sin (M_PI)

[user923005]

Re: sin (M_PI)

On Jul 13, 10:01 pm, "Malcolm McLean" <regniz...@btin ternet.com>
wrote:

"Richard Bos" <r...@hoekstr a-uitgeverij.nlwr ote in message
>
news:46971c4f.1 41298470@news.x s4all.nl...
>
>
>

"Malcolm McLean" <regniz...@btin ternet.comwrote :

>

"Dale Henderson" <nil...@hotpop. comwrote in message
>news:878x9ld0e 3.fsf@hotpop.co m...
>>>>"JT" == Jens Thoms Toerring <j...@toerring. dewrites:

>

JTPi is an irrational number, i.e. you can't write it down
JTexactly without giving an infinite number of digits (what-
JTever number system you use).

>

This isn't so. You can use base pi where pi becomes 10 of course 4 is
a problem. :)

>

More subtly you can use base i (sqrt -1), and allow imaginary digits.

>

No, you can't. And you can't use pi, either.

>

# 5.2.4.2.2 Characteristics of floating types <float.h>
...
# b base or radix of exponent representation (an integer 1)

>

Note: integer. Ternary floating point arithmetic is allowed in C, as is
(probably more commonly; IIRC at least one implementation actually used
it) hexadecimal-based FP. But bases pi and i are not integral.

>

Of course, in mathematics, matters are different. But mathematics deals
with infinite precision, while C deals with the real world, where
everything exists only in quanta.

>
There have been serious proposals to build base phi - the golden ratio OR
1.618... - processors. These have certain advantages because corrupt bits in
integers can be detected. So far no one has implemented a Fibonnaci or base
Phi machine in hardware, to my knowledge.

I have heard of base e and base 3 suggestions but never phi before.
Apparently there is some mathematical reason that base e offers the
most dense compaction of information. Since 3 is close to 2.71828...
base 3 is nearly ideal and can be represented by +1,0,-1 voltage
states. On the other hand, since there is no existing hardware for
base 3 computations, it seems to remain an academic curiosity.

Do you have any citation for the golden ratio base? I would be
curious to read it.

[Keith Thompson]

Re: sin (M_PI)

user923005 <dcorbit@connx. comwrites:
[...]

I have heard of base e and base 3 suggestions but never phi before.
Apparently there is some mathematical reason that base e offers the
most dense compaction of information. Since 3 is close to 2.71828...
base 3 is nearly ideal and can be represented by +1,0,-1 voltage
states. On the other hand, since there is no existing hardware for
base 3 computations, it seems to remain an academic curiosity.

[...]

That's not quite true, though they are mostly an academic curiosity.

See <http://en.wikipedia.or g/wiki/Ternary_compute r>. (Yes, it's a
Wikipedia article, but one of the citations is Knuth.)

--
Keith Thompson (The_Other_Keit h) kst-u@mib.org <http://www.ghoti.net/~kst>
San Diego Supercomputer Center <* <http://users.sdsc.edu/~kst>
"We must do something. This is something. Therefore, we must do this."
-- Antony Jay and Jonathan Lynn, "Yes Minister"

[Malcolm McLean]

Re: sin (M_PI)


"Keith Thompson" <kst-u@mib.orgwrote in message
news:lntzs416xf .fsf@nuthaus.mi b.org...

rlb@hoekstra-uitgeverij.nl (Richard Bos) writes:

>"Malcolm McLean" <regniztar@btin ternet.comwrote :

[...]

>>There have been serious proposals to build base phi - the golden
>>ratio OR 1.618... - processors. These have certain advantages
>>because corrupt bits in integers can be detected. So far no one has
>>implemented a Fibonnaci or base Phi machine in hardware, to my
>>knowledge.

>>
>I'd like to see them do it. It sounds like one of those ideas that work
>like magic in practice, but hit all kinds of snags when you build them
>for real.

