“If I have seen further…”

The erat2 function from the cookbook can be further sped up (by about 20-25%):

erat2a

import itertools as it
def erat2a( ):
    D = {  }
    yield 2
    for q in it.islice(it.count(3), 0, None, 2):
        p = D.pop(q, None)
        if p is None:
            D[q*q] = q
            yield q
        else:
            # old code here:
            # x = p + q
            # while x in D or not (x&1):
            #     x += p
            # changed into:
            x = q + 2*p
            while x in D:
                x += 2*p
            D[x] = p

The not (x&1) check verifies that x is odd. However, since both q and p are odd, by adding 2*p half of the steps are avoided along with the test for oddity.

erat3

If one doesn’t mind a little extra fanciness, erat2 can be sped up by 35-40% with the following changes (NB: needs Python 2.7+ or Python 3+ because of the itertools.compress function):

import itertools as it
def erat3( ):
    D = { 9: 3, 25: 5 }
    yield 2
    yield 3
    yield 5
    MASK= 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0,
    MODULOS= frozenset( (1, 7, 11, 13, 17, 19, 23, 29) )

    for q in it.compress(
            it.islice(it.count(7), 0, None, 2),
            it.cycle(MASK)):
        p = D.pop(q, None)
        if p is None:
            D[q*q] = q
            yield q
        else:
            x = q + 2*p
            while x in D or (x%30) not in MODULOS:
                x += 2*p
            D[x] = p

The erat3 function takes advantage of the fact that all primes (except for 2, 3, 5) modulo 30 result to only eight numbers: the ones included in the MODULOS frozenset. Thus, after yielding the initial three primes, we start from 7 and work only with the candidates.
The candidate filtering uses the itertools.compress function; the “magic” is in the MASK sequence; MASK has 15 elements (there are 15 odd numbers in every 30 numbers, as chosen by the itertools.islice function) with a 1 for every possible candidate, starting from 7. The cycle repeats as specified by the itertools.cycle function.
The introduction of the candidate filtering needs another modification: the or (x%30) not in MODULOS check. The erat2 algorithm processed all odd numbers; now that the erat3 algorithm processes only r30 candidates, we need to make sure that all D.keys() can only be such —false— candidates.

Benchmarks

Results

On an Atom 330 Ubuntu 9.10 server, versions 2.6.4 and 3.1.1+:

$ testit
up to 8192
==== python2 erat2 ====
100 loops, best of 3: 18.6 msec per loop
==== python2 erat2a ====
100 loops, best of 3: 14.5 msec per loop
==== python2 erat3 ====
Traceback (most recent call last):
…
AttributeError: 'module' object has no attribute 'compress'
==== python3 erat2 ====
100 loops, best of 3: 19.2 msec per loop
==== python3 erat2a ====
100 loops, best of 3: 14.1 msec per loop
==== python3 erat3 ====
100 loops, best of 3: 11.7 msec per loop

On an AMD Geode LX Gentoo home server, Python 2.6.5 and 3.1.2:

$ testit
up to 8192
==== python2 erat2 ====
10 loops, best of 3: 104 msec per loop
==== python2 erat2a ====
10 loops, best of 3: 81 msec per loop
==== python2 erat3 ====
Traceback (most recent call last):
…
AttributeError: 'module' object has no attribute 'compress'
==== python3 erat2 ====
10 loops, best of 3: 116 msec per loop
==== python3 erat2a ====
10 loops, best of 3: 82 msec per loop
==== python3 erat3 ====
10 loops, best of 3: 66 msec per loop

Benchmark code

A primegen.py module contains the erat2, erat2a and erat3 functions. Here follows the testing script:

#!/bin/sh
max_num=${1:-8192}
echo up to $max_num
for python_version in python2 python3
do
    for function in erat2 erat2a erat3
    do
        echo "==== $python_version $function ===="
        $python_version -O -m timeit -c \
        -s  "import itertools as it, functools as ft, operator as op, primegen; cmp= ft.partial(op.ge, $max_num)" \
            "next(it.dropwhile(cmp, primegen.$function()))"
    done
done