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Help for a 64-bit RNG

 
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George Marsaglia



Joined: 19 Nov 2009
Posts: 5
Location: United States

PostPosted: Sun Nov 22, 2009 5:49 pm    Post subject: Help for a 64-bit RNG Reply with quote

I have recently, Nov 3:

RNGs: A Super KISS, sci.math, comp.lang.c, sci.crypt.

developed a 32-bit KISS RNG with immense period,
over 10^402575, great speed and excellent
performance on tests of randomness.
But it is not well-suited for extension to
64-bits or adaption to Fortran.

As 64-bit RNGs are becoming more in demand, I
have modified that recently proposed 32-bit one
to produce a 64-bit RNG with similar period,
this time a mere 10^397524, but with instructions
that should serve in diverse programming
languages.

I list a C version below and ask for help from
experts here on adapting it to a Fortran version.

The basic idea is that we add each term from
three sequences,
Congruential, Xorshift and CMWC
(Complimentary-Multiply-With-Carry).
For the latter, we merely take the next Q element,
except if they have all been used, the Q array is
refilled, the index reset and the first Q element
returned.
The refilling process should work for signed
or unsigned integers, as should the Congruential
and Xorshift elements.
The CMWC process requires that we extract the top
and bottom 64 bits of a supposed 128-bit t=a*x+c,
without actually forming t, using only 64-bit operations.

The instructions, operations and assignments themselves
all seem to have Fortran equivalents. The #include
macros are merely sequences of assignments that can be
transformed, and the Q() (nee Q[]) array must be
accessible to all, which I would have done in the
old days with common blocks.

Here is the C listing:

-----------------------------------------------------------
/* SUPRKISS64.c, period 5*2^1320480*(2^64-1) */
#include <stdio.h>
static unsigned long long Q[20632],carry=36243678541LL,
xcng=12367890123456LL,xs=521288629546311LL,indx=20632;

#define CNG ( xcng=6906969069LL*xcng+123 )
#define XS ( xs^=xs<<13,xs^=xs>>17,xs^=xs<<43 )
#define SUPR ( indx<20632 ? Q[indx++] : refill() )
#define KISS SUPR+CNG+XS

unsigned long long refill( )
{int i; unsigned long long z,h;
for(i=0;i<20635;i++){ h=(carry&1);
z=((Q[i]<<41)>>1)+((Q[i]<<39)>>1)+(carry>>1);
carry=(Q[i]>>23)+(Q[i]>>25)+(z>>63);
Q[i]=~((z<<1)+h); }
indx=1; return (Q[0]);
}

int main()
{int i; unsigned long long x;
for(i=0;i<20632;i++) Q[i]=CNG+XS;
for(i=0;i<1000000000;i++) x=KISS;
printf("Does x=-5061657174865032461\n x=%LLd.\n",x);
}
-----------------------------------------------------------


Given the initial seed values, a proper implementation
of the mathematics behind the methods requires that
the 10^9'th call return the 64-bit integer
13385086898844519155, which I have had printed as
-5061657174865032461 for signed-integer implementations.

Can I count on any of you to provide a Fortran version?

George Marsaglia


Here is the 32-bit version of the above:
----------------------------------------------------------
/* SUPRKISS32.c, period 5*2^1320481*(2^32-1) */
#include <stdio.h>
static unsigned long Q[41265],indx=41265,
carry=362,xcng=1236789,xs=521288629;

#define CNG ( xcng=69609*xcng+123 ) /*Congruential*/
#define XS ( xs^=xs<<13,xs^=xs>>17,xs^=xs>>5 )/*Xorshift*/
#define SUPR ( indx<41265 ? Q[indx++] : refill() ) /*CMWC*/
#define KISS SUPR+CNG+XS

int refill( )
{int i; unsigned long z,h;
for(i=0;i<41265;i++){ h=(carry&1);
z=((Q[i]<<9)>>1)+((Q[i]<<7)>>1)+(carry>>1);
carry=(Q[i]>>23)+(Q[i]>>25)+(z>>31);
Q[i]=~((z<<1)+h); }
indx=1; return (Q[0]);
}

int main( )
{unsigned long i,x;
for(i=0;i<41265;i++) Q[i]=CNG+XS;
for(i=0;i<1000000000;i++) x=KISS;
printf("Does x=1493864468?\n x=%d.\n",x);
}

-----------------------------------------------------------
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mecej4



Joined: 31 Oct 2006
Posts: 586

PostPosted: Mon Nov 23, 2009 3:31 am    Post subject: Fortran-90 conversion of Marsaglia SuperKiss64 RNG Reply with quote

Professor Marsaglia,

We appreciate your contributions to the art of RNGs.