>
s/in practice/in theory/
>
(unless you're making some really subtle point).
>

What he means is that the principle is sound, the engineering maybe a bit
more difficult. For instance you would have to design all the logic from
scratch, which might mean a big one-off start up cost. When you add that
venture capitalists want a high return on something so risky - they don't
understand the computer science, but they know it is radically new - it
might simply not be feasible as a commercial proposition.

--
Free games and programming goodies.

http://www.personal.leeds.ac.uk/~bgy1mm

[Richard Bos]

Re: sin (M_PI)

Keith Thompson <kst-u@mib.orgwrote:

rlb@hoekstra-uitgeverij.nl (Richard Bos) writes:

"Malcolm McLean" <regniztar@btin ternet.comwrote :

[...]

There have been serious proposals to build base phi - the golden
ratio OR 1.618... - processors. These have certain advantages
because corrupt bits in integers can be detected. So far no one has
implemented a Fibonnaci or base Phi machine in hardware, to my
knowledge.

I'd like to see them do it. It sounds like one of those ideas that work
like magic in practice, but hit all kinds of snags when you build them
for real.

>
s/in practice/in theory/
>
(unless you're making some really subtle point).

No, that's what I meant. A rather extended typo, that...

Richard

[Richard Bos]

Re: sin (M_PI)

Keith Thompson <kst-u@mib.orgwrote:

user923005 <dcorbit@connx. comwrites:
[...]

I have heard of base e and base 3 suggestions but never phi before.
Apparently there is some mathematical reason that base e offers the
most dense compaction of information. Since 3 is close to 2.71828...
base 3 is nearly ideal and can be represented by +1,0,-1 voltage
states. On the other hand, since there is no existing hardware for
base 3 computations, it seems to remain an academic curiosity.

>
That's not quite true, though they are mostly an academic curiosity.
>
See <http://en.wikipedia.or g/wiki/Ternary_compute r>. (Yes, it's a
Wikipedia article, but one of the citations is Knuth.)

One Knuth citation does not turn anything into the truth, as is
definitively proven by Shaks. Merch.Ven. I,3:94.

Richard

[Richard Heathfield]

Re: sin (M_PI)

Richard Bos said:

<snip>

One Knuth citation does not turn anything into the truth, as is
definitively proven by Shaks. Merch.Ven. I,3:94.

The man is, notwithstanding , sufficient. [op cit]

--
Richard Heathfield <http://www.cpax.org.uk >
Email: -www. +rjh@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999

[Dik T. Winter]

Re: sin (M_PI)

In article <46971c4f.14129 8470@news.xs4al l.nlrlb@hoekstra-uitgeverij.nl (Richard Bos) writes:
....

More subtly you can use base i (sqrt -1), and allow imaginary digits.

>
No, you can't. And you can't use pi, either.
>
# 5.2.4.2.2 Characteristics of floating types <float.h>
...
# b base or radix of exponent representation (an integer 1)
>
Note: integer. Ternary floating point arithmetic is allowed in C, as is
(probably more commonly; IIRC at least one implementation actually used
it) hexadecimal-based FP. But bases pi and i are not integral.

There has been one machine that used ternary floating-point. A host of
machines with hexadecimal floating-points (think IBM and look-alikes).
Also quarternary did occur.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

[Dik T. Winter]

Re: sin (M_PI)

In article <JLBso7.C6x@cwi .nl"Dik T. Winter" <Dik.Winter@cwi .nlwrites:

In article <46971c4f.14129 8470@news.xs4al l.nlrlb@hoekstra-uitgeverij.nl (Richard Bos) writes:
...

More subtly you can use base i (sqrt -1), and allow imaginary digits.

>
No, you can't. And you can't use pi, either.
>
# 5.2.4.2.2 Characteristics of floating types <float.h>
...
# b base or radix of exponent representation (an integer 1)
>
Note: integer. Ternary floating point arithmetic is allowed in C, as is
(probably more commonly; IIRC at least one implementation actually used
it) hexadecimal-based FP. But bases pi and i are not integral.

