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RPNTUTORIAL(1)			    rrdtool			RPNTUTORIAL(1)

       rpntutorial - Reading RRDtool RPN Expressions by Steve Rader

       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 expres
       sion 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 expres
       sion "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" its 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 its important
       to be able to read IF operators along with the LT, LE, GT, GE and EQ

Some Examples
       While compound expressions can look overly complex, they can be consid
       ered 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 its 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, its 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

       Lets use that notation to conveniently solve some complex RPN expres
       sions 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.

       Lets read a complex RPN expression that also has the traditional mul
       tiplication 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 lets 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

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

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

	if input > 10 then 10 else input

       Finally, lets 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)

       Exercise 1:

       Compute "3,2,*,1,+ and "3,2,1,+,*" by hand.  Rewrite them in tradi
       tional 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


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


       Use traditional notation to show these expressions are not the same.
       Write an expression thats 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

       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.)

       Steve Rader 

1.2.15				  2006-07-14			RPNTUTORIAL(1)

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