# Odd Performance

Garner, Robin Robin.Garner@crsrehab.gov.au
Wed, 30 Oct 2002 11:55:51 +1100

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You may also like to compare this definition.

> import List
>
> primes = unfoldr sieve [2..]
> sieve (x:xs) = Just (x,filter (x `notDivisor`) xs)
> x `notDivisor` y = y `mod` x /= 0

Slower, but (IMO) cuter.

Cheers,

-----Original Message-----
From: Tim Otten [mailto:tim@cap.american.edu]
Sent: Wednesday, 23 October 2002 15:10
Cc: scasey@cap.american.edu
Subject: Re: Odd Performance

Tom Pledger writes:

> Probably.  Try replacing this
>    (\z -> z <= (intsqrt x))
> with this
>    (\z -> z^2 <= x)

Yes! This is significantly nicer. Taking 4000 primes, this is about twice
as fast as the original (loose) algorithm, and it appears that it gets
better as n grows. (Call this version #3).

> or moving the O(sqrt(n)) step[1] into a let expression

This does help -- the implementation with this adjustment seems to hover
around 1.1 times the speed of the loose algorithm. (Call this version #4)

> Only if the predicate function (the p in takeWhile p xs) is
> significantly more expensive than a constant-cost piece of arithmetic
> and pattern-matching.

This makes sense and explains the results above: squaring z is easier than
square-rooting x. In fact, the way intsqrt is written, #4's takeWhile
indirectly tests (z*z > x) once for each [z | 1<=z<=sqrt(x)]. #3's
takeWhile expression will only evaluate (z^2 < x) once for each [ z | z <-
primes, z<=sqrt(x) ]. Moreover, #4's takeWhile will do additional tests (z
<= sqrt) for each [z | z <- primes, z <= sqrt(x)]. In retrospect, the
intsqrt-approach seems rather silly. :)

Thank you,
Tim

_______________________________________________

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<TITLE>RE: Odd Performance</TITLE>
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<P><FONT SIZE=3D2>You may also like to compare this definition.</FONT>
</P>

<P><FONT SIZE=3D2>&gt; import List</FONT>
<BR><FONT SIZE=3D2>&gt; </FONT>
<BR><FONT SIZE=3D2>&gt; primes =3D unfoldr sieve [2..]</FONT>
<BR><FONT SIZE=3D2>&gt; sieve (x:xs) =3D Just (x,filter (x =
`notDivisor`) xs)</FONT>
<BR><FONT SIZE=3D2>&gt; x `notDivisor` y =3D y `mod` x /=3D 0</FONT>
</P>

<P><FONT SIZE=3D2>Slower, but (IMO) cuter.</FONT>
</P>

<P><FONT SIZE=3D2>Cheers,</FONT>
</P>

<P><FONT SIZE=3D2>-----Original Message-----</FONT>
<BR><FONT SIZE=3D2>From: Tim Otten [<A =
HREF=3D"mailto:tim@cap.american.edu">mailto:tim@cap.american.edu</A>]</F=
ONT>
<BR><FONT SIZE=3D2>Sent: Wednesday, 23 October 2002 15:10</FONT>
<BR><FONT SIZE=3D2>Cc: scasey@cap.american.edu</FONT>
<BR><FONT SIZE=3D2>Subject: Re: Odd Performance</FONT>
</P>
<BR>

<P><FONT SIZE=3D2>Tom Pledger writes:</FONT>
</P>

<P><FONT SIZE=3D2>&gt; Probably.&nbsp; Try replacing this</FONT>
<BR><FONT SIZE=3D2>&gt;&nbsp;&nbsp;&nbsp; (\z -&gt; z &lt;=3D (intsqrt =
x))</FONT>
<BR><FONT SIZE=3D2>&gt; with this</FONT>
<BR><FONT SIZE=3D2>&gt;&nbsp;&nbsp;&nbsp; (\z -&gt; z^2 &lt;=3D =
x)</FONT>
</P>

<P><FONT SIZE=3D2>Yes! This is significantly nicer. Taking 4000 primes, =
<BR><FONT SIZE=3D2>as fast as the original (loose) algorithm, and it =
appears that it gets</FONT>
<BR><FONT SIZE=3D2>better as n grows. (Call this version #3).</FONT>
</P>

<P><FONT SIZE=3D2>&gt; or moving the O(sqrt(n)) step[1] into a let =
expression</FONT>
</P>

<P><FONT SIZE=3D2>This does help -- the implementation with this =
<BR><FONT SIZE=3D2>around 1.1 times the speed of the loose algorithm. =
(Call this version #4)</FONT>
</P>

<P><FONT SIZE=3D2>&gt; Only if the predicate function (the p in =
takeWhile p xs) is</FONT>
<BR><FONT SIZE=3D2>&gt; significantly more expensive than a =
constant-cost piece of arithmetic</FONT>
<BR><FONT SIZE=3D2>&gt; and pattern-matching.</FONT>
</P>

<P><FONT SIZE=3D2>This makes sense and explains the results above: =
squaring z is easier than</FONT>
<BR><FONT SIZE=3D2>square-rooting x. In fact, the way intsqrt is =
written, #4's takeWhile</FONT>
<BR><FONT SIZE=3D2>indirectly tests (z*z &gt; x) once for each [z | =
1&lt;=3Dz&lt;=3Dsqrt(x)]. #3's</FONT>
<BR><FONT SIZE=3D2>takeWhile expression will only evaluate (z^2 &lt; x) =
once for each [ z | z &lt;-</FONT>
<BR><FONT SIZE=3D2>primes, z&lt;=3Dsqrt(x) ]. Moreover, #4's takeWhile =
<BR><FONT SIZE=3D2>&lt;=3D sqrt) for each [z | z &lt;- primes, z =
&lt;=3D sqrt(x)]. In retrospect, the</FONT>
<BR><FONT SIZE=3D2>intsqrt-approach seems rather silly. :)</FONT>
</P>

<P><FONT SIZE=3D2>Thank you,</FONT>
<BR><FONT SIZE=3D2>Tim</FONT>
</P>
<BR>
<BR>

<P><FONT =
SIZE=3D2>_______________________________________________</FONT>