Subtractive generator

A subtractive generator calculates a sequence of random numbers, where each number is congruent to the subtraction of two previous numbers from the sequence.

The formula is

    $r_{n}=r_{(n-i)}-r_{(n-j)}{\pmod {m}}$

for some fixed values of $i,$ $j,$ and $m$ , all positive integers.

Supposing that $i>j$ , then the state of this generator is the list of the previous numbers from $r_{n-i}$ to $r_{n-1}$ .

Many states generate uniform random integers from $0$ to $m-1$ , but some states are bad. A state, filled with zeros, generates only zeros. If $m$ is even, then a state, filled with even numbers, generates only even numbers. More generally, if $f$ is a factor of $m$ , then a state, filled with multiples of $f$ , generates only multiples of $f$ .

All subtractive generators have some weaknesses. The formula correlates $r_{n}$ , $r_{(n-i)}$ and $r_{(n-j)}$ ; these three numbers are not independent, as true random numbers would be.

Anyone who observes $i$ consecutive numbers can predict the next numbers, so the generator is not cryptographically secure.

The authors of Freeciv (utility/rand.c) and xpat2 (src/testit2.c) knew another problem: the low bits are less random than the high bits.

The subtractive generator has a better reputation than the linear congruential generator, perhaps because it holds more states. A subtractive generator might never multiply numbers: this helps where multiplication is slow. A subtractive generator might also avoid division: the value of $r_{(n-i)}-r_{(n-j)}$ is always between $-m$ and $m$ , so a program only needs to add $m$ to negative numbers.

The choice of $i$ and $j$ affects the period of the generator.

A popular choice is $i=55$ and $j=24$ , so the formula is:

    $r_{n}=r_{(n-55)}-r_{(n-24)}{\pmod {m}}$

The subtractive generator from xpat2 uses:

    $r_{n}=r_{(n-55)}-r_{(n-24)}{\pmod {10^{9}}}$

The implementation is by J. Bentley and comes from program_tools/universal.c of the DIMACS (netflow) archive at Rutgers University. It credits Knuth, TAOCP, Volume 2, Section 3.2.2 (Algorithm A).

Bentley uses this clever algorithm to seed the generator.

Start with a single $seed$ in range $0$ to $10^{9}-1$ .
Set $s_{0}=seed$ and $s_{1}=1$ . The inclusion of $s_{1}=1$ avoids some bad states (like all zeros, or all multiples of 10).
Compute $s_{2},s_{3},...,s_{54}$ using the subtractive formula $s_{n}=s_{(n-2)}-s_{(n-1)}{\pmod {10^{9}}}$
Reorder these 55 values so $r_{0}=s_{34}$ , $r_{1}=s_{13}$ , $r_{2}=s_{47}$ , ..., $r_{n}=s_{(34*(n+1){\pmod {55}})}$
- This is the same order as $s_{0}=r_{54}$ , $s_{1}=r_{33}$ , $s_{2}=r_{12}$ , ..., $s_{n}=r_{((34*n)-1{\pmod {55}})}$
- This rearrangement exploits how 34 and 55 are relatively prime
Compute the next 165 values $r_{55}$ to $r_{219}$ . Store the last 55 values

This generator yields the sequence $r_{220}$ , $r_{221}$ , $r_{222}$ and so on. For example, if the seed is 292929, then the sequence begins with $r_{220}=467478574$ , $r_{221}=512932792$ , $r_{222}=539453717$ . By starting at $r_{220}$ , this generator avoids a bias from the first numbers of the sequence. This generator must store the last 55 numbers of the sequence, so to compute the next $r_{n}$ . Any array or list would work; a ring buffer is ideal but not necessary.

Implement a subtractive generator that replicates the sequences from xpat2.

Ada

subtractive_generator.ads: <lang Ada>package Subtractive_Generator is

  type State is private;
  procedure Initialize (Generator : in out State; Seed : Natural);
  procedure Next (Generator : in out State; N : out Natural);

private

  type Number_Array is array (Natural range <>) of Natural;
  type State is record
     R    : Number_Array (0 .. 54);
     Last : Natural;
  end record;

end Subtractive_Generator;</lang>

subtractive_generator.adb: <lang Ada>package body Subtractive_Generator is

  procedure Initialize (Generator : in out State; Seed : Natural) is
     S : Number_Array (0 .. 1);
     I : Natural := 0;
     J : Natural := 1;
  begin
     S (0) := Seed;
     S (1) := 1;
     Generator.R (54) := S (0);
     Generator.R (33) := S (1);
     for N in 2 .. Generator.R'Last loop
        S (I) := (S (I) - S (J)) mod 10 ** 9;
        Generator.R ((34 * N - 1) mod 55) := S (I);
        I := (I + 1) mod 2;
        J := (J + 1) mod 2;
     end loop;
     Generator.Last := 54;
     for I in 1 .. 165 loop
        Subtractive_Generator.Next (Generator => Generator, N => J);
     end loop;
  end Initialize;

  procedure Next (Generator : in out State; N : out Natural) is
  begin
     Generator.Last := (Generator.Last + 1) mod 55;
     Generator.R (Generator.Last) :=
       (Generator.R (Generator.Last)
        - Generator.R ((Generator.Last - 24) mod 55)) mod 10 ** 9;
     N := Generator.R (Generator.Last);
  end Next;

end Subtractive_Generator;</lang>

Example main.adb: <lang Ada>with Ada.Text_IO; with Subtractive_Generator;

procedure Main is

  Random : Subtractive_Generator.State;
  N      : Natural;

begin

  Subtractive_Generator.Initialize (Generator => Random,
                                    Seed      => 292929);
  for I in 220 .. 222 loop
     Subtractive_Generator.Next (Generator => Random, N => N);
     Ada.Text_IO.Put_Line (Integer'Image (I) & ":" & Integer'Image (N));
  end loop;

end Main;</lang>

Output:

