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Remarkable results with psyco and sieve of Eratosthenes

 
 
Steve Bergman
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      11-29-2006
Just wanted to report a delightful little surprise while experimenting
with psyco.
The program below performs astonoshingly well with psyco.

It finds all the prime numbers < 10,000,000

Processor is AMD64 4000+ running 32 bit.

Non psyco'd python version takes 94 seconds.

psyco'd version takes 9.6 seconds.

But here is the kicker. The very same algorithm written up in C and
compiled with gcc -O3, takes 4.5 seconds. Python is runng half as fast
as optimized C in this test!

Made my day, and I wanted to share my discovery.

BTW, can this code be made any more efficient?


============

#!/usr/bin/python -OO
import math
import sys
import psyco

psyco.full()

def primes():
primes=[3]
for x in xrange(5,10000000,2):
maxfact = int(math.sqrt(x))
flag=True
for y in primes:
if y > maxfact:
break
if x%y == 0:
flag=False
break
if flag == True:
primes.append(x)
primes()

 
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Will McGugan
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      11-29-2006

> #!/usr/bin/python -OO
> import math
> import sys
> import psyco
>
> psyco.full()
>
> def primes():
> primes=[3]
> for x in xrange(5,10000000,2):
> maxfact = int(math.sqrt(x))
> flag=True
> for y in primes:
> if y > maxfact:
> break
> if x%y == 0:
> flag=False
> break
> if flag == True:
> primes.append(x)
> primes()
>


Some trivial optimizations. Give this a whirl.

def primes():
sqrt=math.sqrt
primes=[3]
for x in xrange(5,10000000,2):
maxfact = int(sqrt(x))
for y in primes:
if y > maxfact:
primes.append(x)
break
if not x%y:
break
return primes

--
blog: http://www.willmcgugan.com
 
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Will McGugan
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      11-29-2006
Steve Bergman wrote:
> Just wanted to report a delightful little surprise while experimenting
> with psyco.
> The program below performs astonoshingly well with psyco.
>
> It finds all the prime numbers < 10,000,000


Actualy, it doesn't. You forgot 1 and 2.


Will McGugan
--
blog: http://www.willmcgugan.com
 
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Beliavsky
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      11-29-2006
Will McGugan wrote:
> Steve Bergman wrote:
> > Just wanted to report a delightful little surprise while experimenting
> > with psyco.
> > The program below performs astonoshingly well with psyco.
> >
> > It finds all the prime numbers < 10,000,000

>
> Actualy, it doesn't. You forgot 1 and 2.


The number 1 is not generally considered to be a prime number -- see
http://mathworld.wolfram.com/PrimeNumber.html .

 
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Will McGugan
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      11-29-2006
Beliavsky wrote:

>
> The number 1 is not generally considered to be a prime number -- see
> http://mathworld.wolfram.com/PrimeNumber.html .
>


I stand corrected.

--
blog: http://www.willmcgugan.com
 
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dickinsm@gmail.com
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      11-29-2006
> BTW, can this code be made any more efficient?

I'm not sure, but the following code takes around 6 seconds on my
1.2Ghz iBook. How does it run on your machine?

def smallPrimes(n):
"""Given an integer n, compute a list of the primes < n"""

if n <= 2:
return []

sieve = range(3, n, 2)
top = len(sieve)
for si in sieve:
if si:
bottom = (si*si - 3)//2
if bottom >= top:
break
sieve[bottom::si] = [0] * -((bottom-top)//si)

return [2]+filter(None, sieve)

smallPrimes(10**7)




>
> ============
>
> #!/usr/bin/python -OO
> import math
> import sys
> import psyco
>
> psyco.full()
>
> def primes():
> primes=[3]
> for x in xrange(5,10000000,2):
> maxfact = int(math.sqrt(x))
> flag=True
> for y in primes:
> if y > maxfact:
> break
> if x%y == 0:
> flag=False
> break
> if flag == True:
> primes.append(x)
> primes()


 
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Steve Bergman
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      11-29-2006

Will McGugan wrote:

> Some trivial optimizations. Give this a whirl.



I retimed and got 9.7 average for 3 runs on my version.

Yours got it down to 9.2.

5% improvement. Not bad.

(Inserting '2' at the beginning doesn't seem to impact performance
much. )

BTW, strictly speaking, shouldn't I be adding something to the floating
point sqrt result, before converting to int, to allow for rounding
error? If it is supposed to be 367 but comes in at 366.99999999, don't
I potentially classify a composite as a prime?

How much needs to be added?

 
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Steve Bergman
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      11-29-2006

http://www.velocityreviews.com/forums/(E-Mail Removed) wrote:
> > BTW, can this code be made any more efficient?

>
> I'm not sure, but the following code takes around 6 seconds on my
> 1.2Ghz iBook. How does it run on your machine?
>
>


Hmm. Come to think of it, my algorithm isn't the sieve.

Anyway, this is indeed fast as long as you have enough memory to handle
it for the range supplied.

It comes in at 1.185 seconds average on this box.

Come to think of it, there is a supposedly highly optimized version of
the sieve in The Python Cookbook that I've never bothered to actually
try out. Hmmm...

 
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dickinsm@gmail.com
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      11-30-2006


On Nov 29, 6:59 pm, "Steve Bergman" <(E-Mail Removed)> wrote:
> (E-Mail Removed) wrote:
> > > BTW, can this code be made any more efficient?

>
> > I'm not sure, but the following code takes around 6 seconds on my
> > 1.2Ghz iBook. How does it run on your machine?

>
> Hmm. Come to think of it, my algorithm isn't the sieve.


Right. I guess the point of the sieve is that you don't have to spend
any time
finding that a given odd integer is not divisible by a given prime;
all the
multiplies are done up front, so you save all the operations
corresponding to
the case when x % y != 0 in your code. Or something.

> Anyway, this is indeed fast as long as you have enough memory to handle
> it for the range supplied.


The sieve can be segmented, so that the intermediate space requirement
for
computing the primes up to n is O(sqrt(n)). (Of course you'll still
need
O(n/log n) space to store the eventual list of primes.) Then there
are all sorts
of bells and whistles (not to mention wheels) that you can add to
improve the
running time, most of which would considerably complicate the code.

The book by Crandall and Pomerance (Primes: A Computational
Perspective)
goes into plenty of detail on all of this.

Mark Dickinson

 
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Steven D'Aprano
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      11-30-2006
On Wed, 29 Nov 2006 15:35:39 -0800, Steve Bergman wrote:

> BTW, strictly speaking, shouldn't I be adding something to the floating
> point sqrt result, before converting to int, to allow for rounding
> error?


If you don't mind doing no more than one unnecessary test per candidate,
you can just add one to maxfact to allow for that. Or use round()
rather than int(). Or don't convert it at all, just say:

maxfact = math.sqrt(x)

and compare directly to that.


> If it is supposed to be 367 but comes in at 366.99999999, don't
> I potentially classify a composite as a prime?


Do you fear the math.sqrt() function is buggy? If so, all bets are off


> How much needs to be added?


No more than 1, and even that might lead you to sometimes performing an
unnecessary test.



--
Steven.

 
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