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Infinity + Infinity (or NegInfinity - NegInfinity)

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  Infinity + Infinity (or NegInfinity - NegInfinity) "Skybuck Flying" <Windows7IsOK@DreamPC2006.com> - 2011-10-08 02:54 +0200
    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Robert Myers <rbmyersusa@gmail.com> - 2011-10-07 18:40 -0700
      Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-07 22:39 -0400
    Re: Infinity + Infinity (or NegInfinity - NegInfinity) William Elliot <marsh@rdrop.com> - 2011-10-07 19:56 -0700
      Re: Infinity + Infinity (or NegInfinity - NegInfinity) Jasen Betts <jasen@xnet.co.nz> - 2011-10-08 06:45 +0000
        Re: Infinity + Infinity (or NegInfinity - NegInfinity) gordonb.eug6r@burditt.org (Gordon Burditt) - 2011-10-08 02:37 -0500
          Re: Infinity + Infinity (or NegInfinity - NegInfinity) William Elliot <marsh@rdrop.com> - 2011-10-08 21:40 -0700
            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Kaba <kaba@nowhere.com> - 2011-10-09 12:17 +0300
    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Jasen Betts <jasen@xnet.co.nz> - 2011-10-08 06:33 +0000
    Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Skybuck Flying" <Windows7IsOK@DreamPC2006.com> - 2011-10-08 16:41 +0200
      Re: Infinity + Infinity (or NegInfinity - NegInfinity) kenney@cix.compulink.co.uk - 2011-10-09 04:01 -0500
    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Kaba <kaba@nowhere.com> - 2011-10-08 17:43 +0300
      Re: Infinity + Infinity (or NegInfinity - NegInfinity) "christian.bau" <christian.bau@cbau.wanadoo.co.uk> - 2011-10-08 10:20 -0700
        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Kaba <kaba@nowhere.com> - 2011-10-08 22:35 +0300
      Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-08 18:55 +0100
        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Kaba <kaba@nowhere.com> - 2011-10-08 23:03 +0300
          Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-08 21:27 +0100
            Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-08 17:28 -0400
              Re: Infinity + Infinity (or NegInfinity - NegInfinity) Robert Myers <rbmyersusa@gmail.com> - 2011-10-08 18:15 -0400
              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-08 23:23 +0100
                Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-08 21:29 -0400
                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-09 10:01 +0100
                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Al Grant <algrant@myrealbox.com> - 2011-10-12 06:14 -0700
                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-12 15:25 +0100
                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-12 12:10 -0400
                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-12 18:41 +0100
                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-12 15:32 -0400
                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-12 20:44 +0100
                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) Robert Myers <rbmyersusa@gmail.com> - 2011-10-12 18:11 -0700
                                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) Andrew Reilly <areilly---@bigpond.net.au> - 2011-10-13 03:48 +0000
                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Paul A. Clayton" <paaronclayton@gmail.com> - 2011-10-12 23:15 -0700
                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-13 09:12 +0200
                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Paul A. Clayton" <paaronclayton@gmail.com> - 2011-10-13 00:49 -0700
                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-13 10:37 +0200
                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) EricP <ThatWouldBeTelling@thevillage.com> - 2011-10-13 11:04 -0400
                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) EricP <ThatWouldBeTelling@thevillage.com> - 2011-10-13 17:15 -0400
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-14 09:22 +0200
                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) EricP <ThatWouldBeTelling@thevillage.com> - 2011-10-14 13:37 -0400
                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Andy \"Krazy\" Glew" <andy@SPAM.comp-arch.net> - 2011-10-13 11:11 -0700
                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Al Grant <algrant@myrealbox.com> - 2011-10-14 01:04 -0700
                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-13 09:07 +0100
                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Al Grant <algrant@myrealbox.com> - 2011-10-13 07:00 -0700
                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Robert Myers <rbmyersusa@gmail.com> - 2011-10-14 10:54 -0700
                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) Andrew Reilly <areilly---@bigpond.net.au> - 2011-10-14 22:36 +0000
                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-15 09:26 +0200
                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Paul A. Clayton" <paaronclayton@gmail.com> - 2011-10-15 05:39 -0700
