What's Functional Programming All About?
What's Functional Programming All About?
There are many descriptions floating around the internet, trying to explain functional programming in simple terms. Unfortunately, most discuss details only loosely related to functional programming, while others focus on topics that are completely irrelevant. So of course, I had to write my own!
This post is my own understanding of what is the "core" of "functional programming", how it differs from "imperative" programming, and what the main benefits of the approach are. As a worked example, we will use a kitchen recipe as a proxy for the more-abstract kind of logic you find in program source code, to try and make concrete what is normally a very abstract topic. That recipe is one of my favorite recipes available online, Michael Chu's Classic Tiramisu.About the Author: Haoyi is a software engineer, and the author of many open-source Scala tools such as the Ammonite REPL and the Mill Build Tool. If you enjoyed the contents on this blog, you may also enjoy Haoyi's book Hands-on Scala Programming

  What Functional Programming is Not
    
      Helper Methods
      Writing Things in Haskell
      Compile-time AST Macros
      Static Types
    
  
  Step by Step Imperative Recipes
    
      Kitchen Refactoring
    
  
  Functional Programming Recipes
    
      Tiramisu Diagram to Functional Programming
      Preventing Errors with Functional Programming
      Refactoring a Functional Tiramisu Recipe
      The Core of Functional Programming
    
  
  Conclusion

A topic as broad as "Functional Programming", or "FP" has too many different interpretations and facets to be summarized in one blog post. Nevertheless, this post will discuss what I think is the most core, basic level of functional programming. This will hopefully be something that everyone, from FP newbies to FP "experts", should be able to empathise with and agree is a useful part of functional programming.
It's not surprising that many people have tried to explain functional programming using kitchen/recipe/cookbook examples: learning things "by analogy" of things you already know is one of the easiest ways of learning. However, all explanations I have seen fall short. I will begin by examining some typical, incorrect explanations of what functional programming is about, before discussing how Michael Chu's Classic Tiramisu recipe:

Can provide insight into what I think are the core techniques and benefits of functional programming.What Functional Programming is Not
There are many poor explanations people have given for "what is functional programming". Here is a selection:Helper Methods
One of the most common misconceptions of what FP is is illustrated by the following example:

  FP => I'll have a Sazerac
  Imperative => Excuse me sir, could you take some ice, add rye whiskey, add bitters, add absinthe, shake, strain into a glass, and add a lemon garnish before bringing it to me

While this example was taken from the y-combinator message board, I've seen this attitude in many places: the idea that functional programming is just taking imperative instructions, and wrapping them in a helper. In this case, the messy imperative code will all sit inside a single helper:
def sazerac():
    ... 10000 lines of messy imperative code ...

But even in imperative programming you always end up factoring things into helper methods. Java has helper methods. Write assembly, and it ends up being organized with sub-procedures to encapsulate messes of imperative code. 
Thus, while this is a useful technique, writing helper methods to wrap your messy code in a single method/function/subprocess/subroutine call does not count as functional programming. 
Furthermore, picking an easier/simpler problem, despite making your code look neater, does not count as "Functional Programming" either. Calling a single method that executes a huge blob of code that someone else has written is convenient, but is not functional programming. The point of FP is to face the complexity, own it, and control it, not shove it inside some unmaintained helper function or say it's a problem for some "other department" to deal with.Writing Things in Haskell
sazerac = do
    add ice
    add ryeWhisky
    add bitters
    add absinthe
    shake
    strainInto glass
    add lemonGarnish

