The Unofficial Macro Scheduler Puzzler #11

Anything Really. Just keep it clean!

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JRL
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The Unofficial Macro Scheduler Puzzler #11

Post by JRL » Fri Aug 22, 2014 8:04 pm

I am puzzled. I want to multiply two numbers and get back the result that I see if I do the math using my trusty no.2 yellow pencil.

This code:

Code: Select all

Let>num_1=12345678987654321234567898765432123456789876543212345678987654321
Let>num_2=98765432123456789876543212345678987654321234567898765432123456789
Let>ans=%num_1%*%num_2%
MDL>ans
Gives me an answer that looks like:
"1.21932632007316E129". I want an answer that displays all 130 digits rather than an answer in scientific notation.

The goals for this Puzzler? Write a script that completes the math presented in the script code above and displays all digits of the answer. The multiplication must be accurate and error free. Second the fastest process as determined by my quad core Win7 with 8 gig RAM is the winner. If there is a tie we'll resort to the old standby "fewest lines of code". Hopefully we'll have a winning entry on or before Friday, August 29, 2014. I will test your code with other number combinations. So submitting a formula that somehow presents the correct answer to this particular problem but claims that 2x2 is 57 will negate the accuracy requirement.

What will you win????

Nothing. Only my sincere admiration and possibly a bit of knowledge you didn't have before you started.

I'll present my code next week. As a point of reference mine uses only Macro Scheduler functions and completes the above math in about 10 seconds. Excluding the dialog it is 119 lines of code. I'm guessing this is easily surpassable.

Have fun... good luck.

P.S. Yes Marcus. You can play too. :)

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Meryl
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by Meryl » Mon Aug 25, 2014 4:05 pm

Is everyone stumped???

hagchr
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by hagchr » Tue Aug 26, 2014 3:25 am

Hi, thanks for a great puzzle. I answered the last one too quickly before people had the chance to think it through, which kind of "ruined" the discussion, so will not do that mistake again. I will post my solution (if still worthy) on Thursday. Let's see if I am the only one using PI to solve it.

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JRL
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by JRL » Tue Aug 26, 2014 4:34 am

Using PI? You're going to need to share that no matter what. :) I have resolve PI to xxxxx places on the list of future puzzlers. If you're already doing that you might be getting extra credit.

hagchr
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by hagchr » Thu Aug 28, 2014 3:27 pm

I think you fellow puzzlers have had enough time so here is my attempt for a solution. Without comments etc around 100 or so lines of code. Can get it down to around 60 but then it will be around 1s slower.

Hopefully I did not mislead anyone with my comment about PI. I did use it but maybe not in the way expected. I needed an array to contain the build-up of the result and a fast way to create and initiate an array in MS is to have a string of 0s and use RegEx to create the array. So, to reduce looping, I needed a long string and tried to find any predefined constants in MS and found PI which is defined with 14 decimals, ie 16 characters in total. So to create the string I concatenated PI several times which gives a garbage string but then used RegEx to replace ANY character with a zero. Then use RegEx to get the right number of characters and create the array with the right number of elements, all initiated with zeroes, ready for action.

Another thing I used was that as long as you have control of the size of the numbers you multiply, you don't really have to do it digit by digit but can group them together and do the calculation on the groups.

It gives the right result for 2x2 so hopefully also for the larger numbers.


Any other solutions? Is there a secret MLN (Multiply Large Numbers) function in MS?

Code: Select all

Timer>time0

//Define numbers to multiply
Let>num_1=12345678987654321234567898765432123456789876543212345678987654321
Let>num_2=98765432123456789876543212345678987654321234567898765432123456789

//Create arrays for num_1 and num_2(MA_x, MB_x), and result (MRes_x). Then multiply individual
//numbers, clean MRes, and finally concatenate back to a string/number.
GoSub>Create_Array_for_Numbers
Gosub>Create_Array_for_Result
GoSub>Multiply_individual_Numbers
GoSub>Finalize_Result_Array

