They can, which is disturbing. However, they don't have to (and they have some other advantages, also...).
Thanks, -- Raul On Mon, Apr 8, 2019 at 12:21 PM William Tanksley, Jr <wtanksle...@gmail.com> wrote: > > What's happening is that unexact trumps exact -- floats imply rounding > and other non-arithmetic behavior, so when they appear in data they > always carry a huge warning sign. > > -Wm > > On Mon, Apr 8, 2019 at 8:48 AM 'Mike Day' via Programming > <programm...@jsoftware.com> wrote: > > > > ... also it might be worth noting this, which I had wondered about but not > > confirmed before: > > VERSION_j_ > > 701.1 2 > > datatype 1r3 2r7 > > rational > > datatype 1r3 2r7 1p1 > > floating > > datatype x:1r3 2r7 1p1 > > rational > > > > And I’ve just checked; it’s the same behaviour in J901 beta. > > > > So irrationals trump rationals, as they should, I suppose, just as any > > floats raise type integer to type floating. > > > > x: ensures rational here. > > > > > > Mike > > > > Sent from my iPad > > > > > On 8 Apr 2019, at 14:30, Ian Clark <earthspo...@gmail.com> wrote: > > > > > > Linda wrote > > >> The rational numbers are exact. > > > > > > Yes, programs using them have a nice crisp feel. > > > I used to think the only people who should be using them are number > > > theorists, but now I'm a convert to their general use. > > > But it's worth remembering that in some circumstances you're working with > > > rational approximations, not exact values, > > > Common examples: anything involving π and √2. > > > > > > I'm using Roger Hui's rational replacements for trig based on (o.) -- > > > https://code.jsoftware.com/wiki/Essays/Extended_Precision_Functions#Collected_Definitions > > > …which give 40 decimal places. They're fun to play with. > > > No snags hit yet. But in the course of my investigations, some massively > > > long rationals emerge. > > > Can't see any performance deterioration yet, however, but I've developed > > > code to cut-back a monster rational to (say) 40 decimal places. > > > > > > Ian Clark > > > > > >> On Mon, 8 Apr 2019 at 12:57, Linda Alvord <lindaalvor...@outlook.com> > > >> wrote: > > >> > > >> The rational numbers are exact. > > >> > > >> ([:+/\|.)^:(i.5)1 1 > > >> 1 1 > > >> 1 2 > > >> 2 3 > > >> 3 5 > > >> 5 8 > > >> fr=: 13 :'%/"1([:+/\|.)^:(i.y)1 1' > > >> (,.0j25":"0 fr 20);' ';x:,.fr 20 > > >> ┌───────────────────────────┬─┬──────────┐ > > >> │1.0000000000000000000000000│ │ 1│ > > >> │0.5000000000000000000000000│ │ 1r2│ > > >> │0.6666666666666666300000000│ │ 2r3│ > > >> │0.5999999999999999800000000│ │ 3r5│ > > >> │0.6250000000000000000000000│ │ 5r8│ > > >> │0.6153846153846154200000000│ │ 8r13│ > > >> │0.6190476190476190700000000│ │ 13r21│ > > >> │0.6176470588235294400000000│ │ 21r34│ > > >> │0.6181818181818181700000000│ │ 34r55│ > > >> │0.6179775280898876000000000│ │ 55r89│ > > >> │0.6180555555555555800000000│ │ 89r144│ > > >> │0.6180257510729614300000000│ │ 144r233│ > > >> │0.6180371352785145600000000│ │ 233r377│ > > >> │0.6180327868852458800000000│ │ 377r610│ > > >> │0.6180344478216818200000000│ │ 610r987│ > > >> │0.6180338134001252000000000│ │ 987r1597│ > > >> │0.6180340557275542100000000│ │ 1597r2584│ > > >> │0.6180339631667065600000000│ │ 2584r4181│ > > >> │0.6180339985218034100000000│ │ 4181r6765│ > > >> │0.6180339850173579600000000│ │6765r10946│ > > >> └───────────────────────────┴─┴──────────┘ > > >> > > >> Linda > > >> > > >> > > >> > > >> -----Original Message----- > > >> From: Programming <programming-boun...@forums.jsoftware.com> On Behalf Of > > >> William Tanksley, Jr > > >> Sent: Friday, March 29, 2019 12:23 PM > > >> To: Programming forum <programm...@jsoftware.com> > > >> Subject: Re: [Jprogramming] converting from 'floating' to 'rational' > > >> > > >> Ian Clark <earthspo...@gmail.com> wrote: > > >>> But why should I feel obliged to carry on using lossy methods when > > >>> I've just discovered I don't need to? Methods such as floating point > > >>> arithmetic, plus truncation of infinite series at some arbitrary > > >>> point. The fact that few practical measurements are made to an > > >>> accuracy greater than 0.01% doesn't actually justify lossy methods in > > >>> the calculating machine. It merely condones them, which is something > > >> else entirely. > > >> > > >> There will be a cost, of course. Supporting arbitrarily small and large > > >> numbers changes the time characteristics of the computations in ways that > > >> will depend on the log-size of the numbers -- and of course will blow the > > >> CPU's caching. Also, because the intermediate values are being stored > > >> with > > >> unlimited precision, you may find some surprises, such as values close > > >> to 1 > > >> which have enormous numerators and denominators. > > >> > > >> IMO it's a worthy experiment, especially if you wind up gathering data > > >> about the cost and benefit. > > >> > > >> There's some interesting reflections going on about this on the "unums" > > >> mailing list. The trouble with indefinite precision rationals is that > > >> they > > >> are overkill for all of the problems where they're actually needed, since > > >> the inputs and the solution will normally need to be expressed to only > > >> finite digits. Now, I don't think this makes doing experiments with them > > >> worthless; far from it. By tracking things like the smallest expected > > >> input > > >> (for example the smallest triangle side, or the largest ratio between > > >> sides) and the largest integer generated as intermediate value (perhaps > > >> also tracking the ratio in which this integer appeared), we can wind up > > >> answering how bad things can get (of course, this is the task of > > >> numerical > > >> analysis). > > >> > > >> Ulrich Kulisch developed technology called the "super-accumulator", which > > >> was supposed to function alongside the usual group of floating-point > > >> registers. It stored an overkill number of bits to permit it to > > >> accumulate > > >> multiple additions of products of arbitrary floats, the sort of > > >> operations > > >> you need to evaluate polynomials and linear algebra. Using this, he was > > >> able to show that a large number of operations which were considered > > >> unstable were possible to stabilize by providing this unrounded > > >> accumulator. In the end this wasn't made part of the IEEE standard, but > > >> it's being included in some of the numerical systems being developed in > > >> response to the need for more flexible floating point formats from the > > >> machine-learning world, where smaller-bitwidth floating point numbers > > >> both > > >> make stability a serious concern and also make the required size of the > > >> superaccumulator much smaller. > > >> > > >>> Ian Clark > > >> > > >> -Wm > > >> ---------------------------------------------------------------------- > > >> For information about J forums see > > >> https://eur01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm&data=02%7C01%7C%7Cdd9b991445bd4aee1e7a08d6b462d9c2%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636894733979469996&sdata=0uEWVVEQ3RUboB2xRtJgdxL%2FUSZGkeff1L9HAEGmhYM%3D&reserved=0 > > >> ---------------------------------------------------------------------- > > >> For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
