) 15 4.5527 4.5458 ( 0.15%) 16 4.6952 4.7043 ( 0.19%) 17 4.8530 4.8579 ( 0.10%) 18 4.9912 5.0071 ( 0.32%) 19 5.1578 5.1522 ( 0.11%) 20 5.2992 5.2936 ( 0.11%)
Unicon
<lang unicon>link printf, factors
$define MAX_N 20 $define TIMES 1000000 $define RAND_MAX 2147483647
procedure expected(n)
local sum := 0
every i := 1 to n do
sum +:= factorial(n) / (n ^ i) / factorial(n - i)
return sum
end
procedure test(n, times)
local i, count := 0, x, bits
every i := 0 to times-1 do {
x := 1 bits := 0 while iand(bits, x)=0 do {
count +:= 1
bits := ior(bits, x)
x := ishift(1 , ?n-1)
}
}
return count
end
procedure main(void)
local n, cnt, avg, theory, diff
write(" n\tavg\texp.\tdiff\n", repl("-",29))
every n := 1 to MAX_N do {
cnt := test(n, TIMES)
avg := real(cnt) / TIMES
theory := expected(n)
diff := (avg / theory - 1) * 100
printf("%2d %8.4r %8.4r %6.3r%%\n", n, avg, theory, diff)
}
return 0
end</lang>
- Output:
n avg exp. diff ----------------------------- 1 1.0000 1.0000 0.000% 2 1.5008 1.5000 0.056% 3 1.8879 1.8889 -0.051% 4 2.2208 2.2188 0.091% 5 2.5127 2.5104 0.093% 6 2.7759 2.7747 0.044% 7 3.0175 3.0181 -0.023% 8 3.2425 3.2450 -0.079% 9 3.4571 3.4583 -0.034% 10 3.6613 3.6602 0.029% 11 3.8493 3.8524 -0.081% 12 4.0384 4.0361 0.058% 13 4.2133 4.2123 0.023% 14 4.3804 4.3820 -0.037% 15 4.5475 4.5458 0.038% 16 4.7049 4.7043 0.014% 17 4.8575 4.8579 -0.008% 18 5.0088 5.0071 0.035% 19 5.1533 5.1522 0.021% 20 5.2893 5.2936 -0.081%
zkl
<lang zkl>const N=20;
(" N average analytical (error)").println(); ("=== ========= ============ =========").println(); foreach n in ([1..N]){
a := avg(n);
b := ana(n);
"%3d %9.4f %12.4f (%6.2f%%)".fmt(
n, a, b, ((a-b)/b*100)).println();
}
fcn f(n){ (0).random(n) }
fcn avg(n){
tests := 0d10_000;
sum := 0;
do(tests){
v:=(0).pump(n,List,T(Void,False)).copy();
while(1){
z := f(n);
if(v[z]) break;
v[z] = True; sum += 1;
} } return(sum.toFloat() / tests);
}
fcn fact(n) { (1).reduce(n,fcn(N,n){N*n},1.0) } //-->Float fcn ana(n){
n=n.toFloat();
(1).reduce(n,'wrap(sum,i){ sum+fact(n)/n.pow(i)/fact(n-i) },0.0);
}</lang>
- Output:
N average analytical (error) === ========= ============ ========= 1 1.0000 1.0000 ( 0.00%) 2 1.5053 1.5000 ( 0.35%) 3 1.8899 1.8889 ( 0.05%) 4 2.2384 2.2188 ( 0.89%) 5 2.5090 2.5104 ( -0.06%) 6 2.7824 2.7747 ( 0.28%) 7 3.0449 3.0181 ( 0.89%) 8 3.2430 3.2450 ( -0.06%) 9 3.4744 3.4583 ( 0.47%) 10 3.6693 3.6602 ( 0.25%) 11 3.8833 3.8524 ( 0.80%) 12 4.0225 4.0361 ( -0.34%) 13 4.1899 4.2123 ( -0.53%) 14 4.4135 4.3820 ( 0.72%) 15 4.5807 4.5458 ( 0.77%) 16 4.7304 4.7043 ( 0.56%) 17 4.8437 4.8579 ( -0.29%) 18 4.9838 5.0071 ( -0.46%) 19 5.1767 5.1522 ( 0.48%) 20 5.2723 5.2936 ( -0.40%)