This is a rather mechanical translation of your C code. It is likely to be slightly less efficient.

Code:

module suprkiss64_M
integer,parameter :: I8=selected_int_kind(18)       ! 10^18 fits into 8 bytes
integer(kind=I8) :: Q(20632),carry=36243678541_I8, &
   xcng=12367890123456_I8,xs=521288629546311_I8,indx=20633_I8
contains
function refill() result(s)
   integer(kind=I8) :: s
   integer :: i
   integer(kind=I8) :: z,h
   do i = 1,20635
      h = iand(carry,1_I8)
      z = ishft(ishft(Q(i),41),-1)+ &
          ishft(ishft(Q(i),39),-1)+ &
          ishft(carry,-1)
      carry=ishft(Q(i),-23)+ishft(Q(i),-25)+ishft(z,-63)
      Q(i)=not(ishft(z,1)+h)
   end do
   indx=2
   s=Q(1)
   return
end function refill

end module suprkiss64_M

program tskiss64
use suprkiss64_M
integer :: i
integer(kind=I8) :: x,supr
do i=1,20632
   xcng=xcng*6906969069_I8+123
   xs=ieor(xs,ishft(xs,13))
   xs=ieor(xs,ishft(xs,-17))
   xs=ieor(xs,ishft(xs,43))
   Q(i)=xcng+xs
end do
do i=1,1000000000_I8
   if(indx <= 20632)then
      supr=Q(indx)
      indx=indx+1
   else
      supr=refill()
   endif
   xcng=xcng*6906969069_I8+123
   xs=ieor(xs,ishft(xs,13))
   xs=ieor(xs,ishft(xs,-17))
   xs=ieor(xs,ishft(xs,43))
   x=xcng+xs+supr
end do
write(*,10)x
10 format(' Does x = -5061657174865032461 ?',/,6x,'x = ',I20)
end program tskiss64


Last edited by mecej4 on Sun Dec 04, 2016 6:56 pm; edited 1 time in total
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George Marsaglia



Joined: 19 Nov 2009
Posts: 5
Location: United States

PostPosted: Mon Nov 23, 2009 2:49 pm    Post subject: Reply with quote

Many thanks for that quick response.
From different errors I got after trying to compile
with both Silverfrost, (ftn95) and Nag (f95),
I guessed at a possible solution and changed the three
occurrences of (kind=8) to (kind=I8) (lines 7,9,28),
and got successful compilations for both.
The resulting NAG a.exe file ran and the
10^9th value was the same as that of the C
program and consistent with the underlying
mathematics.

However, on this Windows Vista PC running in
command mode, I was not able to link the .obj
file produced by the Silverfrost compiler.
Is there an easy way to ensure linkability?

As for Fortran versions of this SuperKISS64,
I would like to take advantage of statement
functions---for example, for the Xorshift operations
xs=ieor(xs,ishft(xs,13))
xs=ieor(xs,ishft(xs,-17))
xs=ieor(xs,ishft(xs,43))
I would use a statement function such as
m(y,k)=ieor(ishft(y,k))
so that the code for the Xorshift operation
would become
xs=m(m(m(xs,13),-17),43)

I liked statement functions in the old days when I used
Fortran, but have read that they are currently frowned on.
Would their use here be a bad choice?

If I could link with the Silverfrost compiler I would
have another comparison of speeds. The NAG a.exe file
took around 33 seconds, so that RNGs were produced at
around 30 million per second, whereas the gnu gcc compiler
took 20 seconds, for a rate of 50 million 64-bit random
integers per second.
(The 32-bit RNG took 10 secs for a rate of 100 million/sec.)