Timer>time1
Let>time={%time1%-%time0%}
Mdl>Result: %Res%%CRLF%Time: %time% ms

SRT>Create_Array_for_Numbers
    //One array for each number, MA and MB, each with total elements NMA and NMB
    //Factor determines how many digits are grouped together for the calculation,
    //it is set to 1 for long numbers to avoid size problems for large numbers.
    Length>num_1,NMA
    Length>num_2,NMB
    Let>factor=6
    If>{(%NMA%>1000) or (%NMB%>1000)}
        Let>factor=1
        Else
        If>{(%NMA%>100) or (%NMB%>100)}
            Let>factor=5
        Else
            Let>factor=6
        Endif
    Endif

    //Place 0s to the left of numbers 1&2 to fit an even number of groups,
    //then extract/parse groups.
    Let>x={%NMA% mod %factor%}
    If>%x%<>0
        Let>x=%factor% - %x%
        Let>ct=0
        Let>tmp=
        While>ct<x
            Add>ct,1
            Let>tmp=%tmp%0
        EndWhile
        Let>num_1=%tmp%%num_1%
    Endif
    RegEx>\d{%factor%},num_1,0,MA,NMA,0

    Let>x={%NMB% mod %factor%}
    If>%x%<>0
        Let>x=%factor% - %x%
        Let>ct=0
        Let>tmp=
        While>ct<x
            Add>ct,1
            Let>tmp=%tmp%0
        EndWhile
        Let>num_2=%tmp%%num_2%
    Endif
    RegEx>\d{%factor%},num_2,0,MB,NMB,0

    //Calculate maximum total number of individual digits in the Result
    Let>ResLength={(%NMA%+%NMB%)*%factor%}
END>Create_Array_for_Numbers
SRT>Create_Array_for_Result
    //Concatenate the constant PI several times to create dummy string
    //and then use RegEx to replace characters with 0s and finally extract
    //a string and an array, MRes, with the right number of elements.
    Let>HelpString=%{Pi}%%{Pi}%%{Pi}%%{Pi}%%{Pi}%
    Length>HelpString,nLength
    Let>tmp=ResLength/nLength
    Let>tmp={Int(%tmp%)}+1
    //So if one concatenates HelpString tmp times then the string will long enough
    Let>ct=0
    Let>RefString=
    While>ct<tmp
        Add>ct,1
        Let>RefString=%RefString%%HelpString%
    EndWhile
    RegEx>.,RefString,0,M,NM,1,0,RefString
    RegEx>(?s)^0{%ResLength%},RefString,0,MRes,NMRes,0
    RegEx>\d,MRes_1,0,MRes,NMRes,0
END>Create_Array_for_Result
SRT>Multiply_individual_Numbers
//Multiply by group (size given by factor) and feed into the array, MRes.

    Let>ctB={%NMB%+1}
    While>ctB>1
        Sub>ctB,1
        Let>Ref={(%NMRes%-(%NMB%-%ctB%)*%factor%)}
        Let>ctA={%NMA%+1}
        While>ctA>1
            Sub>ctA,1
            Let>tmpA=MA_%ctA%
            Let>tmpB=MB_%ctB%
            Let>tmpRes=MRes_%Ref%
            Let>MRes_%Ref%=%tmpRes%+{%tmpA%*%tmpB%}
            Let>Ref=%Ref%-%factor%
        EndWhile
    EndWhile

END>Multiply_individual_Numbers
SRT>Finalize_Result_Array
    //If any element in MRes contains a number with more than one digit then
    //leave the final digit and move/add any non-final digits to the array element on the left.
    //Finally concatenate array elements into a string and remove any initial 0s.
    Let>Ref={%NMRes%+1}
    While>Ref>1
        Sub>Ref,1
        //Check the number in the MRes element given by Ref
        Let>tmpRes=MRes_%Ref%
        RegEx>\d,%tmpRes%,0,M,NM,0
        If>{%NM%>=2}
            //Separate the number in a final digit and anything to the left
            RegEx>(\d+)\d,tmpRes,0,M,NM,1,$1,str_non_final_digits
            RegEx>\d+(\d),tmpRes,0,M,NM,1,$1,str_final_digit
            Let>Ref_Left=%Ref%-1
            Let>Restmp=MRes_%Ref_Left%
            //Put the final digit into the position Ref in MRes, add any non-final
            //digits to the position left of Ref.
            Let>MRes_%Ref%=str_final_digit
            Let>MRes_%Ref_Left%=%Restmp%+%str_non_final_digits%
        Else
            //Just one digit, ie already ok.
        Endif
    Endwhile
    //Concatenate the elements in MRes into a string, Res. and remove any initial 0s.
    Let>Res=
    Let>Ref=%NMRes%+1
    While>Ref>1
        Sub>Ref,1
        Let>tmp=MRes_%Ref%
        Let>Res=%tmp%%Res%
    EndWhile
    RegEx>^0+,Res,0,M,NM,1,,Res
END>Finalize_Result_Array

hagchr
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by hagchr » Fri Aug 29, 2014 6:11 am

