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Author
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
 Tomasz Grysztar wrote: I used a simple expression generator (on top of a digit permutation) and fasmg's "eval" to find the above ones.
See if you can find an expression for each of these numbers with your program:

42722830 -- using seven digits, from 1 to 7

I believe this one is doable because I have already worked out a very close value by hand:

(5^6 + 1) x 2734 = 42721484

But I need the exact value.

The pi approximation will be correct to 7 decimal places!

888582403 -- using six digits, either {2, 4, 5, 7, 8, 9} or {3, 4, 5, 6, 7, 8}

This one may not be doable. Still, give it a try.

The pi approximation will be correct to 10 decimal places!

I am happy to share the honors with you!

Please bring me some good news on New Year Day!

31 Dec 2016, 12:23
bitRAKE

Joined: 21 Jul 2003
Posts: 2624
Location: dank orb
3-(51-67)/(24+89) another in the set of 6 correct places, lol.

There are a lot of possible patterns.
02 Jan 2017, 03:37
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
So, the best answers as of now are still correct to 6 decimal places:

T.G.: (2+6*9/743)*8^(1/5) = 3.14159288348 ...
bitRAKE: 3-(51-67)/(24+89) = 3.14159292035 ...

Come on! Something even better is out there!

02 Jan 2017, 04:06
bitRAKE

Joined: 21 Jul 2003
Posts: 2624
Location: dank orb
The power searches take a long time...

Three) 3+4^6^1/(7^2*589)

I can search any pattern you think might be fruitful.

_________________
The generation of random numbers is too important to be left to chance - Robert R Coveyou
02 Jan 2017, 05:05
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
 bitRAKE wrote: 3+4^6^1/(7^2*589)
Just correct to 3 decimal places. Not good enough by the 2017 standard!

 bitRAKE wrote: I can search any pattern you think might be fruitful.
Then try this one:

42722830 -- using seven digits, from 1 to 7

I need the exact value.

The pi approximation will be correct to 7 decimal places!

Thanks!

02 Jan 2017, 05:22
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
Just found one, which has the same value as 355/113:

3^1 + (4^2)/((9+5)*8 + 7 - 6) = 3.14159292035 ...

Still correct to 6 decimal places.

02 Jan 2017, 07:41
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
So, the best answers as of now are still correct to 6 decimal places:

T.G.: (2+6*9/743)*8^(1/5) = 3.14159288348 ...
bitRAKE: 3-(51-67)/(24+89) = 3.14159292035 ...
YONG: 3^1 + (4^2)/((9+5)*8 + 7 - 6) = 3.14159292035 ...