main = serve $ makeCocktail sazerac


  Also from the y-combinator message board

It's often said that you can write COBOL in any language, that you can write Java in any language. Well, you can write any language in Haskell too: the above is basically writing Bash in Haskell
Just because something is implemented in Haskell with Monads, doesn't mean it's functional programming. If it looks like imperative code written in Bash, and it's semantics are like imperative code written in Bash, it's imperative code. This example certainly looks exactly like imperative code written in Bash except it's run using serve $ makeCocktail instead of bash cocktail.sh.Compile-time AST Macros
Some variant of Lisp (or Scheme?) was probably one of the first implemented FP languages; and Lisps tend to have compile-time AST macros that allow you to transform sections of the program at compile-time. 
But compile-time code-transformations are not unique to Lisp; apart from other FP languages that have them, like Template Haskell or Scala Macros, many languages have some sort of compile-time code transformation. From Java Annotation Processors, to my own MacroPy project in Python, it turns out that compile-time ast macros are just as feasible in imperative languages, doing imperative programming. You can manipulate mutable ASTs using imperative Python code just as easily as you can elegantly transform immutable ASTs using Scala.
Furthermore, there are a large set of "obviously" functional programming languages that don't have AST-transforming macros at all. Purescript, non-Template Haskell, Scala 2.9, and many other "obviously" functional languages do not include support for compile-time AST transformations. So whatever is the core of functional programming, it's not AST macros.Static Types
There are a large number of people who use FP together with static types, e.g. in languages like Haskell, Scala, or Ocaml. Thus, if you spend all your time within this world, it might be tempting to think that FP is all about static types. Higher-kinded, Rank-N, Dependent, the fancier the types, the more functional the programming. 
However, there are probably just as many people using FP without static types: in some parts of the Javascript community, Clojure, Scheme or one of the many other Lisps. It turns out, that all those using FP without types still get many of the benefits. And then there are all those people in static-typed languages like Java that use minimal FP in their code. 
So static types, while present in many FP languages, are not the core of FP.Step by Step Imperative Recipes
Now that we've looked at a few common misconceptions of what FP is, let's look at what the core of FP actually is (according to me) in contrast to "imperative" programming, using Michael Chu's Classic Tiramisu:

As an example. To begin with, we'll explore an "imperative" recipe, that is probably familiar to those you already know.
Michael Chu's Classic Tiramisu, like all the other recipe's on his excellent recipe site has roughly four sections on the page:

  
  The backstory of the recipe
  
  A step-by-step guide, with photos, of how to make the Tiramisu
  
  A diagram of the overall process, showing which ingredients are combined with  which others, to create the resultant Tiramisu
  
  A lively and entertaining comments section

For the purpose of this programming blog we will only be looking at parts 2. and 3.: the step by step guide, and the process diagram. The step by step guide details, in order, a series of steps that you can take to make a Tiramisu. At a high level, hiding many of the details, it looks like this:

  
  Begin by assembling four large egg yolks, 1/2 cup sweet marsala wine, 16  ounces mascarpone cheese, 12 ounces espresso, 2 tablespoons cocoa powder, 1  cup heavy cream, 1/2 cup granulated sugar, and enough lady fingers to layer a  12x8 inch pan twice (40).
  
  Stir two tablespoons of granulated sugar into the espresso and put it in  the refrigerator to chill.
  
  Whisk the egg yolks
  
  Pour in the sugar and wine and whisked briefly until it was well blended.
  
  Pour some water into a saucepan and set it over high heat until it began  to boil. 
  
  Lowering the heat to medium, place the heatproof bowl over the water and  stirred as the mixture began to thicken and smooth out. 
  
  Whip the heavy cream until soft peaks.
  
  Beat the mascarpone cheese until smooth and creamy. 
  
  Poured the mixture onto the cheese and beat
  
  Fold in the whipped cream
  
    Assemble the tiramisu. 
    
      
      Give the each ladyfinger cookie  a one second soak on each side and then arrange it on the pan
      
      After the first layer of ladyfingers are done, use a spatula to spread  half the cream mixture over it.
      
      Cover the cream layer with another layer of soaked ladyfingers.
      
      The rest of the cream is spread onto the top and cocoa powder sifted over  the surface to cover the tiramisu.
    
  
  
  The tiramisu was now complete and would require a four hour chill in the refrigerator.

