Thank you, Raul, especially for the "short form". I'll try at least the Wiki article. --Kip
Sent from my iPad On Mar 1, 2013, at 7:50 AM, Raul Miller <[email protected]> wrote: > For readers following along: > > IEEE 754 is a decent search term. > > Or, at least, it is for me - Google apparently customizes search > results so I cannot know for sure that they give good, relevant > results for everyone searching on that phrase (nor can I be sure > whether anyone I care about will get different results from > me).http://steve.hollasch.net/cgindex/coding/ieeefloat.html > > Here are my top four hits when I search for IEEE 754 today: > > http://en.wikipedia.org/wiki/IEEE_floating_point > http://grouper.ieee.org/groups/754/ > http://steve.hollasch.net/cgindex/coding/ieeefloat.html > http://babbage.cs.qc.cuny.edu/IEEE-754/ > > But the short form is: IEEE 754 splits the representation of a number > into a sign (+1 or _1 -- one bit), an exponent and a mantissa. There > are actually several IEEE 754 representations, and current J > implementations use the 64 bit representation. The 64 bit > representation uses 52 bits to represent the mantissa (a number > between 1 and 2, or equal to 1), and and the remaining 11 bits > represent the exponent (a power of 2). In operation we act as though > these elements are multiplied together. > > Obviously, there are some exceptions to the above summary. Without > any exceptions this mechanism could not represent 0. So the lowest > and the highest exponents get treated specially, which is where we get > J's inconsistent numbers (_ __ _.). > > Rather than describe the details of what's going on in deeper detail, > here, I'll let you read other materials. > > FYI, > > -- > Raul > > On Thu, Feb 28, 2013 at 6:11 PM, km <[email protected]> wrote: >> I remember that comparisons with 0 are exact >> >> 0 = N - N + epsilon >> 0 0 0 0 0 0 0 1 1 >> >> which agrees with your first report. No, I am unfamiliar with the IEEE >> structure, and am surprised by these results. >> >> --Kip >> >> Sent from my iPad >> >> >> On Feb 28, 2013, at 4:21 PM, Raul Miller <[email protected]> wrote: >> >>> Are you familiar with the structure of IEEE 754 floating point numbers? >>> >>> Consider, for example: >>> >>> epsilon=: 2^_44 >>> N=: 10^i:4 >>> >>> NB. this result reflects IEEE-754's structure >>> *N+epsilon-N >>> 1 1 1 1 1 1 1 0 0 >>> >>> NB. this result reflects J's heuristic to deal with that structure >>> N=N+epsilon >>> 0 0 0 0 1 1 1 1 1 >>> >>> FYI, >>> >>> -- >>> Raul >>> >>> On Thu, Feb 28, 2013 at 4:28 PM, km <[email protected]> wrote: >>>> Here is what I did >>>> >>>> NB. right hand limit of a function >>>> >>>> lim =: 1 : 0 >>>> value =. u y + (2^_44) >>>> if. value <: - 2^40 do. __ >>>> elseif. value >: 2^40 do. _ >>>> elseif. do. value >>>> end. >>>> ) >>>> >>>> It does "reasonably well" but can be fooled, for example >>>> >>>> ] lim 2^40 >>>> _ >>>> >>>> Here it does better >>>> >>>> *: lim 1000 >>>> 1000000 >>>> >>>> dq =: 1 : (':'; 'y %~ (u x+y) - u x') NB. difference quotient >>>> >>>> 2&(^&3 dq)lim 0 NB. derivative of x^3 at 2 is 12 >>>> 12 >>>> >>>> --Kip >>>> >>>> Sent from my iPad >>>> >>>> >>>> On Feb 28, 2013, at 7:19 AM, Raul Miller <[email protected]> wrote: >>>> >>>>> Here's a model implementation: >>>>> >>>>> lim=: (1 :0)("0) >>>>> tests=. u ((1e_6*1>.|y)*0.5^i.1000)+y >>>>> tests {~{.I.((1 }. 0&~:) * 2 ~:/\ ])(,2:)(*!.0)2 -/\ tests >>>>> ) >>>>> >>>>> My assumptions are: >>>>> >>>>> (1) the limit in question is relatively stable (that my choices for >>>>> epsilon are adequate) >>>>> >>>>> (2) that the result of limit should be a consistent number. >>>>> >>>>> Note that (2) means that _ and __ will typically not be returned, >>>>> since they are inconsistent numbers (but, since they are inconsistent, >>>>> it's impossible to make an entirely consistent guarantee about their >>>>> treatment). >>>>> >>>>> (1&o.%]) lim 0 >>>>> 1 >>>>> % lim 0 >>>>> 2.67877e306 >>>>> -@% lim 0 >>>>> _2.67877e306 >>>>> >>>>> For my purposes, these "e306" values are close enough to infinity to >>>>> be treated as such. >>>>> >>>>> Note also that I'm probably being a bit too aggressive with the number >>>>> of epsilon values I'm using. >>>>> >>>>> If you really want _ and __ results, you could use something like this: >>>>> >>>>> lim=: (1 :0)("0) >>>>> tests=. u ((1e_6*1>.|y)*0.5^i.1000)+y >>>>> 1e_3*1e3* tests {~{.I.((1 }. 0&~:) * 2 ~:/\ ])(,2:)(*!.0)2 -/\ tests >>>>> ) >>>>> >>>>> However, note that this is a heuristic and its ability to force the >>>>> result to be an inconsistent infinity depends on the stability u (and >>>>> also depends on the actual limit value). I place less faith in this >>>>> mechanism than in the ability of the user to recognize that the result >>>>> should be thought of as infinite (and if the user does not understand >>>>> what's going on well enough to make that determination it's hard to >>>>> imagine how this distinction could be useful). >>>>> >>>>> FYI, >>>>> >>>>> -- >>>>> Raul >>>>> >>>>> On Wed, Feb 27, 2013 at 10:55 PM, km <[email protected]> wrote: >>>>>> Can you write an adverb lim so that >>>>>> >>>>>> sin =: 1&o. >>>>>> >>>>>> (sin % ])lim 0 >>>>>> 1 >>>>>> >>>>>> % lim 0 NB. limit is from right >>>>>> _ >>>>>> >>>>>> -@% lim 0 >>>>>> __ >>>>>> >>>>>> >>>>>> Kip Murray >>>>>> >>>>>> Sent from my iPad >>>>>> >>>>>> ---------------------------------------------------------------------- >>>>>> 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 > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
