Month: December 2018

# Self-Employed UK Tax Explained Simply

My 16-yr-old son is about to become self-employed. This is my attempt to explain why we need to keep receipts for expenses, and what impact it will have on the tax he pays.

Note that the UK tax allowance is currently £11,500 (see here), but I made it £10,000 to keep the sums simple.

# Yak shaving

There are some words and phrases whose precise meanings just won’t stick in my head. “Yak shaving” is one of them.

So I want to thank the author of this blog post, Yevgeniy Brikman, first of all for writing a great post about infrastructure, but also for providing a great visual aid to remind me what yak shaving is.

And I’m putting it here so that I can look it up, the next time I forget.

(Apologies, I would have embedded the giphy but I’m still using the web version of WordPress, which doesn’t seem to allow it. I will hopefully sort this out soonish).

# Simultaneous equations and algorithms: Finding the “best fit” of cells in a grid

I got pinged by my colleague @j_f_green this morning with the following message: “Hey Clare. Hope all is well. Was wondering if I could be cheeky and get your input on a (small) thing I’m trying to solve? I am searching for an elegant solution and I’m starting to suspect the answer may involve maths.”

…and of course I was instantly interested.

## The problem

It turned out to be a relatively simple problem, depending on your point of view. If I say “GCSE-level simultaneous equations” your reaction will tell you whether that’s simple for you or not (for people not in theUK, GCSEs are the exams our pupils take when they are 16 years old).

Here is the problem: We want to discover two whole numbers (aka positive integers) that will give us the height and width of a grid in cells, given the following constraints:

1. We have a number, let’s call it capacity, which represents the minimum number of cells to fill the grid. We want at least that many cells. We’re happy if we have more, but not if we have fewer.
2. We also have another number, we’ll call it aspectRatio, which is the aspect ratio of the grid. This is the relationship between height and width. So for instance, if the aspect ratio is 2:1, that means the width is twice the height (it’s twice as wide as it is high). To reduce that to a single number,  aspectRatio is 2.

width = 2 * height = aspectRatio * height

(aspectRatio will often be a decimal or a fraction, so for instance if width is half height, aspectRatio is 0.5, or if the ratio is 2:3 then aspectRatio = 2/3 = 0.66, and width = 0.66 * height)

## A concrete example

Here is a concrete example: Let’s say we have a grid that is twice as wide as it is high, so aspectRatio = 2. Let’s also say that our capacity is 200: We want to fit at least 200 cells into this space. I’ve deliberately chosen an easy example, and the answer in this case would be a width of 20 and a height of 10:

◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️◼️
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Apologies if that’s making your eyes hurt. Is it making your eyes hurt? It’s making my eyes hurt. I’ll try and make a better diagram later. It’s not even got the correct aspect ratio on the page…

If you enjoy attacking maths problems, look away now. Have a go at solving it before you read on. You may come up with a better solution anyway.

## So how do we compute the answer?

One way would be a loop. Here is some code that would do the job:

var capacity = 1000;
var aspectRatio = 1000 / 800 = 1.25;

int width = 0;
int height = 0;

while (width * height < capacity)
{
height += 1;
width = RoundToNearestInteger(height * aspectRatio);
}

Assert.AreEqual(height, 29);
Assert.AreEqual(width, 36);

Here we have a grid that has at least 1000 cells:
36 * 29 = 1044

It has roughly the correct aspect ratio:
36 / 29 = 1.24
1000 / 800 = 1.25

The problem with this is that it does not scale well. The higher the numbers, the longer the loop. But fear not – maths to the rescue.

## A simple use case

Let’s start with a simple use case: imagine the aspect ratio is 2:1, and the width is twice the height. Also let’s say we’re trying to fit 200 pixels into the space.
Let’s call width and height x and y. So:
x * y = 200
and
x = 2 * y = 2y
If x = 2y then x * y is the same as 2y * y, which is 2y²:
2y² = 200
y² = 100
y = 10
x = 20 (because it’s 2y)

## A generalised solution

The first block above gives us our two simultaneous equations.

We can generalise the same approach for all numbers. Let’s call the capacity a, and the ratio b. Using our equations from above :
x * y = a
and
x = b * y
These are our simultaneous equations. Using the same logic as above, we substitute y for x and end up with:
by² = a
Which means that
y² = a/b
and
y = sqrt(a/b)
(where “sqrt” is square root).
Then once you’ve calculated y, you can calculate x:
x = b * y
The only missing piece of the puzzle is that we are dealing with inequality, not equality. So really it’s this:
y >= sqrt(a/b)
This just means that once you get a result, you round it up to the nearest integer. Same applies to the final substitution : if b * y is not a whole number, then round it up.
When you calculate b * y, you should use y BEFORE rounding.
So to finish, let’s stop being mathsy and start being softwarey. Let’s give our variables some meaningful names:
height >= sqrt(capacity / aspectRatio)
width = aspectRatio * height
If we test it with the example we used in the loop above, where aspectRatio = 1.25 and capacity = 1000, we get the same results:
height >= 28.28 [= sqrt(1000 / 1.25)]
width = 36 [= RoundUp(1.25 * height)]
So height = 29, width = 36, aspectRatio = 36 / 29 = 1.24, num cells = 29 * 36 = 1044
Now we can finish with some code:
var heightUnrounded = SquareRoot(capacity / aspectRatio);
var height = RoundUp(heightUnrounded);
var width = RoundUp(heightUnrounded * aspectRatio);
Compared to the loop we started with, this scales brilliantly. It doesn’t matter how big the numbers are, the compute time is basically the same. It has a constant computational complexity, otherwise known as O(1)*. The solution above using a loop gets slower and slower depending on how big the numbers are. It has a computational complexity of O(n), so it’s not as good.
Hopefully that makes sense and I didn’t over-simplify. All feedback welcome.
* I couldn’t remember how to express O(1) so I initially guessed at O(k), because I was thinking of k as a constant. Sadly I was wrong, because I rather like the thumbs-up “Ok” concept.

