

 
Manual Reference Pages  RPNTUTORIAL (1)
NAME
rpntutorial  Reading RRDtool RPN Expressions by Steve Rader
CONTENTS
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 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 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,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.)
AUTHOR
Steve Rader <rader@wiscnet.net>
1.2.30  RPNTUTORIAL (1)  20090119 
Visit the GSP FreeBSD Man Page Interface. Output converted with manServer 1.07. 