                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-15 14:56 +0200
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-17 08:37 +0100
                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-17 12:03 +0200
                                                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-17 11:13 +0100
                                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-17 22:30 +0200
                                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Paul A. Clayton" <paaronclayton@gmail.com> - 2011-10-17 14:47 -0700
                                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-18 11:33 +0200
                                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-18 10:21 +0100
                                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-18 08:16 +0100
                                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-18 14:26 +0200
                                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-18 13:32 +0100
                                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-18 17:47 +0200
                                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-18 17:51 +0100
                                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-19 10:37 +0200
                                                                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-19 09:36 +0100
                                                                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-19 11:46 +0200
                                                                      Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-19 10:45 +0100
                                                                        Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-19 13:11 +0200
                                                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-19 12:12 +0100
                                                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-19 16:22 +0200
                                                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-19 15:16 +0100
                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Andrew Reilly <areilly---@bigpond.net.au> - 2011-10-15 20:16 +0000
                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Robert Myers <rbmyersusa@gmail.com> - 2011-10-15 16:26 -0700
                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-17 08:14 +0200
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) nmm1@cam.ac.uk - 2011-10-17 08:22 +0100
                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-17 11:39 +0200
                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) rpw3@rpw3.org (Rob Warnock) - 2011-10-17 21:00 -0500
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Skybuck Flying" <Windows7IsOK@DreamPC2006.com> - 2011-10-18 05:44 +0200
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Andy \"Krazy\" Glew" <andy@SPAM.comp-arch.net> - 2011-10-17 21:35 -0700
                                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Jasen Betts <jasen@xnet.co.nz> - 2011-10-24 02:40 +0000
                                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) Phil Carmody <thefatphil_demunged@yahoo.co.uk> - 2011-10-27 00:32 +0300
                                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-27 10:41 +0200
                                                Re: Infinity + Infinity (or NegInfinity - NegInfinity) Phil Carmody <thefatphil_demunged@yahoo.co.uk> - 2011-10-27 13:10 +0300
                                                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) Terje Mathisen <"terje.mathisen at tmsw.no"> - 2011-10-27 13:53 +0200
                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Keith Thompson <kst-u@mib.org> - 2011-10-12 12:34 -0700
                          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Al Grant <algrant@myrealbox.com> - 2011-10-13 06:39 -0700
                            Re: Infinity + Infinity (or NegInfinity - NegInfinity) James Kuyper <jameskuyper@verizon.net> - 2011-10-13 10:34 -0400
                              Re: Infinity + Infinity (or NegInfinity - NegInfinity) Al Grant <algrant@myrealbox.com> - 2011-10-14 00:54 -0700
              Re: Infinity + Infinity (or NegInfinity - NegInfinity) Larry Hewitt <larryhwtt@comporium.net> - 2011-10-09 21:35 -0400
                Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Skybuck Flying" <Windows7IsOK@DreamPC2006.com> - 2011-10-10 04:37 +0200
                  Re: Infinity + Infinity (or NegInfinity - NegInfinity) Larry Hewitt <larryhwtt@comporium.net> - 2011-10-10 13:58 -0400
                    Re: Infinity + Infinity (or NegInfinity - NegInfinity) "Skybuck Flying" <Windows7IsOK@DreamPC2006.com> - 2011-10-13 20:17 +0200
          Re: Infinity + Infinity (or NegInfinity - NegInfinity) Arivald <arivald_@AT_interia_DOT_.pl> - 2011-10-13 12:10 +0200

