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flat assembler > Heap > What is the best pie you can get with 9 digits?

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neville



Joined: 13 Jul 2008
Posts: 496
Location: New Zealand

sleepsleep wrote:
the actual pi is what divide by what?

the actual pi is irrational which means that it cannot be the ratio of 2 integers. So no integers a and b exist such that pi = a divided by b
e.g. your "what"s can't both be whole numbers Wink

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Post 08 Feb 2017, 08:09
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YONG



Joined: 16 Mar 2005
Posts: 6712
Location: 22° 15' N | 114° 10' E
Latest submission by me:

(8*2) / (4^(73/(65-9)) - 1) = 3.14159260364 ...

which is correct to 7 decimal places!

Since my expression is based on the result generated by bitRAKE's program and bitRAKE's program is based on the script written by T.G., the honors should be shared between the three of us.

So, the best answer as of now is correct to 7 decimal places:

YONG & bitRAKE & T.G.: (8*2) / (4^(73/(65-9)) - 1) = 3.14159260364 ...

Wink
Post 08 Feb 2017, 09:06
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bitRAKE



Joined: 21 Jul 2003
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Good thinking YONG. I don't feel I should take credit for anything, but I enjoy being included in an interesting discussion. Tomasz helped me see a high level overview of the problem - an effective way to fit the pieces together.

3^(1+((54-6)^(2/7))/8/9) = 3.14159262816...

Smaller absolute error.

We might ask for approximations of any irrational under these constraints:
Euler's number, sqrt(2), etc.
Post 08 Feb 2017, 23:31
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
pi = 3.14159265358979323846264338327950288419716939937510582 ...

So, the latest result generated by bitRAKE's program is a bit closer.

The best answers as of now are correct to 7 decimal places:

bitRAKE & T.G.: 3^(1+((54-6)^(2/7))/8/9) = 3.14159262816 ...
YONG & bitRAKE & T.G.: (8*2) / (4^(73/(65-9)) - 1) = 3.14159260364 ...

Hope that we can have 10 or more decimal places. If so, the old expression found by B. Ziv in 2004 will be superceded:

https://www.futilitycloset.com/2010/05/12/pandigital-approximations/

Wink
Post 09 Feb 2017, 02:03
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revolution
When all else fails, read the source


Joined: 24 Aug 2004
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YONG wrote:

The notes for value for e states:

https://www.futilitycloset.com/2010/05/12/pandigital-approximations/ wrote:
... reproduces e to 18,457,734,525,360,901,453,873,570 decimal places.

That is ~10^25 decimal places. The entire worlds collection of storage doesn't come close to that value. So how was it verified?

Another site claims the world record for digits of e is "only" 5,000,000,000,000 digits. This is about 13 orders of magnitude fewer. And this seems more reasonable.

Did I misunderstand something?
Post 09 Feb 2017, 03:32
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sleepsleep



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neville wrote:

sleepsleep wrote:
the actual pi is what divide by what?

the actual pi is irrational which means that it cannot be the ratio of 2 integers. So no integers a and b exist such that pi = a divided by b
e.g. your "what"s can't both be whole numbers Wink



thanks for the information,
i was thinking there are 2 integers, a / b that produce infinite answer that becoming pi number,

feels very weird,
Post 09 Feb 2017, 04:27
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E

revolution wrote:

YONG wrote:

The notes for value for e states:

https://www.futilitycloset.com/2010/05/12/pandigital-approximations/ wrote:
... reproduces e to 18,457,734,525,360,901,453,873,570 decimal places.

That is ~10^25 decimal places. The entire worlds collection of storage doesn't come close to that value. So how was it verified?

Refer to the following video for how the expression works:

https://www.youtube.com/watch?v=xgBGibfLD-U

I don't think we actually need to store all the decimal places for the verification process.

e = lim[n -> infinity] (1 + 1/n)^n

As n gets bigger and bigger, the calculated value of e gets more and more decimal places. There is a well-studied relationship between the value of n and the number of decimal places of the calculated value of e. In the given pandigital expression, n is just 9^4^42, a very big number.

Refer to:
https://en.wikipedia.org/wiki/E_(mathematical_constant)#Alternative_characterizations

Wink
Post 09 Feb 2017, 05:32
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revolution
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Okay, yeah, 9^4^42 = 3^2^85. Makes sense.

