[Jprogramming] Interger division, the faster one
Roy A. Crabtree
roy.crabtree at gmail.com
Sun Oct 22 20:43:13 HKT 2006
On 10/22/06, Mike Day <mike_liz.day at tiscali.co.uk> wrote:
>
> Getting OT...
>
> Mea culpa (bis)!
>
> NB.
> datatype 1r2
> rational
Certes.
int op int == rat for lower(op)==(max,min,plus,minus,mult,div);
could be for upper(op)==(exp,log,etc.)
by carrying the op the same way rat does for mult/div ...
you would need a more complex polyad structure than just (a/b):
((a/b)^(c/d)) == (a,b,c,d) ... or ((a,b), (c,d))
rat op rat == rat for all op in that case.
(Sounds like a Congressional disgustion on Paging...)
Mike
>
> Roy A. Crabtree wrote:
> > On 10/22/06, Mike Day <mike_liz.day at tiscali.co.uk> wrote:
> >>
> >> (T)ouch(e) !
> >>
> >> Me culpa. I hadn't realised - or checked - that
> >
> >
> > "Mea" (tua) culpa. Certes?
> >
> > integer quotient % integer divisor
> >> is always floating.
> >
> >
> > Sounds like an Enron argument. Two rationals make a float.
> >
> > But why?
> >
> >
> > Qua non?
> >
> > For example, Dyalog APL retains an integer type for
> >> similar arguments:
> >> #DR 17 @ short integer
> >> 83
> >> #DR 548590 556138 1375147 @ long int
> >> 323
> >> #DR 548590 556138 1375147 % 17
> >> 323
> >>
> >> mike
> >>
> >> Oleg Kobchenko wrote:
> >> > Well, that's not exactly an improvement
> >> >
> >> > ts'x <.@% 17'
> >> > 0.19952 2.09723e7
> >> > ts'x (([ - |~) % ]) 17'
> >> > 0.0545567 1.25841e7
> >> >
> >> > ts'x % 17'
> >> > 0.0182892 8.38925e6
> >> > ts'_4 (33 b.) x'
> >> > 0.00982667 4.1952e6
> >> >
> >> > datatype x (([ - |~) % ]) 17
> >> > floating
> >> >
> >> > Maybe a method that represents interger
> >> > division in terms of binary shifts.
> >
> >
> > If J does not do it, you can; but would it be more efficient?
> >
> > Similar optimizations are inherently possible for direct rational
> > arithmetic,
> > and I suspect that if J does not use them, then usually to nearly always
> > it will be much less efficient to attempt to use them;
> >
> > these folks tend to hit it on the nose ...:
> >
> > (ArB) op (CrD) == (op / reorder( A,B,C,D))
> >
> > (postpone the operations and utilize binary equality testing or
> > factoring
> > to unify
> > and reduce later operations).
> >
> > Which blows up in time and space really fast.
> >
> >>
> >> >
> >> > Mike Day mike_liz.day at tiscali.co.uk wrote Sat Oct 21 16:18:53 HKT
> >> 2006
> >> >
> >> > FWIW, pending J6.02, this offers some improvement in
> >> > time and space:
> >> > NB. in J 6.0.1
> >> > NB. reduce x to an integer multiple of y before divide
> >> > NB. does it work for -ve & +ve x
> >> > (x-300000) (<.@% -: ([ - |~) % ]) 17
> >> > 1
> >> > ts'x <.@% 17'
> >> > 0.210536 2.09723e7
> >> > ts'x (([ - |~) % ]) 17'
> >> > 0.0587469 1.25841e7
> >> >
> >> > I expect there's a better way still, and still in 6.01 !
> >
> >
> > Restate the operations as tuple form to keep the arithmetic rational
> > at all
> > times;
> > this avoids division internally entirely, until/unless you ant to float:
> >
> > ((a idiv b) idiv (c idiv d)) == (a imult d) idiv (b imult c)
> >
> > representable as:
> >
> > ((a,b) / (c,d)) == ((a*d),(b*c))
> >
> > which is what I thought rational arithmetic was all about anyway;
> >
> > so I am confused.
> >
> > SImilarly for all other functors that have similar identities.
> >
> >>
> >> > Mike
> >> >
> >> > Roger Hui wrote:
> >> >
> >> >> Preliminary benchmarks:
> >> >>
> >> >> ts=: 6!:2 , 7!:2@]
> >> >> x=: 1e6 ?@$ 2e6
> >> >> ts 'Expression'
> >> >>
> >> >> Expression J6.02x J6.01 Factor
> >> >> x <.@% 512 0.0054822 4.19507e6 0.136416 2.09723e7 23.80 5.00
> >> >> x <.@% 17 0.0213073 4.19507e6 0.154837 2.09723e7 7.27 5.00
> >> >>
> >> >
> >> >
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> >
> >
> >
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