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#4065 — Infinity + Infinity (or NegInfinity - NegInfinity)

Hello,

I was just wondering what happens if "Infinity" is added to "Infinity" when 
it comes to floating point numbers ?

And also what happens if "Negative Infinity" is added to "Negative 
Infinity"... ?

In Delphi these are defined as follows (in the math.pas unit):

const
  Infinity    =  1.0 / 0.0;
  NegInfinity = -1.0 / 0.0;

As far as I know this conforms to some kind of floating point standard.

I haven't tried to add these two numbers together yet, but I will soon try 
it out.

I am just wondering from a theoretical point of view what should happen ?

Should it wrap back from infinity to infinity ?

Should it wrap back from negative infinity to negative infinity ?

Or will perhaps something weird happen ?

Bye,
  Skybuck.

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#4066

On Oct 7, 8:54 pm, "Skybuck Flying"
> I haven't tried to add these two numbers together yet, but I will soon try
> it out.
>
> I am just wondering from a theoretical point of view what should happen ?
>
> Should it wrap back from infinity to infinity ?
>
> Should it wrap back from negative infinity to negative infinity ?
>
> Or will perhaps something weird happen ?
>

Something weird happens, all right.  It's called quantum field theory,
but to add plus and minus infinity and get anything but "Who knows?,"
you have to be a quantum field theorist.

Robert.

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#4067

On 10/07/2011 09:40 PM, Robert Myers wrote:
> On Oct 7, 8:54�pm, "Skybuck Flying"
>> I haven't tried to add these two numbers together yet, but I will soon try
>> it out.
>>
>> I am just wondering from a theoretical point of view what should happen ?
>>
>> Should it wrap back from infinity to infinity ?
>>
>> Should it wrap back from negative infinity to negative infinity ?
>>
>> Or will perhaps something weird happen ?
>>
> 
> Something weird happens, all right.  It's called quantum field theory,
> but to add plus and minus infinity and get anything but "Who knows?,"
> you have to be a quantum field theorist.


#ifdef __STDC_IEC_559__
then IEC 60559 should apply.  I don't have a copy of that standard, but
the C standard indicates that it's closely related to ANSI/IEEE 854,
which I do have. It specifies:

"The invalid operation exception is signaled if an operand is invalid
for the operation being performed. The floating point result delivered
when the exception occurs without a trap shall be a quiet NaN. The
invalid operations are as follows:
...
(2) Addition or subtraction: magnitude subtraction of infinities such as
(+∞) + (-∞)"

The symbols in those parentheses are supposed to be infinities, though
I've found it's hard to predict whether or not symbols will display
properly.
-- 
James Kuyper

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#4071

On Sat, 8 Oct 2011, Skybuck Flying wrote:

> I was just wondering what happens if "Infinity" is added to "Infinity" when 
> it comes to floating point numbers ?
>
> And also what happens if "Negative Infinity" is added to "Negative 
> Infinity"... ?

The infinity of analysis, oo, is not defined by floating point numbers.
There is an oo and -oo defined in the extended reals and within the
extended reals one may consider that oo + oo = oo and -oo + -oo = -oo.

> In Delphi these are defined as follows (in the math.pas unit):
>
Do you mean the Delphic Oracle?  No, the Delphic Oracle doesn't
know math.  As far as computer baloney is concerned, infinity is
simply an over load error message.

> const
> Infinity    =  1.0 / 0.0;
> NegInfinity = -1.0 / 0.0;

Gibberish.

> As far as I know this conforms to some kind of floating point standard.

It does?  In the extended reals, 1/0 and -1/0 are considered undefined.

> I haven't tried to add these two numbers together yet, but I will soon try it 
> out.

You want to add oo and -oo?  In the extend reals oo + -oo is undefined.
If you naively manipulate the symbols, 1/0 + -1/0 = (1 + -1)/0 = 0/0
you are again faced with an undefined indeterminate.

> I am just wondering from a theoretical point of view what should happen ?
>
> Should it wrap back from infinity to infinity ?
> Should it wrap back from negative infinity to negative infinity ?
> Or will perhaps something weird happen ?
>
Doesn't compute.

Speaking of infinite, do you know that the infinity of integers
is smaller than the infinity of real numbers and the infinity
for real numbers is smaller than the infinity of the subsets of
real numbers and that there are more, ever larger infinities -
more infinities than any infinite number?

Conway has constructed a field of infinite numbers, containing all the 
infinite numbers of set theory indicated above.  He even constructs
sqr aleph_0.  I would think in his theory that aleph_eta - aleph_eta = 0.

Here aleph_0 is the first infinite cardinal and aleph_eta is the
eta-th infinite cardinal of a transfinite sequence of all infinite
numbers over all of the ordinal numbers.

