Consistent with the Keep It Simple Stupid (KISS) principle, I have previously suggested 32-bit KISS Random Number Generators (RNGs) that seem to have been frequently adopted.
Having had requests for 64-bit KISSes, and now that 64-bit integers are becoming more available, I will describe here a 64-bit KISS RNG, with comments on implementation for various languages, speed, periods and performance after extensive tests of randomness.
This 64-bit KISS RNG has three components, each nearly good enough to serve alone. The components are: Multiply-With-Carry (MWC), period (2^121+2^63-1) Xorshift (XSH), period 2^64-1 Congruential (CNG), period 2^64
Compact C and Fortran listings are given below. They can be cut, pasted, compiled and run to see if, after 100 million calls, results agree with that provided by theory, assuming the default seeds.
Users may want to put the content in other forms, and, for general use, provide means to set the 250 seed bits required in the variables x,y,z (64 bits) and c (58 bits) that have been given default values in the test versions.
The C version uses #define macros to enumerate the few instructions that MWC, XSH and CNG require. The KISS macro adds MWC+XSH+CNG mod 2^64, so that KISS can be inserted at any place in a C program where a random 64-bit integer is required. Fortran's requirement that integers be signed makes the necessary code more complicated, hence a function KISS().
C version; test by invoking macro KISS 100 million times ----------------------------------------------------------------- #include <stdio.h>
static unsigned long long x=1234567890987654321ULL,c=123456123456123456ULL, y=362436362436362436ULL,z=1066149217761810ULL,t;
int main(void) {int i; for(i=0;i<100000000;i++) t=KISS; (t==1666297717051644203ULL) ? printf("100 million uses of KISS OK"): printf("Fail"); }
--------------------------------------------------------------- Fortran version; test by calling KISS() 100 million times --------------------------------------------------------------- program testkiss implicit integer*8(a-z) do i=1,100000000; t=KISS(); end do if(t.eq.1666297717051644203_8) then print*,"100 million calls to KISS() OK" else; print*,"Fail" end if; end
function KISS() implicit integer*8(a-z) data x,y,z,c /1234567890987654321_8, 362436362436362436_8,& 1066149217761810_8, 123456123456123456_8/ save x,y,z,c m(x,k)=ieor(x,ishft(x,k)) !statement function s(x)=ishft(x,-63) !statement function t=ishft(x,58)+c if(s(x).eq.s(t)) then; c=ishft(x,-6)+s(x) else; c=ishft(x,-6)+1-s(x+t); endif x=t+x y=m(m(m(y,13_8),-17_8),43_8) z=6906969069_8*z+1234567 KISS=x+y+z return; end ---------------------------------------------------------------
Output from using the macro KISS or the function KISS() is MWC+XSH+CNG mod 2^64.
CNG is easily implemented on machines with 64-bit integers, as arithmetic is automatically mod 2^64, whether integers are considered signed or unsigned. The CNG statement is z=6906969069*z+1234567. When I established the lattice structure of congruential generators in the 60's, a search produced 69069 as an easy- to-remember multiplier with nearly cubic lattices in 2,3,4,5- space, so I tried concatenating, using 6906969069 as my first test multiplier. Remarkably---a seemingly one in many hundreds chance---it turned out to also have excellent lattice structure in 2,3,4,5-space, so that's the one chosen. (I doubt if lattice structure of CNG has much influence on the composite 64-bit KISS produced via MWC+XSH+CNG mod 2^64.)
XSH, the Xorshift component, described in www.jstatsoft.org/v08/i14/paper uses three invocations of an integer "xor"ed with a shifted version of itself. The XSH component used for this KISS is, in C notation: y^=(y<<13); y^=(y>>17); y^=(y<<43) with Fortran equivalents y=ieor(y,ishft(y,13)), etc., although this can be effected by a Fortran statement function: f(y,k)=ieor(y,ishft(y,k)) y=f(f(f(y,13),-17),43) As with lattice structure, choice of the triple 13,-17,43 is probably of no particular importance; any one of the 275 full- period triples listed in the above article is likely to provide a satisfactory component XSH for the composite MWC+XSH+CNG.
The choice of multiplier 'a' for the multiply-with-carry (MWC) component of KISS is not so easily made. In effect, a multiply- with-carry sequence has a current value x and current "carry" c, and from each given x,c a new x,c pair is constructed by forming t=a*x+c, then x=t mod b=2^64 and c=floor(t/b). This is easily implemented for 32-bit computers that permit forming a*t+c in 64 bits, from which the new x is the bottom and the new c the top 32-bits.
When a,x and c are 64-bits, not many computers seem to have an easy way to form t=a*x+c in 128 bits, then extract the top and bottom 64-bit segments. For that reason, special choices for 'a' are needed among those that satisfy the general requirement that p=a*b-1 is a prime for which b=2^64 has order (p-1)/2.
My choice---and the only one of this form---is a=2^58+1. Then the top 64 bits of an imagined 128-bit t=a*x+c may be obtained as (using C notation) (x>>6)+ 1 or 0, depending on whether the 64-bit parts of (x<<58)+c+x cause an overflow. Since (x<<58)+c cannot itself cause overflow (c will always be <a), we get the carry as c=(x>>6) plus overflow from (x<<58)+x.
This is easily done in C with unsigned integers, using a different kind of 't': t=(x<<58)+c; c=(x>>6); x=t+x; c=c+(x<t); For Fortran and others that permit only signed integers, more work is needed. Equivalent mod 2^64 versions of t=(x<<58)+c and c=(x>>6) are easy, and if s(x) represents (x>>63) in C or ishft(x,-66) in Fortran, then for signed integers, the new carry c comes from the rule if s(x) equals s(t) then c=(x>>6)+s(x) else c=(x>>6)+1-s(x+t)
Speed: A C version of this KISS RNG takes 18 nanosecs for each 64-bit random number on my desktop (Vista) PC, thus producing KISSes at a rate exceeding 55 million per second. Fortran or other integers-must-be-signed compilers might get "only" around 40 million per second.