 220: 467478574
 221: 512932792
 222: 539453717

AutoHotkey

Works with: AutoHotkey_L

<lang AutoHotkey>r := InitR(292929)

Loop, 10 Out .= (A_Index + 219) ":`t" GetRand(r) "`n"

MsgBox, % Out

GetRand(r) { i := Mod(r["j"], 55) , r[i] := Mod(r[i] - r[Mod(i + 31, 55)], r["m"]) , r["j"] += 1 return, (r[i] < 0 ? r[i] + r["m"] : r[i]) }

InitR(Seed) { r := {"j": 0, "m": 10 ** 9}, s := {0: Seed, 1: 1} Loop, 53 s[A_Index + 1] := Mod(s[A_Index - 1] - s[A_Index], r["m"]) Loop, 55 r[A_Index - 1] := s[Mod(34 * A_Index, 55)] Loop, 165 i := Mod(A_Index + 54, 55) , r[i] := Mod(r[i] - r[Mod(A_Index + 30, 55)], r["m"]) return, r }</lang>

Output:

220:	467478574
221:	512932792
222:	539453717
223:	20349702
224:	615542081
225:	378707948
226:	933204586
227:	824858649
228:	506003769
229:	380969305

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> dummy% = FNsubrand(292929)

     FOR i% = 1 TO 10
       PRINT FNsubrand(0)
     NEXT
     END
     
     DEF FNsubrand(s%)
     PRIVATE r%(), p% : DIM r%(54)
     IF s% = 0 THEN
       p% = (p% + 1) MOD 55
       r%(p%) = r%(p%) - r%((p% + 31) MOD 55)
       IF r%(p%) < 0 r%(p%) += 10^9
       = r%(p%)
     ENDIF
     LOCAL i%
     r%(54) = s% : r%(33) = 1
     p% = 12
     FOR i% = 2 TO 54
       r%(p%) = r%((p%+42) MOD 55) - r%((p%+21) MOD 55)
       IF r%(p%) < 0 r%(p%) += 10^9
       p% = (p% + 34) MOD 55
     NEXT
     FOR i% = 55 TO 219
       IF FNsubrand(0)
     NEXT
     = 0</lang>

Output:

Bracmat

This is a translation of the C example.

<lang bracmat>1000000000:?MOD; tbl$(state,55); 0:?si:?sj;

(subrand-seed=

 i,j,p2

. 1:?p2

 & mod$(!arg,!MOD):?(0$?state)
 & 1:?i
 & 21:?j
 &   whl
   ' ( !i:<55
     & (!j:~<55&!j+-55:?j|)
     & !p2:?(!j$?state)
     & (   !arg+-1*!p2:?p2:<0
         & !p2+!MOD:?p2
       |
       )
     & !(!j$state):?arg
     & !i+1:?i
     & !j+21:?j
     )
 & 0:?s1:?i
 & 24:?sj
 &   whl
   ' ( !i:<165
     & subrand$
     & !i+1:?i
     ));

(subrand=

. (!si:!sj&subrand-seed$0|)

 & (!si:>0&!si+-1|54):?si
 & (!sj:>0&!sj+-1|54):?sj
 & (   !(!si$state)+-1*!(!sj$state):?x:<0
     & !x+!MOD:?x
   |
   )
 & !x:?(!si$?state));

(Main=

. subrand-seed$292929

 & 0:?i
 &   whl
   ' ( !i:<10
     & out$(subrand$)
     & !i+1:?i
     ));

Main$;</lang>

Output:

C

This is basically the same as the reference C code, only differs in that it's C89. <lang c>#include<stdio.h>

define MOD 1000000000

int state[55], si = 0, sj = 0;

int subrand();

void subrand_seed(int p1) { int i, j, p2 = 1;

state[0] = p1 % MOD; for (i = 1, j = 21; i < 55; i++, j += 21) { if (j >= 55) j -= 55; state[j] = p2; if ((p2 = p1 - p2) < 0) p2 += MOD; p1 = state[j]; } si = 0; sj = 24; for (i = 0; i < 165; i++) subrand(); }

int subrand() { int x; if (si == sj) subrand_seed(0);

if (!si--) si = 54; if (!sj--) sj = 54; if ((x = state[si] - state[sj]) < 0) x += MOD;

return state[si] = x; }

int main() { subrand_seed(292929); int i; for (i = 0; i < 10; i++) printf("%d\n", subrand());

return 0; }</lang>

C++

Library: Boost

<lang cpp> // written for clarity not efficiency.

include <iostream>

using std::cout; using std::endl;

include <boost/array.hpp>
include <boost/circular_buffer.hpp>

class Subtractive_generator { private:

   static const int param_i = 55;
   static const int param_j = 24;
   static const int initial_load = 219;
   static const int mod = 1e9;
   boost::circular_buffer<int> r;

public:

   Subtractive_generator(int seed);
   int next(); 
   int operator()(){return next();}

};

Subtractive_generator::Subtractive_generator(int seed)

r(param_i)

{

   boost::array<int, param_i> s;
   s[0] = seed;
   s[1] = 1;
   for(int n = 2; n < param_i; ++n){
       int t = s[n-2]-s[n-1];
       if (t < 0 ) t+= mod;
       s[n] = t;
   }

   for(int n = 0; n < param_i; ++n){

int i = (34 * (n+1)) % param_i;

       r.push_back(s[i]);
   }
   for(int n = param_i; n <= initial_load; ++n) next();

}

int Subtractive_generator::next() {

   int t = r[0]-r[31];
   if (t < 0) t += mod;
   r.push_back(t);
   return r[param_i-1];

}

int main() {

   Subtractive_generator rg(292929);

   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;
   cout << "result = " << rg() << endl;

   return 0;

} </lang>

Output:

result = 467478574
result = 512932792
result = 539453717
result = 20349702
result = 615542081
result = 378707948
result = 933204586