There is a forth type of infinite, the geometric infinity
of the point at infinity where in Euclidean space, parallel
lines meet.  In fact there's even a line at finity of all
the points at infinity, one for each different slope m, where
all the parallel lines with slope m meet.

There are other types of infinity - the philosophical, the theological, 
the science fiction and the computer hack infinities - each progressively
better than the other for infinite nonsense.

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#4075

On 2011-10-08, William Elliot <marsh@rdrop.com> wrote:
> On Sat, 8 Oct 2011, Skybuck Flying wrote:
>
>> I was just wondering what happens if "Infinity" is added to "Infinity" when 
>> it comes to floating point numbers ?
>>
>> And also what happens if "Negative Infinity" is added to "Negative 
>> Infinity"... ?
>
> The infinity of analysis, oo, is not defined by floating point numbers.

the real numbers and IEE 754 floating point are disjoint sets.

with a few exceptions (+Inf, -Inf, NaN) floating point can only
represent a finite subset of the rational numbers)

-- 
⚂⚃ 100% natural

--- Posted via news://freenews.netfront.net/ - Complaints to news@netfront.net ---

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#4076

> the real numbers and IEE 754 floating point are disjoint sets.

I disagree.  Disjoint sets have no members in common.  Both reals
and IEEE 754 include such numbers as -1.0, 0.0, 1.0, and 1.5 .

If your point is that neither set has the other as a subset, point
granted, but "disjoint" isn't the right word here.

> with a few exceptions (+Inf, -Inf, NaN) floating point can only
> represent a finite subset of the rational numbers)

Obviously, otherwise computer floating point would require an
infinite number of bits to represent.

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#4105

On Sat, 8 Oct 2011, Gordon Burditt wrote:

> Obviously, otherwise computer floating point would require an
> infinite number of bits to represent.

Computers are incapable of fathoming infinity for they have finite 
capacity.  Math is better at it for it has a countable capacity.

It's only a desperate illusion that those who know programming,
know something about mathematics.  On the contrary, it appears
knowing programming is a mathematical handicap.

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#4112

William Elliot wrote:
> It's only a desperate illusion that those who know programming,
> know something about mathematics.  On the contrary, it appears
> knowing programming is a mathematical handicap.

A branch of programming known as generic programming is surprisingly 
close to mathematical thinking. With the least amount of assumptions, 
you wish to create data structures and algorithms capable of handling 
the greatest amount of cases. 

For example: you can create a concept named semigroup (corresponding to 
the mathematical concept) by requiring that each type conforming to a 
semigroup have a binary addition operator +, and that operation be 
associative. If you can create an algorithm which only relies on that 
operation, then that algorithm works for all types conforming to the 
semigroup concept (and not just for integers, say).

-- 
http://kaba.hilvi.org

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#4074

On 2011-10-08, Skybuck Flying <Windows7IsOK@DreamPC2006.com> wrote:
> Hello,
>
> I was just wondering what happens if "Infinity" is added to "Infinity" when 
> it comes to floating point numbers ?
>
> And also what happens if "Negative Infinity" is added to "Negative 
> Infinity"... ?

infinity plus infinity is indistinguishable from infinity times two.

> In Delphi these are defined as follows (in the math.pas unit):
>
> const
>   Infinity    =  1.0 / 0.0;
>   NegInfinity = -1.0 / 0.0;

delphi can do that because it only runs on intel compatible hardware
and can rely on the floating point arithmetic to give the
representations it wants.

> As far as I know this conforms to some kind of floating point standard.

IEEE 754

> I haven't tried to add these two numbers together yet, but I will soon try 
> it out.

you should get "NaN" which is "Not a Number" and the same result that
0.0/0.0 gives.

> Or will perhaps something weird happen ?

depends if you're expecting NaN or not.

   ( 1.0 / 0.0 ) + ( -1.0 / 0.0 )
 = ( 1.0 + -1.0 ) / 0.0 
 = 0.0 / 0.0 

http://en.wikipedia.org/wiki/NaN

-- 
⚂⚃ 100% natural

--- Posted via news://freenews.netfront.net/ - Complaints to news@netfront.net ---

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#4082

Hi,

I was mostly interested in +inf + +inf. (Positive infinity + Positive 
infinity)

However perhaps that situation is now unlikely in my algorithm so my 
interest in this special situation is dwindling/decreasing ;)

But maybe this situation might still pop-up and then my interest in it might 
go back up ;) :)

So don't expect me to look to deeply into it for now ;) (If my interest in 
it changes, I'll let you know, could be within a few days, otherwise years, 
or maybe never...)