Setting seeds: Use of KISS or KISS() as a general 64-bit RNG requires specifying 3*64+58=250 bits for seeds, 64 bits each for x,y,z and 58 for c, resulting in a composite sequence with period around 2^250. The actual period is (2^250+2^192+2^64-2^186-2^129)/6 ~= 2^(247.42) or 10^(74.48). We "lose" 1+1.58=2.58 bits from maximum possible period, one bit because b=2^64, a square, cannot be a primitive root of p=ab-1, so the best possible order for b is (p-1)/2. The periods of MWC and XSH have gcd 3=2^1.58, so another 1.58 bits are "lost" from the best possible period we could expect from 250 seed bits.
Some users may think 250 seed bits are an unreasonable requirement. A good seeding procedure might be to assume the default seed values then let the user choose none, one, two,..., or all of x,y,z, and c to be reseeded.
Tests: Latest tests in The Diehard Battery, available at http://i.cs.hku.hk/~diehard/ were applied extensively. Those tests that specifically required 32-bit integers were applied to the leftmost 32 bits (e,g, KISS>>32;), then to the middle 32-bits ((KISS<<16)>>32;) then to the rightmost 32 bits, ( (KISS<<32)>>32). There were no extremes in the more than 700 p-values returned by the tests, nor indeed for similar tests applied to just two of the KISS components: MWC+XSH, then MWC+CNG, then XSH+CNG.
The simplicity, speed, period around 2^250 and performance on tests of randomness---as well as ability to produce exactly the same 64-bit patterns, whether considered signed or unsigned integers---make this 64-bit KISS well worth considering for adoption or adaption to languages other than C or Fortran, as has been done for 32-bit KISSes.
geo <gmarsag...@gmail.com> wrote: >64-bit KISS RNGs
>Consistent with the Keep It Simple Stupid (KISS) principle, >I have previously suggested 32-bit KISS Random Number >Generators (RNGs) that seem to have been frequently adopted.
>Having had requests for 64-bit KISSes, and now that >64-bit integers are becoming more available, I will >describe here a 64-bit KISS RNG, with comments on >implementation for various languages, speed, periods >and performance after extensive tests of randomness.
>This 64-bit KISS RNG has three components, each nearly >good enough to serve alone. The components are: >Multiply-With-Carry (MWC), period (2^121+2^63-1) >Xorshift (XSH), period 2^64-1 >Congruential (CNG), period 2^64
While I hesitate to follow up to George Marsaglia on random numbers, there are a few things here that need at least clarification. I am NOT referring to the generator as such, where I have no disagreement, but to some of the other remarks.
Nobody should EVER use 32-bit generators for more than about ten million numbers in an analysis without careful analysis, because the discreteness will start to show through in at least some real analyses.
No Xorshift generator is good enough to use on its own, because they have some evil properties. We knew about them before 1970 (Knuth refers to it) and at least some of us knew the reason but could not prove it. The person who did was Martin Luescher of CERN, in the 1990s (if I recall).
When using generators in parallel programs, it is as important to worry about their quasi-independence as their serial properties; the common practice of using different seeds to the same generator is NOT a good idea. Multiplicative congruential generators with coprime moduli are quasi-independent over the whole period, as are multiply-with-carry ones with the same modulus but different, 'safe prime' multipliers - unfortunately, complementary multiply-with-carry ones are not.
Most of this is published, but perhaps not the last sentence. More work on the quasi-independence of generators is needed! In particular, my belief is that the best generators for parallel work are multiply-with-carry ones with the same modulus and different, 'safe prime' multipliers, but I have been out of this area for many years and my study of them has been cursory. A quick Web search didn't find anything useful, but there were an awful lot of irrelevant hits, so I may have missed something.
The problem here is that full-period properties do not necessarily map to the properties of shorter sequences (that was the problem with Xorshift generators), and analysing the latter is mathematically evil.
> Consistent with the Keep It Simple Stupid (KISS) principle, > I have previously suggested 32-bit KISS Random Number > Generators (RNGs) that seem to have been frequently adopted.
> Having had requests for 64-bit KISSes, and now that > 64-bit integers are becoming more available, I will > describe here a 64-bit KISS RNG, with comments on > implementation for various languages, speed, periods > and performance after extensive tests of randomness.
> This 64-bit KISS RNG has three components, each nearly > good enough to serve alone. The components are: > Multiply-With-Carry (MWC), period (2^121+2^63-1) > Xorshift (XSH), period 2^64-1 > Congruential (CNG), period 2^64
> Compact C and Fortran listings are given below. They > can be cut, pasted, compiled and run to see if, after > 100 million calls, results agree with that provided > by theory, assuming the default seeds.
> Users may want to put the content in other forms, and, > for general use, provide means to set the 250 seed bits > required in the variables x,y,z (64 bits) and c (58 bits) > that have been given default values in the test versions.
> The C version uses #define macros to enumerate the few > instructions that MWC, XSH and CNG require. The KISS > macro adds MWC+XSH+CNG mod 2^64, so that KISS can be > inserted at any place in a C program where a random 64-bit > integer is required. > Fortran's requirement that integers be signed makes the > necessary code more complicated, hence a function KISS().
> C version; test by invoking macro KISS 100 million times > ----------------------------------------------------------------- > #include <stdio.h>
> static unsigned long long > x=1234567890987654321ULL,c=123456123456123456ULL, > y=362436362436362436ULL,z=1066149217761810ULL,t;
> int main(void) > {int i; > for(i=0;i<100000000;i++) t=KISS; > (t==1666297717051644203ULL) ? > printf("100 million uses of KISS OK"): > printf("Fail"); > }
> --------------------------------------------------------------- > Fortran version; test by calling KISS() 100 million times > --------------------------------------------------------------- > program testkiss > implicit integer*8(a-z) > do i=1,100000000; t=KISS(); end do > if(t.eq.1666297717051644203_8) then > print*,"100 million calls to KISS() OK" > else; print*,"Fail" > end if; end
> function KISS() > implicit integer*8(a-z) > data x,y,z,c /1234567890987654321_8, 362436362436362436_8,& > 1066149217761810_8, 123456123456123456_8/ > save x,y,z,c > m(x,k)=ieor(x,ishft(x,k)) !statement function > s(x)=ishft(x,-63) !statement function > t=ishft(x,58)+c > if(s(x).eq.s(t)) then; c=ishft(x,-6)+s(x) > else; c=ishft(x,-6)+1-s(x+t); endif > x=t+x > y=m(m(m(y,13_8),-17_8),43_8) > z=6906969069_8*z+1234567 > KISS=x+y+z > return; end > ---------------------------------------------------------------
> Output from using the macro KISS or the function KISS() is > MWC+XSH+CNG mod 2^64.