I plan to post a general description of this RNG at
math.sci and sites that seem likely to make it a useful
contribution, along with suggested code.
I am already grateful for the help here, would welcome
further comments that might lead to a better posting.


George Marsaglia


(And I was pleased with the output appearance of that newly
downloaded Silverfrost FTN95,...if I could only figure out
how to link the .obj file.)
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mecej4



Joined: 31 Oct 2006
Posts: 586

PostPosted: Mon Nov 23, 2009 7:54 pm    Post subject: Errors in first Fortran-9x version Reply with quote

Mea culpa.

I wrote up the code with "kind=8" for GFortran in Linux. I then did a manual conversion to kind=I8 for compilers whose kind number for 8 byte integers need not be 8. My conversion was incomplete. However, as you have already noted, the fix is easy -- replace all instances of integer(kind=Cool to integer(kind=ICool.

There is a problem with running the code using Salford Fortran. I have Windows XP rather than Vista. I can compile and link with no problems:

s:\RNG> ftn95 supks64.f90 /optim /p6
s:\RNG> slink supks64.obj

However, the program aborts right away with an integer overflow on the first executable statement

xcng=xcng*6906969069_I8+123

Further examination showed that Salford Fortran-95 implements arithmetic on 8-byte integers using the FPU. Nor could I find the proper compiler switches to turn off the overflow checking.

There are no problems running the code with other compilers. My best Fortran-90 timings are only 10 percent longer than my best C timings.

As for using an ASF, I feel that they make the code more readable; as they are on their way out, I'd stay from them in writing new code.
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mecej4



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Posts: 586

PostPosted: Mon Nov 23, 2009 8:50 pm    Post subject: Another minor bug in C and Fortran versions Reply with quote

The array Q is dimensioned [20632] in your original C version as well as my Fortran translation. The array is overrun in routine refill(), since the loop runs to 20635.

The two limits need to be reconciled.


Last edited by mecej4 on Sun Dec 04, 2016 7:08 pm; edited 1 time in total
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George Marsaglia



Joined: 19 Nov 2009
Posts: 5
Location: United States

PostPosted: Tue Nov 24, 2009 2:57 pm    Post subject: Reply with quote

Yes, my typing slip, 20635 rather than 20632, needs correction.
I have fixed it and got your Fortran version and the C version to
give the same results.
Learning of the -O4 option on NAG's f95,
I get the same +%10% time cost versus C's gcc -O3 compiler.

I am planning to post both C and Fortran versions on sites
that seem suitable, but I am still puzzled on a compatible
Fortran version of the 32-bit C listing.
Here I need only ensure that integer arithmetic is the old
standard mod 2^32, with positive residues for signed integers
and least-absolute residues for signed integers.

Can the default integer*4 still be used, and if not, which
selected_int_kind(?) will---or is likely to---ensure compatibility?
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mecej4



Joined: 31 Oct 2006
Posts: 586

PostPosted: Tue Nov 24, 2009 8:07 pm    Post subject: Fortran version of 32-bit Super-KISS RNG Reply with quote

Just as with the 64-bit RNG, the Fortran-90 version is straightforward. I have checked it out with both 32-bit and 64-bit Fortran compilers (both of which, however, have default 4-byte integers).

Code:

module suprkiss32_M
integer,parameter :: I4=selected_int_kind(9)       ! 10^9 fits into 4 bytes
integer(kind=I4),parameter :: QSIZ=41265_I4, &
                              CMUL=69609_I4, &
                              COFFS=123_I4
integer(kind=I4) :: Q(QSIZ),carry=362_I4, &
   xcng=1236789_I4,xs=521288629_I4,       &
   indx=QSIZ+1
contains
function refill() result(s)
   integer(kind=I4) :: s
   integer :: i
   integer(kind=I4) :: z,h
   do i = 1,QSIZ
      h = iand(carry,1_I4)
      z = ishft(ishft(Q(i),9),-1)+ &
          ishft(ishft(Q(i),7),-1)+ &
          ishft(carry,-1)
      carry=ishft(Q(i),-23)+ishft(Q(i),-25)+ishft(z,-31)
      Q(i)=not(ishft(z,1)+h)
   end do
   indx=2
   s=Q(1)
   return
end function refill

