# Percolation/Mean run density

Percolation/Mean run density
You are encouraged to solve this task according to the task description, using any language you may know.

Percolation Simulation
This is a simulation of aspects of mathematical percolation theory.

For other percolation simulations, see Category:Percolation Simulations, or:
1D finite grid simulation
Mean run density
2D finite grid simulations

Site percolation | Bond percolation | Mean cluster density

Let ${\displaystyle v}$ be a vector of ${\displaystyle n}$ values of either 1 or 0 where the probability of any value being 1 is ${\displaystyle p}$; the probability of a value being 0 is therefore ${\displaystyle 1-p}$. Define a run of 1s as being a group of consecutive 1s in the vector bounded either by the limits of the vector or by a 0. Let the number of such runs in a given vector of length ${\displaystyle n}$ be ${\displaystyle R_{n}}$.

For example, the following vector has ${\displaystyle R_{10}=3}$

[1 1 0 0 0 1 0 1 1 1]
^^^       ^   ^^^^^

Percolation theory states that

${\displaystyle K(p)=\lim _{n\to \infty }R_{n}/n=p(1-p)}$

Any calculation of ${\displaystyle R_{n}/n}$ for finite ${\displaystyle n}$ is subject to randomness so should be computed as the average of ${\displaystyle t}$ runs, where ${\displaystyle t\geq 100}$.

For values of ${\displaystyle p}$ of 0.1, 0.3, 0.5, 0.7, and 0.9, show the effect of varying ${\displaystyle n}$ on the accuracy of simulated ${\displaystyle K(p)}$.

• s-Run on Wolfram mathworld.

## C

#include <stdio.h>
#include <stdlib.h>

// just generate 0s and 1s without storing them
double run_test(double p, int len, int runs)
{
int r, x, y, i, cnt = 0, thresh = p * RAND_MAX;

for (r = 0; r < runs; r++)
for (x = 0, i = len; i--; x = y)
cnt += x < (y = rand() < thresh);

return (double)cnt / runs / len;
}

int main(void)
{
double p, p1p, K;
int ip, n;

puts( "running 1000 tests each:\n"
" p\t n\tK\tp(1-p)\t diff\n"
"-----------------------------------------------");
for (ip = 1; ip < 10; ip += 2) {
p = ip / 10., p1p = p * (1 - p);

for (n = 100; n <= 100000; n *= 10) {
K = run_test(p, n, 1000);
printf("%.1f\t%6d\t%.4f\t%.4f\t%+.4f (%+.2f%%)\n",
p, n, K, p1p, K - p1p, (K - p1p) / p1p * 100);
}
putchar('\n');
}

return 0;
}
Output:
running 1000 tests each:
p         n    K       p(1-p)       diff
-----------------------------------------------
0.1        100  0.0900  0.0900  -0.0001 (-0.06%)
0.1       1000  0.0899  0.0900  -0.0001 (-0.11%)
0.1      10000  0.0902  0.0900  +0.0002 (+0.17%)
0.1     100000  0.0900  0.0900  -0.0000 (-0.03%)

0.3        100  0.2110  0.2100  +0.0010 (+0.46%)
0.3       1000  0.2104  0.2100  +0.0004 (+0.19%)
0.3      10000  0.2100  0.2100  -0.0000 (-0.02%)
0.3     100000  0.2100  0.2100  -0.0000 (-0.01%)

0.5        100  0.2516  0.2500  +0.0016 (+0.66%)
0.5       1000  0.2498  0.2500  -0.0002 (-0.10%)
0.5      10000  0.2500  0.2500  +0.0000 (+0.01%)
0.5     100000  0.2500  0.2500  +0.0000 (+0.01%)

0.7        100  0.2162  0.2100  +0.0062 (+2.93%)
0.7       1000  0.2107  0.2100  +0.0007 (+0.33%)
0.7      10000  0.2101  0.2100  +0.0001 (+0.06%)
0.7     100000  0.2100  0.2100  -0.0000 (-0.02%)

0.9        100  0.0982  0.0900  +0.0082 (+9.07%)
0.9       1000  0.0905  0.0900  +0.0005 (+0.57%)
0.9      10000  0.0901  0.0900  +0.0001 (+0.09%)
0.9     100000  0.0900  0.0900  +0.0000 (+0.03%)

## C++

#include <algorithm>
#include <random>
#include <vector>
#include <iostream>
#include <numeric>
#include <iomanip>
using VecIt = std::vector<int>::const_iterator ;

//creates vector of length n, based on probability p for 1
std::vector<int> createVector( int n, double p ) {
std::vector<int> result( n ) ;
std::random_device rd ;
std::mt19937 gen( rd( ) ) ;
std::uniform_real_distribution<> dis( 0 , 1 ) ;
for ( int i = 0 ; i < n ; i++ ) {
double number = dis( gen ) ;
if ( number <= p )
result[ i ] = 1 ;
else
result[ i ] = 0 ;
}
return result ;
}

//find number of 1 runs in the vector
int find_Runs( const std::vector<int> & numberVector ) {
int runs = 0 ;
VecIt found = numberVector.begin( ) ;
while ( ( found = std::find( found , numberVector.end( ) , 1 ) )
!= numberVector.end( ) ) {
runs++ ;
while ( found != numberVector.end( ) && ( *found == 1 ) )
std::advance( found , 1 ) ;
if ( found == numberVector.end( ) )
break ;
}
return runs ;
}

int main( ) {
std::cout << "t = 100\n" ;
std::vector<double> p_values { 0.1 , 0.3 , 0.5 , 0.7 , 0.9 } ;
for ( double p : p_values ) {
std::cout << "p = " << p << " , K(p) = " << p * ( 1 - p ) << std::endl ;
for ( int n = 10 ; n < 100000 ; n *= 10 ) {
std::vector<double> runsFound ;
for ( int i = 0 ; i < 100 ; i++ ) {
std::vector<int> ones_and_zeroes = createVector( n , p ) ;
runsFound.push_back( find_Runs( ones_and_zeroes ) / static_cast<double>( n ) ) ;
}
double average = std::accumulate( runsFound.begin( ) , runsFound.end( ) , 0.0 ) / runsFound.size( ) ;
std::cout << " R(" << std::setw( 6 ) << std::right << n << ", p) = " << average << std::endl ;
}
}
return 0 ;
}
Output:
t = 100
p = 0.1 , K(p) = 0.09
R(    10, p) = 0.088
R(   100, p) = 0.0931
R(  1000, p) = 0.09013
R( 10000, p) = 0.089947
p = 0.3 , K(p) = 0.21
R(    10, p) = 0.225
R(   100, p) = 0.2089
R(  1000, p) = 0.21043
R( 10000, p) = 0.20991
p = 0.5 , K(p) = 0.25
R(    10, p) = 0.271
R(   100, p) = 0.253
R(  1000, p) = 0.25039
R( 10000, p) = 0.250278
p = 0.7 , K(p) = 0.21
R(    10, p) = 0.264
R(   100, p) = 0.2155
R(  1000, p) = 0.20829
R( 10000, p) = 0.209977
p = 0.9 , K(p) = 0.09
R(    10, p) = 0.167
R(   100, p) = 0.0928
R(  1000, p) = 0.09071
R( 10000, p) = 0.090341