> CNG is easily implemented on machines with 64-bit integers, > as arithmetic is automatically mod 2^64, whether integers > are considered signed or unsigned. The CNG statement is > z=6906969069*z+1234567. > When I established the lattice structure of congruential > generators in the 60's, a search produced 69069 as an easy- > to-remember multiplier with nearly cubic lattices in 2,3,4,5- > space, so I tried concatenating, using 6906969069 as > my first test multiplier. Remarkably---a seemingly one in many > hundreds chance---it turned out to also have excellent lattice > structure in 2,3,4,5-space, so that's the one chosen. > (I doubt if lattice structure of CNG has much influence on the > composite 64-bit KISS produced via MWC+XSH+CNG mod 2^64.)
> XSH, the Xorshift component, described in > www.jstatsoft.org/v08/i14/paper > uses three invocations of an integer "xor"ed with a shifted > version of itself. > The XSH component used for this KISS is, in C notation: > y^=(y<<13); y^=(y>>17); y^=(y<<43) > with Fortran equivalents y=ieor(y,ishft(y,13)), etc., although > this can be effected by a Fortran statement function: > f(y,k)=ieor(y,ishft(y,k)) > y=f(f(f(y,13),-17),43) > As with lattice structure, choice of the triple 13,-17,43 is > probably of no particular importance; any one of the 275 full- > period triples listed in the above article is likely to provide > a satisfactory component XSH for the composite MWC+XSH+CNG.
> The choice of multiplier 'a' for the multiply-with-carry (MWC) > component of KISS is not so easily made. In effect, a multiply- > with-carry sequence has a current value x and current "carry" c, > and from each given x,c a new x,c pair is constructed by forming > t=a*x+c, then x=t mod b=2^64 and c=floor(t/b). > This is easily implemented for 32-bit computers that permit > forming a*t+c in 64 bits, from which the new x is the bottom and > the new c the top 32-bits.
> When a,x and c are 64-bits, not many computers seem to have an easy > way to form t=a*x+c in 128 bits, then extract the top and bottom > 64-bit segments. For that reason, special choices for 'a' are > needed among those that satisfy the general requirement that > p=a*b-1 is a prime for which b=2^64 has order (p-1)/2.
> My choice---and the only one of this form---is a=2^58+1. Then the > top 64 bits of an imagined 128-bit t=a*x+c may be obtained as > (using C notation) (x>>6)+ 1 or 0, depending > on whether the 64-bit parts of (x<<58)+c+x cause an overflow. > Since (x<<58)+c cannot itself cause overflow (c will always be <a), > we get the carry as c=(x>>6) plus overflow from (x<<58)+x.
> This is easily done in C with unsigned integers, using a different > kind of 't': t=(x<<58)+c; c=(x>>6); x=t+x; c=c+(x<t); > For Fortran and others that permit only signed integers, more work > is needed. > Equivalent mod 2^64 versions of t=(x<<58)+c and c=(x>>6) are easy, > and if s(x) represents (x>>63) in C or ishft(x,-66) in Fortran, > then for signed integers, the new carry c comes from the rule > if s(x) equals s(t) then c=(x>>6)+s(x) else c=(x>>6)+1-s(x+t)
> Speed: > A C version of this KISS RNG takes 18 nanosecs for each > 64-bit random number on my desktop (Vista) PC, thus > producing KISSes at a rate exceeding 55 million per second. > Fortran or other integers-must-be-signed compilers might get > "only" around 40 million per second.
> Setting seeds: > Use of KISS or KISS() as a general 64-bit RNG requires specifying > 3*64+58=250 bits for seeds, 64 bits each for x,y,z and 58 for c, > resulting in a composite sequence with period around 2^250. > The actual period is > (2^250+2^192+2^64-2^186-2^129)/6 ~= 2^(247.42) or 10^(74.48). > We "lose" 1+1.58=2.58 bits from maximum possible period, one bit > because b=2^64, a square, cannot be a primitive root of p=ab-1, > so the best possible order for b is (p-1)/2. > The periods of MWC and XSH have gcd 3=2^1.58, so another 1.58 > bits are "lost" from the best possible period we could expect > from 250 seed bits.
> Some users may think 250 seed bits are an unreasonable requirement. > A good seeding procedure might be to assume the default seed > values then let the user choose none, one, two,..., or all > of x,y,z, and c to be reseeded.
> Tests: > Latest tests in The Diehard Battery, available at > http://i.cs.hku.hk/~diehard/ > were applied extensively. Those tests that specifically required > 32-bit integers were applied to the leftmost 32 bits > (e,g, KISS>>32;), then to the middle 32-bits ((KISS<<16)>>32;) > then to the rightmost 32 bits, ( (KISS<<32)>>32). > There were no extremes in the more than 700 p-values returned > by the tests, nor indeed for similar tests applied to just two of the > KISS components: MWC+XSH, then MWC+CNG, then XSH+CNG.
