Distribution of 0 digits in factorial series: Difference between revisions
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{{draft task|Mathematics}}
Large Factorials and the Distribution of '0' in base 10 digits.
; About the task:
We can see that some features of factorial numbers (the series of numbers 1!, 2!, 3!, ...)
come about because such numbers are the product of a series of counting numbers, and so those
products have predictable factors. For example, all factorials above 1! are even numbers,
since they have 2 as a factor. Similarly, all factorials from 5! up end in a 0, because they
have 5 and 2 as factors, and thus have 10 as a factor. In fact, the factorial integers add
another 0 at the end of the factorial for every step of 5 upward: 5! = 120, 10! = 3628800,
15! = 1307674368000, 16! = 20922789888000 and so on.
Because factorial numbers, which quickly become quite large, continue to have another terminal 0
on the right hand side of the number for every factor of 5 added to the factorial product, one might
think that the proportion of zeros in a base 10 factorial number might be close to 1/5. However,
though the factorial products add another terminating 0 every factor of 5 multiplied into the product,
as the numbers become quite large, the number of digits in the factorial product expands exponentially,
and so the number above the terminating zeros tends toward 10% of each digit from 0 to 1 as the factorial
becomes larger. Thus, as the factorials become larger, the proportion of 0 digits in the factorial products
shifts slowly from around 1/5 toward 1/10, since the number of terminating zeros in n! increases only in
proportion to n, whereas the number of digits of n! in base 10 increases exponentially.
; The task:
Create a function to calculate the mean of the proportions of 0 digits out of the total digits found in each factorial
product from 1! to N!. This proportion of 0 digits in base 10 should be calculated using the number as printed as a base 10 integer.
Example: for 1 to 6 we have 1!, 2!, 3!, 4!, 5!, 6!, or (1, 2, 6, 24, 120, 720), so we need the mean of
(0/1, 0/1, 0/1, 0/2, 1/3, 1/3) = (2/3) (totals of each proportion) / 6 (= N), or 0.1111111...
Example: for 1 to 25 the mean of the proportions of 0 digits in the factorial products series of N! with N from 1 to 25 is 0.26787.
Do this task for 1 to N where N is in (100, 1000, and 10000), so, compute the mean of the proportion of 0 digits for each product
in the series of each of the factorials from 1 to 100, 1 to 1000, and 1 to 10000.
; Stretch task:
Find the N in 10000 < N < 50000 where the mean of the proportions of 0 digits in the factorial products from 1 to N
permanently falls below 0.16. This task took many hours in the Python example, though I wonder if there is a faster
algorithm out there.
=={{header|11l}}==
{{trans|Python}}
<lang 11l>F facpropzeros(n, verbose = 1B)
V proportions = [0.0] * n
V (fac, psum) = (BigInt(1), 0.0)
L(i) 0 .< n
fac *= i + 1
V d = String(fac)
psum += sum(d.map(x -> Int(x == ‘0’))) / Float(d.len)
proportions[i] = psum / (i + 1)
print(‘The mean proportion of 0 in factorials from 1 to #. is #..’.format(n, psum / n))
L(n) [100, 1000, 10000]
facpropzeros(n)</lang>
{{out}}
<pre>
The mean proportion of 0 in factorials from 1 to 100 is 0.246753186.
The mean proportion of 0 in factorials from 1 to 1000 is 0.203544551.
The mean proportion of 0 in factorials from 1 to 10000 is 0.173003848.
</pre>
=== Base 1000 version ===
<lang 11l>F zinit()
V zc = [0] * 999
L(x) 1..9
zc[x - 1] = 2
zc[10 * x - 1] = 2
zc[100 * x - 1] = 2
L(y) (10.<100).step(10)
zc[y + x - 1] = 1
zc[10 * y + x - 1] = 1
zc[10 * (y + x) - 1] = 1
R zc
F meanfactorialdigits()
V zc = zinit()
V rfs = [1]
V (total, trail, first) = (0.0, 1, 0)
L(f) 2 .< 50000
V (carry, d999, zeroes) = (0, 0, (trail - 1) * 3)
V (j, l) = (trail, rfs.len)
L j <= l | carry != 0
I j <= l
carry = rfs[j - 1] * f + carry
d999 = carry % 1000
I j <= l
rfs[j - 1] = d999
E
rfs.append(d999)
zeroes += I d999 == 0 {3} E zc[d999 - 1]
carry I/= 1000
j++
L rfs[trail - 1] == 0
trail++
d999 = rfs.last
d999 = I d999 >= 100 {0} E I d999 < 10 {2} E 1
zeroes -= d999
V digits = rfs.len * 3 - d999
total += Float(zeroes) / digits
V ratio = total / f
I f C [100, 1000, 10000]
print(‘The mean proportion of zero digits in factorials to #. is #.’.format(f, ratio))
I ratio >= 0.16
first = 0
E I first == 0
first = f
print(‘The mean proportion dips permanently below 0.16 at ’first‘.’)
meanfactorialdigits()</lang>
{{out}}
<pre>
The mean proportion of zero digits in factorials to 100 is 0.246753186
The mean proportion of zero digits in factorials to 1000 is 0.203544551
The mean proportion of zero digits in factorials to 10000 is 0.173003848
The mean proportion dips permanently below 0.16 at 47332.
