Large factorial numbers

by Stephen R. Schmitt

Computation of N!

For any positive integer N, the factorial is calculated by multiplying together all integers up to and including N, that is,

N! = 1 × 2 × 3 × . . . × (N - 1) × N

0! =      1 note!
1! =      1 
2! =      2 
3! =      6 
4! =     24 
5! =    120
6! =    720 
7! =   5040 
8! =  40320 
9! = 362880 

Factorials start out reasonably small but grow extremely rapidly. A famous approximation, named after the Scottish mathematician James Stirling (1692-1770), can be used in a computer to approximate N factorial.

N! ≈ (2·π·N)1/2(N/e)N

Zeno source code

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const SIZE : int := 200
type iarray : array[SIZE] of int

program

    var a : iarray
    var b : int
   
    put "Enter an integer (> 0) "...
    get b

    if not bigfact( b, a ) then
        put "fail"
    end if

end program

function bigfact( n : int, var a : iarray ) : boolean 

    var c, d, i, j, x : int
    var p : real

    assert n > 0

    for i := 1...SIZE do              % init accumulator array
        a[i] := 0
    end for
   
    p := 0.0                          % count number of digits needed
    for i := 2...n do
        p := p + log10(i)             % log(b!) = log(1)+log(2)+log(3)+...
    end for
    
    d := floor( p + 1 )               % log(9999) = 3.9999566 so need 4 digits
    if d > SIZE then
        put "need array size >= ", d
        return false
    end if
    
    a[1] := 1                         % compute factorial
    c := 0                            % init carry
    for i:= 1...n do                  % 1*2*3*4...
        for j := 1...d do             % do big multiply for each digit
            x    := a[j]*i + c
            a[j] := x mod 10
            c    := x div 10
        end for
    end for
    
    put n, "! = "...                  % print results
    for decreasing i:= d...1 do
        put a[i]...
    end for
    
    return true
     
end function