> The simplicity, speed, period around 2^250 and performance on > tests of randomness---as well as ability to produce exactly > the same 64-bit patterns, whether considered signed or unsigned > integers---make this 64-bit KISS well worth considering for > adoption or adaption to languages other than C or Fortran, > as has been done for 32-bit KISSes.
george!
as chief architect for a leading gaming (gambling) manufacturer in vegas and one of the few engineers with a mathematics background i am often working with our internal rng to provide better tools for our game catalog
for years we had used your 96-bit kiss rng which served us quite well for it's nice distribution speed and fair period
however some of our newest gaming ideas have required rng's with much greater periods (to cover the possibility space of the game) and our latest platform has moved over to use mersenne twisters
(unfortunately it has been an industry standard to just add more rng's (the same ones instantiated multiply!) and i had to fight to ensure my company didn't go that path to ensure unintended correlations were not introduced)
this improvement of yours would have certainly been considered if available at the time but i do fear the period may still have been too small for some of our more extreme future needs (there are already products on the market from several manufacturers that allow a player to play 100 simultaneous poker or keno games)
RANLUX is slow, but at the highest "luxury level" all 24 bits of the mantissa are chaotic. So, one could just stick these together to create numbers containing more bits.
>RANLUX is slow, but at the highest "luxury level" all 24 bits of the >mantissa are chaotic. So, one could just stick these together to create >numbers containing more bits.
That wasn't the issue she (I assume) was addressing - it was one that I did. Yes, that technique works, for both RANLUX and 32-bit KISS. I use my own double-precision generator, of course, which has some theoretical advantages over both and is marginally simpler than (and similar to) KISS.
Galathaea's concern was about the period, and she is very right to be so concerned. While a long period does not guarantee pseudo- randomness, it is a prerequisite for it - in particular, the pseudo- random properties in N dimensions are often limited by the Nth root of the period. And, despite common belief, that is NOT solely true for multiplicative congruential generators.
In article <godlg7$a4...@soup.linux.pwf.cam.ac.uk>, n...@cam.ac.uk writes:
> Galathaea's concern was about the period, and she is very right to > be so concerned. While a long period does not guarantee pseudo- > randomness, it is a prerequisite for it - in particular, the pseudo- > random properties in N dimensions are often limited by the Nth root > of the period. And, despite common belief, that is NOT solely true > for multiplicative congruential generators.
The period of RANLUX is huge---10**164 or something. (In other words, many orders of magnitude larger than the number of distinct bit combinations. Many lesser generators have a period much LESS than the number of distinct bit combinations and some algorithms can have a period at most as long as the number of distinct bit combinations. In these cases, of course, a given number x is always followed by a given number y, which with RANLUX is not the case.)
>> Galathaea's concern was about the period, and she is very right to >> be so concerned. While a long period does not guarantee pseudo- >> randomness, it is a prerequisite for it - in particular, the pseudo- >> random properties in N dimensions are often limited by the Nth root >> of the period. And, despite common belief, that is NOT solely true >> for multiplicative congruential generators.
>The period of RANLUX is huge---10**164 or something. (In other words, >many orders of magnitude larger than the number of distinct bit >combinations. Many lesser generators have a period much LESS than the >number of distinct bit combinations and some algorithms can have a >period at most as long as the number of distinct bit combinations. In >these cases, of course, a given number x is always followed by a given >number y, which with RANLUX is not the case.)
!!!!! ALL such generators have been known to be trash (and, yes, I mean trash) since the late 1960s! One of my papers proves (and I mean mathematically rigorously) that you should never use more than period^(2/3) numbers in a simulation - the rule for never using more than period^(1/2) for cryptographic purposes has been known since time immemorial.
64-bit KISS has a period of about 2^249, according to that posting.
I could explain the potential defects of RANLUX, but would have to rake quite a lot of my memories from where they are archived (on tape, perhaps?) And please note that the word is 'potential' - to know if they were actual needs forms of analysis that I believe are still beyond the state of the art (and well beyond my mathematical ability). However, I believe that it is almost certainly reliable (on mathematical grounds, incidentally).
Similar remarks can be made about KISS and my own generator but, there, I am almost certain that the required analysis is WAY beyond the state of the art.
[ The analysis I am referring to is the one that maps the full-period uniformity, which is a quasi-random property, to the pseudo-randomness of short sequences. The word 'hairy' springs to mind! ]
> --------------------------------------------------------------- > Fortran version; test by calling KISS() 100 million times > --------------------------------------------------------------- > program testkiss > implicit integer*8(a-z) > do i=1,100000000; t=KISS(); end do > if(t.eq.1666297717051644203_8) then > print*,"100 million calls to KISS() OK" > else; print*,"Fail" > end if; end
> function KISS() > implicit integer*8(a-z) > data x,y,z,c /1234567890987654321_8, 362436362436362436_8,& > 1066149217761810_8, 123456123456123456_8/ > save x,y,z,c > m(x,k)=ieor(x,ishft(x,k)) !statement function > s(x)=ishft(x,-63) !statement function > t=ishft(x,58)+c > if(s(x).eq.s(t)) then; c=ishft(x,-6)+s(x) > else; c=ishft(x,-6)+1-s(x+t); endif > x=t+x > y=m(m(m(y,13_8),-17_8),43_8) > z=6906969069_8*z+1234567 > KISS=x+y+z > return; end > ---------------------------------------------------------------
George, this code is not portable Fortran. If you want to specify a kind, you need to use SELECTED_INT_KIND or equivalent action in order for it to be portable.
> The simplicity, speed, period around 2^250 and performance on > tests of randomness---as well as ability to produce exactly > the same 64-bit patterns, whether considered signed or unsigned > integers---make this 64-bit KISS well worth considering for > adoption or adaption to languages other than C or Fortran, > as has been done for 32-bit KISSes.
> George Marsaglia
A PL/I version:
KISS: procedure() returns (fixed binary (64) unsigned) options (reorder); declare (x initial (1234567890987654321), y initial ( 362436362436362436), z initial ( 1066149217761810), c initial ( 123456123456123456), t ) fixed binary (64) unsigned static;
t=isll(x,58)+c; if isrl(x,63) = isrl(t,63) then c=isrl(x,6)+isrl(x, 63); else c=isrl(x,6)+1-isrl(x+t,63); x=t+x; t = ieor(y, isll(y, 13)); t = ieor(t, isrl(t, 17)); y = ieor(t, isll(t, 43)); z=6906969069*z+1234567; return (x+y+z); end KISS;
> >> Galathaea's concern was about the period, and she is very right to > >> be so concerned. While a long period does not guarantee pseudo- > >> randomness, it is a prerequisite for it - in particular, the pseudo- > >> random properties in N dimensions are often limited by the Nth root > >> of the period. And, despite common belief, that is NOT solely true > >> for multiplicative congruential generators.