</pre>
=={{header|C++}}==
{{trans|Phix}}
<lang cpp>#include <array>
#include <chrono>
#include <iomanip>
#include <iostream>
#include <vector>
auto init_zc() {
zc.fill(0);
zc[0] = 3;
for (int x = 1; x <= 9; ++x) {
zc[x] = 2;
zc[10 * x] = 2;
zc[100 * x] = 2;
for (int y = 10; y <= 90; y += 10) {
zc[y + x] = 1;
zc[10 * y + x] = 1;
zc[10 * (y + x)] = 1;
}
}
return zc;
}
template <typename clock_type>
auto elapsed(const std::chrono::time_point<clock_type>& t0) {
auto t1 =
auto duration =
std::chrono::duration_cast<std::chrono::milliseconds>(t1 - t0);
return duration.count();
}
int main() {
auto zc =
auto t0 = std::chrono::high_resolution_clock::now();
int trail = 1,
double total = 0;
std::vector<int> rfs{1};
std::cout << std::fixed << std::setprecision(10);
for (int f = 2; f <= 50000; ++f) {
int carry = 0, d999, zeroes = (trail - 1) * 3, len = rfs.size();
for (int j =
carry += rfs[j] * f;
d999 = carry % 1000;
if (j < len)
rfs[j] = d999;
else
rfs.push_back(d999);
zeroes += zc[d999];
carry /= 1000;
}
while (rfs[trail - 1] == 0)
int digits = rfs.size() * 3 - d999;
total += double(zeroes) / digits;
double ratio = total / f;
if (ratio >= 0.16)
if (f == 100
std::cout << "Mean proportion of zero digits in factorials to " << f
<< " is " << ratio << ". (" << elapsed(t0) << "ms)\n";
}
}
std::cout << "The mean proportion dips permanently below 0.16 at " << first
<< ". (" << elapsed(t0) << "ms)\n";
}</lang>
{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.2467531862. (0ms)
Mean proportion of zero digits in factorials to 1000 is 0.2035445511. (1ms)
Mean proportion of zero digits in factorials to 10000 is 0.1730038482. (152ms)
The mean proportion dips permanently below 0.16 at 47332. (4598ms)
</pre>
=={{header|Go}}==
===
{{libheader|GMP(Go wrapper)}}
{{libheader|Go-rcu}}
Timings here are 2.8 seconds for the basic task and 182.5 seconds for the stretch goal.
<
import (
"fmt"
big "
"
)
func main() {
first :=
firstRatio := 0.0
fmt.Println("The mean proportion of zero digits in factorials up to the following are:")
for n := int64(1); n <= 50000; n++ {
fact.Mul(fact, big.NewInt(n))
bytes := []byte(fact.String())
digits := len(bytes)
zeros := 0
for _, b := range bytes {
if b == '0' {
zeros++
}
}
sum += float64(zeros)/float64(digits)
ratio := sum / float64(n)
if n == 100 || n == 1000 || n == 10000 {
fmt.Printf("%6s = %12.10f\n", rcu.Commatize(int(n)), ratio)
}
if first > 0 && ratio >= 0.16 {
first = 0
firstRatio = 0.0
} else if first == 0 && ratio < 0.16 {
first = n
firstRatio = ratio
}
}
fmt.Printf("%6s = %12.10f", rcu.Commatize(int(first)), firstRatio)
fmt.Println(" (stays below 0.16 after this)")
fmt.Printf("%6s = %12.10f\n", "50,000", sum / 50000)
}</lang>
{{out}}
<pre>
The mean proportion of zero digits in factorials up to the following are:
100 = 0.2467531862
1,000 = 0.2035445511
10,000 = 0.1730038482
47,332 = 0.1599999958 (stays below 0.16 after this)
50,000 = 0.1596200546
</pre>
<br>
==='String math' and base 1000===
{{trans|Phix}}
Much quicker than before with 10,000 now being reached in 0.35 seconds and the stretch goal in about 5.5 seconds.
<lang go>package main
import (
"
"
)
var rfs = []int{1} // reverse factorial(1) in base 1000
var zc = make([]int, 999)
func init() {
for x := 1; x <= 9; x++ {
zc[x-1] = 2 // 00x
zc[10*x-1] = 2 // 0x0
zc[100*x-1] = 2 // x00
var y = 10
for y <= 90 {
zc[y+x-1] = 1 // 0yx
zc[10*y+x-1] = 1 // y0x
zc[10*(y+x)-1] = 1 // yx0
y += 10
}
}
}
func main() {
total := 0.0
first := 0
firstRatio := 0.0
fmt.Println("The mean proportion of zero digits in factorials up to the following are:")
for f := 2; f <= 10000; f++ {
carry := 0
d999 := 0
zeros := (trail - 1) * 3
j := trail
l := len(rfs)
for j <= l || carry != 0 {
if j <= l {
carry = rfs[j-1]*f + carry
}
d999 = carry % 1000
if j <= l {
rfs[j-1] = d999
} else {
rfs = append(rfs, d999)
}
if d999 == 0 {
zeros += 3
} else {
zeros += zc[d999-1]
}
carry /= 1000
j++
}
for rfs[trail-1] == 0 {
trail++
}
// d999 = quick correction for length and zeros
d999 = rfs[len(rfs)-1]
if d999 < 100 {
if d999 < 10 {
d999 = 2
} else {
d999 = 1
}
} else {
d999 = 0
}
zeros -= d999
digits := len(rfs)*3 - d999
total += float64(zeros) / float64(digits)
ratio := total / float64(f)
if ratio >= 0.16 {
first = 0
firstRatio = 0.0
} else if first == 0 {
first = f
firstRatio = ratio
}
if f == 100 || f == 1000 || f == 10000 {
fmt.Printf("%6s = %12.10f\n", rcu.Commatize(f), ratio)
}
}
fmt.Printf("%6s = %12.10f", rcu.Commatize(first), firstRatio)
fmt.Println(" (stays below 0.16 after this)")
fmt.Printf("%6s = %12.10f\n", "50,000", total/50000)
}</
{{out}}
<pre>
Same as 'brute force' version.
