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By: Michael Nunez

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April 20th, 2017

Tail Call Optimization with Fibonacci in Elixir

Languages, Tools, Frameworks

This article orignially appeared on Michael Nunez's personal blog in January 2017.


Today, we’ll discuss Tail Call Optimization, and how it is useful in very complex problems that we may run into on a day to day basis.

According to Wikipedia:

"In computer science, a tail call is a subroutine call performed as the final action of a procedure. If a tail might lead to the same subroutine being called again later in the call chain, the subroutine is said to be tail-recursive, which is a special case of recursion."

Tail recursion is calling a function that will return the same function with different variables. Rather than adding a new function call to the stack, it will return the function with the new variables.

In the example, we will use Fibonacci written in both Ruby and Elixir. We will also optimize the Elixir example to use Tail Call Optimization.

In Ruby, the code looks like this:

def start(n)
  timer = Time.now.to_i
  number = fibonacci(n)
  ender = Time.now.to_i
  amount_of_time_used = ender - timer
  puts "The fibonacci number was #{number}"
  puts "Fibonacci for number placed #{n} finished in #{amount_of_time_used} seconds!"
end

def fibonacci(n)
  raise "You're trying to find a Fibonacci Number under Zero. Please try again." if n < 0
  return 0 if n == 0
  return 1 if n == 1
  return fibonacci(n-1) + fibonacci(n-2)
end

This app simply starts a timer, get’s the Fibonacci number to the n’s place. When we run it to the 40th place, we get following:

2.3.0 :015 > start(40)
The fibonacci number was 102334155
Fibonacci for number placed 40 finished in 20 seconds!

In Elixir, we have a similar solution:

defmodule Fibonacci do
  def start(n) do
    start = :os.system_time(:seconds)
    fibonacciNumber = getNumber(n)
    finish = :os.system_time(:seconds)
    totalTime = finish - start
    IO.puts "The fibonacciNumber was #{fibonacciNumber}."
    IO.puts "Fibonacci for number placed #{n} finished in #{totalTime} seconds!"
  end

  def getNumber(0), do: 0
  def getNumber(1), do: 1
  def getNumber(n) when n < 0, do: raise "You're trying to find a Fibonacci Number under Zero. Please try again."
  def getNumber(n), do: getNumber(n-1) + getNumber(n-2)
end

Very similar to the ruby example, the timer starts, calls the getNumber function, and returns the number.

iex(1)> Fibonacci.start(40)
The fibonacciNumber was 102334155.
Fibonacci for number placed 40 finished in 5 seconds!
:ok

One noticeable difference is that it took 15 longer for the ruby solution to find the same Fibonacci number!

But we can definitely make this faster with Tail Call Optimization. We will make some changes to our getNumber function.

defmodule Fibonacci do
  def start(n) do
    start = :os.system_time(:seconds)
    fibonacciNumber = getNumber(n)
    finish = :os.system_time(:seconds)
    totalTime = finish - start
    IO.puts "The fibonacciNumber was #{fibonacciNumber}."
    IO.puts "Fibonacci for number placed #{n} finished in #{totalTime} seconds!"
  end

  def getNumber(n) when n < 0, do: raise "You're trying to find a Fibonacci Number under Zero. Please try again."
  def  getNumber(n), do: getNumber(n, 1, 0)
  defp getNumber(0, _, result), do: result
  defp getNumber(n, next, result), do: getNumber(n-1, next+result, next)
end

It is very common practice to have an interface of getNumber with an arity 1, that then calls a private function with a greater arity to facilitate the Tail Call Optimization pattern.

The very first function of getNumber(n) calls the private function with a variable signature containing the place in the Fibonacci sequence, the starting number, and it’s previous number. Since the first number is 0, and the next number 1, we can use those to start our Fibonacci function.

Here’s where the magic happens! When we call defp getNumber(n, next, result)we call the very same function, but with different parameters.

do: getNumber(n-1, next+result, next). It subtracts 1 from the sequence, since we’re counting down to zero, it adds the result to the next number, and the next number becomes the result. Since the function is calling itself, it doesn’t have to place a new function in the stack. It calls the same function with new parameters.

defp getNumber(0, _, result), do: result states that when the number has reached zero, return the result.

iex(1)> Fibonacci.start(40)
The fibonacciNumber was 102334155.
Fibonacci for number placed 40 finished in 0 seconds!
:ok

OK speedy, let’s see you try a bigger number!

iex(2)> Fibonacci.start(100)
The fibonacciNumber was 354224848179261915075.
Fibonacci for number placed 100 finished in 0 seconds!
:ok

o.O! Wise guy, eh?!