> >The period of RANLUX is huge---10**164 or something. (In other words, > >many orders of magnitude larger than the number of distinct bit > >combinations. Many lesser generators have a period much LESS than the > >number of distinct bit combinations and some algorithms can have a > >period at most as long as the number of distinct bit combinations. In > >these cases, of course, a given number x is always followed by a given > >number y, which with RANLUX is not the case.)
> !!!!! ALL such generators have been known to be trash (and, yes, I > mean trash) since the late 1960s! One of my papers proves (and I > mean mathematically rigorously) that you should never use more than > period^(2/3) numbers in a simulation - the rule for never using more > than period^(1/2) for cryptographic purposes has been known since > time immemorial.
> 64-bit KISS has a period of about 2^249, according to that posting.
> I could explain the potential defects of RANLUX, but would have to > rake quite a lot of my memories from where they are archived (on > tape, perhaps?) And please note that the word is 'potential' - to > know if they were actual needs forms of analysis that I believe are > still beyond the state of the art (and well beyond my mathematical > ability). However, I believe that it is almost certainly reliable > (on mathematical grounds, incidentally).
> Similar remarks can be made about KISS and my own generator but, > there, I am almost certain that the required analysis is WAY beyond > the state of the art.
> [ The analysis I am referring to is the one that maps the full-period > uniformity, which is a quasi-random property, to the pseudo-randomness > of short sequences. The word 'hairy' springs to mind! ]
>TestU01 is a fabulous battery of tests that has a very broad spectrum >and also implementations for just about every existing, popular prng.
>Highly recommended.
That URL is completely mangled. You don't use (ugh) 'Rich Text' by any chance?
In my view Pierre L'Ecuyer is the leading expert on this area active today[*]. I would do so were I still active in this area myself; it would unquestionably pass most of them, as I have tried. I have some other tests, one of which is harsh enough that it is one of very few that will fail most of Marsaglia's; my generator passed, but that is to be expected :-)
[*} George Marsaglia was the previous holder of that position, as most people will agree, preceded by Donald Knuth.
<n...@cam.ac.uk> wrote in message news:gpegns$ra5$1@soup.linux.pwf.cam.ac.uk... > In my view Pierre L'Ecuyer is the leading expert on this area active > today[*]. I would do so were I still active in this area myself; it > would unquestionably pass most of them, as I have tried. I have some > other tests, one of which is harsh enough that it is one of very few > that will fail most of Marsaglia's;
You are only guessing. Until you have actually run all such tests, your post is just hype.
>> In my view Pierre L'Ecuyer is the leading expert on this area active >> today[*]. I would do so were I still active in this area myself; it >> would unquestionably pass most of them, as I have tried. I have some >> other tests, one of which is harsh enough that it is one of very few >> that will fail most of Marsaglia's;
>You are only guessing. >Until you have actually run all such tests, your post is just hype.
n...@cam.ac.uk wrote: > In article <OvCul.28450$cu.8...@news-server.bigpond.net.au>, > robin <robi...@bigpond.com> wrote:
> >> In my view Pierre L'Ecuyer is the leading expert on this area active > >> today[*]. I would do so were I still active in this area myself; it > >> would unquestionably pass most of them, as I have tried. I have some > >> other tests, one of which is harsh enough that it is one of very few > >> that will fail most of Marsaglia's;
> >You are only guessing. > >Until you have actually run all such tests, your post is just hype.
> Back under your bridge with you!
And take your nonsense about imagined results possibly differing from actual ones with you. Good programmers don't even need to profile/test their code; they just know!
> n...@cam.ac.uk wrote: > > In article <OvCul.28450$cu.8...@news-server.bigpond.net.au>, > > robin <robi...@bigpond.com> wrote:
> > >> In my view Pierre L'Ecuyer is the leading expert on this area active > > >> today[*]. I would do so were I still active in this area myself; it > > >> would unquestionably pass most of them, as I have tried. I have some > > >> other tests, one of which is harsh enough that it is one of very few > > >> that will fail most of Marsaglia's;
> > >You are only guessing. > > >Until you have actually run all such tests, your post is just hype.
> > Back under your bridge with you!
> And take your nonsense about imagined results possibly differing from > actual ones with you. Good programmers don't even need to profile/test > their code; they just know!
On the other hand, Nick Maclaren is a pretty sharp guy. I guess that he has done all the testing he needs to do to determine what he needs to know about his tools.
He had some spot on advice for me in 1999. I recall he said that a particular technique might "run like the clappers", and by golly it did (when I finally got around to thoroughly benching all of the ideas at my disposal, his idea was best). Here is the thread I am referring to: http://groups.google.com/group/comp.theory/browse_thread/thread/c0e22...
At any rate, I guess you have picked the wrong target, though the advice to profile your ideas carefully is excellent advice.
Humorously, I think that there is also some truth in this: "Good programmers don't even need to profile/test their code; they just know!"
If I choose an idea that is O(n) and all the other ideas at my disposal are O(n*log(n)), and my implementation is fast enough when the data is small, then I don't need to bother testing the performance because I can't improve it. When the problem gets large enough, the O(n) solution will dominate, and it is fast enough for small data sets. There is nothing that testing can tell me that I do not already know.
>Humorously, I think that there is also some truth in this: >"Good programmers don't even need to profile/test their code; they >just know!"
>If I choose an idea that is O(n) and all the other ideas at my >disposal are O(n*log(n)), and my implementation is fast enough when >the data is small, then I don't need to bother testing the performance >because I can't improve it. >When the problem gets large enough, the O(n) solution will dominate, >and it is fast enough for small data sets. There is nothing that >testing can tell me that I do not already know.