</pre>
==
'''Works with jq'''
The precision of jq's integer arithmetic is not up to this task, so in the following we borrow from the "BigInt" library and use a string representation of integers.
Unfortunately, although gojq (the Go implementation of jq) does support unbounded-precision integer arithmetic, it is unsuited for the task because of memory management issues.
'''From BigInt.jq'''
<lang jq>
# multiply two decimal strings, which may be signed (+ or -)
def long_multiply(num1; num2):
def stripsign:
.[0:1] as $a
| if $a == "-" then [ -1, .[1:]]
elif $a == "+" then [ 1, .[1:]]
else [1, .]
end;
def adjustsign(sign):
if sign == 1 then . else "-" + . end;
# mult/2 assumes neither argument has a sign
def mult(num1;num2):
(num1 | explode | map(.-48) | reverse) as $a1
| (num2 | explode | map(.-48) | reverse) as $a2
| reduce range(0; num1|length) as $i1
([]; # result
reduce range(0; num2|length) as $i2
(.;
($i1 + $i2) as $ix
| ( $a1[$i1] * $a2[$i2] + (if $ix >= length then 0 else .[$ix] end) ) as $r
| if $r > 9 # carrying
then
.[$ix + 1] = ($r / 10 | floor) + (if $ix + 1 >= length then 0 else .[$ix + 1] end )
| .[$ix] = $r - ( $r / 10 | floor ) * 10
else
.[$ix] = $r
end
)
)
| reverse | map(.+48) | implode;
(num1|stripsign) as $a1
| (num2|stripsign) as $a2
| if $a1[1] == "0" or $a2[1] == "0" then "0"
elif $a1[1] == "1" then $a2[1]|adjustsign( $a1[0] * $a2[0] )
elif $a2[1] == "1" then $a1[1]|adjustsign( $a1[0] * $a2[0] )
else mult($a1[1]; $a2[1]) | adjustsign( $a1[0] * $a2[0] )
end;
</lang>
'''The task'''
<lang jq>
def count(s): reduce s as $x (0; .+1);
def meanfactorialdigits:
def digits: tostring | explode;
def nzeros: count( .[] | select(. == 48) ); # "0" is 48
.
| 0.16 as $goal
| label $out
| reduce range( 1; 1+$N ) as $i ( {factorial: "1", proportionsum: 0.0, first: null };
.factorial = long_multiply(.factorial; $i|tostring)
| (.factorial|digits) as $d
| (.proportionsum / $i) as $propmean
| if .first
then if $propmean > $goal then .first = null else . end
elif $propmean <= $goal then .first = $i
else .
end)
| "Mean proportion of zero digits in factorials to \($N) is \(.proportionsum/$N);" +
(if .first then " mean <= \($goal) from N=\(.first) on." else " goal (\($goal)) unmet." end);
# The task:
100, 1000, 10000 | meanfactorialdigits</lang>
{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.24675318616743216; goal (0.16) unmet.
Mean proportion of zero digits in factorials to 1000 is 0.20354455110316458; goal (0.16) unmet.
Mean proportion of zero digits in factorials to 10000 is 0.17300384824186707; goal (0.16) unmet.
</pre>
=={{header|Julia}}==
<lang julia>function meanfactorialdigits(N, goal = 0.0)
factoril, proportionsum = big"1", 0.0
for i in 1:N
factoril *= i
d = digits(factoril)
zero_proportion_in_fac = count(x -> x == 0, d) / length(d)
proportionsum += zero_proportion_in_fac
println("The mean proportion dips permanently below $goal at $i.")
break
end
if i == N
println("Mean proportion of zero digits in factorials to $N is ", propmean)
end
end
end
@time foreach(meanfactorialdigits, [100, 1000, 10000])
@time meanfactorialdigits(50000, 0.16)
</lang>{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.24675318616743216
Mean proportion of zero digits in factorials to 1000 is 0.20354455110316458
Mean proportion of zero digits in factorials to 10000 is 0.17300384824186707
3.030182 seconds (297.84 k allocations: 1.669 GiB, 0.83% gc time, 0.28% compilation time)
The mean proportion dips permanently below 0.16 at 47332.