Bye,
  Skybuck. 

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#4109

In article <99683$4e9060f9$5419acc3$10591@cache5.tilbu1.nb.home.nl>, 
Windows7IsOK@DreamPC2006.com (Skybuck Flying) wrote:

> I was mostly interested in +inf + +inf. (Positive infinity + Positive 
> infinity)

In which case you should get a book on theoretical mathematics. From 
memory aleph0 plus aleph0 =aleph0.  While infinity theory is important 
in mathematics it is hardly applicable to computer arithmetic.

 Ken Young

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#4083

Skybuck Flying wrote:
> Hello,
> 
> I was just wondering what happens if "Infinity" is added to "Infinity" when 
> it comes to floating point numbers ?
> 
> And also what happens if "Negative Infinity" is added to "Negative 
> Infinity"... ?
> 
> In Delphi these are defined as follows (in the math.pas unit):
> 
> const
>   Infinity    =  1.0 / 0.0;
>   NegInfinity = -1.0 / 0.0;
> 
> As far as I know this conforms to some kind of floating point standard.

Here are some interesting properties of the IEEE floating point 
standard, which is the underlying technology behind most home computers. 
The behaviour of high-level programming languages tend to mirror it 
closely. Let x be some positive, regular, floating point number. The 
basic mathematical model is the affinely extended reals, but the 
standard defines a bit more (see division by zero).

Zeros:

* There are two zeros, +0 and -0.
* -0 = -(+0)
* Floating point comparison is defined so that +0 = -0 is true.

Not-A-Number (NaN):

* Produced as an error value, for example with +0 / +0.
* Multiplying, adding, subtracting or dividing with a NaN produces a 
NaN. The NaN is said to be propagated.
* All comparisons (<, >, =, !=, <=, >=) with a NaN yield false.

Infinities:

* There is infinity (oo) and minus infinity (-oo). 
* -oo = -(oo)
* oo + oo = oo
* -oo + -oo = -oo
* oo + -oo = NaN
* +-x + oo = oo
* +-x + -oo = -oo
* Subtraction similarly...

Division by zero:

* +-x / +0 = +-oo
* +-x / -0 = -+oo
* +-0 / +-0 = NaN

Multiplication by infinities:

* +-x * oo = +-oo
* +-x * -oo = -+oo
* +0 * +-oo = +-0
* -0 * +-oo = -+0

Division by infinities:

* +-x / oo = +-0
* +-x / -oo = -+0

-- 
http://kaba.hilvi.org

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#4087

On Oct 8, 3:43 pm, Kaba <k...@nowhere.com> wrote:

> Multiplication by infinities:
>
> * +-x * oo = +-oo
> * +-x * -oo = -+oo
> * +0 * +-oo = +-0
> * -0 * +-oo = -+0

The 0 times infinity case does not agree with IEEE754.
And note that not all comparisons with NaN yield "false": x != NaN and
NaN != x yields "true".

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#4091

christian.bau wrote:
> On Oct 8, 3:43 pm, Kaba <k...@nowhere.com> wrote:
> 
> > Multiplication by infinities:
> >
> > * +-x * oo = +-oo
> > * +-x * -oo = -+oo
> > * +0 * +-oo = +-0
> > * -0 * +-oo = -+0
> 
> The 0 times infinity case does not agree with IEEE754.
> And note that not all comparisons with NaN yield "false": x != NaN and
> NaN != x yields "true".

Yes, thanks for the fix.

-- 
http://kaba.hilvi.org

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#4089

In article <MPG.28fa8e225336ca499899d3@news.cc.tut.fi>,
Kaba  <kaba@nowhere.com> wrote:
>
>Here are some interesting properties of the IEEE floating point 
>standard, which is the underlying technology behind most home computers. 
>The behaviour of high-level programming languages tend to mirror it 
>closely.