Absolutely. Analyse first. Design second. Code and debug third. Every good engineer knows that rule!
You need to run a couple of simple tests to check that your code matches your analysis, and there isn't a major flaw in the analysis itself, but you don't need to do more than that. The main testing comes where you can't analyse the problem, or where you suspect the analysis may be unreliable.
> In article > <91f7aa39-dfde-4bd2-bc68-702c9cd2a...@o36g2000yqh.googlegroups.com>, > user923005 <dcor...@connx.com> wrote:
> >Humorously, I think that there is also some truth in this: > >"Good programmers don't even need to profile/test their code; they > >just know!"
> >If I choose an idea that is O(n) and all the other ideas at my > >disposal are O(n*log(n)), and my implementation is fast enough when > >the data is small, then I don't need to bother testing the performance > >because I can't improve it. > >When the problem gets large enough, the O(n) solution will dominate, > >and it is fast enough for small data sets. There is nothing that > >testing can tell me that I do not already know.
> Absolutely. Analyse first. Design second. Code and debug third. > Every good engineer knows that rule!
> You need to run a couple of simple tests to check that your code > matches your analysis, and there isn't a major flaw in the analysis > itself, but you don't need to do more than that. The main testing > comes where you can't analyse the problem, or where you suspect the > analysis may be unreliable.
Sometimes you divide your problem into different domains, large, medium, and small, or high and low, or combinations of different ranges for different variables, and you use different algorithms to solve the different domains. Part of the optimal solution then consists of switching to the best algorithm at the right time. Sorting is one of those problems, where with small data sets you might use one algorithm, and you switch to different algorithms as the data sets grow. Also, some sorting algorithms involve partitioning the original large data set into smaller subsets, sometimes doing this recursively, and you might change algorithms as you move up and down the different levels of recursion.
The exact switchover points might be machine dependent, depending say on the relative efficiency of various floating point and integer instructions, or the relative speeds of different parts of the memory hierarchy, or the communication speed with some external device. Much of this must be done in a brute force way where you just search for the switchover points, and then store or interpolate those switchover points for later use. This is the way ATLAS optimizes linear algebra operations.
geo wrote: > Having had requests for 64-bit KISSes, and now that > 64-bit integers are becoming more available, I will > describe here a 64-bit KISS RNG, with comments on > implementation for various languages, speed, periods > and performance after extensive tests of randomness. >. > C version; test by invoking macro KISS 100 million times
Recoded for MS VC (MSC) compilers, which don't understand the 'long long' datatype, and with minor corrections....
#include <stdio.h>
#if _WIN32 typedef unsigned __int64 ullong_t; #else typedef unsigned long long ullong_t; #endif
static ullong_t x = 1234567890987654321/*ULL*/; static ullong_t c = 123456123456123456/*ULL*/; static ullong_t y = 362436362436362436/*ULL*/; static ullong_t z = 1066149217761810/*ULL*/; static ullong_t t;
#define MWC (t = (x<<58)+c, c = (x>>6), x+=t, c+=(x<t), x) #define XSH (y ^= (y<<13), y ^= (y>>17), y ^= (y<<43)) #define CNG (z = 6906969069/*LL*/ * z + 1234567) #define KISS (MWC + XSH + CNG)
int main(void) { int i;
for (i = 0; i < 100000000; i++) t = KISS;
if (t == 1666297717051644203/*ULL*/) printf("100 million uses of KISS OK"); else printf("Fail"); return 0;
David R Tribble wrote: > geo wrote: >> Having had requests for 64-bit KISSes, and now that >> 64-bit integers are becoming more available, I will >> describe here a 64-bit KISS RNG, with comments on >> implementation for various languages, speed, periods >> and performance after extensive tests of randomness. >> . >> C version; test by invoking macro KISS 100 million times
> Recoded for MS VC (MSC) compilers, which don't understand > the 'long long' datatype, and with minor corrections....
> #include <stdio.h>
> #if _WIN32 > typedef unsigned __int64 ullong_t; > #else > typedef unsigned long long ullong_t; > #endif
> static ullong_t x = 1234567890987654321/*ULL*/; > static ullong_t c = 123456123456123456/*ULL*/; > static ullong_t y = 362436362436362436/*ULL*/; > static ullong_t z = 1066149217761810/*ULL*/; > static ullong_t t;
> #define MWC (t = (x<<58)+c, c = (x>>6), x+=t, c+=(x<t), x) > #define XSH (y ^= (y<<13), y ^= (y>>17), y ^= (y<<43)) > #define CNG (z = 6906969069/*LL*/ * z + 1234567) > #define KISS (MWC + XSH + CNG)
> int main(void) > { > int i;
> for (i = 0; i < 100000000; i++) > t = KISS;
> if (t == 1666297717051644203/*ULL*/) > printf("100 million uses of KISS OK"); > else > printf("Fail"); > return 0; > }
n...@cam.ac.uk wrote: > Absolutely. Analyse first. Design second. Code and debug third. > Every good engineer knows that rule!
> You need to run a couple of simple tests to check that your code > matches your analysis, and there isn't a major flaw in the analysis > itself, but you don't need to do more than that. The main testing > comes where you can't analyse the problem, or where you suspect the > analysis may be unreliable.
Which, in practice, is all the time. I mean, have _you_ ever spoken to a customer who knew what he wanted, completely, correctly, and up front? I know I haven't.
>> Absolutely. Analyse first. Design second. Code and debug third. >> Every good engineer knows that rule!
>> You need to run a couple of simple tests to check that your code >> matches your analysis, and there isn't a major flaw in the analysis >> itself, but you don't need to do more than that. The main testing >> comes where you can't analyse the problem, or where you suspect the >> analysis may be unreliable.
>Which, in practice, is all the time. I mean, have _you_ ever spoken to a >customer who knew what he wanted, completely, correctly, and up front? >I know I haven't.