end module suprkiss32_M

program tskiss32
use suprkiss32_M
integer :: i
integer(kind=I4) :: x,supr
do i=1,QSIZ
   xcng=xcng*CMUL+COFFS
   xs=ieor(xs,ishft(xs,13))
   xs=ieor(xs,ishft(xs,-17))
   xs=ieor(xs,ishft(xs,-5))
   Q(i)=xcng+xs
end do
do i=1,1000000000_I4
   if(indx <= QSIZ)then
      supr=Q(indx)
      indx=indx+1
   else
      supr=refill()
   endif
   xcng=xcng*CMUL+COFFS
   xs=ieor(xs,ishft(xs,13))
   xs=ieor(xs,ishft(xs,-17))
   xs=ieor(xs,ishft(xs,-5))
   x=xcng+xs+supr
!   write(*,'(1x,i2,3I12,I14)')i,xcng,xs,supr,x
!   if(i.gt.5)stop
end do
write(*,10)x
10 format(' Does x = 1493864468 ?',/,6x,'x =',I11)
end program tskiss32
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George Marsaglia



Joined: 19 Nov 2009
Posts: 5
Location: United States

PostPosted: Wed Nov 25, 2009 3:14 pm    Post subject: Reply with quote

I have fashioned a 64-bit version patterned after
your model, with a few uses of semicolons to
permit multiple expressions on a line, and with
a KISS64 function in the module.
The 32-bit version is similar, with a KISS32 function.
Both are listed below.

Your suggested use of QSIZ,CMUL,COFFS looked attractive
as a way to make both versions look even more alike,
but I encountered problems, so stuck with specific values
for the 64 and 32 versions.

I plan to make available both 32- and 64-bit versions
for both C and Fortran to various newsgroups.
May I acknowledge your contribution to the Fortran versions?

Do you or other readers of this forum have any suggestions
for improvements or otherwise?

George Marsaglia

The Fortran listings:
[code:1:b0f6883a34]
module suprkiss64_M ! period 5*2^1320480*(2^64-1)
integer,parameter :: I8=selected_int_kind(18)
integer(kind=I8) :: Q(20632),carry=36243678541_I8, &
xcng=12367890123456_I8,xs=521288629546311_I8,indx=20633_I8
contains
function KISS64() result(x)
integer(kind=I8) :: x
if(indx <= 20632)then; x=Q(indx); indx=indx+1
else; x=refill(); endif
xcng=xcng*6906969069_I8+123
xs=ieor(xs,ishft(xs,13))
xs=ieor(xs,ishft(xs,-17))
xs=ieor(xs,ishft(xs,43))
x=x+xcng+xs
return; end function KISS64

function refill() result(s)
integer(kind=I8) :: i,s,z,h
do i=1,20632
h=iand(carry,1_I8)
z = ishft(ishft(Q(i),41),-1)+ &
ishft(ishft(Q(i),39),-1)+ &
ishft(carry,-1)
carry=ishft(Q(i),-23)+ishft(Q(i),-25)+ishft(z,-63)
Q(i)=not(ishft(z,1)+h)
end do
indx=2; s=Q(1)
return; end function refill

end module suprkiss64_M

program testKISS64
use suprkiss64_M
integer(kind=I8) :: i,x
do i=1,20632 !fill Q with Congruential+Xorshift
xcng=xcng*6906969069_I8+123
xs=ieor(xs,ishft(xs,13))
xs=ieor(xs,ishft(xs,-17))
xs=ieor(xs,ishft(xs,43))
Q(i)=xcng+xs
end do
do i=1,1000000000_I8; x=KISS64(); end do
write(*,10) x
10 format(' Does x = 4013566000157423768 ?',/,6x,'x = ',I20)
end program testKISS64


