*Mathematica*

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## July 11, 2018

### Maths in Code

## Programming Equations

Short post this week, with a simple theme: how should you translate equations into code?

## Self Documenting Code

Self-documenting code (aka self-describing code) is code that documents itself, rather like the term 'self-documenting code' already tells you what it means. The idea is to choose function names, variable names and types, and (in relevant languages) Class or Object hierarchies so that they demonstrate what they do. This can mean a lot of things, and tends to mean slightly different things to different people. In general, especially in cases of ambiguity, one aims for the "principle of least surprise" - what is the least unexpected meaning of something with this name? For instance, old fashioned C programmers, might consider "i" to be a perfectly self-documenting loop variable, as the usage is very familiar.

This leads us nicely into the *spirit* of self-documentation. Programming is hard. Choosing sensible names and structures reduces the effort needed to keep track of what things are, as well as everything else in your program. This means that the names you choose should help you, and whoever reads or uses your code, work out what things mean. Imagine coming back to your code after weeks or months not working on it. Can you read it through and guess, correctly, what things mean? If not, consider reworking your namings or structure.

Names shouldn't just repeat information already available though, as that wastes space which could be better used. For example, a boolean or Logical variable already tells you that this is probably some sort of flag, controlling whether or not something should happen or has happened. "write_to_file" is a useful name. "flag" is not. "is_open" is a useful name for a flag associated with a file, "open_flag" is rather less so.

On this note, there is a common naming scheme which uses so-called Hungarian Warts, where some sort of 'type' information is prepended using a few (< 3) letters. Originally these were used to add 'extra' information, such as 'us' for 'unsafe string', which can be very useful. In languages like Python, where definitions don't specify type info, even just tages like 's' for string can be useful, **as long as you make sure this is actually true!**

Don't go mad with descriptions though. Supposedly the average person can keep 7 things in mind at once so you definitely don't want to occupy all that space with your variable names. 2 or 3 pieces is about the limit for most names, so try and use things like "input_file" not "file_to_read_data_from" and avoid linking words as far as practical. Keep names short and sweet, to save typing and brain-space.

By the way, if you use an IDE to program, you might feel that none of this matters, but while it will fill in the names for you, it really doesn't solve everything. You still need to keep track of which variable you want and select the correct one in ambiguous cases, and keep track in your head of what is what. The self-documenting ideal means that when your IDE gives you a list of 5 similar looking variables, you barely have to think to select the one you need in context.

## Self Documenting Equations

Take a nice simple equation, ax^2 + bx + c = 0 and imagine programming something to solve it. If you took the self-documenting advice at face value, you might convert "x" into "polynomial_variable", and "a, b, c" into "coefficient_of_square_term", "coefficient_of_linear_term" and "coefficient_of_constant_term". That leaves the quadratic equation looking pretty complicated, as

polynomial_variable = (- coefficient_of_linear_term +|- sqrt( coefficient_of_linear_term^2 - 4 * coefficient_of_square_term * coefficient_of_constant_term) )/ (2 * coefficient_of_square_term)

instead of the far more familiar

x = (- b +|- sqrt( b^2 - 4 * a * c)) /(2 * a)

Similarly, if you program equations from a paper, it can be much, much easier to use variable names that match the paper, especially when trying to verify them. Sometimes it is even a good idea to mimic the paper's equation format, at least until things are working. More on that in the next section.

## Optimising Equations from Papers

So what about rearranging equations for optimisation purposes? If you ask ten different people, you'll get ten different answers, but in my opinion the following are the keys:

- Only simplify once the code works. Don't bother collecting, hoisting or other tweaks until you need to.
- Keep names generally the same as the paper, but ideally remove ambiguity, so don't use 'a' as well as 'A'. Consider 'a' and 'bigA' for more distinction. Absolutely avoid ambiguities like 'l' and '1'.
- When you do change to rearranged equations, consider leaving the original code present in a comment. This is one of the few places where commented code is a good thing.
- When you combine or collect terms, start by just combining their names or the functions use, so use 'A_squared', 'sin_theta' or 'A_over_B'
- Include a link to the paper or source you used in the code, so that your variable names remain useful. Somebody with a source that uses different namings will find this style really tricky. If you can, consider including a snippet in your docs detailling the original equations and what rearrangements you did.

Find your style, stick to it, and optimise only as much as necessary to get the performance you need!

## Mathematica for Complicated Expressions

If your equations need substantial rearranging or simplifying, you might turn to a tool like Mathematica to do this for you, but did you know that it can even write C or Fortran code directly?

https://reference.wolfram.com/language/tutorial/GeneratingCAndFortranExpressions.html

You can save a lot of typing, and avoid some really easy mistakes by taking advantage of this. Don't forget FullSimplify and to apply suitable restrictions on variables to get a nicer expression before doing this though.