This is, I think, something like what most people would think of when told "imperative recipe". You start with a set of inputs (the bullet 1.) and then perform a series of steps until you have a result at the end. (For now, I'm ignoring the pictures in the recipe, though you could think of them as a sort of assert function for a would-be chef to check some invariants after each step to make sure his tiramisu hasn't gone terribly wrong!)
A simplified Python version of this recipe (ignoring the fact that I'm overloading the same functions to work on different types/number of arguments) may look something like this:
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    dissolve(sugar2, espresso)
    mixture = whisk(eggs)
    beat(mixture, sugar1, wine)
    whisk(mixture) # over steam
    whip(cream)
    beat(cheese)
    beat(mixture, cheese)
    fold(mixture, cream)
    assemble(mixture, fingers)
    sift(mixture, cocoa)
    refrigerate(mixture)
    return mixture # it's now a tiramisu
Kitchen Refactoring
Like most imperative code, it works, but may be hard to understand deeply or difficult to refactor. For example, in cooking terms, you may ask the following questions:

  
  If I have two people to make this tiramisu, which parts can be done in  parallel?
  
    My expresso hasn't arrived yet; can I shift that step down and do other  things first and include the expresso later when it arrives? 
    
      
      What if my eggs haven't arrived? Which steps can I do first before  the eggs turn up?
    
  
  
    At step 9. I screwed up and spilled the bowl onto the floor. Which steps do  I need to re-do (and which ingredients I may have to re-purchase) to recover  and continue the recipe?
    
      
      What if I spilled the bowl at step 10? Or step 8?
    
  
  
    Just before step 10, you realize you forgot to do step 7. How much of  your ingredients have been ruined?
    
      
      What if the forgotten step was step 4? Or step 2?
    
  

All four of these are things that happen regularly in a kitchen, and also happen to correspond to things you do with program code all the time: parallelizing things over the available cores to speed things up, shuffling the order of a computation around, dealing with failures and exceptions, or plain old bugs and mistakes.
The answers to these questions are left as an exercise to the reader; in this case, with 12 steps, it's not terribly hard to figure out. A few minutes carefully studying the recipe and you could probably figure it out, so you should definitely give it a try. 

In a large software project, with a codebase containing thousands or millions of lines of imperative code, that time could easily stretch to days, weeks, or months trying to figure out how to properly recover when one of those imperative steps fails, or how to make your legacy PHP monolith do something faster by using more than 1 of the 32 cores you have available on your beefy server. 
The problem in these cases often isn't that you don't know how to run stuff in a separate process in PHP - the problem is that you don't know enough about your own code to decide what to run in that other process. To move things onto a separate process, you need to know exactly what each bit of code depends on, and who depends on it, so you can pick a set with minimal dependencies to run somewhere else (since inter-process communication is expensive). That's difficult when you have a pile of imperative code and don't even understand it enough to easily move things around within a single process.
The reason that these kinds of analyses are hard on this imperative recipe is the same reason that the analyses are hard when programming in an imperative style: 

  
  There is an ordering of the steps, but the ordering between some steps is  required, e.g. the series 9, 10, 11, while those between other steps is  entirely arbitrary: step 2 could be done anywhere before step 11, and step 7  and 8 could be swapped or done much earlier and nobody would care.
  
  The instructions are based on changing the state of things, e.g. pouring  stuff into mixture, a term that we use repeatedly throughout the recipe  but means a different thing in each step. Even the meaning of cheese and  cream changes as the recipe progresses (e.g. after calling whip(cream)),  but it is entirely hidden from you and not obvious from the code.

Overall, these factors make it hard to decide, given a single step S, what steps S depends on, and what other steps depend on S. Again, it is possible to figure it out, but what is somewhat-tedious to figure out in a 16-line tiramisu recipe becomes painful and difficult in a 1,000,000 line enterprise codebase.
So that's what an imperative Tiramisu recipe looks like. What does a "functional programming" Tiramisu recipe look like?"Functional Programming" Recipes
It turns out, there's a FP version of this recipe right underneath the imperative one! The "process diagram" mentioned above is an excellent illustration of how such a recipe would look like using "Functional Programming":