module suprkiss32_M ! period 5*2^1320481*(2^32-1)
integer,parameter :: I4=selected_int_kind(9)
integer(kind=I4) :: Q(41265),carry=362_I4, &
xcng=1236789_I4,xs=521288629_I4,indx=41266_I4
contains
function KISS32() result(x)
integer(kind=I4):: x
if(indx <= 41265)then;x=Q(indx); indx=indx+1
else; x=refill(); endif
xcng=xcng*69069_I4+123
xs=ieor(xs,ishft(xs,13))
xs=ieor(xs,ishft(xs,-17));
xs=ieor(xs,ishft(xs,5))
x=x+xcng+xs
return; end function KISS32

function refill() result(s)
integer(kind=I4) :: i,s,z,h
do i = 1,41265
h = iand(carry,1_I4)
z = ishft(ishft(Q(i),9),-1)+ &
ishft(ishft(Q(i),7),-1)+ &
ishft(carry,-1)
carry=ishft(Q(i),-23)+ishft(Q(i),-25)+ishft(z,-31)
Q(i)=not(ishft(z,1)+h)
end do
indx=2; s=Q(1)
return; end function refill

end module suprkiss32_M

program testKISS32
use suprkiss32_M
integer(kind=I4) :: i,x
do i=1,41265 !fill Q with Congruential+Xorshift
xcng=xcng*69069_I4+123
xs=ieor(xs,ishft(xs,13))
xs=ieor(xs,ishft(xs,-17)[/c
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mecej4



Joined: 31 Oct 2006
Posts: 586

PostPosted: Sun Dec 04, 2016 8:36 pm    Post subject: George Marsaglia's post revisited, with corrected source cod Reply with quote

Professor George Marsaglia (1924 - 2011), famous for his paper "Random numbers fall mainly in the planes" and other contributions, initiated this thread in 2009, when he was over 85 years old. See https://en.wikipedia.org/wiki/George_Marsaglia .

Today, I was brought back to this thread after a search for some information on random number generators. I noticed that the Forum daemons had truncated his last post, in mid line-of-code even, and that there were still some typos in the programs.

Here are corrected versions of his two programs. Both programs are written in the form of benchmark programs, so they need tweaking if you want to use the contained KISS RNGs in your work. He builds up a table using an RNG, and uses that table as a cache for generating the user's requested random numbers using the RNG along with lots of shuffling of bits and bytes. Note that these programs depend on integer overflow not being trapped, and are NOT standard-conforming. There are, however, useful under adult supervision.

The 32-bit RNG, which uses only 4-byte integers, and can be used with FTN95-32 or FTN95-64, follows.
[continued]


Last edited by mecej4 on Mon Dec 05, 2016 4:56 pm; edited 3 times in total
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PostPosted: Sun Dec 04, 2016 8:37 pm    Post subject: Reply with quote

[continuation] Here is the 32-bit RNG program of Marsaglia.
Code:
! Posted by the late Professor George Marsaglia to the Silverfrost General forum
! in 2009, see http://forums.silverfrost.com/viewtopic.php?t=1480 .
! Works correctly with FTN95 32-bit and 64-bit. With Gfortran 4.9 and later, must
! use the -ftrapv flag to avoid an optimizer bug.
!
    Module suprkiss32_m ! period 5*2^1320481*(2^32-1)
      Implicit None
      Integer, Parameter :: i4 = selected_int_kind(9) ! 10^9 fits into 4 bytes
      Integer (Kind=i4), Parameter :: QSIZ = 41265_i4, CMUL = 69609_i4, &
        coffs = 123_i4
      Integer (Kind=i4) :: q(41265), carry = 362_i4, xcng = 1236789_i4, &
        xs = 521288629_i4, indx = QSIZ+1
    Contains

      Function kiss32() Result (x)
        Implicit None
        Integer (Kind=i4) :: x

        If (indx<=qsiz) Then
          x = q(indx)
          indx = indx + 1
        Else
          x = refill()
        End If
        xcng = xcng*CMUL + COFFS
        xs = ieor(xs, ishft(xs,13))
        xs = ieor(xs, ishft(xs,-17))
        xs = ieor(xs, ishft(xs,-5))
        x = x + xcng + xs
        Return
      End Function kiss32

      Function refill() Result (s)
        Implicit None
        Integer (Kind=i4) :: i, s, z, h