179.157788 seconds (3.65 M allocations: 59.696 GiB, 1.11% gc time)
</pre>
=== Base 1000 version ===
{{trans|Pascal, Phix}}
<lang julia>function init_zc()
zc = zeros(Int, 999)
for x in 1:9
zc[x] = 2 # 00x
zc[10*x] = 2 # 0x0
zc[100*x] = 2 # x00
for y in 10:10:90
zc[y+x] = 1 # 0yx
zc[10*y+x] = 1 # y0x
zc[10*(y+x)] = 1 # yx0
end
end
return zc
end
function meanfactorialzeros(N = 50000, verbose = true)
zc = init_zc()
rfs = [1]
total, trail, first, firstratio = 0.0, 1, 0, 0.0
for f in 2:N
carry, d999, zeroes = 0, 0, (trail - 1) * 3
j, l = trail, length(rfs)
while j <= l || carry != 0
if j <= l
carry = (rfs[j]) * f + carry
end
d999 = carry % 1000
if j <= l
rfs[j] = d999
else
push!(rfs, d999)
end
zeroes += (d999 == 0) ? 3 : zc[d999]
carry ÷= 1000
j += 1
end
while rfs[trail] == 0
trail += 1
end
# d999 = quick correction for length and zeroes:
d999 = rfs[end]
d999 = d999 < 100 ? d999 < 10 ? 2 : 1 : 0
zeroes -= d999
digits = length(rfs) * 3 - d999
total += zeroes / digits
ratio = total / f
if ratio >= 0.16
first = 0
firstratio = 0.0
elseif first == 0
first = f
firstratio = ratio
end
if f in [100, 1000, 10000]
verbose && println("Mean proportion of zero digits in factorials to $f is $ratio")
end
end
verbose && println("The mean proportion dips permanently below 0.16 at $first.")
end
meanfactorialzeros(100, false)
@time meanfactorialzeros()
</lang>{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.24675318616743216
Mean proportion of zero digits in factorials to 1000 is 0.20354455110316458
Mean proportion of zero digits in factorials to 10000 is 0.17300384824186707
The mean proportion dips permanently below 0.16 at 47332.
4.638323 seconds (50.08 k allocations: 7.352 MiB)
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<lang Mathematica>ClearAll[ZeroDigitsFractionFactorial]
ZeroDigitsFractionFactorial[n_Integer] := Module[{m},
m = IntegerDigits[n!];
Count[m, 0]/Length[m]
]
ZeroDigitsFractionFactorial /@ Range[6] // Mean // N
ZeroDigitsFractionFactorial /@ Range[25] // Mean // N
ZeroDigitsFractionFactorial /@ Range[100] // Mean // N
ZeroDigitsFractionFactorial /@ Range[1000] // Mean // N
ZeroDigitsFractionFactorial /@ Range[10000] // Mean // N
fracs = ParallelMap[ZeroDigitsFractionFactorial, Range[50000], Method -> ("ItemsPerEvaluation" -> 100)];
means = Accumulate[N@fracs]/Range[Length[fracs]];
len = LengthWhile[Reverse@means, # < 0.16 &];
50000 - len + 1</lang>
{{out}}
<pre>0.111111
0.267873
0.246753
0.203545
0.173004
47332</pre>
=={{header|Nim}}==
===Task===
{{libheader|bignum}}
<lang Nim>import strutils, std/monotimes
import bignum
let t0 = getMonoTime()
var sum = 0.0
var f = newInt(1)
var lim = 100
for n in 1..10_000:
f *= n
let str = $f
sum += str.count('0') / str.len
if n == lim:
echo n, ":\t", sum / float(n)
lim *= 10
echo()
echo getMonoTime() - t0</lang>
{{out}}
<pre>100: 0.2467531861674322
1000: 0.2035445511031646
10000: 0.1730038482418671
(seconds: 2, nanosecond: 857794404)</pre>
===Stretch task===
{{libheader|bignum}}
At each step, we eliminate the trailing zeroes to reduce the length of the number and save some time. But this is not much, about 8%.
<lang Nim>import strutils, std/monotimes
import bignum
let t0 = getMonoTime()
var sum = 0.0
var first = 0
var f = newInt(1)
var count0 = 0
for n in 1..<50_000:
f *= n
while f mod 10 == 0: # Reduce the length of "f".
f = f div 10
inc count0
let str = $f
sum += (str.count('0') + count0) / (str.len + count0)
if sum / float(n) < 0.16:
if first == 0: first = n
else:
first = 0
echo "Permanently below 0.16 at n = ", first
echo "Execution time: ", getMonoTime() - t0</lang>
{{out}}
<pre>Permanently below 0.16 at n = 47332
Execution time: (seconds: 190, nanosecond: 215845101)</pre>
=={{header|Pascal}}==
Doing the calculation in Base 1,000,000,000 like in [[Primorial_numbers#alternative]].<BR>
The most time consuming is converting to string and search for zeros.<BR>
Therefor I do not convert to string.I divide the base in sections of 3 digits with counting zeros in a lookup table.