## D

Translation of: python
import std.stdio, std.range, std.algorithm, std.random, std.math;

enum n = 100, p = 0.5, t = 500;

double meanRunDensity(in size_t n, in double prob) {
return n.iota.map!(_ => uniform01 < prob)
.array.uniq.sum / double(n);
}

void main() {
foreach (immutable p; iota(0.1, 1.0, 0.2)) {
immutable limit = p * (1 - p);
writeln;
foreach (immutable n2; iota(10, 16, 2)) {
immutable n = 2 ^^ n2;
immutable sim = t.iota.map!(_ => meanRunDensity(n, p))
.sum / t;
writefln("t=%3d, p=%4.2f, n=%5d, p(1-p)=%5.5f, " ~
"sim=%5.5f, delta=%3.1f%%", t, p, n, limit, sim,
limit ? abs(sim - limit) / limit * 100 : sim*100);
}
}
}
Output:
t=500, p=0.10, n= 1024, p(1-p)=0.09000, sim=0.08949, delta=0.6%
t=500, p=0.10, n= 4096, p(1-p)=0.09000, sim=0.08976, delta=0.3%
t=500, p=0.10, n=16384, p(1-p)=0.09000, sim=0.08988, delta=0.1%

t=500, p=0.30, n= 1024, p(1-p)=0.21000, sim=0.20979, delta=0.1%
t=500, p=0.30, n= 4096, p(1-p)=0.21000, sim=0.21020, delta=0.1%
t=500, p=0.30, n=16384, p(1-p)=0.21000, sim=0.21005, delta=0.0%

t=500, p=0.50, n= 1024, p(1-p)=0.25000, sim=0.25016, delta=0.1%
t=500, p=0.50, n= 4096, p(1-p)=0.25000, sim=0.25026, delta=0.1%
t=500, p=0.50, n=16384, p(1-p)=0.25000, sim=0.24990, delta=0.0%

t=500, p=0.70, n= 1024, p(1-p)=0.21000, sim=0.21050, delta=0.2%
t=500, p=0.70, n= 4096, p(1-p)=0.21000, sim=0.20993, delta=0.0%
t=500, p=0.70, n=16384, p(1-p)=0.21000, sim=0.21009, delta=0.0%

t=500, p=0.90, n= 1024, p(1-p)=0.09000, sim=0.09019, delta=0.2%
t=500, p=0.90, n= 4096, p(1-p)=0.09000, sim=0.09047, delta=0.5%
t=500, p=0.90, n=16384, p(1-p)=0.09000, sim=0.09007, delta=0.1%

## EchoLisp

;; count 1-runs - The vector is not stored
(define (runs p n)
(define ct 0)
(define run-1 #t)
(for ([i n])
(if (< (random) p)
(set! run-1 #t) ;; 0 case
(begin ;; 1 case
(when run-1 (set! ct (1+ ct)))
(set! run-1 #f))))
(// ct n))

;; mean of t counts
(define (truns p (n 1000 ) (t 1000))
(// (for/sum ([i t]) (runs p n)) t))

(for ([p (in-range 0.1 1.0 0.2)])
(writeln)
(writeln 'ðŸ”¸ 'p p 'Kp (* p (- 1 p)))
(for ([n '(10 100 1000)])
(printf "\t-- n %5d â†’  %d" n (truns p n)))))

Output:
ðŸ”¸     p     0.1     Kp     0.09
-- n    10 â†’ 0.171
-- n   100 â†’ 0.0974
-- n  1000 â†’ 0.0907
ðŸ”¸     p     0.3     Kp     0.21
-- n    10 â†’ 0.2642
-- n   100 â†’ 0.2161
-- n  1000 â†’ 0.2105
ðŸ”¸     p     0.5     Kp     0.25
-- n    10 â†’ 0.2764
-- n   100 â†’ 0.2519
-- n  1000 â†’ 0.2503
ðŸ”¸     p     0.7     Kp     0.21
-- n    10 â†’ 0.2218
-- n   100 â†’ 0.2106
-- n  1000 â†’ 0.2098
ðŸ”¸     p     0.9     Kp     0.09
-- n    10 â†’ 0.087
-- n   100 â†’ 0.0894
-- n  1000 â†’ 0.0905

## Factor

USING: formatting fry io kernel math math.ranges math.statistics
random sequences ;
IN: rosetta-code.mean-run-density

: rising? ( ? ? -- ? ) [ f = ] [ t = ] bi* and ;

: count-run ( n ? ? -- m ? )
2dup rising? [ [ 1 + ] 2dip ] when nip ;

: runs ( n p -- n )
[ 0 f ] 2dip '[ random-unit _ < count-run ] times drop ;

: rn ( n p -- x ) over [ runs ] dip /f ;

: sim ( n p -- avg )
[ 1000 ] 2dip [ rn ] 2curry replicate mean ;

: theory ( p -- x ) 1 over - * ;

: result ( n p -- )
[ swap ] [ sim ] [ nip theory ] 2tri 2dup - abs
"%.1f  %-5d  %.4f  %.4f  %.4f\n" printf ;

: test ( p -- )
{ 100 1,000 10,000 } [ swap result ] with each nl ;

"1000 tests each:\np n K p(1-p) diff" print ;

: main ( -- ) header .1 .9 .2 <range> [ test ] each ;

MAIN: main
Output:
1000 tests each:
p    n      K       p(1-p)  diff
0.1  100    0.0909  0.0900  0.0009
0.1  1000   0.0902  0.0900  0.0002
0.1  10000  0.0899  0.0900  0.0001

0.3  100    0.2111  0.2100  0.0011
0.3  1000   0.2101  0.2100  0.0001
0.3  10000  0.2100  0.2100  0.0000

