# FreeBSD Manual Pages

```RPNTUTORIAL(1)			    rrdtool			RPNTUTORIAL(1)

NAME

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

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".

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.

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" it's crystal clear: 1 is true so the an-
swer 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 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 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 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

Let's 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 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	de-
scription 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 tradi-
tional notation.	 Explain why they have different answers.

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.

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.

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

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