<lang pascal>program Factorial;
{$IFDEF FPC} {$MODE DELPHI} {$Optimization ON,ALL} {$ENDIF}
uses
sysutils;
type
tMul = array of LongWord;
tpMul = pLongWord;
const
LongWordDec = 1000*1000*1000;
LIMIT = 50000;
var
CountOfZero : array[0..999] of byte;
SumOfRatio :array[0..LIMIT] of extended;
procedure OutMul(pMul:tpMul;Lmt :NativeInt);
// for testing
Begin
write(pMul[lmt]);
For lmt := lmt-1 downto 0 do
write(Format('%.9d',[pMul[lmt]]));
writeln;
end;
procedure InitCoZ;
//Init Lookup table for 3 digits
var
x,y : integer;
begin
fillchar(CountOfZero,SizeOf(CountOfZero),#0);
CountOfZero[0] := 3; //000
For x := 1 to 9 do
Begin
CountOfZero[x] := 2; //00x
CountOfZero[10*x] := 2; //0x0
CountOfZero[100*x] := 2; //x00
y := 10;
repeat
CountOfZero[y+x] := 1; //0yx
CountOfZero[10*y+x] := 1; //y0x
CountOfZero[10*(y+x)] := 1; //yx0
inc(y,10)
until y > 100;
end;
end;
function getFactorialDecDigits(n:NativeInt):NativeInt;
var
res: extended;
Begin
result := -1;
IF (n > 0) AND (n <= 1000*1000) then
Begin
res := 0;
repeat res := res+ln(n); dec(n); until n < 2;
result := trunc(res/ln(10))+1;
end;
end;
function CntZero(pMul:tpMul;Lmt :NativeInt):NativeUint;
//count zeros in Base 1,000,000,000 number
var
q,r : LongWord;
i : NativeInt;
begin
result := 0;
For i := Lmt-1 downto 0 do
Begin
q := pMul[i];
r := q DIV 1000;
result +=CountOfZero[q-1000*r];//q-1000*r == q mod 1000
q := r;
r := q DIV 1000;
result +=CountOfZero[q-1000*r];
q := r;
r := q DIV 1000;
result +=CountOfZero[q-1000*r];
end;
//special case first digits no leading '0'
q := pMul[lmt];
while q >= 1000 do
begin
r := q DIV 1000;
result +=CountOfZero[q-1000*r];
q := r;
end;
while q > 0 do
begin
r := q DIV 10;
result += Ord( q-10*r= 0);
q := r;
end;
end;
function GetCoD(pMul:tpMul;Lmt :NativeInt):NativeUint;
//count of decimal digits
var
i : longWord;
begin
result := 9*Lmt;
i := pMul[Lmt];
while i > 1000 do
begin
i := i DIV 1000;
inc(result,3);
end;
while i > 0 do
begin
i := i DIV 10;
inc(result);
end;
end;
procedure DoChecks(pMul:tpMul;Lmt,i :NativeInt);
//(extended(1.0)* makes TIO.RUN faster // only using FPU?
Begin
SumOfRatio[i] := SumOfRatio[i-1] + (extended(1.0)*CntZero(pMul,Lmt))/GetCoD(pMul,Lmt);
end;
function MulByI(pMul:tpMul;UL,i :NativeInt):NativeInt;
var
prod : Uint64;
j : nativeInt;
carry : LongWord;
begin
result := UL;
carry := 0;
For j := 0 to result do
Begin
prod := i*pMul[0]+Carry;
Carry := prod Div LongWordDec;
pMul[0] := Prod - LongWordDec*Carry;
inc(pMul);
end;
IF Carry <> 0 then
Begin
inc(result);
pMul[0]:= Carry;
End;
end;
procedure getFactorialExact(n:NativeInt);
var
MulArr : tMul;
pMul : tpMul;
i,ul : NativeInt;
begin
i := getFactorialDecDigits(n) DIV 9 +10;
Setlength(MulArr,i);
pMul := @MulArr[0];
Ul := 0;
pMul[Ul]:= 1;
i := 1;
repeat
UL := MulByI(pMul,UL,i);
//Now do what you like to do with i!
DoChecks(pMul,UL,i);
inc(i);
until i> n;
end;
procedure Out_(i: integer);
begin
if i > LIMIT then
EXIT;
writeln(i:8,SumOfRatio[i]/i:18:15);
end;
var
i : integer;
Begin
InitCoZ;
SumOfRatio[0]:= 0;
getFactorialExact(LIMIT);
Out_(100);
Out_(1000);
Out_(10000);
Out_(50000);
i := limit;
while i
Begin
if SumOfRatio[i]/i >0.16 then
dec(i);
end;
inc(i);
writeln('First ratio < 0.16 ', i:8,SumOfRatio[i]/i:20:17);
end.</lang>
{{out}}
<pre> 100 0.246753186167432
1000 0.203544551103165
10000 0.173003848241866
50000 0.159620054602269
First ratio < 0.16 47332 0.15999999579985665
Real time: 4.898 s CPU share: 99.55 % // 2.67s on 2200G freepascal 3.2.2</pre>
=={{header|Perl}}==
{{libheader|ntheory}}
<
use
use ntheory qw/factorial/;
for my $n (100, 1000, 10000) {
my($sum,$f) = 0;
$f = factorial $_ and $sum += ($f =~ tr/0//) / length $f for 1..$n;
printf "%5d: %.5f\n", $n, $sum/$n;
}</lang>
{{out}}
<pre> 100: 0.24675
1000: 0.20354
10000: 0.17300</pre>
=={{header|Phix}}==
Using "string math" to create reversed factorials, for slightly easier skipping of "trailing" zeroes,
but converted to base 1000 and with the zero counting idea from Pascal, which sped it up threefold.