0.5  100    0.2524  0.2500  0.0024
0.5  1000   0.2504  0.2500  0.0004
0.5  10000  0.2501  0.2500  0.0001

0.7  100    0.2149  0.2100  0.0049
0.7  1000   0.2106  0.2100  0.0006
0.7  10000  0.2100  0.2100  0.0000

0.9  100    0.0978  0.0900  0.0078
0.9  1000   0.0905  0.0900  0.0005
0.9  10000  0.0901  0.0900  0.0001

## Fortran

! loosely translated from python. We do not need to generate and store the entire vector at once.
! compilation: gfortran -Wall -std=f2008 -o thisfile thisfile.f08

program percolation_mean_run_density
implicit none
integer :: i, p10, n2, n, t
real :: p, limit, sim, delta
data n,p,t/100,0.5,500/
write(6,'(a3,a5,4a7)')'t','p','n','p(1-p)','sim','delta%'
do p10=1,10,2
p = p10/10.0
limit = p*(1-p)
write(6,'()')
do n2=10,15,2
n = 2**n2
sim = 0
do i=1,t
sim = sim + mean_run_density(n,p)
end do
sim = sim/t
if (limit /= 0) then
delta = abs(sim-limit)/limit
else
delta = sim
end if
delta = delta * 100
write(6,'(i3,f5.2,i7,2f7.3,f5.1)')t,p,n,limit,sim,delta
end do
end do

contains

integer function runs(n, p)
integer, intent(in) :: n
real, intent(in) :: p
real :: harvest
logical :: q
integer :: count, i
count = 0
q = .false.
do i=1,n
call random_number(harvest)
if (harvest < p) then
q = .true.
else
if (q) count = count+1
q = .false.
end if
end do
runs = count
end function runs

real function mean_run_density(n, p)
integer, intent(in) :: n
real, intent(in) :: p
mean_run_density = real(runs(n,p))/real(n)
end function mean_run_density

end program percolation_mean_run_density

$./f t p n p(1-p) sim delta% 500 0.10 1024 0.090 0.090 0.2 500 0.10 4096 0.090 0.090 0.2 500 0.10 16384 0.090 0.090 0.0 500 0.30 1024 0.210 0.210 0.2 500 0.30 4096 0.210 0.210 0.0 500 0.30 16384 0.210 0.210 0.0 500 0.50 1024 0.250 0.250 0.1 500 0.50 4096 0.250 0.250 0.1 500 0.50 16384 0.250 0.250 0.1 500 0.70 1024 0.210 0.210 0.1 500 0.70 4096 0.210 0.210 0.1 500 0.70 16384 0.210 0.210 0.0 500 0.90 1024 0.090 0.090 0.1 500 0.90 4096 0.090 0.090 0.4 500 0.90 16384 0.090 0.090 0.1 ## Go package main import ( "fmt" "math/rand" ) var ( pList = []float64{.1, .3, .5, .7, .9} nList = []int{1e2, 1e3, 1e4, 1e5} t = 100 ) func main() { for _, p := range pList { theory := p * (1 - p) fmt.Printf("\np: %.4f theory: %.4f t: %d\n", p, theory, t) fmt.Println(" n sim sim-theory") for _, n := range nList { sum := 0 for i := 0; i < t; i++ { run := false for j := 0; j < n; j++ { one := rand.Float64() < p if one && !run { sum++ } run = one } } K := float64(sum) / float64(t) / float64(n) fmt.Printf("%9d %15.4f %9.6f\n", n, K, K-theory) } } } Output: p: 0.1000 theory: 0.0900 t: 100 n sim sim-theory 100 0.0883 -0.001700 1000 0.0903 0.000300 10000 0.0898 -0.000242 100000 0.0900 -0.000024 p: 0.3000 theory: 0.2100 t: 100 n sim sim-theory 100 0.2080 -0.002000 1000 0.2106 0.000600 10000 0.2097 -0.000341 100000 0.2100 0.000018 p: 0.5000 theory: 0.2500 t: 100 n sim sim-theory 100 0.2512 0.001200 1000 0.2486 -0.001440 10000 0.2500 0.000021 100000 0.2500 -0.000025 p: 0.7000 theory: 0.2100 t: 100 n sim sim-theory 100 0.2108 0.000800 1000 0.2086 -0.001370 10000 0.2102 0.000247 100000 0.2100 -0.000031 p: 0.9000 theory: 0.0900 t: 100 n sim sim-theory 100 0.0970 0.007000 1000 0.0916 0.001580 10000 0.0905 0.000501 100000 0.0900 0.000050 ## Haskell import Control.Monad.Random import Control.Applicative import Text.Printf import Control.Monad import Data.Bits data OneRun = OutRun | InRun deriving (Eq, Show) randomList :: Int -> Double -> Rand StdGen [Int] randomList n p = take n . map f <$> getRandomRs (0,1)
where f n = if (n > p) then 0 else 1

countRuns xs = fromIntegral . sum $zipWith (\x y -> x .&. xor y 1) xs (tail xs ++ [0]) calcK :: Int -> Double -> Rand StdGen Double calcK n p = (/ fromIntegral n) . countRuns <$> randomList n p

printKs :: StdGen -> Double -> IO ()
printKs g p = do
printf "p= %.1f, K(p)= %.3f\n" p (p * (1 - p))
forM_ [1..5] $\n -> do let est = evalRand (calcK (10^n) p) g printf "n=%7d, estimated K(p)= %5.3f\n" (10^n::Int) est main = do x <- newStdGen forM_ [0.1,0.3,0.5,0.7,0.9]$ printKs x

./percolation
p= 0.1, K(p)= 0.090
n=     10, estimated K(p)= 0.000
n=    100, estimated K(p)= 0.130
n=   1000, estimated K(p)= 0.099
n=  10000, estimated K(p)= 0.090
n= 100000, estimated K(p)= 0.091
p= 0.3, K(p)= 0.210
n=     10, estimated K(p)= 0.200
n=    100, estimated K(p)= 0.250
n=   1000, estimated K(p)= 0.209
n=  10000, estimated K(p)= 0.209
n= 100000, estimated K(p)= 0.211
p= 0.5, K(p)= 0.250
n=     10, estimated K(p)= 0.200
n=    100, estimated K(p)= 0.290
n=   1000, estimated K(p)= 0.252
n=  10000, estimated K(p)= 0.250
n= 100000, estimated K(p)= 0.250
p= 0.7, K(p)= 0.210
n=     10, estimated K(p)= 0.300
n=    100, estimated K(p)= 0.200
n=   1000, estimated K(p)= 0.210
n=  10000, estimated K(p)= 0.209
n= 100000, estimated K(p)= 0.210
p= 0.9, K(p)= 0.090
n=     10, estimated K(p)= 0.200
n=    100, estimated K(p)= 0.090
n=   1000, estimated K(p)= 0.089
n=  10000, estimated K(p)= 0.095
n= 100000, estimated K(p)= 0.090

