A Top Down Operator Precedence parser for algebraic expressions

In my previous post I wrote about a polar curve grapher and its use of Kevin Mehall's tdop_math.js to parse a user input algebraic expression into a javascript eval(uable) string. I ended up spending some time with tdop_math.js - you can find the source here - and I fell into the magical rabbit hole of Top Down Operator Precedence (TDOP) parsers. This post describes how I ended up writing a TDOP parser for algebraic expressions in Ruby.

Some facts: - Vaughan Pratt first wrote about Top Down Operator Precedence parsers in 1973 - Douglas Crockford wrote about them in Beautiful Code, and in an article that includes an implementation of a parser for a Simplified Javascript - he went on to use this in JSlint

Overview of TDOP parsers

In a nutshell, Top Down Operator Precedence parsers are similar to Recursive Descent parsers, but, as claimed by Vaughan Pratt, simpler to understand, implement, and providing of better performance. Borrowing from this excellent article by Eli Bendersky, the main features of a TDOP parser are:

  • A binding power mechanism to handle precedence levels - e.g. the binding power of multiplication '*' is higher than that of addition '+' in arithmetic
  • A means of implementing different functionality of tokens depending on their position relative to their neighbors: prefix or infix. For example, the minus '-' operator is behaving as a prefix operator in -2 but in its infix form, it is subtraction 4 - 2.
  • As opposed to classic Recursive Descent, where semantic actions are associated with grammar rules (BNF), TDOP associates them with tokens. Tokens have a binding power, and know how to handle themselves in infix and/or in prefix situations. Expressions are then evaluated recursively by evaluating them at each 'precedence' level and returning the result to its caller.

Compilers are difficult to understand at the best of times and most articles I have come across on TDOP parsers are by very bright people finishing (or pausing) on their journey to get to know Pratt's work. At my current stage of comfort with TDOP parsers, I cannot come close to explaining the parser any better than Eli Bendersky.

Eli's article, elaborates on the features described above and explains the parser by setting one up to handle simple arithmetic. He then works through a simple example that illustrates it in action as it parses the expression 3 + 1 * 2 * 4 + 5.

For further detail on TDOP parsers, I refer the interested reader to this and Douglas Crockford's article referenced above.

TDOP and Ruby

As I was working on the polar curve grapher I got curious about implementing tdop_math.js in Ruby. I studied the source of tdop_math.js, and its tokeniser tokens.js, and felt that I had a decent understanding of its structure and operation (improving my Javascript skills along the way!). However, a like-for-like implementation in Ruby seemed out of the question. Well, it seemed like a hard proposition that would involve first developing new metaprogramming skills in Ruby!

And then I ran into Glen Vandenburg's project radish, which is now called smithereen.


Smithereen is a result of Glen Vandenburg's meditation on TDOP parsers, during which he decided to write an implementation in Ruby. He has implemented a library for building Top Down Operator Precedence parsers, using Ruby idioms that were completely unfamiliar to me, but that provided an excellent opportunity to learn. As Glen says, he has tried to provide a well-factored set of tools to help with the core TDOP algorithm and several related problems, and so I decided to fork Smithereen with the intention of helping with its development, and, using it to implement tdop_math in Ruby.

Smithereen comes with two samples, an arithmetic parser, and the Simplified Javascript parser. I found it quite difficult to get started with Smithereen and asked Glen a question or two. To my delight, he helped me get started and I found this to be so inspiring that I made rapid progress towards implementing a simple scientific calculator. This is available on my fork of Smithereen as equation_parser.rb

equation_parser.rb will, for instance, take strings such as sin(cos(2pi)) and return 0.84147

By way of explanation, it has hardcoded hashes that declare functions and constants, and these are tokenised as names and given a high binding power.

fns = {
  'sin'  => Proc.new { |x| Math.sin(x) },
  'cos'  => Proc.new { |x| Math.cos(x) },
  'tan'  => Proc.new { |x| Math.tan(x) },
  'sqrt' => Proc.new { |x| Math.sqrt(x) },
  'exp'  => Proc.new { |x| Math.exp(x) }

  'pi' => Math::PI,
  'e'  => Math::E

The infix method on the token ( checks for the presence of a function or a constant (essentially a key in the above hashes), and looks up the corresponding value for the definition of the function. Smithereen takes care of the rest. Well, that is, after tokens for the usual arithmetic operators are defined, with infix and prefix methods as described above.


Completing my goal to create a Ruby equivalent to tdop_math.js, I've taken the work in the example equation_parser.rb and used it to create a Ruby gem called tdop_math. This takes an algebraic expression, parses it, and returns a string of Ruby that represents the expression.

Because the tdop_math gem depends on smithereen (which has yet to be released), it is not available as a built gem.

As far as I know, if you want to install this, you are going to need to pull down my repo and include '<path-to-tdop_math.rb>' manually. Once you have, you use it as follows:

u = "sin(cos(2x))"  ## or some user input algebraic expression
t = TDOPMath::Parser.new(u)
f = t.parse   ## f == "Math.sin(Math.cos(2 * x))"

tdop_math reserves symbols x,y and t as variables. It also provides a basic list of functions, namely, sin, cos, tan, sqrt and exp. If you need to provide an alternate list of variables and functions, you can do so as follows:

u = "baz(foo(2a))"  ## or some user input algrebraic expression
t = TDOPMath::Parser.new(u,
                         :vars => ['a'],
                         :functions => {
                            'foo' => 'Foo::foo',
                            'baz' => 'Baz::baz'
f = t.parse   ## f == "Baz::baz(Foo::foo(2 * a))"

You can use tdop_math wherever you need a user to input an algebraic expression, comprising of a known set of functions and variables. This could find use in apps in maths education or science/data analytics. I hope the community finds it useful, although my motivation was mainly to build it as an exercise, along my path to becoming a better programmer. I've made a gem!