<!--<lang Phix>(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #
<span style="color: #008080;">
<span style="color: #004080;">sequence</span> <span style="color: #000000;">zc</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">999</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">9</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">x</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span> <span style="color: #000080;font-style:italic;">-- 00x</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">10</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span> <span style="color: #000080;font-style:italic;">-- 0x0</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">100</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span> <span style="color: #000080;font-style:italic;">-- x00</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">=</span><span style="color: #000000;">10</span> <span style="color: #008080;">to</span> <span style="color: #000000;">90</span> <span style="color: #008080;">by</span> <span style="color: #000000;">10</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">x</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">-- 0yx</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">10</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">x</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">-- y0x</span>
<span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">10</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #000080;font-style:italic;">-- yx0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">zc</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">zc</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">init_zc</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">
<span style="color: #004080;">integer</span> <span style="color: #000000;">trail</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">:</span><span style="color: #000000;">50000</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d999</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">trail</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">j</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">trail</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rfs</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">j</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">l</span> <span style="color: #008080;">or</span> <span style="color: #000000;">carry</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">j</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">l</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">rfs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])*</span><span style="color: #000000;">f</span><span style="color: #0000FF;">+</span><span style="color: #000000;">carry</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">d999</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">j</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">l</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">rfs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d999</span>
<span style="color: #008080;">else</span>
<span style="color: #000000;">rfs</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">d999</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">+=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d999</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?</span><span style="color: #000000;">3</span><span style="color: #0000FF;">:</span><span style="color: #000000;">zc</span><span style="color: #0000FF;">[</span><span style="color: #000000;">d999</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">carry</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">carry</span><span style="color: #0000FF;">/</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">j</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">rfs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">trail</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span> <span style="color: #008080;">do</span> <span style="color: #000000;">trail</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000080;font-style:italic;">-- d999 := quick correction for length and zeroes:</span>
<span style="color: #000000;">d999</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">rfs</span><span style="color: #0000FF;">[$]</span>
<span style="color: #000000;">d999</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d999</span><span style="color: #0000FF;"><</span><span style="color: #000000;">100</span><span style="color: #0000FF;">?</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d999</span><span style="color: #0000FF;"><</span><span style="color: #000000;">10</span><span style="color: #0000FF;">?</span><span style="color: #000000;">2</span><span style="color: #0000FF;">:</span><span style="color: #000000;">1</span><span style="color: #0000FF;">):</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">d999</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">digits</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rfs</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">3</span><span style="color: #0000FF;">-</span><span style="color: #000000;">d999</span>
<span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">})</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color:
<span style="color: #004080;">string</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"The mean proportion dips permanently below 0.16 at %d. (%s)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">first</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<!--</lang>-->
{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.2467531862 (0s)
Mean proportion of zero digits in factorials to 1000 is 0.2035445511 (0.2s)
Mean proportion of zero digits in factorials to 10000 is 0.1730038482 (2.3s)
The mean proportion dips permanently below 0.16 at 47332. (1 minute and 2s)
</pre>
<small>(stretch goal removed under pwa/p2js since otherwise you'd get a blank screen for 2 or 3 minutes)</small>
=== trailing zeroes only ===
Should you only be interested in the ratio of trailing zeroes, you can do that much faster:
<!--<lang Phix>(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">(),</span>
<span style="color: #000000;">f10</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">log10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">total</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">50000</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">f10</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">log10</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">digits</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">ceil</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f10</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">v</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">5</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">v</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">f</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">zeroes</span> <span style="color: #0000FF;">+=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">/</span><span style="color: #000000;">v</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">v</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">5</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">total</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">zeroes</span><span style="color: #0000FF;">/</span><span style="color: #000000;">digits</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">ratio</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">total</span><span style="color: #0000FF;">/</span><span style="color: #000000;">f</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">ratio</span><span style="color: #0000FF;">>=</span><span style="color: #000000;">0.07</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">elsif</span> <span style="color: #000000;">first</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">first</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1000</span><span style="color: #0000FF;">,</span><span style="color: #000000;">10000</span><span style="color: #0000FF;">})</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Mean proportion of trailing zeroes in factorials to %d is %f\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ratio</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #
<!--</lang>-->
{{out}}
<pre>
Mean proportion of trailing zeroes in factorials to 100 is 0.170338
Mean proportion of trailing zeroes in factorials to 1000 is 0.116334
Mean proportion of trailing zeroes in factorials to 10000 is 0.081267
The mean proportion dips permanently below 0.07 at 31549. (0.1s)
</pre>
=={{header|Python}}==
<lang python>def facpropzeros(N, verbose = True):
proportions = [0.0] * N
fac, psum = 1, 0.0
for i in range(N):
fac *= i + 1
d = list(str(fac))
psum += sum(map(lambda x: x == '0', d)) / len(d)
proportions[i] = psum / (i + 1)
if verbose:
print("The mean proportion of 0 in factorials from 1 to {} is {}.".format(N, psum / N))
return proportions
for n in [100, 1000, 10000]:
props = facpropzeros(47500, False)
n = (next(i for i in reversed(range(len(props))) if props[i] > 0.16))
print("The mean proportion dips permanently below 0.16 at {}.".format(n + 2))
</lang>{{out}}
<pre>
The mean proportion of 0 in factorials from 1 to 100 is 0.24675318616743216.
The mean proportion of 0 in factorials from 1 to 1000 is 0.20354455110316458.
The mean proportion of 0 in factorials from 1 to 10000 is 0.17300384824186707.
The mean proportion dips permanently below 0.16 at 47332.