Oh, yes, but you had better deliver the solution within a couple of days - if not, the requirement will have started drifting ....
On Mar 16, 9:57 am, ralt...@xs4all.nl (Richard Bos) wrote:
> n...@cam.ac.uk wrote: > > Absolutely. Analyse first. Design second. Code and debug third. > > Every good engineer knows that rule!
> > You need to run a couple of simple tests to check that your code > > matches your analysis, and there isn't a major flaw in the analysis > > itself, but you don't need to do more than that. The main testing > > comes where you can't analyse the problem, or where you suspect the > > analysis may be unreliable.
> Which, in practice, is all the time. I mean, have _you_ ever spoken to a > customer who knew what he wanted, completely, correctly, and up front? > I know I haven't.
> Richard
I did. It was wonderful. The University Registrar's Office. My boss had grown up with punchcards and chewed JCL and MARKIV (now called, I think, Vision:Builder) in her sleep. My job was to run database queries to make address labels. NAME STREET APT CITY ST ZIP. Lovely. First I made a WinBatch front-end to automate the TN3270 emulator (hit TAB once instead of 7 times, etc.), then I made a Macro-processor to translate a more sensible query language to MARKIV, then I made a compiler to MARKIV. As long as the labels kept flowing, I basically optimized my job away (but I got to stay, just with massive amounts of paid time).
"blargg" <blargg....@gishpuppy.com> wrote in message news:blargg.ei3-1403091511020001@192.168.1.4... > n...@cam.ac.uk (Nick Maclaren) wrote: > > In article <OvCul.28450$cu.8...@news-server.bigpond.net.au>, > > robin <robi...@bigpond.com> wrote:
> > >> In my view Pierre L'Ecuyer is the leading expert on this area active > > >> today[*]. I would do so were I still active in this area myself; it > > >> would unquestionably pass most of them, as I have tried. I have some > > >> other tests, one of which is harsh enough that it is one of very few > > >> that will fail most of Marsaglia's;
> > >You are only guessing. > > >Until you have actually run all such tests, your post is just hype.
> > Back under your bridge with you!
> And take your nonsense about imagined results possibly differing from > actual ones with you. Good programmers don't even need to profile/test > their code; they just know!
There's no substitute for rigorous testing.
Nor is there any excuse for denigrating an expert's work -- as Maclaren has done -- without having done a single test.
>"blargg" <blargg....@gishpuppy.com> wrote in message news:blargg.ei3-1403091511020001@192.168.1.4... >> n...@cam.ac.uk (Nick Maclaren) wrote: >> > In article <OvCul.28450$cu.8...@news-server.bigpond.net.au>, >> > robin <robi...@bigpond.com> wrote:
>> > >> In my view Pierre L'Ecuyer is the leading expert on this area active >> > >> today[*]. I would do so were I still active in this area myself; it >> > >> would unquestionably pass most of them, as I have tried. I have some >> > >> other tests, one of which is harsh enough that it is one of very few >> > >> that will fail most of Marsaglia's;
>> > >You are only guessing. >> > >Until you have actually run all such tests, your post is just hype.
>> > Back under your bridge with you!
>> And take your nonsense about imagined results possibly differing from >> actual ones with you. Good programmers don't even need to profile/test >> their code; they just know!
>There's no substitute for rigorous testing.
>Nor is there any excuse for denigrating an expert's work >-- as Maclaren has done -- >without having done a single test.
On Feb 28, 8:30 am, geo <gmarsag...@gmail.com> wrote:> 64-bit KISS RNGs> > Consistent with the Keep It Simple Stupid (KISS) principle,> I have previously suggested 32-bit KISS Random Number> Generators (RNGs) that seem to have been frequently adopted.> > Having had requests for 64-bit KISSes, and now that> 64-bit integers are becoming more available, I will> describe here a 64-bit KISS RNG, with comments on> implementation for various languages, speed, periods> and performance after extensive tests of randomness.> > This 64-bit KISS RNG has three components, each nearly> good enough to serve alone. The components are:> Multiply-With-Carry (MWC), period (2^121+2^63-1)> Xorshift (XSH), period 2^64-1> Congruential (CNG), period 2^64> > Compact C and Fortran listings are given below. They> can be cut, pasted, compiled and run to see if, after> 100 million calls, results agree with that provided> by theory, assuming the default seeds.> > Users may want to put the content in other forms, and,> for general use, provide means to set the 250 seed bits> required in the variables x,y,z (64 bits) and c (58 bits)> that have been given default values in the test versions.> > The C version uses #define macros to enumerate the few> instructions that MWC, XSH and CNG require. The KISS> macro adds MWC +XSH+CNG mod 2^64, so that KISS can be> inserted at any place in a C program where a random 64-bit> integer is required.> Fortran's requirement that integers be signed makes the> necessary code more complicated, hence a function KISS().> > C version; test by invoking macro KISS 100 million times> -----------------------------------------------------------------> #include <stdio.h>> > static unsigned long long> x=1234567890987654321ULL,c=123456123456123456ULL,> y=362436362436362436ULL,z=1066149217761810ULL,t;> > #define MWC (t= (x<<58)+c, c=(x>>6), x+=t, c+=(x<t), x)> #define XSH ( y^=(y<<13), y^= (y>>17), y^=(y<<43) )> #define CNG ( z=6906969069LL*z+1234567 )> #define KISS (MWC+XSH+CNG)> > int main(void)> {int i;> for (i=0;i<100000000;i++) t=KISS;> (t==1666297717051644203ULL) ?> printf("100 million uses of KISS OK"):> printf("Fail");> }> > ---------------------------------------------------------------> Fortran version; test by calling KISS() 100 million times> ---------------------------------------------------------------> program testkiss> implicit integer*8(a-z)> do i=1,100000000; t=KISS (); end do> if(t.eq.1666297717051644203_8) then> print*,"100 million calls to KISS() OK"> else; print*,"Fail"> end if; end> > function KISS()> implicit integer*8(a-z)> data x,y,z,c / 1234567890987654321_8, 362436362436362436_8,&> 1066149217761810_8, 123456123456123456_8/> save x,y,z,c> m(x,k)=ieor(x,ishft(x,k)) ! statement function> s(x)=ishft(x,-63) !statement function> t=ishft(x,58)+c> if(s(x).eq.s(t)) then; c=ishft(x,-6)+s(x)