C#

<lang csharp> public class SubtractiveGenerator {

   public static int MAX = 1000000000;
   private int[] state;
   private int pos;

   private int mod(int n) {
       return ((n % MAX) + MAX) % MAX;
   }

   public SubtractiveGenerator(int seed) {
       state = new int[55];

       int[] temp = new int[55];
       temp[0] = mod(seed);
       temp[1] = 1;
       for(int i = 2; i < 55; ++i)
           temp[i] = mod(temp[i - 2] - temp[i - 1]);

       for(int i = 0; i < 55; ++i)
           state[i] = temp[(34 * (i + 1)) % 55];

       pos = 54;
       for(int i = 55; i < 220; ++i)
           next();
   }

   public int next() {
       int temp = mod(state[(pos + 1) % 55] - state[(pos + 32) % 55]);
       pos = (pos + 1) % 55;
       state[pos] = temp;
       return temp;
   }

   static void Main(string[] args) {
       SubtractiveGenerator gen = new SubtractiveGenerator(292929);
       for(int i = 220; i < 230; ++i)
           Console.WriteLine(i.ToString() + ": " + gen.next().ToString());
   }

} </lang>

Output:

220: 467478574
221: 512932792
222: 539453717
223: 20349702
224: 615542081
225: 378707948
226: 933204586
227: 824858649
228: 506003769
229: 380969305

Clojure

<lang clojure>(defn xpat2-with-seed

 "produces an xpat2 function initialized from seed"
 [seed]
 (let [e9 1000000000
       fs (fn i j [j (mod (- i j) e9)])
       s (->> [seed 1] (iterate fs) (map first) (take 55) vec)
       rinit (map #(-> % inc (* 34) (mod 55) s) (range 55))
       r-atom (atom [54 (int-array rinit)])
       update (fn nprev r
                 (let [n (-> nprev inc (mod 55))
                       rx #(get r (-> n (- %) (mod 55)))
                       rn (-> (rx 55) (- (rx 24)) (mod e9))
                       _ (aset-int r n rn)]
                   [n r]))
       xpat2 #(let [[n r] (swap! r-atom update)]
               (get r n))
       _ (dotimes [_ 165] (xpat2))]
   xpat2))

(def xpat2 (xpat2-with-seed 292929))

(println (xpat2) (xpat2) (xpat2)) ; prints: 467478574 512932792 539453717 </lang>

Common Lisp

<lang lisp>(defun sub-rand (state)

 (let ((x (last state)) (y (last state 25)))
   ;; I take "circular buffer" very seriously (until some guru
   ;; points out it's utterly wrong thing to do)
   (setf (cdr x) state)
   (lambda () (setf x (cdr x)

y (cdr y) (car x) (mod (- (car x) (car y)) (expt 10 9))))))

returns an RNG with Bentley seeding

(defun bentley-clever (seed)

 (let ((s (list 1 seed))  f)
   (dotimes (i 53)
     (push (mod (- (cadr s) (car s)) (expt 10 9)) s))
   (setf f (sub-rand

(loop for i from 1 to 55 collect (elt s (- 54 (mod (* 34 i) 55))))))

   (dotimes (x 165) (funcall f))
   f))

test it (output same as everyone else's)

(let ((f (bentley-clever 292929)))

 (dotimes (x 10) (format t "~a~%" (funcall f))))</lang>

D

Translation of: C

<lang d>import std.stdio;

struct Subtractive {

   enum MOD = 1_000_000_000;
   private int[55] state;
   private int si, sj;

   this(in int p1) pure nothrow {
       subrandSeed(p1);
   }

   void subrandSeed(int p1) pure nothrow {
       int p2 = 1;

       state[0] = p1 % MOD;
       for (int i = 1, j = 21; i < 55; i++, j += 21) {
           if (j >= 55)
               j -= 55;
           state[j] = p2;
           if ((p2 = p1 - p2) < 0)
               p2 += MOD;
           p1 = state[j];
       }

       si = 0;
       sj = 24;
       foreach (i; 0 .. 165)
           subrand();
   }

   int subrand() pure nothrow {
       if (si == sj)
           subrandSeed(0);

       if (!si--)
           si = 54;
       if (!sj--)
           sj = 54;

       int x = state[si] - state[sj];
       if (x < 0)
           x += MOD;

       return state[si] = x;
   }

}

void main() {

   auto gen = Subtractive(292_929);
   foreach (i; 0 .. 10)
       writeln(gen.subrand());

}</lang>

Output:

dc

<lang dc>[*

* (seed) lsx --
* Seeds the subtractive generator.
* Uses register R to hold the state.
*]sz

[

[* Fill ring buffer R[0] to R[54]. *]sz
d 54:R SA              [A = R[54] = seed]sz
1 d 33:R SB            [B = R[33] = 1]sz
12 SC                  [C = index 12, into array R.]sz
[55 -]SI
[                      [Loop until C is 54:]sz
 lA lB - d lC:R         [R[C] = A - B]sz
 lB sA sB               [Parallel let A = B and B = A - B]sz
 lC 34 + d 55 !>I d sC  [C += 34 (mod 55)]sz
 54 !=L
]d SL x
[* Point R[55] and R[56] into ring buffer. *]sz
0 55:R                 [R[55] = index 0, of 55th last number.]sz
31 56:R                [R[56] = index 31, of 24th last number.]sz
[* Stir ring buffer. *]sz
165 [                  [Loop 165 times:]sz
 55;R;R 56;R;R - 55;R:R [Discard a random number.]sz
 55;R 1 + d 55 !>I 55:R [R[55] += 1 (mod 55)]sz
 56;R 1 + d 55 !>I 56:R [R[56] += 1 (mod 55)]sz
 1 - d 0 <L
]d sL x
LAsz LBsz LCsz LIsz LLsz

]ss

[*

* lrx -- (random number from 0 to 10^9 - 1)
* Returns the next number from the subtractive generator.
* Uses register R, seeded by lsx.
*]sz

[

55;R;R 56;R;R -        [R[R[55]] - R[R[56]] is next random number.]sz
d 55;R:R               [Put it in R[R[55]]. Also leave it on stack.]sz
[55 -]SI
55;R 1 + d 55 !>I 55:R [R[55] += 1 (mod 55)]sz
56;R 1 + d 55 !>I 56:R [R[56] += 1 (mod 55)]sz
[1000000000 +]sI
1000000000 % d 0 >I    [Random number = it (mod 10^9)]sz
LIsz

]sr

[* Seed with 292929 and print first three random numbers. *]sz 292929 lsx lrx psz lrx psz lrx psz</lang>

This program prints 467478574, 512932792, 539453717.