</pre>
The means can be plotted, showing a jump from 0 to over 0.25, followed by a slowly dropping curve:
<lang python>import matplotlib.pyplot as plt
plt.plot([i+1 for i in range(len(props))], props)
</lang>
=== Base 1000 version ===
{{trans|Go via Phix via Pascal}}
<lang python>def zinit():
zc = [0] * 999
for x in range(1, 10):
zc[x - 1] = 2 # 00x
zc[10 * x - 1] = 2 # 0x0
zc[100 * x - 1] = 2 # x00
for y in range(10, 100, 10):
zc[y + x - 1] = 1 # 0yx
zc[10 * y + x - 1] = 1 # y0x
zc[10 * (y + x) - 1] = 1 # yx0
return zc
def meanfactorialdigits():
zc = zinit()
rfs = [1]
total, trail, first = 0.0, 1, 0
for f in range(2, 50000):
carry, d999, zeroes = 0, 0, (trail - 1) * 3
j, l = trail, len(rfs)
while j <= l or carry != 0:
if j <= l:
carry = rfs[j-1] * f + carry
d999 = carry % 1000
if j <= l:
rfs[j-1] = d999
else:
zeroes += 3 if d999 == 0 else zc[d999-1]
carry //= 1000
while rfs[trail-1] == 0:
trail += 1
# d999 is a quick correction for length and zeros
d999 = rfs[-1]
d999 = 0 if d999 >= 100 else 2 if d999 < 10 else 1
zeroes -= d999
if f in [100, 1000, 10000]:
print("The mean proportion of zero digits in factorials to {} is {}".format(f, ratio))
if ratio >= 0.16:
first = 0
elif first == 0:
first = f
print("The mean proportion dips permanently below 0.16 at {}.".format(first))
import time
TIME0 = time.perf_counter()
meanfactorialdigits()
print("\nTotal time:", time.perf_counter() - TIME0, "seconds.")
</lang>{{out}}
<pre>
The mean proportion of zero digits in factorials to 100 is 0.24675318616743216
The mean proportion of zero digits in factorials to 1000 is 0.20354455110316458
The mean proportion of zero digits in factorials to 10000 is 0.17300384824186707
The mean proportion dips permanently below 0.16 at 47332.
Total time: 648.3583232999999 seconds.
</pre>
=={{header|Raku}}==
Works, but depressingly slow for 10000.
<lang perl6>sub postfix:<!> (Int $n) { ( constant factorial = 1, 1, |[\*] 2..* )[$n] }
sink 10000!; # prime the iterator to allow multithreading
sub zs ($n) { ( constant zero-share = (^Inf).race(:32batch).map: { (.!.comb.Bag){'0'} / .!.chars } )[$n+1] }
.say for (
100
,1000
,10000
).map: -> \n { "{n}: {([+] (^n).map: *.&zs) / n}" }</lang>
{{out}}
<pre>100: 0.24675318616743216
1000: 0.20354455110316458
10000: 0.17300384824186605
</pre>
=={{header|REXX}}==
<lang rexx>/*REXX program computes the mean of the proportion of "0" digits a series of factorials.*/
parse arg $ /*obtain optional arguments from the CL*/
if $='' | $="," then $= 100 1000 10000 /*not specified? Then use the default.*/
#= words($) /*the number of ranges to be used here.*/
numeric digits 100 /*increase dec. digs, but only to 100. */
big= word($, #); != 1 /*obtain the largest number in ranges. */
do i=1 for big /*calculate biggest ! using 100 digs.*/
!= ! * i /*calculate the factorial of BIG. */
end /*i*/
if pos('E', !)>0 then do /*In exponential format? Then get EXP.*/
parse var ! 'E' x /*parse the exponent from the number. */
numeric digits x+1 /*set the decimal digits to X plus 1.*/
end /* [↑] the +1 is for the dec. point.*/
title= ' mean proportion of zeros in the (decimal) factorial products for N'
say ' N │'center(title, 80) /*display the title for the output. */
say '───────────┼'center("" , 80, '─') /* " a sep " " " */
do j=1 for #; n= word($, j) /*calculate some factorial ranges. */
say center( commas(n), 11)'│' left(0dist(n), 75)... /*show results for above range.*/
end /*j*/
say '───────────┴'center("" , 80, '─') /*display a foot sep for the output. */
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
0dist: procedure; parse arg z; != 1; y= 0
do k=1 for z; != ! * k; y= y + countstr(0, !) / length(!)
end /*k*/
return y/z</lang>
{{out|output|text= when using the default inputs:}}
<pre>
N │ mean proportion of zeros in the (decimal) factorial products for N
───────────┼────────────────────────────────────────────────────────────────────────────────
100 │ 0.2467531861674322177784158871973526991129407033266153063813195937196095976...
1,000 │ 0.2035445511031646356400438031711455302985741167890402203486699704599684047...
10,000 │ 0.1730038482418660531800366428930706156810278809057883361518852958446868172...
