Path: ...!not-for-mail From: geo Newsgroups: sci.math,comp.lang.c,comp.lang.fortran Subject: 64-bit KISS RNGs Date: Sat, 28 Feb 2009 04:30:48 -0800 (PST) Organization: http://groups.google.com Lines: 196 Message-ID: NNTP-Posting-Host: 69.254.163.220 Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: posting.google.com 1235824248 24083 127.0.0.1 (28 Feb 2009 12:30:48 GMT) X-Complaints-To: groups-abuse@google.com NNTP-Posting-Date: Sat, 28 Feb 2009 12:30:48 +0000 (UTC) Complaints-To: groups-abuse@google.com Injection-Info: j8g2000yql.googlegroups.com; posting-host=69.254.163.220; posting-account=lMPBJwoAAADEIw-zQg-LUcxd542NXZ4c User-Agent: G2/1.0 X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 6.0; GTB5; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) 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 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>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) 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 >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>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 ========== REMAINDER OF ARTICLE TRUNCATED ==========