This implementation never uses multiplication, but it does use modulus (remainder from division) to put each random number in range from 0 to 10^9 - 1.

Elixir

Translation of: Ruby

<lang elixir>defmodule Subtractive do

 def new(seed) when seed in 0..999_999_999 do
   s = Enum.reduce(1..53, [1, seed], fn _,[a,b|_]=acc -> [b-a | acc] end)
       |> Enum.reverse
       |> List.to_tuple
   state = for i <- 1..55, do: elem(s, rem(34*i, 55))
   {:ok, _pid} = Agent.start_link(fn -> state end, name: :Subtractive)
   Enum.each(1..220, fn _ -> rand end) # Discard first 220 elements of sequence.
 end
 
 def rand do
   state = Agent.get(:Subtractive, &(&1))
   n = rem(Enum.at(state, -55) - Enum.at(state, -24) + 1_000_000_000, 1_000_000_000)
   :ok = Agent.update(:Subtractive, fn _ -> tl(state) ++ [n] end)
   hd(state)
 end

end

Subtractive.new(292929) for _ <- 1..10, do: IO.puts Subtractive.rand</lang>

Output:

F#

Similar to Haskell, using lazy evaluation.

<lang fsharp>[<EntryPoint>] let main argv =

   let m = 1000000000
   let init = Seq.unfold (fun ((i, s2, s1)) -> Some((s2,i), (i+1, s1, (m+s2-s1)%m))) (0, 292929, 1)
           |> Seq.take 55
           |> Seq.sortBy (fun (_,i) -> (34*i+54)%55)
           |> Seq.map fst
   let rec r = seq {
       yield! init
       yield! Seq.map2 (fun u v -> (m+u-v)%m) r (Seq.skip 31 r)
   }
       
   r |> Seq.skip 220 |> Seq.take 3
   |> Seq.iter (printfn "%d")
   0</lang>

Output:

467478574
512932792
539453717

Fortran

Works with: Fortran version 90 and later

<lang fortran>module subgenerator

 implicit none

 integer, parameter :: modulus = 1000000000
 integer :: s(0:54), r(0:54)

contains

subroutine initgen(seed)

 integer :: seed
 integer :: n, rnum

 s(0) = seed
 s(1) = 1

 do n = 2, 54
   s(n) = mod(s(n-2) - s(n-1), modulus)
   if (s(n) < 0) s(n) = s(n) + modulus
 end do
  
 do n = 0, 54
   r(n) = s(mod(34*(n+1), 55))
 end do

 do n = 1, 165
   rnum = subrand()
 end do

end subroutine initgen

integer function subrand()

 integer, save :: p1 = 0
 integer, save :: p2 = 31

 r(p1) = mod(r(p1) - r(p2), modulus)
 if (r(p1) < 0) r(p1) = r(p1) + modulus
 subrand = r(p1)
 p1 = mod(p1 + 1, 55)
 p2 = mod(p2 + 1, 55)

end function subrand end module subgenerator

program subgen_test

 use subgenerator
 implicit none

 integer :: seed = 292929
 integer :: i
 
 call initgen(seed)
 do i = 1, 10
   write(*,*) subrand()
 end do

end program</lang>

Output:

Go

<lang go>package main

import (

   "fmt"
   "os"

)

// A fairly close port of the Bentley code, but parameterized to better // conform to the algorithm description in the task, which didn't assume // constants for i, j, m, and seed. also parameterized here are k, // the reordering factor, and s, the number of intial numbers to discard, // as these are dependant on i. func newSG(i, j, k, s, m, seed int) func() int {

   // check parameters for range and mutual consistency
   assert(i > 0, "i must be > 0")
   assert(j > 0, "j must be > 0")
   assert(i > j, "i must be > j")
   assert(k > 0, "k must be > 0")
   p, q := i, k
   if p < q {
       p, q = q, p
   }
   for q > 0 {
       p, q = q, p%q
   }
   assert(p == 1, "k, i must be relatively prime")
   assert(s >= i, "s must be >= i")
   assert(m > 0, "m must be > 0")
   assert(seed >= 0, "seed must be >= 0")
   // variables for closure f
   arr := make([]int, i)
   a := 0
   b := j
   // f is Bently RNG lprand
   f := func() int {
       if a == 0 {
           a = i
       }
       a--
       if b == 0 {
           b = i
       }
       b--
       t := arr[a] - arr[b]
       if t < 0 {
           t += m
       }
       arr[a] = t
       return t
   }
   // Bentley seed algorithm sprand
   last := seed
   arr[0] = last
   next := 1
   for i0 := 1; i0 < i; i0++ {
       ii := k * i0 % i
       arr[ii] = next
       next = last - next
       if next < 0 {
           next += m
       }
       last = arr[ii]
   }
   for i0 := i; i0 < s; i0++ {
       f()
   }
   // return the fully initialized RNG
   return f

}

func assert(p bool, m string) {

   if !p {
       fmt.Println(m)
       os.Exit(1)
   }

}

func main() {

   // 1st test case included in program_tools/universal.c.
   // (2nd test case fails.  A single digit is missing, indicating a typo.)
   ptTest(0, 1, []int{921674862, 250065336, 377506581})

