


     RRRRPPPPNNNNTTTTUUUUTTTTOOOORRRRIIIIAAAALLLL((((1111))))         1111....2222....11113333 ((((2222000000006666----00005555----00004444))))         RRRRPPPPNNNNTTTTUUUUTTTTOOOORRRRIIIIAAAALLLL((((1111))))



     NNNNAAAAMMMMEEEE
          rpntutorial - Reading RRDtool RPN Expressions by Steve Rader

     DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN
          This tutorial should help you get to grips with RRDtool RPN
          expressions as seen in CDEF arguments of RRDtool graph.

     RRRReeeeaaaaddddiiiinnnngggg CCCCoooommmmppppaaaarrrriiiissssoooonnnn OOOOppppeeeerrrraaaattttoooorrrrssss
          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, then 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 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.

     RRRReeeeaaaaddddiiiinnnngggg tttthhhheeee IIIIFFFF OOOOppppeeeerrrraaaattttoooorrrr
          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 bizarre 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
          operators.



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     SSSSoooommmmeeee EEEExxxxaaaammmmpppplllleeeessss
          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 and is replaced by 5.
          This results in the stack "1,5,+".  Finally, "1,5,+" is
          evaluated resulting in the answer 6.  For convenience, 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 conveniently 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  ( lets call this 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 with it's
          verbose description againg and--voila!--you have a easily



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          readable description of the expression:

           if input > 10 then 10 else input

          Finally, let's go back to 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)

     EEEExxxxeeeerrrrcccciiiisssseeeessss
          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,*

          Use traditional 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.



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          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.  Parentheses 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 was 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.)

     AAAAUUUUTTTTHHHHOOOORRRR
          Steve Rader <rader@wiscnet.net>



















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