Just to avoid confusion for anyone new to MS, to initiate an array I usually use RegEx or a loop. Involving PI was just for this particular problem to increase the speed.

Also, for anyone new to RegEx, the command:

Code: Select all

RegEx>0,0000000000,0,M,NM,0
will search for the first string (0) in the second string (0000000000) and if it finds any, will put each one found into
an array (M) and the total number of items found into a variable (NM). So all in all just one line of code will produce an array, here called M with 10 elements (M_1, M_2, … M_10), all initiated to zero.

If you have any string you can convert it to a string with eg. zeroes with:

Code: Select all

RegEx>.,asdfhkjeuf,0,M,NM,1,0,Result
which will search for any (.=dot in RegEx language) character in the second string (asdfhkjeuf) and replace it with a zero. The array M will contain garbage but the variable Result will contain the string after replacements i.e. (0000000000).

MS help file has all details of the various RegEx parameters.

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JRL
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by JRL » Fri Aug 29, 2014 7:45 pm

hagchr,

Looking like you've got the only entry this time and it is amazingly fast. I think it would have been difficult for anyone to write a faster script. My script (which, much to my embarrassment, I will post) performs the prescribed calculation in about 10 seconds. Your script does the calculation in 2 to 3 tenths of a second. My script does the math exactly the way one would do it with a pencil and paper. I have not yet taken the time to study your script to ascertain what mathematical magic you're using.

I am impressed.





















My script:

Code: Select all


Dialog>Dialog1
object Dialog1: TForm
  Left = 644
  Top = 113
  HelpContext = 5000
  BorderIcons = [biSystemMenu]
  Caption = 'Multiply Long Numbers'
  ClientHeight = 397
  ClientWidth = 1056
  Color = clBtnFace
  Position = poScreenCenter
  Font.Charset = DEFAULT_CHARSET
  Font.Color = clWindowText
  Font.Height = -11
  Font.Name = 'MS Sans Serif'
  Font.Style = []
  OldCreateOrder = True
  ShowHint = True
  OnTaskBar = False
  DesignSize = (
    1056
    397)
  PixelsPerInch = 96
  TextHeight = 13
  object Label1: TLabel
    Left = 40
    Top = 8
    Width = 59
    Height = 13
    Caption = 'First Number'
  end
  object Label2: TLabel
    Left = 40
    Top = 56
    Width = 77
    Height = 13
    Caption = 'Second Number'
  end
  object Label3: TLabel
    Left = 40
    Top = 200
    Width = 37
    Height = 13
    Caption = 'Product'
  end
  object Label4: TLabel
    Left = 368
    Top = 128
    Width = 63
    Height = 13
    Caption = 'Product Time'
  end
  object Label5: TLabel
    Left = 976
    Top = 128
    Width = 77
    Height = 13
    Caption = 'Character Count'
  end
  object Label6: TLabel
    Left = 376
    Top = 152
    Width = 6
    Height = 13
    Caption = '0'
  end
  object Label7: TLabel
    Left = 984
    Top = 152
    Width = 6
    Height = 13
    Caption = '0'
  end
  object Label8: TLabel
    Left = 840
    Top = 128
    Width = 47
    Height = 13
    Caption = 'Sum Time'
  end
  object Label9: TLabel
    Left = 848
    Top = 152
    Width = 6
    Height = 13
    Caption = '0'
  end
  object Label10: TLabel
    Left = 904
    Top = 128
    Width = 47
    Height = 13
    Caption = 'TotalTime'
  end
  object Label11: TLabel
    Left = 912
    Top = 152
    Width = 6
    Height = 13
    Caption = '0'
  end
  object Edit1: TEdit
    Left = 14
    Top = 28
    Width = 1034
    Height = 21
    TabOrder = 0
  end
  object Edit2: TEdit
    Left = 14
    Top = 74
    Width = 1030
    Height = 21
    TabOrder = 1
  end
  object MSButton1: tMSButton
    Left = 476
    Top = 361
    Width = 75
    Height = 25
    Anchors = [akLeft, akBottom]
    Caption = 'Multiply'
    TabOrder = 2
    DoBrowse = False
    BrowseStyle = fbOpen
  end
  object MSMemo1: tMSMemo
    Left = 14
    Top = 216
    Width = 1027
    Height = 130
    WordWrap = True
    VertScrollBarStyles.ButtonSize = 16
    HorzScrollBarStyles.ButtonSize = 16
    Anchors = [akLeft, akTop, akBottom]
    TabOrder = 3
  end
  object ProgressBar1: TProgressBar
    Left = 128
    Top = 177
    Width = 785
    Height = 16
    TabOrder = 4
  end
end
EndDialog>Dialog1