Oh, no, they don't!  Not even if you had got it correct.

>   Let x be some positive, regular, floating point number. The 
>basic mathematical model is the affinely extended reals, but the 
>standard defines a bit more (see division by zero).

Its signed zero handling is incompatible with that model in several
respects where traditional floating-point isn't.

Regards,
Nick Maclaren.

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#4093

 wrote:
> In article <MPG.28fa8e225336ca499899d3@news.cc.tut.fi>,
> Kaba  <kaba@nowhere.com> wrote:
> >
> >Here are some interesting properties of the IEEE floating point 
> >standard, which is the underlying technology behind most home computers. 
> >The behaviour of high-level programming languages tend to mirror it 
> >closely.
> 
> Oh, no, they don't!  Not even if you had got it correct.

Simply asserting that isn't particularly helpful. Maybe you could step 
in and make a better statement, on where the behaviour agreees and where 
it differs.

> >   Let x be some positive, regular, floating point number. The 
> >basic mathematical model is the affinely extended reals, but the 
> >standard defines a bit more (see division by zero).
> 
> Its signed zero handling is incompatible with that model in several
> respects where traditional floating-point isn't.

I disagree. Mathematics has one zero, the IEEE floating point has two. 
However, they negative and positive zero can not be differentiated by 
their action, expect when dividing by zero. But dividing by zero is not 
defined in the affinely extended reals. Thus, unless I am missing 
something, the IEEE floating point is a fine model for the affinely 
extended reals.

-- 
http://kaba.hilvi.org

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#4094

In article <MPG.28fad914bc1e280d9899d5@news.cc.tut.fi>,
Kaba  <kaba@nowhere.com> wrote:
>> >
>> >Here are some interesting properties of the IEEE floating point 
>> >standard, which is the underlying technology behind most home computers. 
>> >The behaviour of high-level programming languages tend to mirror it 
>> >closely.
>> 
>> Oh, no, they don't!  Not even if you had got it correct.
>
>Simply asserting that isn't particularly helpful. Maybe you could step 
>in and make a better statement, on where the behaviour agreees and where 
>it differs.

The IEEE 754 model is incompatible with most language's arithmetic
models, so it isn't possible (even in theory) to answer that question
for the vast majority of languages.  If you really want to know the
details, you will have to start by studying various language standards
and specifications - i.e. the definitions of the languages, not
someone's summary of them.

>> >   Let x be some positive, regular, floating point number. The 
>> >basic mathematical model is the affinely extended reals, but the 
>> >standard defines a bit more (see division by zero).
>> 
>> Its signed zero handling is incompatible with that model in several
>> respects where traditional floating-point isn't.
>
>I disagree. Mathematics has one zero, the IEEE floating point has two. 
>However, they negative and positive zero can not be differentiated by 
>their action, expect when dividing by zero. But dividing by zero is not 
>defined in the affinely extended reals. Thus, unless I am missing 
>something, the IEEE floating point is a fine model for the affinely 
>extended reals.

You are.  IEEE 754 also has a sign test function, and that means
that -(x-y) is distinguishable from y-x, whereas -(x-y) = y-x for
in that form of mathematics.


Regards,
Nick Maclaren.

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#4095

On 10/08/2011 04:27 PM, nmm1@cam.ac.uk wrote:
> In article <MPG.28fad914bc1e280d9899d5@news.cc.tut.fi>,
> Kaba  <kaba@nowhere.com> wrote:
...
> The IEEE 754 model is incompatible with most language's arithmetic
> models,

That claim would be more useful if you would at least hint at what the
typical incompatibilities are. I doubt that the writers of that standard
intended it to be incompatible with those languages, and I doubt that
the designers of all of those languages went out of their way to be
incompatible with that standard, which suggests that any such
incompatibility is likely to be subtle. If so, reading the IEEE 754
standard and comparing it with the language standards is not likely to
be fruitful without at least a hint of what incompatibilities to look for.