To read this, the raw ingredients are on the left, and each of the boxes represents a single operation that transforms and combines the ingredients. After all the combinations have taken place, you end up on the right with a single, complete Tiramisu. While this "2D" format is not how people write program source code, the underlying structure is not too different from how people structure "FP" programs, which I will demonstrate below.
This diagram leaves out some the detail that the full imperative recipe provides, even compared to the abridged version I transcribed above. For example, chilling the expresso or explicitly boiling the water are left out, and the details of assemble are not included. Nevertheless, it contains the same high-level steps of how to build the tiramisu I abridged above. We're not leaving out large numbers of operations or hiding things behind high-level instructions: all the same steps are still there, just organized slightly differently.
But even if this diagram has the same "content" as the imperative instruction-list I discussed earlier, what about this makes this presentation of the recipe more "functional"?Tiramisu Diagram to Functional Programming
While nobody actually writes their code in a 2D table-flowchart-thing like this tiramisu diagram is, it turns out underneath the 2D format the "core" of this diagram is the dependency graph between elements:

Where each box takes in some "inputs" from the left, and results in an "output" that can be used by more-rightward boxes. This can be straightforwardly represented in code by treating the boxes as functions, e.g. in the following Python code:
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
                 
    return refrigerate(
        sift(
            assemble(
                fold(
                    beat(
                        whisk( # over steam
                            beat(beat(eggs), sugar1, wine)
                        ), 
                        beat(cheese)
                    ), 
                    whip(cream)
                ), 
                soak2seconds(fingers, dissolve(sugar2, espresso))
            ), 
            cocoa
        )
    )

(Again, forgive the fact that I'm overloading the same functions to work on different types and numbers of arguments)
If it's not immediately clear how this code relates to the "functional programming dependency diagram" I discussed above, we can draw the dependency graph of this code: showing where the input variables go, where the return value of each function goes, all the way into the "final" result that gets returned:

It might look like a bit of a mess, but if you look carefully, you will see that although the graphs are laid out differently, the fundamental structure of the two graphs is identical! That is what I mean when I say the 2D box-diagram is a "FP Recipe": although people don't tend to write code in 2D box-diagrams, the underlying structure that the diagram represents is totally equivalent to some "FP"-ish Python code, not too dissimilar to what people do write. 
This code looks very unlike code you are likely to see in a Python project, "in the wild", but we can fix that! If you prefer to have intermediate named values instead of one big expression, it's straightforward to pull out each function call into it's own statement: 
# FP         
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    beat_eggs = beat(eggs)
    mixture = beat(beat_eggs, sugar1, wine)
    whisked = whisk(mixture)
    beat_cheese = beat(cheese)
    cheese_mixture = beat(whisked, beat_cheese)
    whipped_cream = whip(cream)
    folded_mixture = fold(cheese_mixture, whipped_cream)
    sweet_espresso = dissolve(sugar2, espresso)
    wet_fingers = soak2seconds(fingers, sweet_espresso)
    assembled = assemble(folded_mixture, wet_fingers)
    complete = sift(assembled, cocoa)
    ready_tiramisu = refrigerate(complete)
    return ready_tiramisu

That makes it look entirely "pythonic", indistinguishable from the code you might find in any random project on Github
Moving every expression into a separate statement is a straightforward transformation, at least for FP programs, and is the kind of thing that compilers regularly do automatically. Thus, although that block-flow-chart diagram may have looked a bit foreign at first, it really isn't that different from the code people write day to day, all year round.
In fact, it looks not too unlike the "Imperative" version we came up with earlier!
# Imperative
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    dissolve(sugar2, espresso)
    mixture = whisk(eggs)
    beat(mixture, sugar1, wine)
    whisk(mixture) # over steam
    whip(cream)
    beat(cheese)
    beat(mixture, cheese)
    fold(mixture, cream)
    soak2seconds(fingers, espresso)
    assemble(mixture, fingers)
    sift(mixture, cocoa)
    refrigerate(mixture)
    return mixture # it's now a tiramisu

These two snippets of code look very similar, but the top one is "Functional Programming" while the bottom one is "Imperative Programming". The difference between them? 

  
  In the first, you can see that beat(cheese) must come  before beat(whisked, beat_cheese), because beat_cheese  is defined by the beat(cheese) and used by beat(whisked, beat_cheese).  Even if you know nothing about beat, cheese or whisked, it is clear  from the code that if you tried to reverse the order - and  beat(whisked, beat_cheese) before beat(cheese), it wouldn't work.
  
