# Why can adding bracket make an arithmetic expression much faster?

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### #1 Nallo

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# Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 08:27 AM

I looked at the two following expressions. And I don't understand why adding brackets make the second one 10000 times faster. Anyone an idea?

```>>> import timeit
>>> expr1 = "3 ** 10 ** 5"
>>> expr2 = "(3 ** 10) ** 5"
>>> timer1 = timeit.Timer("3 ** 10 ** 5")
>>> timer2 = timeit.Timer("(3 ** 10) ** 5")
>>> time1 = timer1.timeit(1)
>>> time2 = timer2.timeit(2)
>>> print time1
0.0190041065216
>>> print time2
1.90734863281e-06
>>> print time1 / time2
9963.625

```

This post has been edited by atraub: 03 August 2012 - 11:16 AM

Is This A Good Question/Topic? 1

## Replies To: Why can adding bracket make an arithmetic expression much faster?

### #2 sepp2k

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 08:54 AM

Because the exponentiation operator is right-associativity. So a ** b ** c is the same as a ** (b ** c), not (a ** b) ** c.

If you actually print the results of the expressions in your sample code, you'll see that expr1 evaluates to a much larger number than expr2, so of course it takes longer to evaluate.

### #3 Nallo

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 09:32 AM

Oops, thanks sepp2k. Completely forgot that ** is right associative. Now I feel really dumb.

### #4 jon.kiparsky

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 09:39 AM

Nallo, on 03 August 2012 - 11:32 AM, said:

Now I feel really dumb.

Which is weird, because really, you just got a bit smarter.

### #5 atraub

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 11:10 AM

sepp2k, on 03 August 2012 - 11:54 AM, said:

Because the exponentiation operator is right-associativity. So a ** b ** c is the same as a ** ( b ** c), not (a ** ** c.

If you actually print the results of the expressions in your sample code, you'll see that expr1 evaluates to a much larger number than expr2, so of course it takes longer to evaluate.

I have to admit, this one would have fooled me too! What a great demonstration of operator precedence working in a correct, yet counter intuitive, way!

EDIT:
I really like this post. Not only for what it shows about the operators, but also as a mini demonstration of the timeit module. FEATURED!

This post has been edited by atraub: 03 August 2012 - 11:18 AM

### #6 Dogstopper

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 11:14 AM

Excellent! That makes so much sense.
I guess it's like abc, where b^c is executed then before a is raised to that power.

This, on the other hand, is a much larger number:
(ab)c

### #7 Nallo

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 12:01 PM

atraub, you are not trying to hook me up on writing a mini tutorial on the timeit module, do you?

• MrCupOfT

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 12:57 PM

Nallo why not write that tutorial and earn yourself some kudos (50pts)

### #9 atraub

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 03 August 2012 - 01:26 PM

I'm with Speight, there are no tutorials on the timeit module. Until you mentioned it recently, I didn't even know it existed! If you don't write one soon, I'm going to snake it and keep all those delicious Kudos for myself.

This post has been edited by atraub: 03 August 2012 - 01:27 PM

### #10 ziggyfish

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 05 August 2012 - 01:18 AM

Nallo, on 03 August 2012 - 08:27 AM, said:

I looked at the two following expressions. And I don't understand why adding brackets make the second one 10000 times faster. Anyone an idea?

```>>> import timeit
>>> expr1 = "3 ** 10 ** 5"
>>> expr2 = "(3 ** 10) ** 5"
>>> timer1 = timeit.Timer("3 ** 10 ** 5")
>>> timer2 = timeit.Timer("(3 ** 10) ** 5")
>>> time1 = timer1.timeit(1)
>>> time2 = timer2.timeit(2)
>>> print time1
0.0190041065216
>>> print time2
1.90734863281e-06
>>> print time1 / time2
9963.625

```

This is because ** is not constant time operator, depending on the implementation it will take O(n) to complete.

So we all know how to expand a exponent, for example 10 ** 5 = 10 * 10 * 10 * 10 * 10 = 100000. So this in this case n=5, now if we take the first part 3 ** 100000 = 3 * 3 * 3 * ... = 1.334971414×10^47712. So this in this part n = 100000. So the total time taken will be 100000+5 = 100005

The second formula forces the first part first, so 3 ** 10 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 59049. So in this case n = 10. So we now take the second part 59049 ** 5 = 59049 ** 59049 ** 59049 ** 59049 ** 59049 = 7.178979877×10^23.

