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

Reading Comparison Operators

       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 evalu­
       ates 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 evalu­
       ates to 0.

Reading the IF Operator

       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 Con­
       dition then Y else Z".  So it's important to be able to
       read IF operators along with the LT, LE, GT, GE and EQ
       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 tra­
       ditional 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

        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

       read "if A then 10 else input".  Now replace A it's ver­
       bose 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

       Exercise 1:

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

       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


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


       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 }

       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

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