  In the second, it's not so clear: does beat(cheese) need to come before  beat(mixture, cheese)? Or does beat(mixture, cheese) need to come before  beat(cheese)? In this case, we have a link to the "docs" (the original  recipe) so we can look it up, but which one depends on the other - and  whether they are currently in the right order - is not clear from the code.

But how does this seemingly-trivial difference affect the way you build software?Preventing Errors with Functional Programming
The difference between the two Python snippets, the # FP and # Imperative snippets, will become clear with the following thought experiment: what if we try to make changes to the code?
Changing code is something we do all day, and sometimes we do it incorrectly. It would be a nice property of a codebase if changes tended to be easier to make correctly, and incorrect changes were easier to spot. We'll discuss the latter first.
If I try to tidy things up and accidentally move the statement
beat_cheese = beat(cheese)

below
cheese_mixture = beat(whisked, beat_cheese)

It should be clear to me that something is wrong, because there will be no beat_cheese in scope to create the cheese_mixture. Even if it's not clear to me, it's probably clear to my linter and editor:

As you can see, not only does the usage of beat_cheese raise an error because no such variable is defined, the definition of beat_cheese also raises a visual warning: it is greyed out since it is dead code! This makes it very hard to miss when you make such trivial error, and saves you time: rather than waiting 10s for your test suite to run, within less than 1s your linter would have lit up and flagged the lines as invalid. Over the days, months and years, this adds up to a significant productivity boost
However, in the Imperative case, it's not clear how 
beat(mixture, cheese)

Relates to the things before or after it. If I remove the beat(cheese) earlier, I still have a cheese to pass in. If I remove the beat(mixture, cheese) entirely, I still have a mixture I can use in later steps of the recipe. So how do I know, from looking at the code, that removing a step or re-ordering them so that beat(cheese) comes after beat(mixture, cheese) is a problem? 
The answer is, you often don't, and neither does your computer, or your editor and linter, who aren't going to help you spot the fact that you accidentally swapped two of the imperative statements:

Fundamentally, in the "FP" example, the code is laid out in a way that the "correct" usage is obvious: each function, e.g. beat, only depends on the things that are passed into it, and it's output is only depended upon by whoever uses it's return value. In the "Imperative" example, it's not clear who depends on who: you have to memorize the fact that beat(cheese) must come before beat(mixture, cheese), and not the other way around. 
While this is not difficult assuming we are looking at already-correct code (the current order is the correct order!), when mistakes are made, and code happens to be incorrect, "FP" code makes the mistakes much easier for you (or your linter) to spot so you can correct them.

While this example may seem contrived, the basic problem exists in all large codebases I've worked with. For example, maybe you've bumped into code similar to the following three functions:
def initialize():
   ... 1000 lines of messy code, no return value...
   
def make_app():
   ... 2000 lines of messy code, no return value...

def start_server():
   ... 4000 lines of messy code, no return value...

Which transitively depend on a 1 million line codebase ("The App"). How could I know that start_server() needs to be called before make_app(), which itself needs to be called before initialize(), when all of them are global functions which don't take arguments or return anything? I have certainly spent countless days of my career puzzling over such mysteries in large codebases, and I am sure others have too. If start_server returned something I needed to pass to make_app, which returned something I needed to pass to initialize, that would make it clear from the outset which one needs to come before the other.
Re-ordering or shuffling around statements is not uncommon. When you are refactoring a piece of code to let you re-use it in a different place, a lot of time is spent shifting bits of code up and down small amounts, just like the example I showed above, so that the code you want to re-use is all in one place and you can extract it into a helper. 
Perhaps you just want to tidy up what was previous a messy function to organize the code a bit better than it already is, grouping related lines so they can be read together easily, without changing any behavior at all.
Or perhaps, as mentioned earlier, someone made a mistake and the code that already exists is incorrect, and your job is to figure out which of the statements is out of order so you can fix it.
All of these are things that software engineers do day in, day out. And often, we make mistakes when doing so. With functional programming, whether in a typed language or not, it tends to be much more clear when you've made a trivial, dumb error. That means you get feedback quicker: you get corrected quietly by your linter in the privacy of your own laptop, and can quickly fix it and make progress, rather than waiting a long time only to be loudly yelled at by Jenkins CI in front of your entire team.Refactoring a Functional Tiramisu Recipe
Even if you haven't already-made a mistake, and are just thinking of making a change to a codebase, the # FP version of the code is a lot easier to think about than the # Imperative version. The same often applies whether you're writing dealing with Python, Javascript, Scala, or a Tiramisu recipe!
I have already shown above how the 2D-block-diagram version of this recipe is exactly equivalent in semantics to a "FP" Python function. For this section I will use the 2D-block-diagrams to illustrate my points, as it is much clearer visually, but the same kind of reasoning applies to "FP" code in Python or any other programming language. While working with an FP style, you quickly get used to performing the same analyses in your head, just as quickly, but on lines of source code rather than 2D-block-diagrams.
What is interesting is that this structure lets us very easily answer some of the questions we asked above:

  
    If I have two people to make this tiramisu, which parts can be done in parallel?
  

This one is easy: 

Anything vertically separated can be done in parallel. For example, preparing the ladies fingers and preparing the eggs/sugar/wine are separate and can be done independently, as can whipping the cream and mascarpone cheese. Thus, if you have three people, you might assign: 

  one person to be the egg/wine/sugar mixture czar,
  one to be the mascarpone/cream czar, and
  one to be the expresso/ladyfingers czar.

On the other hand, anything horizontally separated has to be done sequentially:

Thus even if you parallelize the early bits, the later beat-fold-assembly-refrigerate steps all have to be done sequentially, and how much time you can save on your Tiramisu is limited by the length of the Critical Path. 
Working with the "FP" representation of the recipe doesn't shorten the critical path, and thus doesn't affect how much you can "theoretically" speed up your recipe with parallelism. What it does do is make clear exactly which parts of the recipe can be parallelized and which can't, so you can more quickly organize your work to get maximum parallelism given the constraints of the recipe, and then move on to other things.
Again, while we're looking at a 2D-block-diagram, the same applies to FP-style code in Python, Javascript, Scala, or any other programming language.

  
    My expresso hasn't arrived yet; can I shift that step down and do other things first and include the expresso later when it arrives?
  


If you expresso hasn't arrived, anything depending on it can't be done, but anything else involving eggs/sugar/wine/cheese/cream can be prepared: the sections marked in red make it clear which parts of the recipe depend on expresso; the rest can be done while waiting for the expresso to arrive

  
    What if my eggs haven't arrived? Which steps can I do first before  the eggs turn up?
  


In this case, the top block can't be done but you can prepare the bottom and middle blocks: preparing the expresso, beating the cream and mascarpone cheese. Again, this is obvious from looking at the diagram

  
    At step 9. I screwed up and spilled the bowl onto the floor. Which steps do I need to re-do (and which ingredients I may have to re-purchase) to recover and continue the recipe?
  

Step 9 is when you beat the Mascarpone cheese into the egg mixture. Once we find it on the diagram, it's clear what we need to do:

You will need to get some new eggs/sugar/wine/cheese and beat/beat/whisk/beat them all over again 

  
    What if I spilled the bowl at step 10? Or step 8?
  


Spilling the bowl at step 10 (folding the whipped cream into the main mixture) is the same as spilling the bowl at step 9, except you need to get new cream too.

Spilling the bowl at step 8 (beating Mascarpone cheese) and you just need to get new mascarpone cheese and beat it. The rest of your ingredients are fine. 

  
    Just before step 10, you realize you forgot to do step 7. How much of your ingredients have been ruined?
  


In the diagram above, the red boxes represent the steps we've already done, up to step 10 (folding in the whipped cream). As you can see, not having done step 7 (whipping the heavy cream) is no big deal; we haven't needed to done it up to now, so we can do it and continue with step 10 

  
    What if the forgotten step was step 4? Or step 2?
  