Thanks for the explanation.
Post 09 Feb 2017, 05:44
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bitRAKE



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Post 09 Feb 2017, 06:16
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revolution
When all else fails, read the source


Joined: 24 Aug 2004
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So one of those pandigital approximations does comply with the rules given here and gives 9 digits:

E. Pegg: 3+(1-(9-8^(-5))^(-6))/(7+2^(-4)) = 3.1415926539165...
Post 09 Feb 2017, 06:33
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YONG



Joined: 16 Mar 2005
Posts: 6712
Location: 22° 15' N | 114° 10' E

revolution wrote:
So one of those pandigital approximations does comply with the rules given here and gives 9 digits:

E. Pegg: 3+(1-(9-8^(-5))^(-6))/(7+2^(-4)) = 3.1415926539165...

No, it does not, as negative powers are not allowed.

a^(-b) = 1 / a^b

At least one digit is saved if we allow negative powers.

Wink
Post 09 Feb 2017, 06:49
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revolution
When all else fails, read the source


Joined: 24 Aug 2004
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YONG wrote:
...negative powers are not allowed.

I don't see that from the rules:

YONG wrote:
The rules are:

- Every digit from 1 to 9 must be used exactly once.
- No decimal point.
- Only basic arithmetic operations are allowed: +, -, *, /, (), power.

Post 09 Feb 2017, 06:51
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
Refer to:

https://board.flatassembler.net/topic.php?p=192799#192799

I am just following the "teachings" of a wise forum member!

Wink
Post 09 Feb 2017, 07:01
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revolution
When all else fails, read the source


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Yes, I am aware of that post. But you never explicitly disallowed unary minus. You just said "allowing unary minus ...". And indeed you said in that post you are keeping the same rules. And the way I read the rules unary minus is allowed. It is a basic arithmetic operation.

If you want to alter the rules then that is a different matter. So perhaps you want to state that you are modifying the rules, and what the new rules are?

E. Pegg will be so disappointed. Sad
Post 09 Feb 2017, 07:18
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
The challenge is to find the best zeroless pandigital approximation of pi.

The rules are:

- Every digit from 1 to 9 must be used exactly once.
- No decimal point can be used.
- Only basic arithmetic operations are allowed: +, -, *, /, (), power.
- No radical sign can be used.
- The unary minus operation is disallowed.

The rationale is to keep the rules as restrictive as possible.

As of now, the best answers are correct to 7 decimal places:

bitRAKE & T.G.: 3^(1+((54-6)^(2/7))/8/9) = 3.14159262816 ...
YONG & bitRAKE & T.G.: (8*2) / (4^(73/(65-9)) - 1) = 3.14159260364 ...

The goal is to have 10 or more decimal places. If achieved, the old expression found by B. Ziv in 2004 will be superceded:

https://www.futilitycloset.com/2010/05/12/pandigital-approximations/

Good luck.

Wink


Last edited by YONG on 09 Feb 2017, 08:42; edited 1 time in total
Post 09 Feb 2017, 07:52
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revolution
When all else fails, read the source


Joined: 24 Aug 2004
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YONG wrote:
The rules are:

- Every digit from 1 to 9 must be used exactly once.
- No decimal point can be used.
- Only basic arithmetic operations are allowed: +, -, *, /, (), power.
- No radical sign can be used.
- The unary minus operation is disallowed.

Hehe, okay I think E. Pegg is still going to be happy:

E. Pegg: 3+(1-(9-8^(0-5))^(0-6))/(7+2^(0-4)) = 3.1415926539165...

You didn't say anything about the digit zero, so I assume it is allowed. Razz
Post 09 Feb 2017, 08:22
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
Thanks for pointing out the omission.

Added "zeroless" before "pandigital approximation of pi".

Wink
Post 09 Feb 2017, 09:22
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neville



Joined: 13 Jul 2008
Posts: 496
Location: New Zealand

sleepsleep wrote:
feels very weird,

You can think of the irrationality of pi this way: if the diameter of a circle is any exact number of whole units, then it's circumference is never an exact number of whole units.


Code:
Some rational approximations of pi ................correct to:

22/7 = 3 + 1/7 = 3.142857 142857 142857...         2 decimal places only!

355/113 = 3 + 16/113 = 3.14159292035398...         6 decimal places

103993/33102 = 3 + 4687/33102 = 3.1415926530119... 9 decimal places


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Post 09 Feb 2017, 22:54
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YONG



Joined: 16 Mar 2005
Posts: 6712
Location: 22° 15' N | 114° 10' E
A spinoff challenge:

Using this set of four digits, {1, 2, 5, 9}, form an expression that gives the exact value of 33102.

Same rules apply.

2^15 + 9 = 32777 ...... Close but not good enough!

Wink
Post 10 Feb 2017, 03:28
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revolution
When all else fails, read the source


Joined: 24 Aug 2004
Posts: 14483
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YONG wrote:
A spinoff challenge:

Using this set of four digits, {1, 2, 5, 9}, form an expression that gives the exact value of 33102.

I see what you did there.

3+4687/Get33102From(1,2,5,9) gives 9 DP.
Post 10 Feb 2017, 03:41
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