AddDialogHandler>Dialog1,MSButton1,OnClick,Multiply
Show>Dialog1,
Exit>

SRT>Multiply
Timer>StartTime
SetDialogProperty>Dialog1,Label9,Caption,0
SetDialogProperty>Dialog1,Label7,Caption,0
SetDialogProperty>Dialog1,Label6,Caption,0
SetDialogProperty>Dialog1,Label11,Caption,0
SetDialogProperty>Dialog1,MSMemo1,Text,

GetDialogProperty>Dialog1,Edit1,Text,no1
GetDialogProperty>Dialog1,Edit2,Text,no2




GoSub>Works
Timer>EndTime
Let>ProdTime={(%EndTime%-%StartTime%)/1000}
SetDialogProperty>Dialog1,Label6,Caption,ProdTime

Timer>StartTime
Let>CycleCount=0
GoSub>NewAdd
Length>Answer,CharCount

Let>dotpos=%CharCount%-%DecimalPlaces%
Midstr>Answer,1,dotpos,integer
Midstr>Answer,dotpos,DecimalPlaces,decimal
If>decimal=
Else
  Let>Answer=%Integer%.%decimal%
EndIf

Timer>EndTime
Let>SumTime={(%EndTime%-%StartTime%)/1000}
Let>TotalTime=%SumTime%+%ProdTime%

SetDialogProperty>Dialog1,Label9,Caption,SumTime
SetDialogProperty>Dialog1,Label7,Caption,CharCount
SetDialogProperty>Dialog1,Label11,Caption,TotalTime

SetDialogProperty>Dialog1,MSMemo1,Text,%Answer%
END>Multiply



SRT>Works
  Position>.,no1,1,dotpos1
  If>dotpos1>0
    StringReplace>no1,.,,no1
    RegEx>.,No1,0,Digit_1,Digit_1_count,0
    Let>decimalPlaces1={%Digit_1_Count%-%dotpos1%+1}
  Else
    Let>decimalplaces1=0
  EndIf
  Position>.,no2,1,dotpos2
  If>Dotpos2>0
    StringReplace>no2,.,,no2
    RegEx>.,No2,0,Digit_2,Digit_2_count,0
    Let>decimalPlaces2={%Digit_2_Count%-%dotpos2%+1}
  Else
    Let>decimalplaces2=0
  EndIf