## Icon and Unicon

The following works in both languages:

procedure main(A)
t := integer(A[2]) | 500

write(left("p",8)," ",left("n",8)," ",left("p(1-p)",10)," ",left("SimK(p)",10))
every (p := 0.1 | 0.3 | 0.5 | 0.7 | 0.9, n := 1000 | 2000 | 3000) do {
Ka := 0.0
every !t do {
every (v := "", !n) do v ||:= |((?0.1 > p,"0")|"1")
R := 0
v ? while tab(upto('1')) do R +:= (tab(many('1')), 1)
Ka +:= real(R)/n
}
write(left(p,8)," ",left(n,8)," ",left(p*(1-p),10)," ",left(Ka/t, 10))
}
end

Output:

->pmrd
p        n        p(1-p)     SimK(p)
0.1      1000     0.09000000 0.09021400
0.1      2000     0.09000000 0.08984799
0.1      3000     0.09000000 0.08993666
0.3      1000     0.21       0.21080999
0.3      2000     0.21       0.209953
0.3      3000     0.21       0.210564
0.5      1000     0.25       0.250024
0.5      2000     0.25       0.25007399
0.5      3000     0.25       0.24975266
0.7      1000     0.21       0.21098799
0.7      2000     0.21       0.20987700
0.7      3000     0.21       0.21047333
0.9      1000     0.08999999 0.09016400
0.9      2000     0.08999999 0.09004800
0.9      3000     0.08999999 0.09023200
->

## J

NB. translation of python

NB. 'N P T' =: 100 0.5 500 NB. hypothetical example values, to aid comprehension...

newv =: ([email protected](#&0))~ NB. generate a random binary vector. Use: N newv P
runs =: {: + [: +/ 1 0&E. NB. add the tail to the sum of 1 0 occurrences Use: runs V
mean_run_density =: [ %~ [: runs newv NB. perform experiment. Use: N mean_run_density P

main =: 3 : 0 NB.Usage: main T
T =. y
smoutput' T P N P(1-P) SIM DELTA%'
for_P. 10 %~ >: +: i. 5 do.
LIMIT =. (* -.) P
smoutput ''
for_N. 2 ^ 10 + +: i. 3 do.
SIM =. T %~ +/ (N mean_run_density P"_)^:(<T) 0
smoutput 4 5j2 6 6j3 6j3 4j1 ": T, P, N, LIMIT, SIM, SIM (100 * [`(|@:(- % ]))@.(0 ~: ])) LIMIT
end.
end.
EMPTY
)

Session:

main 500
T  P    N    P(1-P) SIM   DELTA%

500 0.10  1024 0.090 0.090 0.1
500 0.10  4096 0.090 0.090 0.2
500 0.10 16384 0.090 0.090 0.2

500 0.30  1024 0.210 0.210 0.2
500 0.30  4096 0.210 0.209 0.3
500 0.30 16384 0.210 0.210 0.1

500 0.50  1024 0.250 0.250 0.2
500 0.50  4096 0.250 0.250 0.1
500 0.50 16384 0.250 0.250 0.2

500 0.70  1024 0.210 0.210 0.0
500 0.70  4096 0.210 0.210 0.2
500 0.70 16384 0.210 0.210 0.2

500 0.90  1024 0.090 0.091 1.1
500 0.90  4096 0.090 0.090 0.1
500 0.90 16384 0.090 0.090 0.1

## Julia

Works with: Julia version 0.6
Translation of: Python
using Distributions, IterTools

newv(n::Int, p::Float64) = rand(Bernoulli(p), n)
runs(v::Vector{Int}) = sum((a & ~b) for (a, b) in zip(v, IterTools.chain(v[2:end], v[1])))

mrd(n::Int, p::Float64) = runs(newv(n, p)) / n

nrep = 500

for p in 0.1:0.2:1
lim = p * (1 - p)

println()
for ex in 10:2:14
n = 2 ^ ex
sim = mean(mrd.(n, p) for _ in 1:nrep)
@printf("nrep = %3i\tp = %4.2f\tn = %5i\np Â· (1 - p) = %5.3f\tsim = %5.3f\tÎ” = %3.1f%%\n",
nrep, p, n, lim, sim, lim > 0 ? abs(sim - lim) / lim * 100 : sim * 100)
end
end
Output:
nrep = 500	p = 0.10	n =  1024
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.4%
nrep = 500	p = 0.10	n =  4096
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.2%
nrep = 500	p = 0.10	n = 16384
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.0%

nrep = 500	p = 0.30	n =  1024
p Â· (1 - p) = 0.210	sim = 0.211	Î” = 0.5%
nrep = 500	p = 0.30	n =  4096
p Â· (1 - p) = 0.210	sim = 0.210	Î” = 0.1%
nrep = 500	p = 0.30	n = 16384
p Â· (1 - p) = 0.210	sim = 0.210	Î” = 0.0%

nrep = 500	p = 0.50	n =  1024
p Â· (1 - p) = 0.250	sim = 0.250	Î” = 0.0%
nrep = 500	p = 0.50	n =  4096
p Â· (1 - p) = 0.250	sim = 0.250	Î” = 0.1%
nrep = 500	p = 0.50	n = 16384
p Â· (1 - p) = 0.250	sim = 0.250	Î” = 0.0%

nrep = 500	p = 0.70	n =  1024
p Â· (1 - p) = 0.210	sim = 0.209	Î” = 0.3%
nrep = 500	p = 0.70	n =  4096
p Â· (1 - p) = 0.210	sim = 0.210	Î” = 0.1%
nrep = 500	p = 0.70	n = 16384
p Â· (1 - p) = 0.210	sim = 0.210	Î” = 0.0%

nrep = 500	p = 0.90	n =  1024
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.0%
nrep = 500	p = 0.90	n =  4096
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.0%
nrep = 500	p = 0.90	n = 16384
p Â· (1 - p) = 0.090	sim = 0.090	Î” = 0.1%