BIGGER!

iex(3)> Fibonacci.start(10000)
The fibonacciNumber was 33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875.
Fibonacci for number placed 10000 finished in 0 seconds!
:ok

This is getting out of hand…

In fact, I was able to go up to 100,000! Check it out!!

iiex(1)> Fibonacci.start(100000)
The fibonacciNumber was 25974069347221724166155034021275915414880485386517696584724770703952534543511273686265556772836716744754637587223074432111638399473875091030965697382188304493052287638531334921353026792789567010512765782716356080730505322002432331143839865161378272381247774537783372999162146340500546698603908627509966393664092118901252719601721050603003505868940285581036751176582513683774386849364134573388343651587754253719124105003321959913300622043630352137565254218239986908485563740801792517616293917549634585586163007628199160811098365263529954406942842065710460449038056471363460330005208522777075544467947237090309790190148604328468198579610159510018506082649192345873133991501339199323631023018641725364771362664750801339824312317034314529641817900511879573167668349799016820118499077566864568450662873924856039140476051995500662888263458771894106803700918793650017330117100283104739474562560914449328213748555738640805798130282666402703542944121049199958031318768058991865134251759599115205631553377039969410355182752749199598022575079020377981030899229849963044962558140455170002502997643221934621653662108418767454282982613982344783665815880408190033073829395000821320093747154851310272208173054322648669496309879147143629255542526240439996153269798768075106468190687921182991679644091782718685617029181022126792674013626504997849688436809752547001310045741864064482994858725517447466956518791269169932445648176733222571493149677633458466238303338202397024368594782876418757885729107101337003000942293335972927791914092128049015459762627910570552481588840517794181929052167695766087488155678601288183543542923073978101547857013284386127286201766539534449930019800629538936985500723286651317181135886613537472684585432548981137176605194616937916884425342594781263103889520479565943807153019112539648471126389007133628569101551453423329441284357220996286746119420951661002309740709965531900508158669911445442647882872642845017253320486483194578920399848938236367456182203750973485668474338872490493370316338265717607297788917989136673251906232471180372801739215723908227692280772924566627505383375006926077210593619421268920302567443565378008318306375933345023502569729065152853271943677560156660399164048825639676930792905029514886934137991251748566670747175149389790386533381395346848378086126737554383821108448976538368483182588363399173104558509056638462025014631311831087429077292622159430204291594740306101839816855066950261973761508571761199475875722129872053120607918649803615960923395941041186351688548839119185179061511562752936158490008721501922265117853150892510275280451512386037921846921215338292871369243215273327141574788295902601571954853164447945467502858402360002383447905203451080332820138038807089807348326201227952633606773669875783326254859449060219173688677862411205621098369850197290177157801120404586491539351157834995461006366357454485082418882790675313599505192062229760153765297973085881648731173082370598284894044874039320535929359764541655607954724778620299692329561389719894679422187273605123365595211331087787582288795975803204596084790245063851941743126163775104599211024868794963417068620929088930685252348056925998333775103901013166178123051145719327066291671254465121517468025481903583516889717075706778656188008220346836321018130262329960275994035799977740462449521145315883703579044832931500072461734173558055678321534543411700202585608091662941986374015145695722728369219632295111877625307534025947814482046574602884855000628069348113982760168555840795421620575435572915106415375929390228843561207926437055600623679865443824643739469724719459965557955058380348255978396827760847315302517889517186307227611036305093600742622617173630586132915440246954329046162586917746305785076749374879923291817501634840688134655343709975893536074051729094126976575932951568186247471276364688365517570183534172746626073065104511957628663499228486787805910851189856535554349587616640164475880286336297040462890970677362565843002353147494612339120686321466370878446992104275415694109122465685712047172411333784898167640969249816334211768571503116710400681753031921154156119580425706586931272762137106974722260296555246110537155545324997508432752001992143019105053629960070429632978051030666506387862681576587726837451289768507963663710593809112254288358391941211547737599813019216509521401333060709873137329265181692268450634439540567298120315463923249817937804691037934221694952291007930299492375072993250630509428139027930841344730614116433556147640931044259184813639305423693789765205264563476483182726333715121120306292338892864879492097378478618848682608046473195392008403983080088038690495574197562192939221108257663976813610444900247209483403267967688376213967440757138872928630798218493143438797780887379588968409461434159271317578365114578289355818599029235343888888465874521308381377794436361197628390368945957601203165022798579015453447473527069728514545998614229027372911314637820455162254475353567736227936485450357102086445412089842350389087702230398493802147348096874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Fibonacci for number placed 100000 finished in 0 seconds!

Pretty amazing stuff!!

Finally getting the hang of Tail Call Optimization! Let me know what you think, and what other real world solutions would you use it in!


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About Michael Nunez

Born and raised in the Bronx, New York, Michael serves Stride as a Senior Consultant with over 6 years in software development and information technology. He has worked in a range of industries, developing applications in both Java and Ruby and using frameworks such as Rails, and React. A fun fact about Michael is that he went to LaGuardia High School for Music -- He once played the trumpet very well, but had to put it down when he got braces. Since then, he hasn't picked it up. He can still read music though!

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