  Let>DecimalPlaces=%decimalPlaces1%+%decimalPlaces2%

  RegEx>.,No1,0,Digit_1,Digit_1_count,0
  RegEx>.,No2,0,Digit_2,Digit_2_count,0
  Let>AnswerStr=
  Let>Progressadd={30/%Digit_1_Count%}
  Let>kk1={%Digit_1_count%+1}
  Let>Cycles=digit_1_count
  Repeat>kk1
    Sub>kk1,1
    Let>AnswerStr_%kk1%=
    Let>PrevStr=
    Let>CarryOver=0
    Let>kk2={%Digit_2_count%+1}
    Repeat>kk2
      Sub>kk2,1
      If>kk2=0
        If>CarryOver<>0
          Let>AnswerStr_%kk1%=%CarryOver%%PrevStr%
        EndIf
      Else
        Let>product=Digit_1_%kk1%*Digit_2_%kk2%
        Add>product,%CarryOver%
        If>Product>9
          RegEx>.,Product,0,Char,Char_count,0
          Let>AnswerStr_%kk1%=%Char_2%%PrevStr%
          Let>CarryOver=%Char_1%
        Else
          Let>AnswerStr_%kk1%=%Product%%PrevStr%
          Let>CarryOver=0
        EndIf
        Let>PrevStr=AnswerStr_%kk1%
      EndIf
    Until>kk2=0
    Let>Value=AnswerStr_%kk1%
    Let>Places={%Digit_1_count%-%kk1%}
    If>Places>0
      Let>kkp=0
      Repeat>kkp
        Add>kkp,1
        Let>Value=%value%0
      Until>kkp=Places
    EndIf
   Let>AnswerStr_%kk1%=Value
   Let>Progress={round(%Progressadd%*((%Digit_1_Count%+1)-%kk1%))}
   SetDialogProperty>Dialog1,ProgressBar1,Position,Progress
  Until>kk1=1
  Let>Progress=30
  SetDialogProperty>Dialog1,ProgressBar1,Position,Progress
END>Works


SRT>NewAdd
  Repeat>CycleCount
  Add>CycleCount,1
  If>CycleCount=1
    If>Cycles=1
      Let>Answer=AnswerStr_1
      Goto>NewAddComplete
    Else
      Let>Term_1=AnswerStr_1
      Let>Term_2=AnswerStr_2
      Add>CycleCount,1
    EndIf
  Else
    Let>Term_1=Answer
    Let>Term_2=AnswerStr_%CycleCount%
  EndIf
  Let>ProgressAdd={70/%Cycles%}
  VBEval>StrReverse("%Term_1%"),Term_1
  VBEval>StrReverse("%Term_2%"),Term_2
  RegEx>.,Term_1,0,Digit_1,Digit_1_count,0
  RegEx>.,Term_2,0,Digit_2,Digit_2_count,0
  If>{%Digit_1_Count%>=%digit_2_Count%}
    Let>MaxLength=Digit_1_Count
    If>{%Digit_1_Count%=%digit_2_Count%}
    Else
      Let>Places=%Digit_2_count%
      If>Places>0
        Repeat>Places
          Add>Places,1
          Let>Digit_2_%Places%=0
        Until>Places=Digit_1_Count
      EndIf
    EndIf
  Else
    Let>MaxLength=Digit_2_Count
    Let>Places=%Digit_1_count%
    If>Places>0
      Repeat>Places
        Add>Places,1
        Let>Digit_1_%Places%=0
      Until>Places=Digit_2_Count
    EndIf
  EndIf
  Let>CarryOver=0
  Let>DD=0
  Repeat>DD
    Add>DD,1
    Let>Sum_%DD%=Digit_1_%DD%+Digit_2_%DD%
    Let>Sum_%DD%=Sum_%DD%+%CarryOver%
    If>Sum_%DD%>9
      RegEx>.,Sum_%DD%,0,Char,Char_count,0
      Let>Sum_%DD%=Char_2
      Let>CarryOver=Char_1
    Else
      Let>CarryOver=0
    EndIf
  Until>DD=%MaxLength%
  If>CarryOver>0
    Add>DD,1
    Let>Sum_%DD%=CarryOver
  EndIf
  Let>Answer=
  Let>FF=0
  Repeat>FF
    Add>FF,1
    Let>Value=Sum_%FF%
    Let>Answer=%Value%%Answer%
  Until>FF=DD
  Let>Progress={30+(round(%Progressadd%*%CycleCount%))}
  SetDialogProperty>Dialog1,ProgressBar1,Position,Progress
  Until>CycleCount=Cycles
  Label>NewAddComplete
  Let>Progress=100
  SetDialogProperty>Dialog1,ProgressBar1,Position,Progress
END>NewAdd

ABIVEN
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Posts: 67
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Re: Puzzler #11 MULTIPLICATION

Post by ABIVEN » Tue Jul 18, 2017 9:32 pm

I just discovered this post.
Some years ago I had to make out a routine for AxB ; Lot slower than hagchr !
But , in fact what I needed was mostly B(Unltd) x A(<= 14 digits)
So I did a "lite" routine [ 80 lines] five time faster than the first one

