Guessing "exact" values 
Szabolcs 
5/16/07 2:25 AM 
"Another computer algebra system" has a function, identify(), which attempts to guess the exact expression that evaluates to a particular numerical value. Example: In[1]:= N[3Pi+3/2,10] Out[1]= 10.92477796 > identify(10.92477796); 3  + 3 Pi 2 Is there a package with similar functionality for Mathematica? Szabolcs 
Re: Guessing "exact" values 
dimitris 
5/17/07 2:57 AM 
I think there is a package of Ted Ersek but I am not sure! Check www.wolfram.com and its archives. Dimitris =CF/=C7 Szabolcs =DD=E3=F1=E1=F8=E5: 
Re: Guessing "exact" values 
Murray Eisenberg 
5/17/07 3:10 AM 
As I remember, at IMS'06 (Avignon) Stephen Wolfram (via remote linkup) played with a function to do just that. Did I remember correctly? If so, I cannot recall the name of the function, so I don't know whether the function made it into 6.0. The closest thing I can find is RootApproximant, but that doesn't seem to "recognize" an expression involving a transcendental number. Szabolcs wrote: > "Another computer algebra system" has a function, identify(), which > attempts to guess the exact expression that evaluates to a particular > numerical value. > > Example: > > In[1]:= N[3Pi+3/2,10] > > Out[1]= 10.92477796 > > > identify(10.92477796); > 3 >  + 3 Pi > 2 > > Is there a package with similar functionality for Mathematica? > > Szabolcs >  Murray Eisenberg mur...@math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 5491020 (H) University of Massachusetts 413 5452859 (W) 710 North Pleasant Street fax 413 5451801 Amherst, MA 010039305 
Re: Guessing "exact" values 
JeanMarc Gulliet 
5/17/07 3:35 AM 
Hi, The builtin function *Rationalize* and the addon package NumberTheory`Rationalize` should be the closest equivalent to the function identify(). In[1]:= x = N[3*Pi + 3/2] Out[1]= 10.92477796076938 In[2]:= Rationalize[x, 0] Out[2]= 569958623/52171186 In[3]:= x  % Out[3]= 0. In[4]:= $Version Out[4]= "5.2 for Microsoft Windows (June 20, 2005)" HTH, JeanMarc 
Re: Guessing "exact" values 
Dana DeLouis 
5/18/07 3:17 AM 
Hi. This is not the best solution, but here's one idea I use. It's not the best solution, because I can't find a way to make Mathematica's Hyperlinks dynamic. In other words, once a hyperlink is made (Entered), it appears the address is locked in stone. Maybe an expert can jump in and make this dynamic. A number we know nothing about...(??) n = 6.283185307179586 Set the variable to something you would like to use. For me... NumberToSearch = n; ReEnter this equation (Shift Enter), and then click the link. Hyperlink["Click Here: Plouffe's Inverter", StringReplace["http://bootes.math.uqam.ca/cgibin/ipcgi/lookup.\ pl?Submit=GO+&number=#&lookup_type=simple", "#" :> ToString[Evaluate[NumberToSearch], InputForm, NumberMarks > False]]] The above click shows it might be 2 Pi. For your example, I took the full value: n=10.92477796076938 This number didn't work. Sometimes it won't work if the number is large. I don't know what "large" means, but the program really works with fractions. I decided to divided the number by 3. NumberToSearch = 10.92477796076938/3 3.641592653589793 If I reenter the Hyperlink, and click the link, the solution is Pi+1/2. Multiply by 3 to get your equation. Reference: http://pi.lacim.uqam.ca/eng/  HTH :>) Dana DeLouis Windows XP & Mathematica 6.0 & Help files 5.2 :>~ "Szabolcs" <szho...@gmail.com> wrote in message news:f2eim0$t2f$1@smc.vnet.net... > "Another computer algebra system" has a function, identify(), which > attempts to guess the exact expression that evaluates to a particular > numerical value. > > Example: > > In[1]:= N[3Pi+3/2,10] > > Out[1]= 10.92477796 > > > identify(10.92477796); > 3 >  + 3 Pi > 2 > > Is there a package with similar functionality for Mathematica? > > Szabolcs >

Re: Guessing "exact" values 
Dana DeLouis 
5/18/07 3:30 AM 
As soon as I hit send, I thought of another way. Don't know why I didn't think of this sooner. I just modified mine to the following... RealLookup[NumberToSearch_Real] := Module[ { s1 = "Click Here: Plouffe's Inverter on: ", s2 = ToString[NumberToSearch, InputForm, NumberMarks > False], link = "http://bootes.math.uqam.ca/c\ gibin/ipcgi/lookup.pl?Submit=GO+&number=#&lookup_type=simple" }, Hyperlink[StringJoin[s1, s2], StringReplace[link, "#" :> s2]] ] RealLookup[NumberToSearch_] := StringJoin["Number must be Real, not: ", ToString[NumberToSearch, InputForm]]
RealLookup[2.*Pi] <...Link here...>
n = 3*Pi + 3/2; RealLookup[n/3.] <...Link here...> RealLookup[n/3]
"Number must be Real, not: (3/2 + 3*Pi)/3" RealLookup["Dog"] "Number must be Real, not: Dog"  HTH :>) Dana DeLouis
Windows XP & Mathematica 6, and 5.2 Help. :>~ "Szabolcs" <szho...@gmail.com> wrote in message news:f2eim0$t2f$1@smc.vnet.net... > "Another computer algebra system" has a function, identify(), which > attempts to guess the exact expression that evaluates to a particular > numerical value. > > Example: > > In[1]:= N[3Pi+3/2,10] > > Out[1]= 10.92477796 > > > identify(10.92477796); >
> Is there a package with similar functionality for Mathematica? > > Szabolcs >