> Output from using the macro KISS or the function KISS() is> MWC+XSH
+CNG mod 2^64.> > CNG is easily implemented on machines with 64-bit integers,> as arithmetic is automatically mod 2^64, whether integers> are considered signed or unsigned. The CNG statement is> z=6906969069*z+1234567.> When I established the lattice structure of congruential> generators in the 60's, a search produced 69069 as an easy-> to-remember multiplier with nearly cubic lattices in 2,3,4,5-> space, so I tried concatenating, using 6906969069 as> my first test multiplier. Remarkably---a seemingly one in many> hundreds chance---it turned out to also have excellent lattice> structure in 2,3,4,5-space, so that's the one chosen.> (I doubt if lattice structure of CNG has much influence on the> composite 64-bit KISS produced via MWC+XSH+CNG mod 2^64.)> > XSH, the Xorshift component, described in> www.jstatsoft.org/v08/i14/paper> uses three invocations of an integer "xor"ed with a shifted> version of itself.> The XSH component used for this KISS is, in C notation:> y^=(y<<13); y^=(y>>17); y^=(y<<43)> with Fortran equivalents y=ieor(y,ishft(y,13)), etc., although> this can be effected by a Fortran statement function:> f(y,k)=ieor (y,ishft(y,k))> y=f(f(f(y,13),-17),43)> As with lattice structure, choice of the triple 13,-17,43 is> probably of no particular importance; any one of the 275 full-> period triples listed in the above article is likely to provide> a satisfactory component XSH for the composite MWC+XSH+CNG.> > The choice of multiplier 'a' for the multiply-with-carry (MWC)> component of KISS is not so easily made. In effect, a multiply-> with-carry sequence has a current value x and current "carry" c,> and from each given x,c a new x,c pair is constructed by forming> t=a*x+c, then x=t mod b=2^64 and c=floor(t/ b).> This is easily implemented for 32-bit computers that permit> forming a*t+c in 64 bits, from which the new x is the bottom and> the new c the top 32-bits.> > When a,x and c are 64-bits, not many computers seem to have an easy> way to form t=a*x+c in 128 bits, then extract the top and bottom> 64-bit segments. For that reason, special choices for 'a' are> needed among those that satisfy the general requirement that> p=a*b-1 is a prime for which b=2^64 has order (p-1)/ 2.> > My choice---and the only one of this form---is a=2^58+1. Then the> top 64 bits of an imagined 128-bit t=a*x+c may be obtained as> (using C notation) (x>>6)+ 1 or 0, depending> on whether the 64-bit parts of (x<<58)+c+x cause an overflow.> Since (x<<58)+c cannot itself cause overflow (c will always be <a),> we get the carry as c=(x>>6) plus overflow from (x<<58)+x.> > This is easily done in C with unsigned integers, using a different> kind of 't': t=(x<<58)+c; c= (x>>6); x=t+x; c=c+(x<t);> For Fortran and others that permit only signed integers, more work> is needed.> Equivalent mod 2^64 versions of t=(x<<58)+c and c=(x>>6) are easy,> and if s(x) represents (x>>63) in C or ishft(x,-66) in Fortran,> then for signed integers, the new carry c comes from the rule> if s(x) equals s(t) then c=(x>>6)+s(x) else c=(x>>6)+1-s(x+t)> > Speed:> A C version of this KISS RNG takes 18 nanosecs for each> 64-bit random number on my desktop (Vista) PC, thus> producing KISSes at a rate exceeding 55 million per second.> Fortran or other integers-must-be-signed compilers might get> "only" around 40 million per second.> > Setting seeds:> Use of KISS or KISS() as a general 64-bit RNG requires specifying> 3*64+58=250 bits for seeds, 64 bits each for x,y,z and 58 for c,> resulting in a composite sequence with period around 2^250.> The actual period is> (2^250+2^192+2^64-2^186-2^129)/6 ~= 2^(247.42) or 10^(74.48).> We "lose" 1+1.58=2.58 bits from maximum possible period, one bit> because b=2^64, a square, cannot be a primitive root of p=ab-1,> so the best possible order for b is (p-1)/2.> The periods of MWC and XSH have gcd 3=2^1.58, so another 1.58> bits are "lost" from the best possible period we could expect> from 250 seed bits.> > Some users may think 250 seed bits are an unreasonable requirement.> A good seeding procedure might be to assume the default seed> values then let the user choose none, one, two,..., or all> of x,y,z, and c to be reseeded.> > Tests:> Latest tests in The Diehard Battery, available at> http://i.cs.hku.hk/~diehard/> were applied extensively. Those tests that specifically required> 32-bit integers were applied to the leftmost 32 bits> (e,g, KISS>>32;), then to the middle 32-bits ((KISS<<16)>>32;)> then to the rightmost 32 bits, ( (KISS<<32)>>32).> There were no extremes in the more than 700 p-values returned> by the tests, nor indeed for similar tests applied to just two of the> KISS components: MWC+XSH, then MWC+CNG, then XSH+CNG.> > The simplicity, speed, period around 2^250 and performance on> tests of randomness--- as well as ability to produce exactly> the same 64-bit patterns, whether considered signed or unsigned> integers---make this 64-bit KISS well worth considering for> adoption or adaption to languages other than C or Fortran,> as has been done for 32-bit KISSes.> > George MarsagliaAssembling a as an equality function is a fairly appliable algorithm method.It appears stalworth. The seed appears indivisible in code. Memory overuns cause inability to address memory.z= f(z) is in there. Please consider taking it out. Running through all random numbers by using f(z) is possible.It is a simple function that can map seed to series. Z= F(seed) WAS NOT TO BE NOT USED.Is it critical to the whole series?