Come on! Something even better is out there!
02 Jan 2017, 07:43
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6685
Location: Kraków, Poland
My simple "eval" script was able to search only through specific types of expressions, so to give it a try I wrote a proof-of-concept script in fasmg that does a brute force search through all possible expressions in RPN notation:
 Code: aim = 3.141592653589793238462643383279502884197169399375105820974 delta = 1 ; start showing results when they are within this distance from the aim digits db '1234' digits_count = \$-digits operators db '+-*/^' operators_count = \$-operators ln2 = 0.693147180559945309417232121458176568075500134360255254120 while 1         repeat 1 shl (digits_count-1), separators:0                 ; Create numbers from digits:                 position = 0                 repeat digits_count, index:0                         load digit:byte from digits+position                         position = position + 1                         number#index = digit-'0'                         while separators shr (position-1) and 1                                 load digit:byte from digits+position                                 position = position + 1                                 number#index = number#index*10 + digit-'0'                         end while                         if position >= digits_count                                 numbers_count = %                                 break                         end if                 end repeat                 variations_count = 1                 repeat numbers_count-1                         variations_count = variations_count*operators_count                 end repeat                 repeat variations_count, variation:0                         ; Prepare distribution of operators:                         operators_count0 = 0                         counter = numbers_count-1                         while 1                                 repeat counter                                         operators_count#% = 1                                 end repeat                                 ; Evaluate:                                 variation_cursor = variation                                 repeat numbers_count, index:0                                         stack =: number#index                                         repeat operators_count#index                                                 tmp = stack                                                 restore stack                                                 operator_index = variation_cursor mod operators_count                                                 variation_cursor = variation_cursor / operators_count                                                 load operator:1 from operators+operator_index                                                 if operator = '+'                                                         stack = stack + tmp                                                 else if operator = '-'                                                         stack = stack - tmp                                                 else if operator = '*'                                                         stack = stack * tmp                                                 else if operator = '/'                                                         stack = float stack / tmp                                                 else if operator = '^'                                                         if tmp < 20 & tmp > -20 & tmp = trunc tmp                                                                 tmp = trunc tmp                                                                 inv = 0                                                                 if tmp < 0                                                                         inv = 1                                                                         tmp = -tmp                                                                 end if                                                                 sq = stack                                                                 stack = 1                                                                 while tmp                                                                         if tmp and 1                                                                                 stack = stack * sq                                                                         end if                                                                         sq = sq * sq                                                                         tmp = tmp shr 1                                                                 end while                                                                 if inv                                                                         stack = float 1/stack                                                                 end if                                                         else                                                                 x = float stack                                                                 if x > 0                                                                         ln = 2*(x-1)/(x+1)                                                                         repeat 4                                                                                 k = trunc (ln/ln2)                                                                                 r = ln - k*ln2                                                                                 term = 1                                                                                 exp = term                                                                                 repeat 12                                                                                         term = term*r/(float %)                                                                                         exp = exp + term                                                                                 end repeat                                                                                 exp = exp shl k                                                                                 ln = ln + 2*(x-exp)/(x+exp)                                                                         end repeat                                                                         x = ln*tmp                                                                         k = trunc (x/ln2)                                                                         r = x - k*ln2                                                                         term = 1                                                                         stack = term                                                                         repeat 12                                                                                 term = term*r/(float %)                                                                                 stack = stack + term                                                                         end repeat                                                                         stack = stack shl k                                                                 else                                                                         stack = 0                                                                 end if                                                         end if                                                 else                                                         err 'unknown operator'                                                 end if                                         end repeat                                 end repeat                                 result = stack                                 restore stack                                 d = result-aim                                 if (d > 0 & d < delta) | (d < 0 & d > -delta)                                         if d > 0                                                 delta = d                                         else                                                 delta = -d                                         end if                                         ; Display generated expression:                                         variation_cursor = variation                                         repeat numbers_count, index:0                                                 if % > 1                                                         display ' '                                                 end if                                                 repeat 1, x: number#index                                                         display `x                                                 end repeat                                                 repeat operators_count#index                                                         operator_index = variation_cursor mod operators_count                                                         variation_cursor = variation_cursor / operators_count                                                         load operator:1 from operators+operator_index                                                         display ' ',operator                                                 end repeat                                         end repeat                                         out showfloat result                                         display ' = ',out,13,10                                 end if                                 ; Next distribution of operators:                                 counter = -1                                 repeat numbers_count-1, index:0                                         if operators_count#index > 0                                                 operators_count#% = operators_count#% + 1                                                 counter = operators_count#index - 1                                                 operators_count#index = 0                                                 break                                         end if                                 end repeat                                 if counter < 0                                         break                                 end if                         end while                 end repeat         end repeat         ; Next permutation of digits:         i = digits_count-1         while i >= 0                 load b:byte from digits+i                 i = i - 1                 if i >= 0                         load a:byte from digits+i                         if a < b                                 break                         end if                 end if         end while         if i < 0                 break         end if         j = i + 1         while j < digits_count-1                 load c:byte from digits+j+1                 if c > a                         b = c                         j = j + 1                 else                         break                 end if         end while         store b:byte at digits+i         store a:byte at digits+j         i = i + 1         j = digits_count-1         while i < j                 load x:byte from digits+i                 load y:byte from digits+j                 store x:byte at digits+j                 store y:byte at digits+i                 i = i + 1                 j = j - 1         end while end while
As you can see, I set up the sample above so that it searches through expressions composed from digits 1-4 only. It gets very slow quickly for larger sets of digits (to monitor the script I made a tweaked version of fasmg that shows the messages generated with DISPLAY immediately instead of buffering them*). The next step should be to rewrite this prototype script in assembly, though I do not have time for this now. With a native implementation exhaustive search through expressions generated from a sets of 6 or even 7 digits may be doable. But 8 and more is probably going to be unattainable with this method.

