Code-Golf: Lattice Points inside a Circle

J, 21 19 18

+/@,@(>:|@j./~@i:)

Builds complexes from -x-xj to x+xj and takes magnitude.

Edit: With >:

Edit 2: With hook and monadic ~. Runs a few times slower for some reason, but still 10-ish seconds for f(1000).

3 :'+/,y>:%:+/~*:i:y'

Very brutal: computes sqrt(x²+y²) over the [-n,n] range and counts items ≤n. Still very acceptable times for 1000.

Edit: i:y is quite a bit shorter than y-i.>:+:y. Thanks Jesse Millikan!

f=->r{1+4*eval((0..r).map{|i|"%d"%(r*r-i*i)**0.5}*?+)}

Example:

f[1001]
=> 3147833

t=Time.now;f[1001];Time.now-t
=> 0.003361411