


rrdtool                                            RPNTUTORIAL(1)



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     rpntutorial - Reading RRDTtool RPN Expressions by Steve
     Rader

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     This tutorial should help you get to grips with rrdtool RPN
     expressions as seen in CDEF arguments of rrdtool graph.

RRRReeeeaaaaddddiiiinnnngggg CCCCoooommmmppppaaaarrrriiiissssoooonnnn OOOOppppeeeerrrraaaattttoooorrrrssss
     The LT, LE, GT, GE and EQ RPN logic operators are not as
     tricky as they appear.  These operators act on the two
     values on the stack preceding them (to the left).  Read
     these two values on the stack from left to right inserting
     the operator in the middle.  If the resulting statement is
     true, the replace the three values from the stack with "1".
     If the statement if false, replace the three values with
     "0".

     For example think about "2,1,GT".  This RPN expression could
     be read as "is two greater than one?"  The answer to that
     question is "true".  So the three values should be replaced
     with "1".  Thus the RPN expression 2,1,GT evaluates to 1.

     Now also consider "2,1,LE".  This RPN expression could be
     read as "is two less than or equal to one?".   The natural
     response is "no" and thus the RPN expression 2,1,LE
     evaluates to 0.

RRRReeeeaaaaddddiiiinnnngggg tttthhhheeee IIIIFFFF OOOOppppeeeerrrraaaattttoooorrrr
     The IF RPN logic operator can be straightforward also.  The
     key to reading IF operators is to understand that the
     condition part of the traditional "if X than Y else Z"
     notation has *already* been evaluated.  So the IF operator
     acts on only one value on the stack: the third value to the
     left of the IF value.  The second value to the left of the
     IF corresponds to the true ("Y") branch.  And the first
     value to the left of the IF corresponds to the false ("Z")
     branch.  Read the RPN expression "X,Y,Z,IF" from left to
     right like so: "if X then Y else Z".

     For example, consider "1,10,100,IF".  It looks bizzare to
     me.  But when I read "if 1 then 10 else 100" it's crystal
     clear: 1 is true so the answer is 10.  Note that only zero
     is false; all other values are true.  "2,20,200,IF" ("if 2
     then 20 else 200") evaluates to 20.  And "0,1,2,IF" ("if 0
     then 1 else 2) evaluates to 2.

     Notice that none of the above examples really simulate the
     whole "if X then Y else Z" statement.  This is because
     computer programmers read this statement as "if Some
     Condition then Y else Z".  So it's important to be able to
     read IF operators along with the LT, LE, GT, GE and EQ



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rrdtool                                            RPNTUTORIAL(1)



     operators.

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     While compound expressions can look overly complex, they can
     be considered elegantly simple.  To quickly comprehend RPN
     expressions, you must know the the algorithm for evaluating
     RPN expressions: iterate searches from the left to the right
     looking for an operator, when it's found, apply that
     operator by popping the operator and some number of values
     (and by definition, not operators) off the stack.

     For example, the stack "1,2,3,+,+" gets "2,3,+" evaluated
     (as "2+3") during the first iteration which is replaced by
     5.  This results in the stack "1,5,+".  Finally, "1,5,+" is
     evaluated resulting in the answer 6.  For convenience sake,
     it's useful to write this set of operations as:

      1) 1,2,3,+,+    eval is 2,3,+ = 5    result is 1,5,+
      2) 1,5,+        eval is 1,5,+ = 6    result is 6
      3) 6

     Let's use that notation to conviently solve some complex RPN
     expressions with multiple logic operators:

      1) 20,10,GT,10,20,IF  eval is 20,10,GT = 1     result is 1,10,20,IF

     read the eval as pop "20 is greater than 10" so push 1

      2) 1,10,20,IF         eval is 1,10,20,IF = 10  result is 10

     read pop "if 1 then 10 else 20" so push 10.  Only 10 is left
     so 10 is the answer.

     Let's read a complex RPN expression that also has the
     traditional multiplication operator:

      1) 128,8,*,7000,GT,7000,128,8,*,IF  eval 128,8,*       result is 1024
      2) 1024,7000,GT,7000,128,8,*,IF     eval 1024,7000,GT  result is 0
      3) 0,128,8,*,IF                     eval 128,8,*       result is 1024
      4) 0,7000,1024,IF                                      result is 1024

     Now let's go back to the first example of multiple logic
     operators but replace the value 20 with the variable
     "input":

      1) input,10,GT,10,input,IF  eval is input,10,GT  result is A

     Read eval as "if input > 10 then true" and replace
     "input,10,GT" with "A:

      2) A,10,input,IF            eval is A,10,input,IF




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rrdtool                                            RPNTUTORIAL(1)



     read "if A then 10 else input".  Now replace A it's verbose
     description and--voila!--you have a easily readable
     description of the expression:

      if input > 10 then 10 else input

     Lastly, let's to back the first most complex example and
     replace the value 128 with "input":

      1) input,8,*,7000,GT,7000,input,8,*,IF  eval input,8,*     result is A

     where A is "input * 8"

      2) A,7000,GT,7000,input,8,*,IF          eval is A,7000,GT  result is B

     where B is "if ((input * 8) > 7000) then true"

      3) B,7000,input,8,*,IF                  eval is input,8,*  result is C

     where C is "input * 8"

      4) B,7000,C,IF

     At last we have a readable decoding of the complex RPN
     expression with a variable:

      if ((input * 8) > 7000) then 7000 else (input * 8)


EEEExxxxeeeerrrrcccciiiisssseeeessss
     Exercise 1:

     Compute "3,2,*,1,+ and "3,2,1,+,*" by hand.  Rewrite them in
     traditional notation.  Explain why they have different
     answers.

     Answer 1:

         3*2+1 = 7 and 3*(2+1) = 9.  These expressions have
         different answers because the altering of the plus and
         times operators alter the order of their evaluation.

     Exercise 2:

     One may be tempted to shorten the expression

      input,8,*,56000,GT,56000,input,*,8,IF

     by removing the redundant use of "input,8,*" like so:

      input,56000,GT,56000,input,IF,8,*




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rrdtool                                            RPNTUTORIAL(1)



     Use tradition notation to show these expressions are not the
     same.  Write an expression that's equivalent to the first
     expression but uses the LE and DIV operators.

     Answer 2:

         if (input <= 56000/8 ) { input*8 } else { 56000 }
         input,56000,8,DIV,LT,input,8,*,56000,IF

     Exercise 3:

     Briefly explain why traditional mathematic notation requires
     the use of parentheses.  Explain why RPN notation does not
     require the use of parentheses.

     Answer 3:

         Traditional mathematic expressions are evaluated by
         doing multiplication and division first, then addition and
         subtraction.  Perentences are used to force the evaluation of
         addition before multiplication (etc).  RPN does not require
         parentheses because the ordering of objects on the stack
         can force the evaluation of addition before multiplication.

     Exercise 4:

     Explain why it is desirable for the RRDtool developers to
     implement RPN notation instead of traditional mathematical
     notation.

     Answer 4:

         The algorithm that implements traditional mathematical
         notation is more complex then algorithm used for RPN.
         So implementing RPN allowed Tobias Oetiker to write less
         code!  (The code is also less complex and therefore less
         likely to have bugs.)


AAAAUUUUTTTTHHHHOOOORRRR
     steve rader <rader@wiscnet.net>














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