Thanks

Brendan

### #11 atraub

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 06 August 2012 - 08:24 AM

Everyone understands why it takes longer to compute 3**(10**5) than (3**10)**5 lol. The question was more revolved around the parsing of the problem in the first place. Hell, by virtue of the fact that one problem need only allocate enough memory to store
```717897987691852588770249
```

where as the other would need to allocate enough room for
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10651302278135270887403087321530046460312357988524
47012260817864082180431402136059248549489112477275
66289913273441809238040802387058773562765141503378
16432268184999321673894277627015958098281315365832
92971242307846759150927159012701864028118631218801
28610103755943887061752742445724616537396077303195
93639094452907668815118912039668443580791938645373
88945457653627280378482553149854828483809325751596
07276463328760306668403558229494845328177547545133
75766541635663236893532169753493257682034426160136
76145534449112910305562227378744505167776520163901
02436811661314789440197945030580604606481813964231
72896717835446897094354737305722548000966457425066
9865522000001
```

would be a source of slowdown on its own.

EDIT:
That's 47,713 digits

This post has been edited by atraub: 06 August 2012 - 09:26 AM
Reason for edit:: WHEW, FINALLY GOT THE NUMBER DISPLAYING NICELY!!

### #12 ziggyfish

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## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 06 August 2012 - 09:29 AM

atraub, on 06 August 2012 - 08:24 AM, said:

Everyone understands why it takes longer to compute 3**(10**5) than (3**10)**5 lol. The question was more revolved around the parsing of the problem in the first place. Hell, by virtue of the fact that one problem need only allocate enough memory to store

Often there are a lot of contributing factors to performance issues. So rather than be rude and suggest that was the only problem, I wanted to give an alternative answer.

Anyway doing this on a core i7.

I get different results to the OP:
```import timeit
def preventOptimizations( x, y, z ):
return x ** y ** z;
def preventOptimizationsWithBracket( x, y, z ):
return (x ** y) ** z;
timer1 = timeit.Timer("preventOptimizations(0.8, 100, 100)", "from __main__ import preventOptimizations")
timer1 = timeit.Timer("preventOptimizationsWithBracket(0.8, 100, 100)", "from __main__ import preventOptimizationsWithBracket")
timer2 = timeit.Timer("1 ** (3 ** 10) ** 5")
time1 = timer1.timeit(1)
time2 = timer2.timeit(2)
print "preventOptimizations(0.8, 100, 100) = ", preventOptimizations(0.8, 100, 100)
print "time preventOptimizations(0.8, 100, 100)", time1
print "time preventOptimizationsWithBracket(0.8, 100, 100)", time2
print time1 / time2
print "============================================================="
timer1 = timeit.Timer("preventOptimizations(3, 10, 5)", "from __main__ import preventOptimizations")
timer1 = timeit.Timer("preventOptimizationsWithBracket(3, 10, 5)", "from __main__ import preventOptimizationsWithBracket")
time1 = timer1.timeit(1)
time2 = timer2.timeit(2)

print "preventOptimizations(3, 10, 5) = ", preventOptimizations(3, 10, 5)
print "time preventOptimizations(3, 10, 5)", time1
print "time preventOptimizationsWithBracket(3, 10, 5)", time2
print time1 / time2

```

Outputs:

```preventOptimizations(0.8, 100, 100) =  0.0
time preventOptimizations(0.8, 100, 100) 2.19345092773e-05
time preventOptimizationsWithBracket(0.8, 100, 100) 1.19209289551e-05
1.84
=============================================================
preventOptimizations(3, 10, 5) =  13349714142304014694589143904897822922452485076062773553456279939841274146215896869522305980255798233239149889425419590524332428873537083020782279802984670783174192973884875772106202489810965195750670540017729104653253124906754225088733426295710089412993001722330331661295444568362994984525794865040154552147363813207209965356768946943602282911944326154630492113002579642323048858295434770136138180753150615828831261940501650012085725391052575622010755977649358077483459506643914763503008295365339548656044817120009527050408028288030864446330435819349233745903249195549325378269055546256255906921806226622155440527634094668066864913662376758390987340208907851764726460451616058172047554721044836608196713744799182264426519285296998864927038780736553082694994579374849302380335292984474443305592397659315237841028937215223851003498757650130412927359617600578030416190379629830290362486696343529892391498918169043371620434554192522038249180947265608151009858387951861581947756044043981510670879539942910694256596626795929246341714202829694934465304337015784361135937440940664714154737267519782717648220624437729749531136313911718284652684739601110025664196472691584052902613335044057215287732505585026553121689863435648710646543348275130053326487223367985783551212746606106199891326833236919894249821574280609481988804468986557501778462928403152880725103465924306130045709815180283855532895934988218337241145274956916036957147433610106567088144668073515362596624861696341133128835679623951071274725034116042262749977206507262437895238456653520219653531978179499712629802187572575395668729737618555635727168853113989057965388330448277178948303138842135748100850985659359217667819174634578587692306462315845540888641745296842571424109732223649817540134839891873620127918668509134869736194958920383674812649484543522305782210030259264680757572757819426071718154393548287040335918074923307771615303400117935893619244758582724650083947293717152256662390112441619439508345839860146997174686693257680063934668354311100491354219636350945161847892966360191444321041581453853452923897050987659679889616437760976850921718682205364271288156051226089311343949176457285645271120756498992435885098961897778719822624908417733714563282280705117960125913614359871691670603977668478901681124754489210554082390746894419702864471352116539332018940148495574571759288871693950249476997814208572794165892615358025892272169479945101558169714879671102845757093354494145314661487827312807482491775681483079035986821162264588779661180105819077546376651029290771199711062697429534178304613634429363432074053861168972297326077166299627902905946932587202905279625157429485419086296159203877027132844055322219126395805057934699127235739861503038989250127304616711372778706981341392639654455238652201759721823876118739158659966206241353921351891864551357833885049339610031984466052086707447491444300749100900595376599498429978267285770396497730695448276118667756543639772410681399983929270452748169361509813391108129985672725008227765388521363898370115108211658702915208809959767686699918728617436653531520802216039774386262115948415561569610180934727841358866792048475047369735775727018812749016750931462912955125594990141946585753506471581028434134030334253795501120437081554489348281267964300340702546149261758433349553443616586778838250466294453904884775564679059806343071479179073240102312163538900058615581472276172865913602757323014454633136379106944743507849778763582887527437449170717108602424789427423435553801671600206695745150428043471456453112384569120096903178329882927413119995868243277991757712924788257773771686400347046399412661923709542272363661203265890314172863777680215765908406658926806646326817842515365371167556944030615284341857951862721561578231817486905498217743895367054995929228193579563481630718696004411487869728307585432187923207127600357354878480714218752438508012798453491057359195756540776837477244064221930590143121983577298182656237987162764092981528923429253386877359962569294501631045721511755432506942787120817891759234131779070559106034198794187251387824661921949042262945957068396059323374312334946294356669650495836863646061179558326934103380974998937492264627350053185617549472518215461705602125114147952657770963143594638938255348900250723484000556441997983976116369585895825630676154243140915938352423587621070954427793165042162893765245650686307707212763130705657093765605944832005356656517514369489808699210259077860781370469328465898127056123546044871844886852604902001821067807846544727699328119684922325802937704865234083191521371703740711123270315703275144874305194888994126792022738384085792232919851738923726460019795291895781083785682532241066019705320752114796228872683259890669093806679550398968426724049719994416201025870023890556670708103167866014585829415703593737083155257804012414300702488730288149452569380882923587421428258970340997214682459415086807761182313990330758232299757829957865736348493038237318819914908472697993369543135217780612641552478562753911332170838692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time preventOptimizations(3, 10, 5) 3.09944152832e-06
time preventOptimizationsWithBracket(3, 10, 5) 1.09672546387e-05
0.282608695652

```

This post has been edited by atraub: 06 August 2012 - 09:31 AM
Reason for edit:: fixed code tag

### #13 atraub

• Pythoneer

Reputation: 820
• Posts: 2,197
• Joined: 23-December 08

## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 06 August 2012 - 09:35 AM

You'll have to help me out here... what exactly are you trying to prove?

This post has been edited by atraub: 06 August 2012 - 09:44 AM

### #14 ziggyfish

Reputation: 0
• Posts: 4
• Joined: 05-August 12

## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 06 August 2012 - 09:54 AM

atraub, on 06 August 2012 - 09:35 AM, said:

You'll have to help me out here... what exactly are you trying to prove?

Not trying to prove anything just suggesting a different answer, and I don't think "Everyone understands why it takes longer ". When you reply to my post can you actually read it?

If "Everyone understands why it takes longer ", then the original poster wouldn't have actually posted the question, right? I am sure there are others on this site that don't really understand how x ** y is actually evaluated by python.

Why does it matter that I posted a different answer to what had already been given, is this really how this community normally behaves to new members???????????????????????????????

### #15 atraub

• Pythoneer

Reputation: 820
• Posts: 2,197
• Joined: 23-December 08

## Re: Why can adding bracket make an arithmetic expression much faster?

Posted 06 August 2012 - 10:41 AM

Quote

Why does it matter that I posted a different answer to what had already been given, is this really how this community normally behaves to new members???????????????????????????????
Don't be one of those members. No one is being rude; I was being blunt. No one has insulted you; you've taken offense to what I've said. So, yeah, this is no different than I'd treat any member regardless of level of experience.

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If "Everyone understands why it takes longer ", then the original poster wouldn't have actually posted the question, right?
You clearly don't know Nallo. He's a far better programmer than most. I'm a professional in my field and assisted teaching Python at the collegiate level for 2 years, but even I have the humility to admit that I'm years behind him.

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I am sure there are others on this site that don't really understand how x ** y is actually evaluated by python.
This is the advanced discussion forum. A general understanding of all of python's operators is taken for granted here.

I think you misunderstand why the question was asked, so let me explain. Nallo had noticed that the parenthesis appeared to make the expression compute faster. He had a (rare) D'oh! moment and didn't realize that Python's interpreter was parsing the exponents in a correct but counter-intuitive way. He mentioned it on here and sepp2k alerted him to his goof up.

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let me also take this moment to point out that no one on this forum has any sort of "beef" with you. I believe I can safely say that no one here thinks lowly of you or thinks you're not smart. I chalked it up to you misunderstanding the initial problem and excitedly responding without reading the rest of the thread... it's understandable; we've all done it. I sincerely hope that you do contribute to DIC because it's always nice to have new faces in the Python forum. Try not to let this rocky start sour your opinion of DIC.

This post has been edited by atraub: 06 August 2012 - 02:29 PM