>
There has been one machine that used ternary floating-point. A host of
machines with hexadecimal floating-points (think IBM and look-alikes).
Also quarternary did occur.

And I did forget base 8, decimal; and Illiac was base 16384 (IIRC).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

[Malcolm McLean]

Re: sin (M_PI)


"user923005 " <dcorbit@connx. comwrote in message

Do you have any citation for the golden ratio base? I would be
curious to read it.
>

Phi is 1.62... when you square it you obtain Phi+1. The other number which
has this property, phi, is 0.62... or Phi -1, and is also 1/Phi.

So Phi^0 = 1s
Phi^1 = 1.62s
Phi^2 = 1 + 1.62s
Phi^3 = 1 + 2 * 1.62s
...
Phi^N = Phi^(N-2) + Phi^(N-1)

What it mean is that if we ever have two consecutive 1s, we can turn them
into zeroes and set the following digit.

So you can detect bit corruption.

Also, neatly, you can represent integers. 1 is easy - 1. 2 is 10.01. Why?
because 0.11 = 1, remembering our rule that Phi^N = Phi^(N-1) + Phi(N-2). So
2 is 1.11, but we don't like consecutive set digits, so it becomes 10.01.

Every integer can be represented with the same logic.

--
Free games and programming goodies.

http://www.personal.leeds.ac.uk/~bgy1mm

[Richard Bos]

Re: sin (M_PI)

"Dik T. Winter" <Dik.Winter@cwi .nlwrote:

In article <JLBso7.C6x@cwi .nl"Dik T. Winter" <Dik.Winter@cwi .nlwrites:

In article <46971c4f.14129 8470@news.xs4al l.nlrlb@hoekstra-uitgeverij.nl (Richard Bos) writes:
...

More subtly you can use base i (sqrt -1), and allow imaginary digits.
>
No, you can't. And you can't use pi, either.
>
# 5.2.4.2.2 Characteristics of floating types <float.h>
...
# b base or radix of exponent representation (an integer 1)
>
Note: integer. Ternary floating point arithmetic is allowed in C, as is
(probably more commonly; IIRC at least one implementation actually used
it) hexadecimal-based FP. But bases pi and i are not integral.

>
There has been one machine that used ternary floating-point. A host of
machines with hexadecimal floating-points (think IBM and look-alikes).
Also quarternary did occur.

>
And I did forget base 8, decimal; and Illiac was base 16384 (IIRC).

Ok, ternary I can accept. It's silly, but OK. Hex, quat, octal, fine.
Decimal, I can see the reason. But 16384? 2**14? Come on, they must have
been taking the piss with that one...

Richard

[Dik T. Winter]

Re: sin (M_PI)

In article <469e0520.73085 912@news.xs4all .nlrlb@hoekstra-uitgeverij.nl (Richard Bos) writes:

"Dik T. Winter" <Dik.Winter@cwi .nlwrote:

In article <JLBso7.C6x@cwi .nl"Dik T. Winter" <Dik.Winter@cwi .nlwrites:

In article <46971c4f.14129 8470@news.xs4al l.nlrlb@hoekstra-uitgeverij.nl (Richard Bos) writes:

....

There has been one machine that used ternary floating-point. A host of
machines with hexadecimal floating-points (think IBM and look-alikes).
Also quarternary did occur.

And I did forget base 8, decimal; and Illiac was base 16384 (IIRC).

>
Ok, ternary I can accept. It's silly, but OK. Hex, quat, octal, fine.
Decimal, I can see the reason. But 16384? 2**14? Come on, they must have
been taking the piss with that one...

Yes, my bad memory, but it is from an e-mail I received some 13 years ago.
It was Maniac and the base was 65536 (it had 68 bit words, presumably 64
for the mantissa and 4 for the exponent).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/