## Kotlin

Translation of: C
// version 1.2.10

import java.util.Random

val rand = Random()
const val RAND_MAX = 32767

// just generate 0s and 1s without storing them
fun runTest(p: Double, len: Int, runs: Int): Double {
var cnt = 0
val thresh = (p * RAND_MAX).toInt()
for (r in 0 until runs) {
var x = 0
var i = len
while (i-- > 0) {
val y = if (rand.nextInt(RAND_MAX + 1) < thresh) 1 else 0
if (x < y) cnt++
x = y
}
}
return cnt.toDouble() / runs / len
}

fun main(args: Array<String>) {
println("running 1000 tests each:")
println(" p\t n\tK\tp(1-p)\t diff")
println("------------------------------------------------")
val fmt = "%.1f\t%6d\t%.4f\t%.4f\t%+.4f (%+.2f%%)"
for (ip in 1..9 step 2) {
val p = ip / 10.0
val p1p = p * (1.0 - p)
var n = 100
while (n <= 100_000) {
val k = runTest(p, n, 1000)
println(fmt.format(p, n, k, p1p, k - p1p, (k - p1p) / p1p * 100))
n *= 10
}
println()
}
}

Sample output:

running 1000 tests each:
p	   n	K	p(1-p)	     diff
------------------------------------------------
0.1	   100	0.0908	0.0900	+0.0008 (+0.93%)
0.1	  1000	0.0900	0.0900	+0.0000 (+0.02%)
0.1	 10000	0.0899	0.0900	-0.0001 (-0.08%)
0.1	100000	0.0900	0.0900	-0.0000 (-0.05%)

0.3	   100	0.2112	0.2100	+0.0012 (+0.56%)
0.3	  1000	0.2096	0.2100	-0.0004 (-0.21%)
0.3	 10000	0.2101	0.2100	+0.0001 (+0.05%)
0.3	100000	0.2101	0.2100	+0.0001 (+0.03%)

0.5	   100	0.2522	0.2500	+0.0022 (+0.90%)
0.5	  1000	0.2504	0.2500	+0.0004 (+0.15%)
0.5	 10000	0.2500	0.2500	-0.0000 (-0.00%)
0.5	100000	0.2500	0.2500	+0.0000 (+0.00%)

0.7	   100	0.2162	0.2100	+0.0062 (+2.95%)
0.7	  1000	0.2106	0.2100	+0.0006 (+0.29%)
0.7	 10000	0.2101	0.2100	+0.0001 (+0.03%)
0.7	100000	0.2100	0.2100	+0.0000 (+0.01%)

0.9	   100	0.0982	0.0900	+0.0083 (+9.17%)
0.9	  1000	0.0911	0.0900	+0.0011 (+1.17%)
0.9	 10000	0.0902	0.0900	+0.0002 (+0.18%)
0.9	100000	0.0900	0.0900	-0.0000 (-0.02%)

## Mathematica

meanRunDensity[p_, len_, trials_] :=
Mean[Length[Cases[[email protected]#, {1, ___}]] & /@
Unitize[Chop[RandomReal[1, {trials, len}], 1 - p]]]/len

[email protected][
Grid[Join[{{p, n, K, diff}},
Table[{q, n, x = meanRunDensity[q, n, 100] // N,
q (1 - q) - x}, {n, {100, 1000, 10000, 100000}}], {}],
Alignment -> Left], {q, {.1, .3, .5, .7, .9}}]
Output:
p	n	K	diff
0.1	100	0.0905	-0.0005
0.1	1000	0.0900	-0.00001
0.1	10000	0.0902	-0.00015
0.1	100000	0.0901	-0.0001265

p	n	K	diff
0.3	100	0.2088	 0.0012
0.3	1000	0.2101	-0.00011
0.3	10000	0.2099	 0.000049
0.3	100000	0.2100	-0.0000352

p	n	K	diff
0.5	100	0.2533	-0.0033
0.5	1000	0.2515	-0.00146
0.5	10000	0.2501	-0.000131
0.5	100000	0.2500	-0.0000425

p	n	K	diff
0.7	100	0.2172 	-0.0072
0.7	1000	0.2106	-0.0006
0.7	10000	0.2098	 0.000194
0.7	100000	0.2102	-0.0002176

p	n	K	diff
0.9	100	0.0924	-0.0024
0.9	1000	0.0895	 0.00049
0.9	10000	0.0899	 0.00013
0.9	100000	0.0900	-0.0000144