# Martin Fowler webinar, new Refactoring book

These may well be the notiest notes I’ve ever published, but just in case they’re of any use to anyone… if nothing else they may whet your appetite for the new edition of Martin Fowler’s Refactoring book.

I confess I never read it first time round (not on purpose, just there are so many books in the world and so little time…), so I’m looking forward to reading it this time. It hasn’t come out yet in the UK but should be some time in the next few weeks. Mine is already on order [drums fingers impatiently].

So anyway, Martin did a webinar today on the topic of his new book, and here are my notes:

## Refactoring should be done via small changes

• v small semantics-preserving changes
• So small they’re not worth doing on their own
• String together small changes to make a big change

• Alternate between refactoring and adding functionality
• Often it’s easier to add new functionality if you refactor first
• Balance between adding new functionality and refactoring – it’s a matter of judgement

## V1 of Martin’s Refactoring book vs v2

• Some things that were included in the first book he decided not to include I the second
• Eg Unidirectional vs bidirectional – not important
• Some things he thought were too trivial for the first book, he has now changed his mind about and included
• Eg moving statements within a function
• Most notable changes = no longer centrally object-oriented
• “Extract function” used throughout instead of extract method
• One of the refactorings is around the choice between OO and non-OO
• The most visible thing here is the choice of language (javascript)
• But language is not necessarily relevant anyway
• OO vs functional
• He doesn’t see that as a huge shift
• OO should still have functional elements – referentially transparent
• They are overlapping paradigms, not distinct

## Favourite refactorings?

• “Split phase”
• Hunk of computation which can sensibly be divided into two phases with a data structure communicating between them
• Eg parsing – separate the tokenising out – deal with a series of tokens instead of a stream of text
• But if you split a loop, what if you are introducing performance problems? See below…

## Performance and refactoring

• Split loop is something people often worry about because it will run the loop through twice
• Most of the time it doesn’t matter – a push for clarity will not change the performance of the code
• Most of the time by refactoring you open up an opportunity for performance improvements you would never have noticed otherwise
• Then again you should run performance tests frequently
• Not necessarily with every build, but every day or two
• But most micro-changes have no impact on performance
• You will only find real performance impact by performance testing

## Refactoring architecture:

• Always have to ask, how do you make the smallest possible change?
• Eg extracting a data-intensive microservice from a larger system
• Most of the moves are happening within the monolith BEFORE you start firing up the external system
• Then you move stuff in really small pieces
• Things that change for similar reasons should be together – you need to see the relationship between them

## Branching / trunk-based development

• Develop on a branch and merge at the end, with branch being long-lived – days, weeks or months
• Still a really common pattern
• But if something has been refactored, you don’t find out until you merge – this can blow up on you
• It means you become scared to refactor! There’s a disincentive.
• Continuous integration: Integrate with each other fully at regular intervals – at least once a day
• Every day you receive everyone else’s code changes and push your own changes back
• Removes the big painful merge problem – not saving up for a big horrible crunch
• Adage of Apollo – it hurts if you do it more often (?)
• Sorry, I wrote this down but I’ve since Googled it and I don’t really know what it’s about – I may have misheard!
• Open source development:
• Actually this CAN be a good place to use branches, because everyone working separately and you can’t guarantee all changes should be merged
• People call it trunk-based development but “continuous integration” is a better name because that’s what it’s really about
• People worry about continuous integration but it’s less painful than they think
• If you do integration every day it becomes easier and less painful until you stop even noticing

## Responsibility

• Keep code base in healthy state
• If not you are stealing from colleagues, clients and employers!
• But no point saying you should do it for moral reasons, that won’t motivate people to do it
• It’s economic: Want to get more changes out more quickly
• It’s professional: We’re being paid to produce features rapidly
• It’s not up to managers or business: They don’t know how to keep code healthy – that’s what they pay us to do

## Refactoring and design patterns – are they incompatible?

• With a lot of patterns, you shouldn’t introduce them right away cos they’re not appropriate and you don’t require the full strength of them yet, particularly the more complicated ones
• Introduce patterns as a result of refactoring: Refactor towards them
• This was recognised by the original Design Patterns authors (Gamma, Helm, Johnson, Vlissides)
• Also Josh Kerievsky: Refactoring to Patterns

## Code smells

• Code smell of the week!
• “This week we will look at long methods” – whenever we find them we try to refactor them

## When not to refactor!

• Code that is never touched
• Sitting behind an API, working quite happily, doesn’t need to change – doesn’t matter if it’s a mess!
• Don’t clean code all in one go – just a little bit more every time you go in
• Over time that will concentrate your efforts where they are most needed – ie areas modified most frequently

## Refactor as a step in the red-green-refactor cycle

• You should only refactor when tests are green
• Sometimes that means you have to remove a failing test, refactor, then put the failing test back in again
• Ralph Johnson, John Brant, Don Roberts wrote the first Smalltalk refactoring book (A refactoring tool for smalltalk?)

## If you don’t have tests you can’t refactor?

• “Probably safe” refactorings? Interesting route to go? Not a great thing for most people to focus on, requires a lot of expertise
• For most people you should have tests as well – after all, how do you know whether you’re breaking what’s currently there?

## Final takeaway:

• Small steps, commit frequently.
• Then you can always roll back.