NAME

       rpntutorial - Reading RRDtool RPN Expressions by Steve Rader

DESCRIPTION

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

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, 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.

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 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.

Some Examples

       While compound expressions can look overly complex, they can be considered elegantly
       simple.  To quickly comprehend RPN expressions, you must know 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,              7000,128,8,*,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 again
       and--voila!--you have an easily 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)

Exercises

       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.

       Answer 2:

           if (input <= 56000/8 ) { input*8 } else { 56000 }
           input,56000,8,DIV,LE,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.)

AUTHOR

       Steve Rader <rader@wiscnet.net>