(The script uses the "showfloat" macro I once shared in the other thread.)

Some interesting results from my test searches with small sets of digits: with digits 1-4 the best possible approximation of pi is 3+2/14 = 3.1428571429..., interestingly digits 1-5 yield no better result. For a set of digits 1-6 some better results start coming in, for example 1*(35-4)^(2/6)=3.141380652... but I have not yet finished this search.

 YONG wrote: See if you can find an expression for each of these numbers with your program: (...)
With this script I can now try to search through expressions generated with 6 digits. With a native implementation perhaps the exhaustive search with 7 digits would also be possible.

___
* This is now in the official fasmg releases as a hidden "-v2" option.

Last edited by Tomasz Grysztar on 08 Sep 2017, 17:04; edited 9 times in total
02 Jan 2017, 09:38
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
Am I dreaming?

How come the above script -- written by T.G. -- actually has some comments?

02 Jan 2017, 11:49
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6685
Location: Kraków, Poland
And another bug in fasmg found thanks to these scripts...
02 Jan 2017, 15:26
bitRAKE

Joined: 21 Jul 2003
Posts: 2624
Location: dank orb
 Code: 5)     296^5/(81-4)^7+3     (278^4+-1)/59^6+3     (6/8)^5 * (3/9)^1 * (7/4)^2     [6^(4-3)+7^(9-2)]/8^(1+5)
...some lesser solutions.
02 Jan 2017, 18:53
bitRAKE

Joined: 21 Jul 2003
Posts: 2624
Location: dank orb
 Tomasz Grysztar wrote: And another bug in fasmg found thanks to these scripts...
Does it also cover multiple operators? Specifically, A*-B comes to mind. I was thinking "-/" could be replaced by a monotonic negation - actually, I think it's called something else. Where each value can be subtracted from zero by itself. This probably increases the complexity rather than the intent.

I like how you choose to iterate between the operators - very concise.

_________________
The generation of random numbers is too important to be left to chance - Robert R Coveyou
02 Jan 2017, 19:58
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6685
Location: Kraków, Poland

bitRAKE wrote:
 Tomasz Grysztar wrote: And another bug in fasmg found thanks to these scripts...
Does it also cover multiple operators? Specifically, A*-B comes to mind. I was thinking "-/" could be replaced by a monotonic negation - actually, I think it's called something else. Where each value can be subtracted from zero by itself. This probably increases the complexity rather than the intent.

I like how you choose to iterate between the operators - very concise.

My implementation does not have unary operators. You could insert any number of unary operators in any place in a RPN expression. For unary minus it does not make sense to put more than one in a row anywhere, but for example if we had allowed square root (which is also an unary operator), the number of potential expressions would become infinite.

Whether the formulation of problem allows to unary minus is another issue. This probably should be clarified.
02 Jan 2017, 20:06
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
 Tomasz Grysztar wrote: but for example if we had allowed square root (which is also an unary operator) ... Whether the formulation of problem allows to unary minus is another issue. This probably should be clarified.
Allowing square root means freeing up at least two digits:

^(1/2) becomes √

Allowing the radical sign of the higher roots means freeing up at least one digit:

^(1/n) becomes ^n √

Allowing unary minus in the power means freeing up at least one digit:

(1/x)^n becomes x^(-n)

A wise forum member once said, "Something something keeping up the standards something something."