        Do i = 1, qsiz
          h = iand(carry, 1_i4)
          z = ishft(ishft(q(i),9), -1) + ishft(ishft(q(i),7), -1) + &
            ishft(carry, -1)
          carry = ishft(q(i), -23) + ishft(q(i), -25) + ishft(z, -31)
          q(i) = not(ishft(z,1)+h)
        End Do
        indx = 2
        s = q(1)
        Return
      End Function refill

    End Module suprkiss32_m

    Program testkiss32
      Use suprkiss32_m
      Implicit None
      Integer (Kind=i4) :: i, x, supr

      Do i = 1, qsiz !fill Q with Congruential+Xorshift
        xcng = xcng*CMUL + COFFS
        xs = ieor(xs, ishft(xs,13))
        xs = ieor(xs, ishft(xs,-17))
        xs = ieor(xs, ishft(xs,-5))
        q(i) = xcng + xs
      End Do
      Do i = 1, 1000000000_i4
        x = kiss32()
      End Do
      Write (*, 100) x
100   Format (' Does x =  1493864468 ?', /, 6X, 'x =', I12)
    End Program testkiss32

[continued]


Last edited by mecej4 on Mon Dec 05, 2016 12:53 pm; edited 2 times in total
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PostPosted: Sun Dec 04, 2016 8:42 pm    Post subject: Reply with quote

[continuation]
Here is the 64 bit version of George Marsaglia's KISS RNG. It employs 8-byte integers, and works with FTN95-64. It does not work with FTN95-32 because INTEGER*8 arithmetic is implemented with X87 instructions and integer overflows cause FPU exceptions.
Code:
! Posted by the late Professor George Marsaglia to the Silverfrost General forum
! in 2009, see http://forums.silverfrost.com/viewtopic.php?t=1480 .
! Does not work with FTN95 32-bit, since it uses the X87 to process 8-byte integers
! Works correctly with FTN95 64-bit. With Gfortran 4.9 and later, must
! use the -ftrapv flag to avoid an optimizer bug.
!
    Module suprkiss64_m ! period 5*2^1320480*(2^64-1)
      Implicit None
      Integer, Parameter :: i8 = selected_int_kind(18)
      Integer, Parameter :: QSIZ = 20632
      Integer (Kind=i8), Parameter :: CMUL = 6906969069_i8, COFFS = 123_i8
      Integer (Kind=i8) :: q(QSIZ), carry = 36243678541_i8, &
        xcng = 12367890123456_i8, xs = 521288629546311_i8, indx = QSIZ+1_i8
    Contains

      Function kiss64() Result (x)
        Implicit None
        Integer (Kind=i8) :: x

        If (indx<=QSIZ) Then
          x = q(indx)
          indx = indx + 1
        Else
          x = refill()
        End If
        xcng = xcng*CMUL + COFFS
        xs = ieor(xs, ishft(xs,13))
        xs = ieor(xs, ishft(xs,-17))
        xs = ieor(xs, ishft(xs,43))
        x = x + xcng + xs
        Return
      End Function kiss64

      Function refill() Result (s)
        Implicit None
        Integer (Kind=i8) :: i, s, z, h

        Do i = 1, QSIZ
          h = iand(carry, 1_i8)
          z = ishft(ishft(q(i),41), -1) + ishft(ishft(q(i),39), -1) + &
            ishft(carry, -1)
          carry = ishft(q(i), -23) + ishft(q(i), -25) + ishft(z, -63)
          q(i) = not(ishft(z,1)+h)
        End Do
        indx = 2
        s = q(1)
        Return
      End Function refill

    End Module suprkiss64_m

    Program testkiss64
      Use suprkiss64_m
      Implicit None
      Integer (Kind=i8) :: i, x

      Do i = 1, qsiz !fill Q with Congruential+Xorshift
        xcng = xcng*CMUL + COFFS
        xs = ieor(xs, ishft(xs,13))
        xs = ieor(xs, ishft(xs,-17))
        xs = ieor(xs, ishft(xs,43))
        q(i) = xcng + xs
      End Do
      Do i = 1, 1000000000_i8
        x = kiss64()
      End Do
      Write (*, 100) x
100   Format (' Does x =  4013566000157423768 ?', /, 6X, 'x = ', I20)
    End Program testkiss64
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