If you forgot step 4 (whisking in wine and sugar to the beaten eggs) you've ruined your eggs/sugar/wine/cheese:

As you can see, the stuff we've been whisking and beating was not prepared properly before being whisked and beaten, since we forgot to mix in the wine and sugar. Assuming we don't know enough kitchen chemistry to incorporate the wine/sugar in at this stage (Our eggs may well have turned into omelettes by now without the additional liquid from the wine...) we will need to re-do all the steps in the upper red box.
If you forgot step 2 (dissolving sugar into expresso) you're fine. The expresso hasn't been needed yet:

According to the imperative recipe above, we should have done the expresso mixing first before starting on the egg/wine/cheese. But even though we didn't do it, it is trivial to see from the FP-style recipe that there really isn't any loss: no other steps so far depended on that, no other ingredients were ruined.

As you can see, many of the questions that were non-trivial to answer when dealing with the imperative code back in Kitchen Refactoring are now trivial to answer when working with the FP-style 2D-block-diagrams.
Again, while nobody actually codes in 2D-block-diagrams (except skilled engineers running recipe blogs) the 2D-block-diagrams are equivalent to a relatively straightforward snippet as shown above. With some experience dealing with FP code, you can often perform the same analyses just as easily when working directly with the equivalent Python code we showed earlier. And it's not just about programmers: automated tools linters or IDEs often perform the same analysis on the fly, as shown earlier, quickly alerting you if you make a mistake that means the recipe can no longer be completed successfully:
The Core of Functional Programming
The core of Functional Programming is thinking about data-flow rather than control-flow. Although, by virtue of editing plain text, you are forced to order your code in a linear sequence of statements, those statements are a thin skin over what you really care about: the shape and structure of the data-flow graph within your program.
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    beat_eggs = beat(eggs)
    mixture = beat(beat_eggs, sugar1, wine)
    whisked = whisk(mixture)
    beat_cheese = beat(cheese)
    cheese_mixture = beat(whisked, beat_cheese)
    whipped_cream = whip(cream)
    folded_mixture = fold(cheese_mixture, whipped_cream)
    sweet_espresso = dissolve(sugar2, espresso)
    wet_fingers = soak2seconds(fingers, sweet_espresso)
    assembled = assemble(folded_mixture, wet_fingers)
    complete = sift(assembled, cocoa)
    ready_tiramisu = refrigerate(complete)
    return ready_tiramisu

 
Similarly, when executing a "functional program" in a single thread, you are forced to pick a linear order in which you execute each individual instruction, which e.g. might be the same as the order in which it is written down in the code. But since we know what really matters is the shape of the data-flow graph, we can freely re-arrange the statements in the code, and the order of execution, as long as the graph shape is preserved. Since the data-flow graph matches the graph of definitions and usages, even your editors and linters understand it enough to warn you if you re-arrange things in an invalid order. In fact, if you have multiple cores (or multiple cooks!) you can execute parts of it in parallel, not in any linear order at all! Exactly in what order the program-counter proceeds from instruction to instruction is irrelevant.
This is in contrast to an imperative program, where the exact order in which the program-counter executes each statement, going in and out of loops, in and out of sub-routines, is the key to understanding the program. In an imperative program, you tend to think in terms of steps that must happen "before" and "after", and make sure that the control-flow of the program executes the commands in the right order for the program to work.
Note that none of the FP examples here are "less complex" than the "imperative" recipe we discussed above. It's about the same number of lines:
def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    dissolve(sugar2, espresso)
    mixture = whisk(eggs)
    beat(mixture, sugar1, wine)
    whisk(mixture) # over steam
    whip(cream)
    beat(cheese)
    beat(mixture, cheese)
    fold(mixture, cream)
    assemble(mixture, fingers)
    sift(mixture, cocoa)
    refrigerate(mixture)
    return mixture # it's now a tiramisu

def make_tiramisu(eggs, sugar1, wine, cheese, cream, fingers, espresso, sugar2, cocoa):
    beat_eggs = beat(eggs)
    mixture = beat(beat_eggs, sugar1, wine)
    whisked = whisk(mixture)
    beat_cheese = beat(cheese)
    cheese_mixture = beat(whisked, beat_cheese)
    whipped_cream = whip(cream)
    folded_mixture = fold(cheese_mixture, whipped_cream)
    sweet_espresso = dissolve(sugar2, espresso)
    wet_fingers = soak2seconds(fingers, sweet_espresso)
    assembled = assemble(folded_mixture, wet_fingers)
    complete = sift(assembled, cocoa)
    ready_tiramisu = refrigerate(complete)
    return ready_tiramisu