So, I would say we better stick to those restrictive rules and maintain the difficulty of the problem.

03 Jan 2017, 03:48
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E

bitRAKE wrote:
 Code: 5)     296^5/(81-4)^7+3     (278^4+-1)/59^6+3     (6/8)^5 * (3/9)^1 * (7/4)^2     [6^(4-3)+7^(9-2)]/8^(1+5)
...some lesser solutions.

In (278^4+-1)/59^6+3, there is no need to write "+-1". Just write "-1".

The expression ( 6 / 8 )^5 * (3/9)^1 * (7/4)^2 gives 0.24224853515. Something is wrong with your program.

03 Jan 2017, 03:57
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6685
Location: Kraków, Poland
 YONG wrote: 888582403 -- using six digits, either {2, 4, 5, 7, 8, 9} or {3, 4, 5, 6, 7, 8} This one may not be doable. Still, give it a try.
I've been running my fasmg script in the background searching with these two sets of digits, but as you suspected no expression yielded such result. The closest number that it found is:
 Code: [RPN] 5 2 * 9 4 78 / - ^ = (5*2)^(9-(4/78)) = 888623816.274...

If we raise this to the power 1/(3*6), which is what I suspect you wanted to do, we get an approximation of pi, but it is much worse than the ones we already have:
 Code: ((5*2)^(9-(4/78)))^(1/(3*6)) = 3.1416007876788536...

The search for 42722830 that you also asked for is practically out of reach with fasmg script, because the number of expressions for seven digits is much larger. But I we write a native implementation of the searcher, then such search may become possible. But brute force scans with 8 and 9 digits are going to be out of reach anyway.
05 Jan 2017, 09:12
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
@T.G.: Thank you very much.

So, six digits is basically the limit for your script. Then give the following a try:

924269 -- using six digits:

{2, 5, 6, 7, 8, 9},
{3, 4, 5, 7, 8, 9},
{2, 4, 5, 6, 7, 8},
{2, 3, 5, 6, 7, 9}, or
{2, 3, 4, 6, 8, 9}.

(924269)^(1/(3*4))
= (924269)^(1/(2*6))
= (924269)^(1/(3+9))
= (924269)^(1/( 4 + 8 ))
= (924269)^(1/(5+7))
= 3.14159260217 ... (correct to 7 decimal places)

If I can use one more digit (4), here is the solution:

(6^3 * 4279 + 5)^(1/(8+4)) = 924269^(1/12) = 3.14159260217 ...

Please give me some good news this time!

05 Jan 2017, 10:20
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 6685
Location: Kraków, Poland
I have continued to look for 888582403 with other sets of digits, because there are also other ways to make 1/18 power. And I found this one:
 Code: [RPN]41 5 ^ 9 7 ^ - 8 * = (41^5-9^7)*8 = 888585856
Still not perfect, but a bit better, and the pi approximation it gives is:
 Code: ((41^5-9^7)*8)^(2/36)=3.14159333180...
05 Jan 2017, 13:56
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
 Tomasz Grysztar wrote: there are also other ways to make 1/18 power
Good point!

1/12 = 2/24 = 2/( 3 * 8 ) = 2/(4*6) = 4/48 = 4/( 6 * 8 ) = 8/96

{1, 4, 5, 6, 7, 9},
{1, 3, 5, 7, 8, 9},
{1, 2, 3, 5, 7, 9}, and
{1, 2, 3, 4, 5, 7}

for 924269.

Thanks!

06 Jan 2017, 01:33
YONG

Joined: 16 Mar 2005
Posts: 8000
Location: 22° 15' N | 114° 10' E
Any number n within the range

888582131 =< n =< 888582639

can give a pi approximation correct to 7 (or more) decimal places.

Come on!

06 Jan 2017, 04:57
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