## Pascal

Translation of: C
Works with: Free Pascal

{$MODE objFPC}//for using result,parameter runs becomes for variable.. uses sysutils;//Format const MaxN = 100*1000; function run_test(p:double;len,runs: NativeInt):double; var x, y, i,cnt : NativeInt; Begin result := 1/ (runs * len); cnt := 0; for runs := runs-1 downto 0 do Begin x := 0; y := 0; for i := len-1 downto 0 do begin x := y; y := Ord(Random() < p); cnt := cnt+ord(x < y); end; end; result := result *cnt; end; //main var p, p1p, K : double; ip, n : nativeInt; Begin randomize; writeln( 'running 1000 tests each:'#13#10, ' p n K p(1-p) diff'#13#10, '-----------------------------------------------'); ip:= 1; while ip < 10 do Begin p := ip / 10; p1p := p * (1 - p); n := 100; While n <= MaxN do Begin K := run_test(p, n, 1000); writeln(Format('%4.1f %6d %6.4f %6.4f %7.4f (%5.2f %%)', [p, n, K, p1p, K - p1p, (K - p1p) / p1p * 100])); n := n*10; end; writeln; ip := ip+2; end; end. Output running 1000 tests each: p n K p(1-p) diff ----------------------------------------------- 0.1 100 0.0894 0.0900 -0.0006 (-0.70 %) 0.1 1000 0.0898 0.0900 -0.0002 (-0.17 %) 0.1 10000 0.0900 0.0900 0.0000 ( 0.02 %) 0.1 100000 0.0900 0.0900 0.0000 ( 0.04 %) 0.3 100 0.2112 0.2100 0.0012 ( 0.57 %) 0.3 1000 0.2101 0.2100 0.0001 ( 0.04 %) 0.3 10000 0.2099 0.2100 -0.0001 (-0.04 %) 0.3 100000 0.2099 0.2100 -0.0001 (-0.03 %) 0.5 100 0.2516 0.2500 0.0016 ( 0.66 %) 0.5 1000 0.2497 0.2500 -0.0003 (-0.14 %) 0.5 10000 0.2501 0.2500 0.0001 ( 0.03 %) 0.5 100000 0.2500 0.2500 0.0000 ( 0.01 %) 0.7 100 0.2144 0.2100 0.0044 ( 2.08 %) 0.7 1000 0.2107 0.2100 0.0007 ( 0.32 %) 0.7 10000 0.2101 0.2100 0.0001 ( 0.02 %) 0.7 100000 0.2100 0.2100 0.0000 ( 0.01 %) 0.9 100 0.0978 0.0900 0.0078 ( 8.69 %) 0.9 1000 0.0909 0.0900 0.0009 ( 0.96 %) 0.9 10000 0.0901 0.0900 0.0001 ( 0.10 %) 0.9 100000 0.0900 0.0900 0.0000 ( 0.02 %) ## Perl Translation of: Perl 6 sub R { my ($n, $p) = @_; my$r = join '',
map { rand() < $p ? 1 : 0 } 1 ..$n;
0+ $r =~ s/1+//g; } use constant t => 100; printf "t= %d\n", t; for my$p (qw(.1 .3 .5 .7 .9)) {
printf "p= %f, K(p)= %f\n", $p,$p*(1-$p); for my$n (qw(10 100 1000)) {
my $r;$r += R($n,$p) for 1 .. t; $r /=$n;
printf " R(n, p)= %f\n", $r / t; } } Output: t= 100 p= 0.100000, K(p)= 0.090000 R(n, p)= 0.095000 R(n, p)= 0.088100 R(n, p)= 0.089420 p= 0.300000, K(p)= 0.210000 R(n, p)= 0.225000 R(n, p)= 0.208800 R(n, p)= 0.210020 p= 0.500000, K(p)= 0.250000 R(n, p)= 0.289000 R(n, p)= 0.249900 R(n, p)= 0.248980 p= 0.700000, K(p)= 0.210000 R(n, p)= 0.262000 R(n, p)= 0.213200 R(n, p)= 0.209690 p= 0.900000, K(p)= 0.090000 R(n, p)= 0.177000 R(n, p)= 0.096200 R(n, p)= 0.091730 ## Perl 6 sub R($n, $p) { [+] ((rand <$p) xx $n).squish } say 't= ', constant t = 100; for .1, .3 ... .9 ->$p {
say "p= $p, K(p)= {$p*(1-$p)}"; for 10, 100, 1000 ->$n {
printf " R(%6d, p)= %f\n", $n, t R/ [+] R($n, $p)/$n xx t
}
}
Output:
t= 100
p= 0.1, K(p)= 0.09
R(    10, p)= 0.088000
R(   100, p)= 0.085600
R(  1000, p)= 0.089150
p= 0.3, K(p)= 0.21
R(    10, p)= 0.211000
R(   100, p)= 0.214600
R(  1000, p)= 0.211160
p= 0.5, K(p)= 0.25
R(    10, p)= 0.279000
R(   100, p)= 0.249200
R(  1000, p)= 0.250870
p= 0.7, K(p)= 0.21
R(    10, p)= 0.258000
R(   100, p)= 0.215400
R(  1000, p)= 0.209560
p= 0.9, K(p)= 0.09
R(    10, p)= 0.181000
R(   100, p)= 0.094500
R(  1000, p)= 0.091330

## Phix

Translation of: zkl
function run_test(atom p, integer len, runs)
integer count = 0
for r=1 to runs do
bool v, pv = false
for l=1 to len do
v = rnd()<p
count += pv<v
pv = v
end for
end for
return count/runs/len
end function

procedure main()
printf(1,"Running 1000 tests each:\n")
printf(1," p n K p(1-p) delta\n")
printf(1,"--------------------------------------------\n")
for ip=1 to 10 by 2 do
atom p = ip/10,
p1p = p*(1-p)
integer n = 100
while n<=100000 do
atom K = run_test(p, n, 1000)
printf(1,"%.1f  %6d  %6.4f  %6.4f  %+7.4f (%+5.2f%%)\n",
{p, n, K, p1p, K-p1p, (K-p1p)/p1p*100})
n *= 10
end while
printf(1,"\n")
end for
end procedure
main()
Output:
Running 1000 tests each:
p        n  K       p(1-p)       delta
--------------------------------------------
0.1     100  0.0889  0.0900  -0.0011 (-1.20%)
0.1    1000  0.0896  0.0900  -0.0004 (-0.45%)
0.1   10000  0.0900  0.0900  -0.0000 (-0.02%)
0.1  100000  0.0900  0.0900  -0.0000 (-0.04%)

0.3     100  0.2112  0.2100  +0.0012 (+0.57%)
0.3    1000  0.2101  0.2100  +0.0001 (+0.07%)
0.3   10000  0.2101  0.2100  +0.0001 (+0.06%)
0.3  100000  0.2100  0.2100  -0.0000 (-0.01%)

0.5     100  0.2528  0.2500  +0.0028 (+1.13%)
0.5    1000  0.2500  0.2500  +0.0000 (+0.01%)
0.5   10000  0.2500  0.2500  +0.0000 (+0.00%)
0.5  100000  0.2500  0.2500  -0.0000 (-0.00%)

0.7     100  0.2174  0.2100  +0.0074 (+3.50%)
0.7    1000  0.2105  0.2100  +0.0005 (+0.26%)
0.7   10000  0.2101  0.2100  +0.0001 (+0.06%)
0.7  100000  0.2100  0.2100  +0.0000 (+0.01%)

0.9     100  0.0986  0.0900  +0.0086 (+9.53%)
0.9    1000  0.0908  0.0900  +0.0008 (+0.88%)
0.9   10000  0.0901  0.0900  +0.0001 (+0.11%)
0.9  100000  0.0900  0.0900  +0.0000 (+0.03%)