───────────┴────────────────────────────────────────────────────────────────────────────────
</pre>
=={{header|Rust}}==
{{trans|Phix}}
<lang rust>fn init_zc() -> Vec<usize> {
let mut zc = vec![0; 1000];
zc[0] = 3;
for x in 1..=9 {
zc[x] = 2;
zc[10 * x] = 2;
zc[100 * x] = 2;
let mut y = 10;
while y <= 90 {
zc[y + x] = 1;
zc[10 * y + x] = 1;
zc[10 * (y + x)] = 1;
y += 10;
}
}
zc
}
fn main() {
use std::time::Instant;
let zc = init_zc();
let t0 = Instant::now();
let mut first = 0;
let mut total: f64 = 0.0;
let mut rfs = vec![1];
for f in 2..=50000 {
let mut carry = 0;
let mut d999: usize;
let len = rfs.len();
let mut j = trail - 1;
while j < len || carry != 0 {
if j < len {
carry += rfs[j] * f;
}
d999 = carry % 1000;
if j < len {
rfs[j] = d999;
} else {
rfs.push(d999);
}
zeroes += zc[d999];
carry /= 1000;
j += 1;
}
}
d999 = if d999
if d999 < 10 {
2
} else {
1
}
} else {
0
};
zeroes -= d999;
let digits = rfs.len() * 3 - d999;
total += (zeroes as f64) / (digits as f64);
let ratio = total / (f as f64);
if ratio >= 0.16 {
first = 0;
} else if first == 0 {
first = f;
}
if f == 100 || f == 1000 || f == 10000 {
let duration = t0.elapsed();
println!(
"Mean proportion of zero digits in factorials to {} is {:.10}. ({}ms)",
ratio,
duration.as_millis()
);
}
}
let duration = t0.elapsed();
println!(
"The mean proportion dips permanently below 0.16 at {}. ({}ms)",
first,
duration.as_millis()
);
}</lang>
{{out}}
<pre>
Mean proportion of zero digits in factorials to 100 is 0.2467531862. (0ms)
Mean proportion of zero digits in factorials to 1000 is 0.2035445511. (1ms)
Mean proportion of zero digits in factorials to 10000 is 0.1730038482. (149ms)
The mean proportion dips permanently below 0.16 at 47332. (4485ms)
</pre>
=={{header|Sidef}}==
<lang ruby>func mean_factorial_digits(n, d = 0) {
var v = 1
var total = 0.float
for k in (1..n) {
total += v.digits.count(d)/v.len
}
total / n
}
say mean_factorial_digits(100)
say mean_factorial_digits(1000)
say mean_factorial_digits(10000)</lang>
{{
<pre>
0.246753186167432217778415887197352699112940703327
0.203544551103164635640043803171145530298574116789
0.173003848241866053180036642893070615681027880906
</pre>
=={{header|Wren}}==
===Brute force===
{{libheader|
{{libheader|Wren-fmt}}
Very slow indeed, 10.75 minutes to reach N = 10,000.
<lang ecmascript>import "/big" for BigInt
import "/fmt" for Fmt
var fact = BigInt.one
var sum = 0
System.print("The mean proportion of zero digits in factorials up to the following are:")
for (n in 1..10000) {
fact = fact * n
var bytes = fact.toString.bytes
var digits = bytes.count
var zeros = bytes.count { |b| b == 48 }
sum = sum + zeros / digits
if (n == 100 || n == 1000 || n == 10000) {
Fmt.print("$,6d = $12.10f", n, sum / n)
}
}</lang>
{{out}}
<pre>
The mean proportion of zero digits in factorials up to the following are:
100 = 0.2467531862
1,000 = 0.2035445511
10,000 = 0.1730038482
</pre>
<br>
==='String math' and base 1000===
{{trans|Phix}}
Around 60 times faster than before with 10,000 now being reached in about 10.5 seconds. Even the stretch goal is now viable and comes in at 5 minutes 41 seconds.
<lang ecmascript>import "/fmt" for Fmt
var rfs = [1] // reverse factorial(1) in base 1000
var init = Fn.new { |zc|
for (x in 1..9) {
zc[x-1] = 2 // 00x
zc[10*x - 1] = 2 // 0x0
zc[100*x - 1] = 2 // x00
var y = 10
while (y <= 90) {
zc[y + x - 1] = 1 // 0yx
zc[10*y + x - 1] = 1 // y0x
zc[10*(y + x) - 1] = 1 // yx0
y = y + 10
}
}
}
var zc = List.filled(999, 0)
init.call(zc)
var total = 0
var trail = 1
var first = 0
var firstRatio = 0
System.print("The mean proportion of zero digits in factorials up to the following are:")
for (f in 2..50000) {
var d999 = 0
var zeros = (trail-1) * 3
var j = trail
var l = rfs.count
if (j <= l) carry = rfs[j-1]*f + carry
d999 = carry % 1000
if (j <= l) {
rfs[j-1] = d999
} else {
rfs.add(d999)
}
zeros = zeros + ((d999 == 0) ? 3 : zc[d999-1])
carry = (carry/1000).floor
}
while (rfs[trail-1] == 0) trail = trail + 1
// d999 = quick correction for length and zeros
d999 = rfs[-1]
d999 = (d999 < 100) ? ((d999 < 10) ? 2 : 1) : 0
total = total + zeros/digits
var ratio = total / f
} else if (first == 0) {
first = f
firstRatio = ratio
}
if (f == 100 || f == 1000 || f == 10000) {
Fmt.print("$,6d = $12.10f", f, ratio)
}
}
Fmt.write("$,6d = $12.10f", first, firstRatio)
System.print(" (stays below 0.16 after this)")
Fmt.print("$,6d = $12.10f", 50000, total/50000)</lang>
{{out}}
<pre>
The mean proportion of zero digits in factorials up to the following are:
1,000 = 0.2035445511
10,000 = 0.1730038482
47,332 = 0.1599999958 (stays below 0.16 after this)
50,000 = 0.1596200546
</pre>
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