whether as multiple statements, one big expression, or as a 2D block diagram. All the same operations are present: beating, whiping, folding, etc.. Functional Programming is not about hiding ugly code in helper methods and hoping nobody notices: it's about managing the same complexity in a way that makes the dependencies between each piece of code obvious, by following the graph of where function arguments come from and where return values end up.
When you have a working program, having the dependency graph of function return values being passed into other functions as arguments makes it really easy to analyze code. For example, if we were curious what exactly is required to get our wet_fingers_mixture, we can see:

  wet_fingers comes from soak2seconds(fingers, sweet_espresso)
  sweet_espresso comes from dissolve(sugar2, espresso)
  sugar2, fingers, espresso are the initial ingredients of the recipe

An there you have it: just a few steps, entirely mechanical, and we can see exactly what wet_fingers needs. We need no understanding of what dissolve does, or what a sugar2 is: just from the structure of the code we can already see what wet_fingers requires. Just as importantly, we can also see that it does not depend on folded_mixture, whipped_cream, or any of the other steps that are above it in the code: while those steps "come before" the operations that give us a wet_fingers, it's clear from this analysis that their ordering is entirely accidental, and that we could e.g. prepare the wet_fingers before the other steps if we so desired.
It's not hard to do this yourself, but any IDE with jump-to-definition should be able to do this for you, and so can automated linters and code analysis tools. And understanding the code is the first step in changing it, without bugs.
When you have a broken program, having the dependencies be easy to analyze means it's easier to spot when you make a mistake or do something out of order: even in a dynamic language like python, a subtly bad copy-paste job can get called out by your editor so you can fix it before needing to run any code:

Whether you're working in a dynamic language like Python or a static language like Scala, whether your code is currently working or broken, Functional Programming's data-flow-centric approach helps you understand your code faster, easier and with more tooling help than a Imperative, mutation-heavy approach.Conclusion
The core of Functional Programming is thinking about data-flow rather than control-flow

While this may seem a trivial definition of "Functional Programming", I think it is really the core of the idea. While there are many further steps, from the simple (immutability, referential transparency, ...) to the more advanced (monads, lenses, ...) this core of should be something that everyone, from newbies to old hands, whether using Scala or Clojure or Haskell or React.js, should be able to empathise with. Even in a language like Python, as I have used for the examples, it is possible to program in a more "Functional" style, and reap some of the benefits of functional programming.
Those more advanced topics don't really fit this worked example anyway: kitchen ingredients tend to be very, very mutable (and perishable!).
Though it's growing, this baseline-level of FP is not yet widespread in industry. 

  
  Whole languages, such as Bash, make it a pain in the neck to take  non-trivial function arguments or return non-trivial results, resulting in  people's code writing things to the filesystem, hopefully "before" someone  else needs to read them.
  
  Languages like Java encourage  patterns where you instantiate a half-baked object and then set the fields  later, praying that nobody accidentally tries to use it "too early" it in  it's half-baked state while it's internal variables are garbage. 

In all of these cases, the order in which things run - exactly how the program-counter progresses from statement to statement, in and out of for-loops, in and out of sub-routines - is critical. 
Even in the kitchen, having a "FP-style" recipe like the block diagram I showed above is helpful, because when the person bringing your Marsala Wine is stuck in traffic, it makes it easier to re-organize your recipe so you can get as much work done immediately. When that person arrives, it helps you figure out how to parallelize the work over the people you have available. When someone screws up, it helps you figure out exactly which ingredients you need to re-purchase and steps you need to re-do. 
This widespread applicability, even to fields outside the software world, and to every "FP" language within the software world, is why I think this is truly what functional programming is all about.About the Author: Haoyi is a software engineer, and the author of many open-source Scala tools such as the Ammonite REPL and the Mill Build Tool. If you enjoyed the contents on this blog, you may also enjoy Haoyi's book Hands-on Scala Programming