## Python

from __future__ import division
from random import random
from math import fsum

n, p, t = 100, 0.5, 500

def newv(n, p):
return [int(random() < p) for i in range(n)]

def runs(v):
return sum((a & ~b) for a, b in zip(v, v[1:] + [0]))

def mean_run_density(n, p):
return runs(newv(n, p)) / n

for p10 in range(1, 10, 2):
p = p10 / 10
limit = p * (1 - p)
print('')
for n2 in range(10, 16, 2):
n = 2**n2
sim = fsum(mean_run_density(n, p) for i in range(t)) / t
print('t=%3i p=%4.2f n=%5i p(1-p)=%5.3f sim=%5.3f delta=%3.1f%%'
% (t, p, n, limit, sim, abs(sim - limit) / limit * 100 if limit else sim * 100))
Output:
t=500 p=0.10 n= 1024 p(1-p)=0.090 sim=0.090 delta=0.2%
t=500 p=0.10 n= 4096 p(1-p)=0.090 sim=0.090 delta=0.0%
t=500 p=0.10 n=16384 p(1-p)=0.090 sim=0.090 delta=0.1%

t=500 p=0.30 n= 1024 p(1-p)=0.210 sim=0.210 delta=0.0%
t=500 p=0.30 n= 4096 p(1-p)=0.210 sim=0.210 delta=0.0%
t=500 p=0.30 n=16384 p(1-p)=0.210 sim=0.210 delta=0.0%

t=500 p=0.50 n= 1024 p(1-p)=0.250 sim=0.251 delta=0.3%
t=500 p=0.50 n= 4096 p(1-p)=0.250 sim=0.250 delta=0.0%
t=500 p=0.50 n=16384 p(1-p)=0.250 sim=0.250 delta=0.0%

t=500 p=0.70 n= 1024 p(1-p)=0.210 sim=0.210 delta=0.0%
t=500 p=0.70 n= 4096 p(1-p)=0.210 sim=0.210 delta=0.1%
t=500 p=0.70 n=16384 p(1-p)=0.210 sim=0.210 delta=0.0%

t=500 p=0.90 n= 1024 p(1-p)=0.090 sim=0.091 delta=0.6%
t=500 p=0.90 n= 4096 p(1-p)=0.090 sim=0.090 delta=0.2%
t=500 p=0.90 n=16384 p(1-p)=0.090 sim=0.090 delta=0.0%

## Racket

#lang racket
(require racket/fixnum)
(define t (make-parameter 100))

(define (Rn v)
(define (inner-Rn rv idx b-1)
(define b (fxvector-ref v idx))
(define rv+ (if (and (= b 1) (= b-1 0)) (add1 rv) rv))
(if (zero? idx) rv+ (inner-Rn rv+ (sub1 idx) b)))
(inner-Rn 0 (sub1 (fxvector-length v)) 0))

(define ((make-random-bit-vector p) n)
(for/fxvector
#:length n ((i n))
(if (<= (random) p) 1 0)))

(define (Rn/n l->p n) (/ (Rn (l->p n)) n))

(for ((p (in-list '(1/10 3/10 1/2 7/10 9/10))))
(define l->p (make-random-bit-vector p))
(define Kp (* p (- 1 p)))
(printf "p = ~a\tK(p) =\t~a\t~a~%" p Kp (real->decimal-string Kp 4))
(for ((n (in-list '(10 100 1000 10000))))
(define sum-Rn/n (for/sum ((i (in-range (t)))) (Rn/n l->p n)))
(define sum-Rn/n/t (/ sum-Rn/n (t)))
(printf "mean(R_~a/~a) =\t~a\t~a~%"
n n sum-Rn/n/t (real->decimal-string sum-Rn/n/t 4)))
(newline))

(module+ test
(require rackunit)
(check-eq? (Rn (fxvector 1 1 0 0 0 1 0 1 1 1)) 3))
Output:
p = 1/10	K(p) =	9/100	0.0900
mean(R_10/10) =	3/40	0.0750
mean(R_100/100) =	221/2500	0.0884
mean(R_1000/1000) =	4469/50000	0.0894
mean(R_10000/10000) =	90313/1000000	0.0903

p = 3/10	K(p) =	21/100	0.2100
mean(R_10/10) =	231/1000	0.2310
mean(R_100/100) =	1049/5000	0.2098
mean(R_1000/1000) =	131/625	0.2096
mean(R_10000/10000) =	209873/1000000	0.2099

p = 1/2	K(p) =	1/4	0.2500
mean(R_10/10) =	297/1000	0.2970
mean(R_100/100) =	1263/5000	0.2526
mean(R_1000/1000) =	24893/100000	0.2489
mean(R_10000/10000) =	124963/500000	0.2499

p = 7/10	K(p) =	21/100	0.2100
mean(R_10/10) =	131/500	0.2620
mean(R_100/100) =	2147/10000	0.2147
mean(R_1000/1000) =	1049/5000	0.2098
mean(R_10000/10000) =	210453/1000000	0.2105

p = 9/10	K(p) =	9/100	0.0900
mean(R_10/10) =	169/1000	0.1690
mean(R_100/100) =	119/1250	0.0952
mean(R_1000/1000) =	4503/50000	0.0901
mean(R_10000/10000) =	89939/1000000	0.0899

## REXX

Translation of: Fortran
/* REXX */
Numeric Digits 20
Call random(,12345) /* make the run reproducable */
pList = '.1 .3 .5 .7 .9'
nList = '1e2 1e3 1e4 1e5'
t = 100
Do While plist<>''
Parse Var plist p plist
theory=p*(1-p)
Say ' '
Say 'p:' format(p,2,4)' theory:'format(theory,2,4)' t:'format(t,4)
Say ' n sim sim-theory'
nl=nlist
Do While nl<>''
Parse Var nl n nl
sum=0
Do i=1 To t
run=0
Do j=1 To n
one=random(1000)<p*1000
If one & (run=0) Then
sum=sum+1
run=one
End
End
sim=sum/(n*100)
Say format(n,10)' ' format(sim,2,4)' 'format(sim-theory,2,6)
End
End
Output:
p:  0.1000  theory: 0.0900  t: 100
n          sim     sim-theory
100          0.0875 -0.002500
1000          0.0894 -0.000560
10000          0.0902  0.000237
100000          0.0899 -0.000112

p:  0.3000  theory: 0.2100  t: 100
n          sim     sim-theory
100          0.2088 -0.001200
1000          0.2116  0.001570
10000          0.2101  0.000056
100000          0.2099 -0.000120

p:  0.5000  theory: 0.2500  t: 100
n          sim     sim-theory
100          0.2557  0.005700
1000          0.2513  0.001280
10000          0.2497 -0.000267
100000          0.2501  0.000107

p:  0.7000  theory: 0.2100  t: 100
n          sim     sim-theory
100          0.2171  0.007100
1000          0.2095 -0.000490
10000          0.2099 -0.000137
100000          0.2103  0.000321

p:  0.9000  theory: 0.0900  t: 100
n          sim     sim-theory
100          0.0999  0.009900
1000          0.0898 -0.000240
10000          0.0906  0.000568
100000          0.0908  0.000775