   // reproduce 3 values given in task description
   skip := 220
   sg := newSG(55, 24, 21, skip, 1e9, 292929)
   for n := skip; n <= 222; n++ {
       fmt.Printf("r(%d) = %d\n", n, sg())
   }

}

func ptTest(nd, s int, rs []int) {

   sg := newSG(55, 24, 21, 220+nd, 1e9, s)
   for _, r := range rs {
       a := sg()
       if r != a {
           fmt.Println("Fail")
           os.Exit(1) 
       }
   }

}</lang>

Output:

r(220) = 467478574
r(221) = 512932792
r(222) = 539453717

Haskell

<lang haskell>subtractgen seed = drop 220 out where out = mmod $ r ++ zipWith (-) out (drop 31 out) where r = take 55 $ shuffle $ cycle $ take 55 s shuffle x = head xx:shuffle xx where xx = drop 34 x s = mmod $ seed:1:zipWith (-) s (tail s) mmod = map (`mod` 10^9)

main = mapM_ print $ take 10 $ subtractgen 292929</lang>

Output:

Icon and Unicon

<lang Icon>procedure main()

  every 1 to 10 do 
     write(rand_sub(292929))

end

procedure rand_sub(x) static ring,m

  if /ring then {
     m := 10^9
     every (seed | ring) := list(55)
     seed[1] := \x | ?(m-1)
     seed[2] := 1
     every seed[n := 3 to 55] := (seed[n-2]-seed[n-1])%m
     every ring[(n := 0 to 54) + 1] := seed[1 + (34 * (n + 1)%55)]
     every  n := *ring to 219 do {
        ring[1] -:= ring[-24]    
        ring[1] %=  m
        put(ring,get(ring))     
        }
  }
  ring[1] -:= ring[-24]
  ring[1] %:= m
  if ring[1] < 0 then ring[1] +:= m
  put(ring,get(ring))
  return ring[-1]

end</lang>

Output:

J

sg.ijs

Loops are hidden in a generalized power conjunction ^: . f^:n y evaluates f n times, as in f(f(f(...f(y)))...) . Yes! f^:(-1) IS the inverse of f . When known.

<lang J>came_from_locale_sg_=: coname cocurrent'sg' NB. install the state of rng sg into locale sg

SEED=: 292929 'I J M first_Bentley_number B2'=: 55 24 1e9 34 165 SG=: 1 : 'M&|@:-/@:(m&{)' r=: (I|(first_Bentley_number*>:i.I)) { (, _2 _1 SG)^:(I-2) 1,~SEED

sg=: 3 : 0 t=. (, (-I,J)SG)^:y r r=: y }. t t {.~ -y ) discard=. sg B2

cocurrent came_from_locale NB. return to previous locale sg=: sg_sg_ NB. make a local name for sg in locale sg </lang>

Use: <lang sh>$ jconsole

  load'sg.ijs'
  sg 2

467478574 512932792

  sg 4

539453717 20349702 615542081 378707948

</lang>

Java

Translation of C via D

Works with: Java version 8

<lang java>import java.util.function.IntSupplier; import static java.util.stream.IntStream.generate;

public class SubtractiveGenerator implements IntSupplier {

   static final int MOD = 1_000_000_000;
   private int[] state = new int[55];
   private int si, sj;

   public SubtractiveGenerator(int p1) {
       subrandSeed(p1);
   }

   void subrandSeed(int p1) {
       int p2 = 1;

       state[0] = p1 % MOD;
       for (int i = 1, j = 21; i < 55; i++, j += 21) {
           if (j >= 55)
               j -= 55;
           state[j] = p2;
           if ((p2 = p1 - p2) < 0)
               p2 += MOD;
           p1 = state[j];
       }

       si = 0;
       sj = 24;
       for (int i = 0; i < 165; i++)
           getAsInt();
   }

   @Override
   public int getAsInt() {
       if (si == sj)
           subrandSeed(0);

       if (si-- == 0)
           si = 54;
       if (sj-- == 0)
           sj = 54;

       int x = state[si] - state[sj];
       if (x < 0)
           x += MOD;

       return state[si] = x;
   }

   public static void main(String[] args) {
       generate(new SubtractiveGenerator(292_929)).limit(10)
               .forEach(System.out::println);
   }

}</lang>

Mathematica

<lang Mathematica>initialize[n_] :=

Module[{buffer},
 buffer = 
  Join[Nest[Flatten@{#, Mod[Subtract @@ #-2 ;;, 10^9]} &, {n, 1},
      53][[1 + Mod[34 Range@54, 55]]], {n}];
 Nest[nextValue, buffer, 165]]
 
 nextValue[buffer_] := 
Flatten@{Rest@buffer, Mod[Subtract @@ buffer[[{1, 32}]], 10^9]}</lang>

buffer = initialize[292929];
Do[Print@Last[buffer = nextValue[buffer]], {10}]

467478574
512932792
539453717
20349702
615542081
378707948
933204586
824858649
506003769
380969305

OCaml

Translation of: C

<lang ocaml>let _mod = 1_000_000_000 let state = Array.create 55 0 let si = ref 0 let sj = ref 0

let rec subrand_seed _p1 =

 let p1 = ref _p1 in
 let p2 = ref 1 in
 state.(0) <- !p1 mod _mod;
 let j = ref 21 in
 for i = 1 to pred 55 do
   if !j >= 55 then j := !j - 55;
   state.(!j) <- !p2;
   p2 := !p1 - !p2;
   if !p2 < 0 then p2 := !p2 + _mod;
   p1 := state.(!j);
   j := !j + 21;
 done;
 si := 0;
 sj := 24;
 for i = 0 to pred 165 do ignore (subrand()) done

and subrand() =

 if !si = !sj then subrand_seed 0;
 decr si;  if !si < 0 then si := 54;
 decr sj;  if !sj < 0 then sj := 54;
 let x = state.(!si) - state.(!sj) in
 let x = if x < 0 then x + _mod else x in
 state.(!si) <- x;
 (x)

let () =

 subrand_seed 292929;
 for i = 1 to 10 do Printf.printf "%d\n" (subrand()) done</lang>