Code: Select all

Let>DECIMAL_SEPARATOR=.
let>MSG_HEIGHT=600
let>MSG_WIDTH=900

let>A=12345678987654
let>B=9876543212345678987654321234567898765432123456789876543212345678956789

len>A,lenA
len>B,lenB
let>B°=B
Timer>startTime
GoSub>BxA-14Digits
let>B=B°
Timer>endTime
Let>elapsed_seconds={(%endTime%-%startTime%)}
mdl>%CRLF% A = %A%              lenA= %lenA%%CRLF% B = %B%       lenB= %lenB%%CRLF% Result P°= %P°%%CRLF%     Time:  --- %elapsed_seconds% ms ---
//------------********SRT B(Unltd)*A(<= 14 Digits)********------------
SRT>BxA-14Digits
let>mlennn=15-lenA
if>{(%lenA%+%lenB%)<16}
let>P°=A*B
// If A*B compatible with Macro Scheduler (little numbers)
goto>BxA-14Digits fin
else
endif
MidStr>000000000000000,1,{(%mlennn%)},FOR1
concat>B,FOR1
len>B,newlenB
let>newlenB={(Trunc(%newlenB%/%mlennn%)*%mlennn%)}
MidStr>B,1,newlenB,B
let>lensupp=newlenB-lenB
let>newlenB={(%newlenB%/%mlennn%)}
//[1]Gives number of blocks - so gives start j
//*************------------*****
let>j=newlenB
let>ADD=0
let>FIR1=FOR1
midStr>B,{(%mlennn%*%j%-(%mlennn%-1))},mlennn,Bj
let>Pj=%Bj%*A
LEN>Pj,lenPj
//[2] Adding some 0s
concat>FIR1,Pj
let>Pj=FIR1
midStr>Pj,{(%lenPj%+1)},mlennn,Pj
midStr>FIR1,1,lenPj,ADD
let>P°=Pj
len>P°,lenP°
let>j=newlenB-1
//*************------------*****
//start j decreasing to j=2
While>j>1
let>FIR1=FOR1
midStr>B,{(%mlennn%*%j%-(%mlennn%-1))},mlennn,Bj
let>Pj=%Bj%*A
let>Pj=Pj+ADD
LEN>Pj,lenPj
// Works out step by step P° from start j to 2 (with perhaps some 0s added)
concat>FIR1,Pj
let>Pj=FIR1
midStr>Pj,{(%lenPj%+1)},mlennn,Pj
midStr>FIR1,1,lenPj,ADD
concat>Pj,P°
let>P°=Pj
let>j=j-1
Endwhile
//*************------------*****
// last with j=1
midStr>B,{(%mlennn%*%j%-(%mlennn%-1))},mlennn,Bj
let>Pj=%Bj%*A
let>Pj=Pj+ADD
LEN>Pj,lenPj
concat>Pj,P°
let>P°=Pj
len>P°,lenP°
let>lenresultat=lenP°-lensupp
MidStr>P°,1,lenresultat,P°
//[3] 0s added are now deleted
label>BxA-14Digits fin
END>BxA-14Digits
//------------********SRT B(Unltd)*A(<= 14 Digits)********------------
Hagchr's routine [with A = 12345678987654] gives on my computer 90 ms
What about a hagchr similar "lite"routine ? !

Regards

User avatar
Djek
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Posts: 134
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Location: Holland
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Re: The Unofficial Macro Scheduler Puzzler #11

Post by Djek » Fri Aug 04, 2017 10:31 pm

Thanks for the puzzle, JRL

i just saw it, and i do not think i can do it faster then Hagchr solution.

But i guess i could go for the most simple solution?
I know, im a bit cheating but it does supply you the answer:

Code: Select all

Let>num_1=12345678987654321234567898765432123456789876543212345678987654321
Let>num_2=98765432123456789876543212345678987654321234567898765432123456789
Timer>startTime
HTTPRequest>http://apfloat.appspot.com,,POST,input=%num_1%*%num_2%,result,,,,
Separate>result,#10;,answ
Separate>answ_2,%LF%,ans
Timer>endTime
Let>elapsed_seconds={(%endTime%-%startTime%)}
MessageModal>elapsed_seconds
MessageModal>ans_1
and it shows how MS gives the possibilities to solve this puzzle, without having the brains of Hagchr...
:D
kind regards,
Djek

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