Pi approximation games

On 02/05/2012 12:12, Uwe Hercksen wrote:
>
>
> David Brown schrieb:
>
>> At university I remember a project that involved calculating all the
>> digits of pi.
>
> Hello,
>
> it is known since centuries that calculating all the
> digits of pi is not possible. Pi has an infinite number of digits.
>
> Bye
>

<http://en.wikipedia.org/wiki/Lazy_evaluation>

The most relevant section, "working with infinite data structures", is 
missing - but I hope you get the point anyway.

mvh.,

David

On 02/05/2012 09:07, Martin Brown wrote:

<snip>

> As a physicist I found the classic approximations
>
> pi^2 = g
> pi x 10^7 seconds in a year
>
> quite handy to within 1% slide rule accuracy

Which is why a 1m pendulum has a half period of ~1 second using the 
classic formula two pi root ell over gee or:

l g / sqrt pi * 2 *

in RPN

Cheers
-- 
Syd

In article <jnq291$5dj$1@speranza.aioe.org>,
glen herrmannsfeldt  <gah@ugcs.caltech.edu> wrote:
>In comp.dsp Tim Wescott <tim@seemywebsite.com> wrote:
>> Instead of doing productive work, I just spent a few enjoyable minutes
>> with Scilab finding approximations to pi of the form m/n.
>
>> Because I'm posting to a couple of nerd groups, I can be confident
>> that most of you probably know 22/7 off the tops of your heads.
>
>> What interested me is how spotty things are -- after 22/7, the error
>> drops for a bit until you get down to 355/113 (which, if you're at an
>> equal level of nerdiness to me will ring a bell, but not have been
>> swimming around in your brain to be found).
>
>Yes. It was the first problem in the book for the HP 25C calculator
>that I got many years ago.
>
>> But what's _really_ interesting, is that the next better fit
>> isn't found until you get up to 52163/16604.  Then things get
>> steadily better until you hit 104348/33215 -- at which point
>> the next lowest ratio which improves anything is 208341/66317,
>> then 312689/99532.  At this point I decided that I would post
>> my answers for your amusement, and get back to being productive.
>
>There is an algorithm that some calculators use for converting a
>decimal result to a fraction. If I remember, that one easily finds
>successively better fractions approximating any given value.
>
>I don't remember the details, but I do remember how funny it is,
>in that at one point it takes two fractions, and adds their numerators
>and denominators, before goint to the next step.

That is standard fare in continued fractions. Everybody interested in
these kind of approximations should take a look at this fascinating
subject.

>
>I believe it is described in the manual for one of the HP
>calculators that does that conversion.
>
>Otherwise, I have the TI-92, which will generate fraction (rational)
>results, then you ask for an approximate result. Some calculations
>will give a symbolic pi result.
>
>> Discrete math is so fun.  And these newfangled chips are just
>> destroying the joy, by making floating point efficient and
>> cheap enough that you don't need to know little tricks
>> like pi = (almost) 355/113.
>
>-- glen

It depends. Pi has been calculated to billions of decimals.
Simple floating point doesn't get you there.

Groetjes Albert

--
-- 
Albert van der Horst, UTRECHT,THE NETHERLANDS
Economic growth -- being exponential -- ultimately falters.
albert@spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst

On Wed, 02 May 2012 11:10:53 +0200, David Brown
<david@westcontrol.removethisbit.com> wrote:

>On 02/05/2012 01:16, Tim Wescott wrote:
>> Instead of doing productive work, I just spent a few enjoyable minutes
>> with Scilab finding approximations to pi of the form m/n.
>>
>> Because I'm posting to a couple of nerd groups, I can be confident that
>> most of you probably know 22/7 off the tops of your heads.
>>
>> What interested me is how spotty things are -- after 22/7, the error
>> drops for a bit until you get down to 355/113 (which, if you're at an
>> equal level of nerdiness to me will ring a bell, but not have been
>> swimming around in your brain to be found).
>>
>> But what's _really_ interesting, is that the next better fit isn't found
>> until you get up to 52163/16604.  Then things get steadily better until
>> you hit 104348/33215 -- at which point the next lowest ratio which
>> improves anything is 208341/66317, then 312689/99532.  At this point I
>> decided that I would post my answers for your amusement, and get back to
>> being productive.
>>
>> Discrete math is so fun.  And these newfangled chips are just destroying
>> the joy, by making floating point efficient and cheap enough that you
>> don't need to know little tricks like pi = (almost) 355/113.
>>
>
>Wikipedia is often a great starting point for these sorts of things.  It 
>typically has enough information to give you some hints - but not so 
>much that you can't have fun finding out more:
>
><http://en.wikipedia.org/wiki/Pi#Continued_fractions>
>
>
>At university I remember a project that involved calculating all the 
>digits of pi.  It was written using a functional programming language 
>(similar to Haskell) - the result was an unending list of the digits of 
>pi.  But since the language used lazy evaluation, it didn't bother 
>calculating the entries until you tried to print them out.  I used 
>polynomial expansions of arctan() to do the sums.

There was a short PDP-8 assembly program that printed the digits of e
forever.


