3 rpntutorial - Reading RRDtool RPN Expressions by Steve Rader
5 =for html <div align="right"><a href="rpntutorial.pdf">PDF</a> version.</div>
9 This tutorial should help you get to grips with RRDtool RPN expressions
10 as seen in CDEF arguments of RRDtool graph.
12 =head1 Reading Comparison Operators
14 The LT, LE, GT, GE and EQ RPN logic operators are not as tricky as
15 they appear. These operators act on the two values on the stack
16 preceding them (to the left). Read these two values on the stack
17 from left to right inserting the operator in the middle. If the
18 resulting statement is true, the replace the three values from the
19 stack with "1". If the statement if false, replace the three values
22 For example think about "2,1,GT". This RPN expression could be
23 read as "is two greater than one?" The answer to that question is
24 "true". So the three values should be replaced with "1". Thus the
25 RPN expression 2,1,GT evaluates to 1.
27 Now also consider "2,1,LE". This RPN expression could be read as "is
28 two less than or equal to one?". The natural response is "no"
29 and thus the RPN expression 2,1,LE evaluates to 0.
31 =head1 Reading the IF Operator
33 The IF RPN logic operator can be straightforward also. The key
34 to reading IF operators is to understand that the condition part
35 of the traditional "if X than Y else Z" notation has *already*
36 been evaluated. So the IF operator acts on only one value on the
37 stack: the third value to the left of the IF value. The second
38 value to the left of the IF corresponds to the true ("Y") branch.
39 And the first value to the left of the IF corresponds to the false
40 ("Z") branch. Read the RPN expression "X,Y,Z,IF" from left to
41 right like so: "if X then Y else Z".
43 For example, consider "1,10,100,IF". It looks bizarre to me.
44 But when I read "if 1 then 10 else 100" it's crystal clear: 1 is true
45 so the answer is 10. Note that only zero is false; all other values
46 are true. "2,20,200,IF" ("if 2 then 20 else 200") evaluates to 20.
47 And "0,1,2,IF" ("if 0 then 1 else 2) evaluates to 2.
50 Notice that none of the above examples really simulate the whole
51 "if X then Y else Z" statement. This is because computer programmers
52 read this statement as "if Some Condition then Y else Z". So it's
53 important to be able to read IF operators along with the LT, LE,
54 GT, GE and EQ operators.
58 While compound expressions can look overly complex, they can be
59 considered elegantly simple. To quickly comprehend RPN expressions,
60 you must know the the algorithm for evaluating RPN expressions:
61 iterate searches from the left to the right looking for an operator,
62 when it's found, apply that operator by popping the operator and some
63 number of values (and by definition, not operators) off the stack.
65 For example, the stack "1,2,3,+,+" gets "2,3,+" evaluated (as "2+3")
66 during the first iteration which is replaced by 5. This results in
67 the stack "1,5,+". Finally, "1,5,+" is evaluated resulting in the
68 answer 6. For convenience sake, it's useful to write this set of
71 1) 1,2,3,+,+ eval is 2,3,+ = 5 result is 1,5,+
72 2) 1,5,+ eval is 1,5,+ = 6 result is 6
75 Let's use that notation to conveniently solve some complex RPN expressions
76 with multiple logic operators:
78 1) 20,10,GT,10,20,IF eval is 20,10,GT = 1 result is 1,10,20,IF
80 read the eval as pop "20 is greater than 10" so push 1
82 2) 1,10,20,IF eval is 1,10,20,IF = 10 result is 10
84 read pop "if 1 then 10 else 20" so push 10. Only 10 is left so
87 Let's read a complex RPN expression that also has the traditional
88 multiplication operator:
90 1) 128,8,*,7000,GT,7000,128,8,*,IF eval 128,8,* result is 1024
91 2) 1024,7000,GT,7000,128,8,*,IF eval 1024,7000,GT result is 0
92 3) 0,128,8,*,IF eval 128,8,* result is 1024
93 4) 0,7000,1024,IF result is 1024
96 Now let's go back to the first example of multiple logic operators
97 but replace the value 20 with the variable "input":
99 1) input,10,GT,10,input,IF eval is input,10,GT result is A
101 Read eval as "if input > 10 then true" and replace "input,10,GT"
104 2) A,10,input,IF eval is A,10,input,IF
106 read "if A then 10 else input". Now replace A it's verbose
107 description and--voila!--you have a easily readable description
110 if input > 10 then 10 else input
112 Lastly, let's to back the first most complex example and replace
113 the value 128 with "input":
115 1) input,8,*,7000,GT,7000,input,8,*,IF eval input,8,* result is A
117 where A is "input * 8"
119 2) A,7000,GT,7000,input,8,*,IF eval is A,7000,GT result is B
121 where B is "if ((input * 8) > 7000) then true"
123 3) B,7000,input,8,*,IF eval is input,8,* result is C
125 where C is "input * 8"
129 At last we have a readable decoding of the complex RPN expression with
132 if ((input * 8) > 7000) then 7000 else (input * 8)
138 Compute "3,2,*,1,+ and "3,2,1,+,*" by hand. Rewrite them in
139 traditional notation. Explain why they have different answers.
143 3*2+1 = 7 and 3*(2+1) = 9. These expressions have
144 different answers because the altering of the plus and
145 times operators alter the order of their evaluation.
150 One may be tempted to shorten the expression
152 input,8,*,56000,GT,56000,input,*,8,IF
154 by removing the redundant use of "input,8,*" like so:
156 input,56000,GT,56000,input,IF,8,*
158 Use tradition notation to show these expressions are not the same.
159 Write an expression that's equivalent to the first expression but
160 uses the LE and DIV operators.
164 if (input <= 56000/8 ) { input*8 } else { 56000 }
165 input,56000,8,DIV,LT,input,8,*,56000,IF
170 Briefly explain why traditional mathematic notation requires the
171 use of parentheses. Explain why RPN notation does not require
172 the use of parentheses.
176 Traditional mathematic expressions are evaluated by
177 doing multiplication and division first, then addition and
178 subtraction. Parentheses are used to force the evaluation of
179 addition before multiplication (etc). RPN does not require
180 parentheses because the ordering of objects on the stack
181 can force the evaluation of addition before multiplication.
186 Explain why it is desirable for the RRDtool developers to implement
187 RPN notation instead of traditional mathematical notation.
191 The algorithm that implements traditional mathematical
192 notation is more complex then algorithm used for RPN.
193 So implementing RPN allowed Tobias Oetiker to write less
194 code! (The code is also less complex and therefore less
195 likely to have bugs.)
200 Steve Rader E<lt>rader@wiscnet.netE<gt>