Output:

$ ocaml sub_gen.ml
467478574
512932792
539453717
20349702
615542081
378707948
933204586
824858649
506003769
380969305

PARI/GP

<lang parigp>sgv=vector(55,i,random(10^9));sgi=1; sg()=sgv[sgi=sgi%55+1]=(sgv[sgi]-sgv[(sgi+30)%55+1])%10^9</lang>

Perl

<lang perl>use 5.10.0; use strict;

{ # bracket state data into a lexical scope my @state; my $mod = 1_000_000_000;

sub bentley_clever { my @s = ( shift() % $mod, 1); push @s, ($s[-2] - $s[-1]) % $mod while @s < 55; @state = map($s[(34 + 34 * $_) % 55], 0 .. 54); subrand() for (55 .. 219); }

sub subrand() { bentley_clever(0) unless @state; # just incase

my $x = (shift(@state) - $state[-24]) % $mod; push @state, $x; $x; } }

bentley_clever(292929); say subrand() for (1 .. 10);</lang>

Output:

Perl 6

Translation of: Perl

Works with: niecza

Works with: rakudo version nom

<lang perl6>sub bentley_clever($seed) {

   constant $mod = 1_000_000_000;
   my @seeds = ($seed % $mod, 1, (* - *) % $mod ... *)[^55];
   my @state = @seeds[ 34, (* + 34 ) % 55 ... 0 ];

   sub subrand() {
       push @state, (my $x = (@state.shift - @state[*-24]) % $mod);
       $x;
   }

   subrand for 55 .. 219;

   &subrand ... *;

}

my @sr := bentley_clever(292929); .say for @sr[^10];</lang> Here we just make the seeder return the random sequence as a lazy list.

Output:

PicoLisp

Using a circular list (as a true "ring" buffer). <lang PicoLisp>(setq

  *Bentley (apply circ (need 55))
  *Bentley2 (nth *Bentley 32) )

(de subRandSeed (S)

  (let (N 1  P (nth *Bentley 55))
     (set P S)
     (do 54
        (set (setq P (nth P 35)) N)
        (when (lt0 (setq N (- S N)))
           (inc 'N 1000000000) )
        (setq S (car P)) ) )
  (do 165 (subRand)) )

(de subRand ()

  (when (lt0 (dec *Bentley (pop '*Bentley2)))
     (inc *Bentley 1000000000) )
  (pop '*Bentley) )</lang>

Test: <lang PicoLisp>(subRandSeed 292929) (do 7 (println (subRand)))</lang>

Output:

PL/I

<lang PL/I> subtractive_generator: procedure options (main);

  declare (r, s) (0:54) fixed binary (31);
  declare (i, n, seed)  fixed binary (31);

  /* Bentley's initialization */
  seed = 292929;
  s(0) = seed; s(1) = 1;

  /* Compute s2,s3,...,s54 using the subtractive formula sn = s(n-2) - s(n-1)(mod 10**9). */
  do n = 2 to hbound(s,1);
     s(n) = mod ( s(n-2) - s(n-1), 1000000000);
  end;

  /* Rearrange initial values. */
  do n = 0 to hbound(r,1);
     r(n) = s( mod(34*(n+1), 55));
  end;

  do n = 55 to 219;
     i = mod (n, 55);
     r(i) = mod ( r(mod(n-55, 55)) - r(mod(n-24, 55)), 1000000000);
  end;

  do n = 220 to 235;
     i = mod(n, 55);
     r(i) = mod ( r(mod(n-55, 55)) - r(mod(n-24, 55)), 1000000000);
     put skip list (r(i));
  end;

end subtractive_generator; </lang>

Required 3 results:
     467478574
     512932792 
     539453717
Subsequent values:
      20349702 
     615542081 
     378707948 
     933204586 
     824858649 
     506003769 
     380969305 
     442823364 
     994162810 
     261423281 
     139610325 
      80746560 
     563900213

PowerShell

Works with: PowerShell version 2

The so-called modulus operator in PowerShell (%) returns a remainder not a modulus. Hence the need for the custom Mod function when working with negative numbers. ( X % M + M ) % M can be replaced with ( X + M ) % M when X is always between -M and M, as is the case in this task, but the former is used for clarity. The first 55 generated values are placed directly into their reordered slots in the ring. An array object is used along with a rotating index object to simulate a ring. <lang PowerShell> function Get-SubtractiveRandom ( [int]$Seed )

   {
   function Mod ( [int]$X, [int]$M = 1000000000 ) { ( $X % $M + $M ) % $M }

   If ( $Seed )
       {
       $R = New-Object int[] 55

       $N1 = 55 - 1
       $N2 = ( $N1 + 34 ) % 55

       $R[$N1] = $Seed
       $R[$N2] = 1

       ForEach ( $x in 2..(55-1) )
           {
           $N0, $N1, $N2 = $N1, $N2, ( ( $N2 + 34 ) % 55 )
           $R[$N2] = Mod ( $R[$N0] - $R[$N1] )
           }

       $i = -55 - 1
       $j = -24 - 1

       ForEach ( $x in 55..219 )
           {
           $i = ++$i % 55
           $j = ++$j % 55
           $R[$i] = Mod ( $R[$i] - $R[$j] )
           }

       $Script:RandomRing  = $R
       $Script:RandomIndex = $i
       }

   $i = $Script:RandomIndex = ++$Script:RandomIndex % 55
   $j = ( $i + 55 - 24 ) % 55

   return ( $Script:RandomRing[$i] = Mod ( $Script:RandomRing[$i] - $Script:RandomRing[$j] ) )
   }

