# A Dream to good to be true

A few years back I was doing mostly computational geometry for a while. In that field, floating point errors are often of great concern. Some algorithms will simply crash or fail when it’s not taken into account. Back then, the idea of doing all the required math using rationals seemed very alluring.

For the uninitiated: a good rational type based on two integers, a numerator and a denominator allows you to perform the basic math operations of addition, subtraction, multiplication and division without any loss of precision. Doing all the math without any loss of precision, without fuzzy comparisons, without imperfection.

Alas, I didn’t have a good rational type available at the time, so the thought remained in the realm of ideas.

# A Dream come true?

Fast forward a couple of years to just two months ago. We were starting a new project and set ourselves the requirement of not introducing floating point errors. Naturally, I immediately thought of using rationals. That project is written in java and already using jscience, which happens to have a nice Rational type. I expected the whole thing to be a bit slower than math using build-in types. But not like this.

It seemed like a part that was averaging about 2000 “count rate” rationals was extremely slow. It seemed to take about 13 seconds, which we thought was way too much. Curiously, the problem never appeared when the count rate was zero. Knowing a little about the internal workings of rational, I quickly suspected the summation to be the culprit. But the code was doing a few other things to, so naturally my colleagues demanded proof that that was indeed the problem. Hence I wrote a small demo application to benchmark the problem.

The code that I measured was this:

Rational sum = Rational.ZERO; for (final Rational each : list) { sum = sum.plus(each); } return sum;

Of course I needed some test data, that I generated like this:

final List<Rational> list = new ArrayList<>(); for (int i=0; i<2000; ++i) { list.add(Rational.valueOf(i, 100)); } return list;

This took about 10ms. Bad, but not 13s catastrophic.

Now from using rational numbers in school, we remember that summing up numbers with equal denominators is actually quite easy. You just leave the denominator as is and add the two numerators. But what if the denominators are different? We need to find a common multiple of the two denominators before we can add. Usually we want the smallest such number, which is called the lowest common multiple (lcm). This is so that the numbers don’t just explode, i.e. get larger and larger with each addition. The algorithm to find this is to just multiply the two numbers and divide by their greatest common divisor (gcd). Whenever I held the debugger during my performance problems, I’d see the thread in a function called gcd. The standard algorithm to determine the gcd is the Euclidean Algorithm. I’m not sure if jscience uses it, but I suspect it does. Either way, it successively reduces the problem via a division to a smaller instance.

# What does this all mean?

This means that much of the complexity involved happens only when there’s variation in the denominator. Looking at my actual data, I saw that this was the case for our problem. The numbers were actually close to one, but with the numerator and the denominator each close to about 4 million. This happened because the counts that we based this data on where “normalized” by a time value that was close, but not equal to one. So let’s try another input sequence:

final Random randomGenerator = new Random(); final List<Rational> list = new ArrayList<>(); for (int i=0; i<2000; ++i) { list.add(Rational.valueOf(4000000, 4000000 + randomGenerator.nextInt(2000))); } return list;

That already takes 10 seconds. Wow. Here’s the rational number it produced:

10925412090387826443356493315570970692092187751160448231723307006165619476894539076955047153673774291312083131288405937659569868481689058438177131164590263653191192222748118270107450103646129518975243330010671567374030224485257527751326361244902048868408897125206116588469496776748796898821150386049548562755608855373511995533133184126260366718312701383461146530915166140229600694518624868861494033238710785848962470254060147575168902699542775933072824986578365798070786548027487694989976141979982648898126987958081197552914965070718213744773755869969060335112209538431594197330895961595759802183116845875558763156976148877035872955887060171489805289872602037240200456259054999832744341644730504414071499368916837445036074038038173507351044919790626190261603929855676606292397018669058312742095714513746321779160111694908027299104638873374887446030780299702571350702255334922413606738293755688345624599836921568658488773148103958376515598663301740183540590772327963247869780883754669368812549202207109506869618137936835948373483789482539362351437914427056800252076700923528652746231774096814984445889899312297224641143778818898785578577803614153163690077765243456672395185549445788345311588933624794815847867376081561699024148931189645066379838249345071569138582032485393376417849961802417752153599079098811674679320452369506913889063163196412025628880049939111987749980405089109506513898205693912239150357818383975619592689319025227977609104339564104111365559856023347326907967378614602690952506049069808017773270860885025279401943711778677651095917727518548067748519579391709794743138675921116461404265591335091759686389002112580445715713768865942326646771624461518371508718346301286775279265940739820780922411618115665915206028180761758701198283575402598963356532479352810604578392844754856057089349811569436655814012237637615544417676166890247526742765145909088354349593431829508073735508662766171346365854920116894738553593715805698326801840647472004571022201012455368883190600587502030947401749733901881425019359516340993849314997522931836068574283213181677667615770392454157899894789963788314779707393082602321025304730355204512687710695657016587562258289968709342507303760359107314805479150337790244385189611378805094282650120553138575380568150214510972734241803176908917697662914714188030879994734853772797322420241420911735874903926141598416992690859929943631826094723456317312589265104334870907579391696178556354299428366394819280011410287891113591176612795009226826412471238783334239148961082442565804292473501012401378940718084589859443350905260282342990350362981901637062679381912861429756544396701574099199222399937752826106312708211791773562169940745686837853342547182813438086856565980815543626740277913678365142830117575847966404149038892476111835346566933160119385992791677587359063277202990220629004309670865867774206252830200897207368966439730136012024728717701204793182480513620275549665094200202565592742030772102704751736850897665353297536494739059325582661212315355306787427752670613324951121097833683795311514392922347268374097451268196257308005629903372871471809591087849716533132440301432155867780938535327925645340832637372702171777123816397448399703780105396941226655424025197472384099218081468916864256472238808237005121132164363385877692234230678011184351921814453560033879491735351402997266882544304106997065987376103362395437737475217181551336569975031721614790499945872209261769951117223344186839969922893394319287462384028859822057955389124951467203432571737865201780344423642467187208636881135573636815083891626138564337634176587161231028307776960866522346008589607259041199676560090157817882260300414906572885890188984036234226505815367029839231023461597364977306399898603903392434756572392816540125771578189640871020070756539777101197151773304409519870643142190955018579914630314940373832858007535828153361236115553577694543503842444481599944319287815162136101362705211937257383677282014480487759786222801447548899760241829116959865698836386442016721709983097509675552750221989521551169512674725876581185837611167980363615880958917421251873901289922888492447507837290336628975165062036681599909052030653421736716061426079882106810502703095803882805916960831442634085856041781093664688754713907512226706324967656091109936101526173370212867073380662492009726657437921033740063367290862521594119329592938626114166263957511012256023777676569002181500977475083845756500926631153405264250959378833667206532373995888322137324027620266863005721216133252342921697663864807284554205674829658250755046340838031118227643145562001361542532622713886266813492926885236832665609571019479812713355021295737820773552735161701716018010606040731647943600206193923458996150345093898644748170519757957470535978378479854546255651200511536560142431948781377187548218601919108870420102025378751015728281345799655926856602543107729659372984539588835599345223921737022220676709028150797109091782506736145801340069563865839397272145141831011878720095142353543406658905222847479419799336972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216266239656298115575018443700688208844387102582445812545858014862427316785193110884751319467750425511273629581416588296942433219322216784262821066075700266485645060935996743388485096744598169509920624971167075214788838073469621687619355816573640360338756385397428237445511332615921133308459043700086925442114337299109227541364012352140312577797531234832237279430947774637533319353713938562360646441591033255036

I kid you not, that’s over 10000 digits! In the editor I’m writing this in, that’s roughly 3 pages. No wonder it took that long. Let’s use even more variation in the numbers:

final Random randomGenerator = new Random(); final List<Rational> list = new ArrayList<>(); for (int i=0; i<2000; ++i) { list.add(Rational.valueOf(4000000 + randomGenerator.nextInt(5000), 4000000 + randomGenerator.nextInt(20000))); }

Now that already takes 16 s, with about 14000 digits. Oh boy. Now the maximum number of values I expected to do this averaging for was about 4000, so let’s scale that up:

final Random randomGenerator = new Random(); final List<Rational> list = new ArrayList<>(); for (int i=0; i<4000; ++i) { list.add(Rational.valueOf(4000000 + randomGenerator.nextInt(5000), 4000000 + randomGenerator.nextInt(20000))); } return list;

That took 77 seconds! More than 4 times as long as for half the amount of data. The resulting number has over 26000 digits. Obviously, this scales way worse than linear.

# An Explanation

By now it was pretty clear what was happening: The ever so slightly not-1 values were causing an “explosion” in the denominator after all. When two denominators are coprime, i.e. their greatest common divisor is 1, the length of the denominators just adds up. The effect also happens when the gcd is very small, such as 2 or 3. This can happen quite a lot with huge numbers in a sufficiently large range. So when things go bad for your input data, the length of the denominator just keeps growing linearly with the number of input values, making each successive addition slower and slower. Your rationals just exploded.

# Conclusion

After this, it became apparent that using rationals was not a great idea after all. You should be very careful when doing series of additions with them. Ironically, we were throwing away all the precision anyways before presenting the number to a user. There’s no way for anyone to grok a 26000 digit number anyways, especially if the result is basically 4000.xx. I learned my lesson and buried the dream of perfect arithmetic. I’m now using fixed point arithmetic instead.