rrdtool RPNTUTORIAL(1)
NAME
rpntutorial - Reading RDtool RPN Expressions by Steve Rader
DESCRIPTION
This tutorial should help you get to grips with RDtool RPN
expressions as seen in CDEF arguments of RDtool 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.
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rrdtool RPNTUTORIAL(1)
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
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rrdtool RPNTUTORIAL(1)
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.
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rrdtool RPNTUTORIAL(1)
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 RDtool 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
1.3.5 Last change: 2008-03-15 4
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