## Sidef

Translation of: Perl 6
func R(n,p) {
n.of { 1.rand < p ? 1 : 0}.sum;
}

const t = 100;
say ('t=', t);

range(.1, .9, .2).each { |p|
printf("p= %f, K(p)= %f\n", p, p*(1-p));
[10, 100, 1000].each { |n|
printf (" R(n, p)= %f\n", t.of { R(n, p) }.sum/n / t);
}
}
Output:
t=100
p= 0.100000, K(p)= 0.090000
R(n, p)= 0.099000
R(n, p)= 0.105000
R(n, p)= 0.099810
p= 0.300000, K(p)= 0.210000
R(n, p)= 0.301000
R(n, p)= 0.289800
R(n, p)= 0.300720
p= 0.500000, K(p)= 0.250000
R(n, p)= 0.481000
R(n, p)= 0.501800
R(n, p)= 0.498260
p= 0.700000, K(p)= 0.210000
R(n, p)= 0.695000
R(n, p)= 0.698400
R(n, p)= 0.701220
p= 0.900000, K(p)= 0.090000
R(n, p)= 0.910000
R(n, p)= 0.898500
R(n, p)= 0.899080

## Tcl

proc randomString {length probability} {
for {set s ""} {[string length $s] <$length} {} {
append s [expr {rand() < $probability}] } return$s
}

# By default, [regexp -all] gives the number of times that the RE matches
proc runs {str} {
regexp -all {1+} $str } # Compute the mean run density proc mrd {t p n} { for {set i 0;set total 0.0} {$i < $t} {incr i} { set run [randomString$n $p] set total [expr {$total + double([runs $run])/$n}]
}
return [expr {$total /$t}]
}

# Parameter sweep with nested [foreach]
set runs 500
foreach p {0.10 0.30 0.50 0.70 0.90} {
foreach n {1024 4096 16384} {
set theory [expr {$p * (1 -$p)}]
set sim [mrd $runs$p $n] set diffpc [expr {abs($theory-$sim)*100/$theory}]
puts [format "t=%d, p=%.2f, n=%5d, p(1-p)=%.3f, sim=%.3f, delta=%.2f%%" \
$runs$p $n$theory $sim$diffpc]
}
puts ""
}
Output:
t=500, p=0.10, n= 1024, p(1-p)=0.090, sim=0.090, delta=0.07%
t=500, p=0.10, n= 4096, p(1-p)=0.090, sim=0.090, delta=0.06%
t=500, p=0.10, n=16384, p(1-p)=0.090, sim=0.090, delta=0.17%

t=500, p=0.30, n= 1024, p(1-p)=0.210, sim=0.210, delta=0.23%
t=500, p=0.30, n= 4096, p(1-p)=0.210, sim=0.210, delta=0.09%
t=500, p=0.30, n=16384, p(1-p)=0.210, sim=0.210, delta=0.01%

t=500, p=0.50, n= 1024, p(1-p)=0.250, sim=0.250, delta=0.10%
t=500, p=0.50, n= 4096, p(1-p)=0.250, sim=0.250, delta=0.07%
t=500, p=0.50, n=16384, p(1-p)=0.250, sim=0.250, delta=0.08%

t=500, p=0.70, n= 1024, p(1-p)=0.210, sim=0.211, delta=0.33%
t=500, p=0.70, n= 4096, p(1-p)=0.210, sim=0.210, delta=0.00%
t=500, p=0.70, n=16384, p(1-p)=0.210, sim=0.210, delta=0.01%

t=500, p=0.90, n= 1024, p(1-p)=0.090, sim=0.091, delta=1.61%
t=500, p=0.90, n= 4096, p(1-p)=0.090, sim=0.090, delta=0.08%
t=500, p=0.90, n=16384, p(1-p)=0.090, sim=0.090, delta=0.09%

## zkl

Translation of: C
fcn run_test(p,len,runs){
cnt:=0; do(runs){
pv:=0; do(len){
v:=0 + ((0.0).random(1.0)<p); // 0 or 1, value of V[n]
cnt += (pv<v); // if v is 1 & prev v was zero, inc cnt
pv = v;
}
}
return(cnt.toFloat() / runs / len);
}
println("Running 1000 tests each:\n"
" p\t n\tK\tp(1-p)\t diff\n"
"-----------------------------------------------");
foreach p in ([0.1..0.9,0.2]) {
p1p:=p*(1.0 - p);
n:=100; while(n <= 100000) {
K:=run_test(p, n, 1000);
"%.1f\t%6d\t%.4f\t%.4f\t%+.4f (%+.2f%%)".fmt(
p, n, K, p1p, K - p1p, (K - p1p) / p1p * 100).println();
n *= 10;
}
println();
}
Output:
Running 1000 tests each:
p	   n	K	p(1-p)	     diff
-----------------------------------------------
0.1	   100	0.0903	0.0900	+0.0003 (+0.36%)
0.1	  1000	0.0900	0.0900	-0.0000 (-0.01%)
0.1	 10000	0.0901	0.0900	+0.0001 (+0.16%)
0.1	100000	0.0900	0.0900	+0.0000 (+0.01%)

0.3	   100	0.2115	0.2100	+0.0015 (+0.73%)
0.3	  1000	0.2105	0.2100	+0.0005 (+0.23%)
0.3	 10000	0.2098	0.2100	-0.0002 (-0.07%)
0.3	100000	0.2100	0.2100	+0.0000 (+0.00%)

0.5	   100	0.2521	0.2500	+0.0021 (+0.86%)
0.5	  1000	0.2503	0.2500	+0.0003 (+0.13%)
0.5	 10000	0.2500	0.2500	-0.0000 (-0.01%)
0.5	100000	0.2500	0.2500	-0.0000 (-0.00%)

0.7	   100	0.2151	0.2100	+0.0051 (+2.41%)
0.7	  1000	0.2103	0.2100	+0.0003 (+0.16%)
0.7	 10000	0.2100	0.2100	+0.0000 (+0.00%)
0.7	100000	0.2100	0.2100	-0.0000 (-0.01%)

0.9	   100	0.0979	0.0900	+0.0079 (+8.74%)
0.9	  1000	0.0911	0.0900	+0.0011 (+1.17%)
0.9	 10000	0.0902	0.0900	+0.0002 (+0.18%)
0.9	100000	0.0900	0.0900	-0.0000 (-0.00%)