Re: Guessing "exact" values 
Roman 
5/18/07 3:42 AM 

Re: Guessing "exact" values 
Andrzej Kozlowski 
5/18/07 3:47 AM 
Mathematica can "recognize" algebraic numbers, by using the function RootApproximant (or the function Recognize in the Legacy NumberTherory package): ToRadicals[RootApproximant[N[Sqrt[2] + Sqrt[3], 30]]] Sqrt[5 + 2*Sqrt[6]] This may not look so impressive until you check: FullSimplify[Sqrt[2] + Sqrt[3]  Sqrt[5 + 2*Sqrt[6]]] 0 Anyway, doing this for algebraic numbers has a solid mathematical basis: the so called LLL Lattice Reduction) algorithm. I don=92t know, = however, of any mathematical basis for "recognizing" transcendentals, = except by means of stored values or some other "ad hoc" approach (essentially "sophisticated guessing") Andrzej Kozlowski On 17 May 2007, at 18:59, Murray Eisenberg wrote:
> As I remember, at IMS'06 (Avignon) Stephen Wolfram (via remote link= > up) > played with a function to do just that. Did I remember correctly? > > If so, I cannot recall the name of the function, so I don't know > whether > the function made it into 6.0. The closest thing I can find is > RootApproximant, but that doesn't seem to "recognize" an expression > involving a transcendental number. >
> Szabolcs wrote: >> "Another computer algebra system" has a function, identify(), which >> attempts to guess the exact expression that evaluates to a particular >> numerical value. >> >> Example: >> >> In[1]:= N[3Pi+3/2,10] >> >> Out[1]= 10.92477796 >> >>> identify(10.92477796); >> 3 >>  + 3 Pi >> 2 >> >> Is there a package with similar functionality for Mathematica? >> >> Szabolcs >> >
>  > Murray Eisenberg mur...@math.umass.edu > Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 413 5491020 (H) > University of Massachusetts 413 5452859 (W) > 710 North Pleasant Street fax 413 5451801 > Amherst, MA 010039305 >

Re: Guessing "exact" values 
Szabolcs 
5/19/07 1:35 AM 
dimitris wrote: > I think there is a package of Ted Ersek but I am not sure! > Check www.wolfram.com and its archives. Thanks for everyone who replied! I learned a lot today. I did not find the package by Ted Ersek, but I found another one by Eric Weisstein: http://library.wolfram.com/infocenter/MathSource/5087/ The relevant file is Simplify.m Interesting functions are ToExact and TranscendentalRecognize. In another thread someone mentioned a website which can be used for similar purposes: http://pi.lacim.uqam.ca/eng/ 
Re: Re: Guessing "exact" values 
Murray Eisenberg 
5/19/07 1:51 AM 
When I run that code and click the link, my browser shows a "File not found!" error page. 
Murray Eisenberg mur...@math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 5491020 (H) University of Massachusetts 413 5452859 (W) 710 North Pleasant Street fax 413 5451801 Amherst, MA 010039305 
Re: Guessing "exact" values 
Daniel Lichtblau 
5/19/07 2:10 AM 
>> >>Murray Eisenberg mur...@math.umass.edu >>Mathematics & Statistics Dept. >>Lederle Graduate Research Tower phone 413 5491020 (H) >>University of Massachusetts 413 5452859 (W) >>710 North Pleasant Street fax 413 5451801 >>Amherst, MA 010039305 >> A common approach is to use a predefined basis of transcendentals, e.g certain powers of e, pi, gamma, and maybe some set of radicals of "small" integers. One then seeks rational combinations of these that give the input value to close approximation. This step typically uses LLL or PSLQ. Simple code to demonstrate the concept was presented in TMJ back in 1996 (and based on some email and maybe news group correspondence from late 1995). The reference is below. Transcendental Recognition. In: Tricks of the Trade, Paul Abbott editor. The Mathematica Journal 6(2):2930 (1996). To obtain electronically go to: http://www.mathematicajournal.com/issue/v6i2/ Scroll to Tricks of the Trade (under Tutorials), click to download the notebook, go to "Trancendental Recognition" section. I'll mention that a perusal of documentation suggests the identify() function for the most part uses this approach. Daniel Lichtblau Wolfram Research

Re: Guessing "exact" values 
Szabolcs 
5/19/07 11:51 PM 
Thanks again for all the replies! I think that I need to explain that I sent this message before I received the replies from Dana DeLouis and Roman, but for some reason it arrived with a delay of 2 days. 
Re: Guessing "exact" values 
Szabolcs 
5/21/07 3:09 AM 
Murray Eisenberg wrote: > When I run that code and click the link, my browser shows a "File not > found!" error page. > It is because of the line break in the URL. Make sure that after you paste it, you remove the spaces which got inserted between ... "ipcgi/lookup." and "pl?Submit" ... Szabolcs 