Get-SubtractiveRandom 292929 Get-SubtractiveRandom Get-SubtractiveRandom Get-SubtractiveRandom Get-SubtractiveRandom </lang>

Output:

Python

Python: With explanation

Uses collections.deque as a ring buffer

<lang python> import collections s= collections.deque(maxlen=55)

Start with a single seed in range 0 to 10**9 - 1.

seed = 292929

Set s0 = seed and s1 = 1.
The inclusion of s1 = 1 avoids some bad states
(like all zeros, or all multiples of 10).

s.append(seed) s.append(1)

Compute s2,s3,...,s54 using the subtractive formula
sn = s(n - 2) - s(n - 1)(mod 10**9).

for n in xrange(2, 55):

   s.append((s[n-2] - s[n-1]) % 10**9)

Reorder these 55 values so r0 = s34, r1 = s13, r2 = s47, ...,
rn = s(34 * (n + 1)(mod 55)).

r = collections.deque(maxlen=55) for n in xrange(55):

   i = (34 * (n+1)) % 55
   r.append(s[i])

This is the same order as s0 = r54, s1 = r33, s2 = r12, ...,
sn = r((34 * n) - 1(mod 55)).
This rearrangement exploits how 34 and 55 are relatively prime.
Compute the next 165 values r55 to r219. Store the last 55 values.

def getnextr():

   """get next random number"""
   r.append((r[0]-r[31])%10**9)
   return r[54]

rn = r(n - 55) - r(n - 24)(mod 10**9) for n >= 55

for n in xrange(219 - 54):

   getnextr()

now fully initilised
print first five numbers

for i in xrange(5):

   print "result = ", getnextr()

</lang>

Python: As a class within a module

Python 2 and 3 compatable. <lang python>import collections

_ten2nine = 10**9

class Subtractive_generator():

   def __init__(self, seed=292929):
       self.r = collections.deque(maxlen=55)
       s = collections.deque(maxlen=55)
       s.extend([seed, 1])
       s.extend((s[n-2] - s[n-1]) % _ten2nine for n in range(2, 55))
       self.r.extend(s[(34 * (n+1)) % 55] for n in range(55)) 
       for n in range(219 - 54):
           self()
    
   def __call__(self):
       r = self.r
       r.append((r[0] - r[31]) % _ten2nine)
       return r[54]

if __name__ == '__main__':

   srand = Subtractive_generator()
   print([srand() for i in range(5)])</lang>

Output:

[467478574, 512932792, 539453717, 20349702, 615542081]

Racket

<lang Racket>#lang racket (define (make-initial-state a-list max-i)

 (for/fold ((state a-list))
           ((i (in-range (length a-list) max-i)))
   (append state (list (- (list-ref state (- i 2)) (list-ref state (- i 1))))))) ;from the seed and 1 creates the initial state

(define (shuffle a-list)

 (for/list ((i (in-range (length a-list))))
   (list-ref a-list (modulo (* 34 (add1 i)) 55))))  ;shuffles the state

(define (advance-state state (times 1))

 (cond ((= 0 times) state)
       (else (advance-state
              (cdr (append state
                           (list (modulo (- (list-ref state 0) (list-ref state 31))
                                         (expt 10 9)))))
                            (sub1 times)))))  ;takes a state and the times it must be advanced, and returns the new state

(define (create-substractive-generator s0)

 (define s1 1)
 (define first-state (make-initial-state (list s0 s1) 55))
 (define shuffled-state (shuffle first-state))
 (define last-state (advance-state shuffled-state 165))
 (lambda ((m (expt 10 9)))
   (define new-state (advance-state last-state))
   (set! last-state new-state)
   (modulo (car (reverse last-state)) m)))                    ;the lambda is a function with an optional argument
                                                              ;that returns a new random number each time it's called

(define rand (create-substractive-generator 292929)) (build-list 3 (lambda (_) (rand))) ;returns a list made from the 3 wanted numbers</lang>

REXX

Translation of: PL/I

<lang rexx>/*REXX program uses a subtractive generator, and creates a sequence of random numbers. */ s.0=292929; s.1=1; billion=10**9 /* ◄────────┐ */ numeric digits 20; billion=1e9 /*same as─►─┘ */ cI=55; do i=2 to cI-1

                  s.i=mod(s(i-2) - s(i-1), billion)
                  end   /*i*/

Cp=34

                  do j=0    to cI-1
                  r.j=s(mod(cP*(j+1), cI))
                  end   /*j*/

m=219; Cj=24

                  do k=cI   to m;     _=k//cI
                  r._=mod(r(mod(k-cI, cI)) - r(mod(k-cJ, cI)), billion)
                  end   /*m*/

t=235

                  do n=m+1  to t;     _=n//cI
                  r._=mod(r(mod(n-cI, cI)) - r(mod(n-cJ, cI)), billion)
                  say   right(r._, 40)
                  end   /*n*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ mod: procedure; parse arg a,b; return ((a // b) + b) // b r: parse arg #; return r.# s: parse arg #; return s.#</lang> output when using the default input:

Ruby

This implementation aims for simplicity, not speed. SubRandom#rand pushes to and shifts from an array; this might be slower than a ring buffer. The seeding method must call rand 55 extra times (220 times instead of 165 times). The code also calls Ruby's modulus operator, which always returns a non-negative integer if the modulus is positive.

<lang ruby># SubRandom is a subtractive random number generator which generates

the same sequences as Bentley's generator, as used in xpat2.

class SubRandom

 # The original seed of this generator.
 attr_reader :seed

 # Creates a SubRandom generator with the given _seed_.
 # The _seed_ must be an integer from 0 to 999_999_999.
 def initialize(seed = Kernel.rand(1_000_000_000))
   (0..999_999_999).include? seed or
     raise ArgumentError, "seed not in 0..999_999_999"

   # @state = 55 elements.
   ary = [seed, 1]
   53.times { ary << ary[-2] - ary[-1] }
   @state = []
   34.step(1870, 34) {|i| @state << ary[i % 55] }

   220.times { rand }  # Discard first 220 elements of sequence.