-- 

John Larkin                  Highland Technology Inc
www.highlandtechnology.com   jlarkin at highlandtechnology dot com   

Precision electronic instrumentation
Picosecond-resolution Digital Delay and Pulse generators
Custom timing and laser controllers
Photonics and fiberoptic TTL data links
VME  analog, thermocouple, LVDT, synchro, tachometer
Multichannel arbitrary waveform generators

John Larkin wrote:

> There was a short PDP-8 assembly program that printed the digits of e
> forever.

In order ?

	Mel.

On Tuesday, May 1, 2012 7:16:25 PM UTC-4, Tim Wescott wrote:
> Instead of doing productive work, I just spent a few enjoyable minutes 
> with Scilab finding approximations to pi of the form m/n.
> 
> Because I'm posting to a couple of nerd groups, I can be confident that 
> most of you probably know 22/7 off the tops of your heads.
> 
> What interested me is how spotty things are -- after 22/7, the error 
> drops for a bit until you get down to 355/113 (which, if you're at an 
> equal level of nerdiness to me will ring a bell, but not have been 
> swimming around in your brain to be found).
> 
> But what's _really_ interesting, is that the next better fit isn't found 
> until you get up to 52163/16604.  Then things get steadily better until 
> you hit 104348/33215 -- at which point the next lowest ratio which 
> improves anything is 208341/66317, then 312689/99532.  At this point I 
> decided that I would post my answers for your amusement, and get back to 
> being productive.
> 
> Discrete math is so fun.  And these newfangled chips are just destroying 
> the joy, by making floating point efficient and cheap enough that you 
> don't need to know little tricks like pi = (almost) 355/113.
> 
> -- 
> My liberal friends think I'm a conservative kook.
> My conservative friends think I'm a liberal kook.
> Why am I not happy that they have found common ground?
> 
> Tim Wescott, Communications, Control, Circuits & Software
> http://www.wescottdesign.com

Looking at a slide rule with a folded scale, a reasonable approximation to pi is immediately seen to be sqrt(10). rel. err less than 1%.
Clay

<mwilson@the-wire.com> wrote in message news:jnrgfo$ccj$1@dont-email.me...
> John Larkin wrote:
> 
>> There was a short PDP-8 assembly program that printed the digits of e
>> forever.
> 
> In order ?
> 
> Mel.
>

LOL

On Wed, 02 May 2012 10:31:20 -0400, mwilson@the-wire.com wrote:

>John Larkin wrote:
>
>> There was a short PDP-8 assembly program that printed the digits of e
>> forever.
>
>In order ?
>
>	Mel.

Yes. 


-- 

John Larkin                  Highland Technology Inc
www.highlandtechnology.com   jlarkin at highlandtechnology dot com   

Precision electronic instrumentation
Picosecond-resolution Digital Delay and Pulse generators
Custom timing and laser controllers
Photonics and fiberoptic TTL data links
VME  analog, thermocouple, LVDT, synchro, tachometer
Multichannel arbitrary waveform generators

On May 1, 8:26=A0pm, John Larkin <jlar...@highlandtechnology.com> wrote:
> On Tue, 01 May 2012 18:16:25 -0500, Tim Wescott <t...@seemywebsite.com>
> wrote:
>
>
>
>
>
> >Instead of doing productive work, I just spent a few enjoyable minutes
> >with Scilab finding approximations to pi of the form m/n.
>
> >Because I'm posting to a couple of nerd groups, I can be confident that
> >most of you probably know 22/7 off the tops of your heads.
>
> >What interested me is how spotty things are -- after 22/7, the error
> >drops for a bit until you get down to 355/113 (which, if you're at an
> >equal level of nerdiness to me will ring a bell, but not have been
> >swimming around in your brain to be found).
>
> >But what's _really_ interesting, is that the next better fit isn't found
> >until you get up to 52163/16604. =A0Then things get steadily better unti=
l
> >you hit 104348/33215 -- at which point the next lowest ratio which
> >improves anything is 208341/66317, then 312689/99532. =A0At this point I
> >decided that I would post my answers for your amusement, and get back to
> >being productive.
>
> >Discrete math is so fun. =A0And these newfangled chips are just destroyi=
ng
> >the joy, by making floating point efficient and cheap enough that you
> >don't need to know little tricks like pi =3D (almost) 355/113.
>
> My old HP35 calculators have a key for pi. The newer ones hide it, a
> tiny pastel shift key thing. So I just key in 3.14. Rob down the hall
> uses 3.

Grin... I always just let 2*pi =3D 10, so pi =3D 5!

(and then remember there's a 1.59 floating around)

George H.
>
> We are increasingly using floats in embedded stuff. Our ARM LPC3250
> has SIMD hardware FP operations.
>
> --
>
> John Larkin =A0 =A0 =A0 =A0 Highland Technology, Inc
>
> jlarkin at highlandtechnology dot comhttp://www.highlandtechnology.com
>
> Precision electronic instrumentation
> Picosecond-resolution Digital Delay and Pulse generators
> Custom laser drivers and controllers
> Photonics and fiberoptic TTL data links
> VME thermocouple, LVDT, synchro =A0 acquisition and simulation- Hide quot=
ed text -
>
> - Show quoted text -

David Brown schrieb:

> <http://en.wikipedia.org/wiki/Lazy_evaluation>
> 
> The most relevant section, "working with infinite data structures", is 
> missing - but I hope you get the point anyway.

Hello,

even with infinite data structures, with finite time and finite RAM, it 
is not possible to compute all digits of pi.

Bye