...
>> I disagree. Mathematics has one zero, the IEEE floating point has two. 
>> However, they negative and positive zero can not be differentiated by 
>> their action, expect when dividing by zero. But dividing by zero is not 
>> defined in the affinely extended reals. Thus, unless I am missing 
>> something, the IEEE floating point is a fine model for the affinely 
>> extended reals.
> 
> You are.  IEEE 754 also has a sign test function, and that means
> that -(x-y) is distinguishable from y-x, whereas -(x-y) = y-x for
> in that form of mathematics.

That doesn't matter unless a sign test function, with conflicting
semantics, is part of the definition of affinely extended reals.
-- 
James Kuyper

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#4098

On 10/8/2011 5:28 PM, James Kuyper wrote:
> On 10/08/2011 04:27 PM, nmm1@cam.ac.uk wrote:
>> In article<MPG.28fad914bc1e280d9899d5@news.cc.tut.fi>,
>> Kaba<kaba@nowhere.com>  wrote:
> ...
>> The IEEE 754 model is incompatible with most language's arithmetic
>> models,
>
> That claim would be more useful if you would at least hint at what the
> typical incompatibilities are. I doubt that the writers of that standard
> intended it to be incompatible with those languages, and I doubt that
> the designers of all of those languages went out of their way to be
> incompatible with that standard, which suggests that any such
> incompatibility is likely to be subtle. If so, reading the IEEE 754
> standard and comparing it with the language standards is not likely to
> be fruitful without at least a hint of what incompatibilities to look for.
>
> ...
>>> I disagree. Mathematics has one zero, the IEEE floating point has two.
>>> However, they negative and positive zero can not be differentiated by
>>> their action, expect when dividing by zero. But dividing by zero is not
>>> defined in the affinely extended reals. Thus, unless I am missing
>>> something, the IEEE floating point is a fine model for the affinely
>>> extended reals.
>>
>> You are.  IEEE 754 also has a sign test function, and that means
>> that -(x-y) is distinguishable from y-x, whereas -(x-y) = y-x for
>> in that form of mathematics.
>
> That doesn't matter unless a sign test function, with conflicting
> semantics, is part of the definition of affinely extended reals.

Thank you for introducing me to the affinely extended reals.

I struggled vainly to come to grips with many odd objects in theoretical 
physics, until I finally grasped that, for someone with a pedestrian 
mind like mine, these discussions make sense only if you are willing to 
dig into the implied limiting processes in detail.

Appealing to arbitrary standards is well past silly.  If you can't 
define the limiting processes you are assuming in pedestrian terms, you 
are either a genius or you don't know what you are talking about.

Robert.

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#4099

In article <j6qf8u$hji$1@dont-email.me>,
James Kuyper  <jameskuyper@verizon.net> wrote:
>...
>> The IEEE 754 model is incompatible with most language's arithmetic
>> models,
>
>That claim would be more useful if you would at least hint at what the
>typical incompatibilities are. I doubt that the writers of that standard
>intended it to be incompatible with those languages, and I doubt that
>the designers of all of those languages went out of their way to be
>incompatible with that standard, which suggests that any such
>incompatibility is likely to be subtle. If so, reading the IEEE 754
>standard and comparing it with the language standards is not likely to
>be fruitful without at least a hint of what incompatibilities to look for.

Aargh!  You have got the whole thing arsey-versey.  If there isn't
any match between the basic concepts, what on earth is the point
in looking for detailed incompatibilities?  I said that the MODELS
were incompatible and not that the DETAILS were.

The clearest standard in this respect is Fortran: look at 7.1.8.3
in Fortran 2003 "Once the interpretation has been established in
accordance with those rules, the processor may evaluate any
mathematically equivalent expression, provided that the integrity of
parentheses is not violated."  But that is NOT the only relevant
specification of the model, even in Fortran.

Look, when I raised this on the IEEE 754R mailing list, and in
personal contact, I was told that the languages would simply have
to change their arithmetic model to fit.  I told them "no chance"
but said that I would raise it in some ISO working groups - which,
of course, said "no chance".  Why ON EARTH do you think that there
is no full support for IEEE 754 in any language whatsoever after
27 years?

And, no, I am NOT going to describe what C99 really specifies,
even though I could in all its gory detail (as I think you know).


Regards,
Nick Maclaren.

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