   @seed = seed        # Save original seed.
 end

 # Duplicates internal state so SubRandom#dup never shares state.
 def initialize_copy(orig)
   @state = @state.dup
 end

 # Returns the next random integer, from 0 to 999_999_999.
 def rand
   @state << (@state[-55] - @state[-24]) % 1_000_000_000
   @state.shift
 end

end

rng = SubRandom.new(292929) p (1..3).map { rng.rand }</lang>

[467478574, 512932792, 539453717]

Seed7

<lang seed7>$ include "seed7_05.s7i";

const integer: MOD is 1000000000;

const type: subtractiveGenerator is new struct

   var array integer: state is [0 .. 54] times 0;
   var integer: si is 0;
   var integer: sj is 24;
 end struct;

const func integer: subrand (inout subtractiveGenerator: generator) is forward;

const func subtractiveGenerator: subrandSeed (in var integer: p1) is func

 result
   var subtractiveGenerator: generator is subtractiveGenerator.value;
 local
   var integer: p2 is 1;
   var integer: i is 0;
   var integer: j is 21;
 begin
   generator.state[0] := p1 mod MOD;
   for i range 1 to 54 do
     generator.state[j] := p2;
     p2 := (p1 - p2) mod MOD;
     p1 := generator.state[j];
     j := (j + 21) mod 55;
   end for;
   for i range 1 to 165 do
     ignore(subrand(generator));
   end for;
 end func;

const func integer: subrand (inout subtractiveGenerator: generator) is func

 result
   var integer: subrand is 0;
 begin
   if generator.si = generator.sj then
     generator := subrandSeed(0);
   end if;
   generator.si := pred(generator.si) mod 55;
   generator.sj := pred(generator.sj) mod 55;
   subrand := (generator.state[generator.si] - generator.state[generator.sj]) mod MOD;
   generator.state[generator.si] := subrand;
 end func;

const proc: main is func

 local
   var subtractiveGenerator: gen is subrandSeed(292929);
   var integer: i is 0;
 begin
   for i range 1 to 10 do
     writeln(subrand(gen));
   end for;
 end func;</lang>

Output:

Sidef

<lang ruby>class SubRandom(seed, state=[]) {

   const mod = 1_000_000_000;

   method init {
       var s = [seed % mod, 1];
       53.times {
           s.append((s[-2] - s[-1]) % mod);
       }
       state = s.range.map {|i| s[(34 + 34*i) % 55] };
       range(55, 219).each { self.subrand };
   }

   method subrand {
       var x = ((state.shift - state[-24]) % mod);
       state.append(x);
       return x;
   }

}

var r = SubRandom(292929); 10.times { say r.subrand };</lang>

Output:

Tcl

Translation of: C

<lang tcl>package require Tcl 8.5 namespace eval subrand {

   variable mod 1000000000 state [lrepeat 55 0] si 0 sj 0

   proc seed p1 {

global subrand::mod subrand::state subrand::si subrand::sj set p2 1 lset state 0 [expr {$p1 % $mod}] for {set i 1; set j 21} {$i < 55} {incr i; incr j 21} { if {$j >= 55} {incr j -55} lset state $j $p2 if {[set p2 [expr {$p1 - $p2}]] < 0} {incr p2 $mod} set p1 [lindex $state $j] } set si 0 set sj 24 for {set i 0} {$i < 165} {incr i} { gen }

   proc gen {} {

global subrand::mod subrand::state subrand::si subrand::sj if {$si == $sj} {seed 0} if {[incr si -1] < 0} {set si 54} if {[incr sj -1] < 0} {set sj 54} set x [expr {[lindex $state $si] - [lindex $state $sj]}] if {$x < 0} {incr x $mod} lset state $si $x return $x

}

subrand::seed 292929 for {set i 0} {$i < 10} {incr i} {

   puts [subrand::gen]

}</lang>

uBasic/4tH

<lang>Push 292929 : Gosub 100 : d = Pop()

For i = 1 To 10

 Push 0 : Gosub 100
 Print Pop()

100 s = Pop()

   If s = 0 Then
      p = (p + 1) % 55
      @(p) = @(p) - @((p + 31) % 55)
      If @(p) < 0 Then
         @(p) = @(p) + 1000000000
      Endif
      Push (@(p)) : Return
   Endif

   @(54) = s : @(33) = 1
   p = 12

   For i = 2 To 54
     @(p) = @((p + 42) % 55) - @((p + 21) % 55)
     If @(p) < 0 Then
         @(p) = @(p) + 1000000000
     Endif
     p = (p + 34) % 55
   Next

   For i = 55 To 219
       Push 0 : Gosub 100 : d = Pop()
   Next

   Push 0 : Return</lang>

Output:

467478574
512932792
539453717
20349702
615542081
378707948
933204586
824858649
506003769
380969305

0 OK, 0:864

zkl

Translation of: Icon and Unicon

Translation of: Python

Translation of: C

<lang zkl>fcn rand_sub(x){

  var ring=L(),m=(1e9).toInt();
  mod:='wrap(n){ if(n<0) n+m else n };
  if(not ring){
     seed:=L( (if(vm.numArgs) x else m-1), 1);
     foreach n in ([2 .. 54]){ seed.append((seed[n-2]-seed[n-1]):mod(_)) }
     foreach n in (55){ ring.append(seed[(34*(n+1))%55]) }
     do(220-ring.len()){ self.fcn() } // 165
  }
  ring.append((ring.pop(0)-ring[-24]):mod(_));
  return(ring[-1]);

}</lang> <lang zkl>do(4){ println(rand_sub(292929)) } //seed ignored after first call</lang>

Output: