# Segmented wheel Sieve of Eratosthenes in Python

Here is a new version of a segmented and wheel factorized Sieve of Eratosthenes. It currently uses mod 30 wheel factorization to eliminate multiples of 2, 3, and 5 in the sieve data structure to gain speed. It wraps the wheel with segmentation in order to reduce its memory footprint so it can scale up to N in the billions and beyond. (yeah, I know, Buzz Lightyear)

This is a follow on to an earlier version. Thanks to @GZ0 for comments including warning me about how soon Python 2.7 will go unsupported, and a huge thanks to @QuantumChris for the thorough code review, especially for encouraging me to use OOP for modularity.

I decided to use a class for everything related to the mod 30 wheel. I hope that makes the design more clear, since the wheel and segmentation code is now separate.

The performance degraded by ~1.5%. I think that is fine, since:

• perhaps more people will read it. More eyeballs on any code is a Good Thing in my opinion.
• cProfile output is more helpful because the code is more granular. Woo-hoo! It now shows that cull_one_multiple is the hot spot followed by segmentedSieve.
• it will allow replacing the multiple culling code easily, such as a mod 210 wheel (to also eliminate multiples of 7), with only tiny changes outside of the wheel class. This may make up for the degradation if done carefully.

Please let me know what you think.

#!/usr/bin/python3 -Wall
"""program to find all primes <= n, using a segmented wheel sieve"""

from sys  import argv
from math import log
from time import time

# non standard packages
from bitarray import bitarray

# tuning parameters
CUTOFF      = 1e4           # small for debug
SIEVE_SIZE  = 2 ** 20       # in bytes, tiny (i.e. 1) for debug
CLOCK_SPEED = 1.6           # in GHz, on my i5-6285U laptop

def progress(current, total):
"""Display a progress bar on the terminal."""
size = 60
x = size * current // total
print(f'\rSieving: [{"#" * x}{"." * (size - x)}] {current}/{total}', end="")

def seg_wheel_stats(n):
"""Returns only the stats from the segmented sieve."""
return(segmentedSieve(n, statsOnly=True))

def print_sieve_size(sieve):
print("sieve size:", end=' ')
ss = len(memoryview(sieve))
print(ss//1024, "KB") if ss > 1024 else print(ss, "bytes")

def prime_gen_wrapper(n):
"""
Decide whether to use the segmented sieve or a simpler version.
Stops recursion.
"""
return smallSieve(n + 1) if n < CUTOFF else segmentedSieve(n)
# NB: rwh_primes1 (a.k.a. smallSieve) returns primes < N.
# We need sieving primes <= sqrt(limit), hence the +1

def smallSieve(n):
"""Returns a list of primes less than n."""
# a copy of Robert William Hanks' odds only rwh_primes1
#     used to get sieving primes for smaller ranges
#     from https://stackoverflow.com/a/2068548/11943198
sieve = [True] * (n // 2)
for i in range(3, int(n ** 0.5) + 1, 2):
if sieve[i // 2]:
sieve[i * i // 2::i] = [False] * ((n - i * i - 1) // (2 * i) + 1)
return [2] + [2 * i + 1 for i in range(1, n // 2) if sieve[i]]

class PrimeMultiple:
"""Contains information about sieving primes and their multiples"""
__slots__ = ['prime', 'multiple', 'wheel_index']

def __init__(self, prime):
self.prime = prime

def update(self, multiple, wheel_index):
self.multiple = multiple
self.wheel_index = wheel_index

def update_new_mult(self, multiple, wheel_index, wheel):
self.update(multiple, wheel_index)
wheel.inc_mults_in_use()

class m30_wheel:
"""Contains all methods and data unique to a mod 30 (2, 3, 5) wheel"""
# mod 30 wheel factorization based on a non-segmented version found here
#     https://programmingpraxis.com/2012/01/06/pritchards-wheel-sieve/
#  in a comment by Willy Good

def __init__(self, sqrt):
# mod 30 wheel constant arrays
self.skipped_primes   = [2, 3, 5]      # the wheel skips multiples of these
self.wheel_primes     = [7, 11, 13, 17, 19, 23, 29, 31]
self.wheel_primes_m30 = [7, 11, 13, 17, 19, 23, 29, 1]

self.gaps             = [4,2,4,2,4,6,2,6, 4,2,4,2,4,6,2,6]  # 2 loops for overflow

self.wheel_indices    = [0,0,0,0,1,1,2,2,2,2, 3,3,4,4,4,4,5,5,5,5, 5,5,6,6,7,7,7,7,7,7]
self.round2wheel      = [7,7,0,0,0,0,0,0,1,1, 1,1,2,2,3,3,3,3,4,4, 5,5,5,5,6,6,6,6,6,6]

# get sieving primes recursively,
#   skipping over those eliminated by the wheel
self.mults = [PrimeMultiple(p) for p in prime_gen_wrapper(sqrt)[len(self.skipped_primes):]]
self.mults_in_use = 0

def inc_mults_in_use(self):
self.mults_in_use += 1

def get_skipped_primes(self):
"""Returns tiny primes which this wheel ignores otherwise"""
return self.skipped_primes

def num2ix(self, n):
"""Return the wheel index for n."""
n = n - 7  # adjust for wheel starting at 7 vs. 0
return (n//30 << 3) + self.wheel_indices[n % 30]

def ix2num(self, i):
"""Return the number corresponding wheel index i."""
return 30 * (i >> 3) + self.wheel_primes[i & 7]

def cull_one_multiple(self, sieve, lo_ix, high, pm):
"""Cull one prime multiple from this segment"""
p = pm.prime
wx = pm.wheel_index
mult = pm.multiple - 7     # compensate for wheel starting at 7 vs. 0
p8 = p << 3
for j in range(8):
cull_start = ((mult // 30 << 3)
+ self.wheel_indices[mult % 30] - lo_ix)
sieve[cull_start::p8] = False
mult += p * self.gaps[wx]
wx += 1

# calculate the next multiple of p and its wheel index

# f = next factor of a multiple of p past this segment
f = (high + p - 1)//p
f_m30 = f % 30
# round up to next wheel index to eliminate multiples of 2,3,5
wx = self.round2wheel[f_m30]
# normal multiple of p past this segment
mult = p * (f - f_m30 + self.wheel_primes_m30[wx])
pm.update(mult, wx)         # save multiple and wheel index

def cull_segment(self, sieve, lo_ix, high):
"""Cull all prime multiples from this segment"""
# generate new multiples of sieving primes and wheel indices
#   needed in this segment
for pm in self.mults[self.mults_in_use:]:
p = pm.prime
psq = p * p
if psq > high:
break
pm.update_new_mult(psq, self.num2ix(p) & 7, self)

# sieve the current segment
for pm in self.mults[:self.mults_in_use]:
# iterate over all prime multiples relevant to this segment
if pm.multiple <= high:
self.cull_one_multiple(sieve, lo_ix, high, pm)

def segmentedSieve(limit, statsOnly=False):
"""
Sieves potential prime numbers up to and including limit.

statsOnly (default False) controls the return.
when False, returns a list of primes found.
when True,  returns a count of the primes found.
"""
# segmentation originally based on Kim Walisch's
#   simple C++ example of segmantation found here:
#   https://github.com/kimwalisch/primesieve/wiki/Segmented-sieve-of-Eratosthenes

assert(limit > 6)
sqrt = int(limit ** 0.5)
wheel = m30_wheel(sqrt)
lim_ix = wheel.num2ix(limit)
sieve_bits = SIEVE_SIZE * 8
while (sieve_bits >> 1) >= max(lim_ix, 1):
sieve_bits >>= 1          # adjust the sieve size downward for small N

sieve = bitarray(sieve_bits)
num_segments = (lim_ix + sieve_bits - 1) // sieve_bits  # round up
show_progress = False
if statsOnly:   # outer loop?
print_sieve_size(sieve)
if limit > 1e8:
show_progress = True

outPrimes = wheel.get_skipped_primes()  # these may be needed for output
count = len(outPrimes)

# loop over all the segments
for lo_ix in range(0, lim_ix + 1, sieve_bits):
high = wheel.ix2num(lo_ix + sieve_bits) - 1
sieve.setall(True)
if show_progress:
progress(lo_ix // sieve_bits, num_segments)

wheel.cull_segment(sieve, lo_ix, high)

# handle any extras in the last segment
top = lim_ix - lo_ix + 1 if high > limit else sieve_bits

# collect results from this segment
if statsOnly:
count += sieve[:top].count()  # a lightweight way to get a result
else:
for i in range(top):  # XXX not so lightweight
if sieve[i]:
x = i + lo_ix
# ix2num(x) inlined below, performance is sensitive here
p = 30 * (x >> 3) + wheel.wheel_primes[x & 7]
outPrimes.append(p)

if show_progress:
progress(num_segments, num_segments)
print()

return count if statsOnly else outPrimes

if __name__ == '__main__':
a = '1e8' if len(argv) < 2 else argv[1]

n = int(float(a))

start = time()
count = segmentedSieve(n, statsOnly=True)
elapsed = time() - start

BigOculls = n * log(log(n, 2), 2)
cycles = CLOCK_SPEED * 1e9 * elapsed
cyclesPerCull = cycles/BigOculls

print(f"pi({a}) = {count}")
print(f"{elapsed:.3} seconds, {cyclesPerCull:.2} cycles/N log log N)")

if count < 500:
print(segmentedSieve(n))


Performance Data:

$./v51_segwheel.py 1e6 sieve size: 64 KB pi(1e6) = 78498 0.00406 seconds, 1.5 cycles/N log log N)$ ./v51_segwheel.py 1e7
sieve size: 512 KB
pi(1e7) = 664579
0.0323 seconds, 1.1 cycles/N log log N)
$./v51_segwheel.py 1e8 sieve size: 1024 KB pi(1e8) = 5761455 0.288 seconds, 0.97 cycles/N log log N)$ ./v51_segwheel.py 1e9
sieve size: 1024 KB
Sieving: [############################################################] 32/32
pi(1e9) = 50847534
2.79 seconds, 0.91 cycles/N log log N)


The cycles per N log log N shrink as the sieve size grows, probably due to a higher ratio of optimized sieving code to initialization and everything else. The sieve size is capped at 1MB; that produces the fastest results for N in the billions perhaps due to how it almost fits in the L2 0.5MB CPU cache. For the smaller sieve sizes, there should only be one segment. The progress bar starts appearing - possible ADD issues here :-( .

N = 1e9 (one billion) is the performance sweet spot at present. Beyond that, you can see the cycles per N log log N starting to creep up:

$./v51_segwheel.py 1e10 sieve size: 1024 KB Sieving: [############################################################] 318/318 pi(1e10) = 455052511 35.3 seconds, 1.1 cycles/N log log N)  I've run the earlier version up to 1e12 (1 trillion). But that's no fun for someone with mild ADD. It takes a good part of a day. The progress bar starts to be very useful. I had to keep my eye on the laptop to prevent it from hibernating as much as possible. Once when it did hibernate and I woke it up, my WSL Ubuntu bash terminal froze, but I was able to hit various keys to salvage the run. The hot spots: $ python3 -m cProfile -s 'tottime' ./v51_segwheel.py 1e9 | head -15
...
ncalls  tottime  percall  cumtime  percall filename:lineno(function)
77125    1.664    0.000    1.736    0.000 v51_segwheel.py:112(cull_one_multiple)
2/1    1.188    0.594    3.049    3.049 v51_segwheel.py:153(segmentedSieve)
33    0.083    0.003    1.837    0.056 v51_segwheel.py:136(cull_segment)
80560    0.075    0.000    0.075    0.000 v51_segwheel.py:64(update)
32    0.012    0.000    0.012    0.000 {method 'count' of 'bitarray._bitarray' objects}
3435    0.009    0.000    0.015    0.000 v51_segwheel.py:68(update_new_mult)


WHAT I'M LOOKING FOR

• Performance enhancements.
• I'm using a bitarray as the sieve. If you know of something that performs better as a sieve, please answer.
• Help here:
        # collect results from this segment
if statsOnly:
count += sieve[:top].count()  # a lightweight way to get a result
else:
for i in range(top):  # XXX not so lightweight
if sieve[i]:
x = i + lo_ix
# ix2num(x) inlined below, performance is sensitive here
p = 30 * (x >> 3) + wheel.wheel_primes[x & 7]
outPrimes.append(p)


The statsOnly leg is great because bitarray is doing the work in optimized C no doubt. I think the else leg could be shrunk. It would be fantastic to change the else into a generator, i.e. yield the primes. I tried that, but then had problems getting it to return the count when the recursion unwound to the top level. It seemed to be stuck in generator mode and didn't want to be bi-modal.

• algorithmic advice. I chose a mod 30 wheel vs. mod 210 because the former has 8 teeth allowing shifts and & ops to replace divide and mod. But I see that there are only a couple of places where the bit hacks are used in the critical paths, so eliminating multiples of 7 from the data structure/culling code may be a win.

• Ways to shrink, clarify, or further modularize the code.

• Help with the class stuff. This is my first voluntary OOP effort. I did dabble in JUnit back when I worked for {bigCo}. That gave me a bad taste for objects, but in retrospect, the badness was probably due to the JVM. Not a problem in Python.

EDIT

• Updated the code with a new version which adds the PrimeMultiple class in place of three separate arrays. No noticeable change in performance.
• Added the performance info and "what I want" sections.
• Minor wording tweaks to the original post

## smallSieve

PEP 8 recommends using snake_case for function names, so the function should be named small_sieve.

You've imported bitarray, but do not use it in this function. In addition to reducing the memory requirement of the function, it could really clean up the code (and perhaps speed it up). The key is the slice assignment,

            sieve[i * i // 2::i] = False


which will assign a single scalar value to every element in the slice. This means you don't have to calculate how many False values to assign into the slice, nor allocate an entire list of False values, just to set each entry of the slice to False.

Finally, the return statement repeatedly indexes into the sieve list, sieve[i], which is inefficient. It is better to iterate over the sieve list directly, fetching the sieve's primality flags from the iterator. Since you need the indices as well, for i, flag in enumerate(sieve) is the preferred list comprehension construct:

def small_sieve(n):
sieve = bitarray.bitarray(n // 2)

sieve.setall(True)
sieve[0] = False    # 1 is not prime

for i in range(3, int(n ** 0.5) + 1, 2):
if sieve[i // 2]:
sieve[i * i // 2::i] = False

return [2] + [2 * i + 1 for i, flag in enumerate(sieve) if flag]


## m30_wheel.__init__

The m30_wheel is only constructed once, so its performance is not critical. Instead of hand-coded constants, have you considered computing the constants? It would make building the mod 210 wheel much easier!

As an example:

self.wheel_primes_m30 = [ wheel_prime % 30 for wheel_prime in self.wheel_primes ]


Also, instead of spelling out the gaps twice, after computing the gaps, use list multiplication:

temp = self.wheel_primes + [self.wheel_primes[0] + 30]
self.gaps = [ b - a for a, b in zip(temp[:-1], temp[1:]) ] * 2


There are various hard-coded numbers in the wheel that could be made into member values ... 30, 7, 8 ... but hard-coded integers will be faster than member access. So, despite computing the initialization data instead of using hard-coded numbers, I'd be inclined to leave the numbers as numbers in the various member functions which are called multiple times.

## Use computed assignments

Python cannot optimize a statement like:

n = n - 7


into:

n -= 7


due to its interpreted nature, where the meaning of the various operations depends on type(n), which can be different every time the statement is executed. So in the former case, the Python interpreter will search its dictionary for the variable n, subtract 7, and then search its dictionary for the variable n to store the value into. In the latter case, the variable n is only searched for once; the value is retrieved, modified, and stored without needing to consult the variable dictionary a second time.

## Unused variables

In the loop:

for j in range(8):


the variable j is never used. By convention, the _ variable should be used when it is needed for syntactical purposes only:

for _ in range(8):


## XXX not so lightweight

As noted, this code is not lightweight:

        for i in range(top):
if sieve[i]:
x = i + li_ix
...


due to the repeated indexing into the sieve list. The lighter weight approach is to use iteration over the contents of the list directly:

        for i, flag in enumerate(sieve[:top]):
if flag:
x = i + li_ix
...


Even better, since i is only used to compute x, which is a constant offset from i, we can start the enumeration at the value li_ix, and avoid the addition:

        for x, flag in enumerate(sieve[:top], li_ix):
if flag:
...

• I'm on it! I was thinking using bitarray in small_sieve since I already use it in the big sieve. fwiw I just cut and pasted the code from the referenced SO post, but I've already changed it to use math.sqrt() per @Reinderien and made note that it's slightly modified for anyone who cares. enumerate(): D'oh, should have known, the reviewer of my last version already suggested that elsewhere. computing wheel constants at init: yep, @ Reinderien already nudged me in that direction. I hope I can figure out a way to do it just once for all the recursive calls. – Greg Ames Oct 8 '19 at 14:00
• avoiding constant addition of li_ix: most excellent – Greg Ames Oct 8 '19 at 14:09
• your bitarray verion of small_sieve rocks! It's twice as fast @ 1e8 when isolated – Greg Ames Oct 9 '19 at 1:03
• Thank-you. I was happy to see you were using bitarray but then horrified that you weren’t using the scalar-to-slice assignment... and then confused when I saw you were using it in sieve[cull_start::p8] = False. I’m happy you like the improvements; it is why I’m here in Code Review. Enjoy! – AJNeufeld Oct 9 '19 at 2:29
• yeah it's just because I dropped in existing code for small_sieve then pretty much ignored it (except for PEP8) and worked on the big sieve – Greg Ames Oct 9 '19 at 2:36

Prepare for a random grab-bag of solicited and unsolicited advice.

## Shebang

It's typically preferred to use

#!/usr/bin/env python3


so that a non-system, e.g. a virtualenv-based, Python binary can kick in automatically when needed. The script can be opinionated about which version of Python it's running, but shouldn't be when it comes to which interpreter binary should be used.

## Clock speed

Firstly: as you'll no doubt already know, it's not meaningful to hard-code the clock speed. You could do a trivial parse of /proc/cpuinfo which would tie you to Linux, or you could import a third-party library which is able to do this in a platform-agnostic manner.

Even then: once you have the processor frequency, that's only loosely correlated with actual execution speed. Python is a multi-architecture interpreter. Different CPUs have very different capabilities in terms of branch lookahead, etc. which make it so that an advanced 1GHz CPU will beat the pants off of a cheap, consumer-grade 2GHz CPU ten times out of ten.

Another big factor is the entire idea of how much gets done in one instruction cycle based on the instruction set - x86_64 (CISC) versus Arm (RISC) being a huge gap.

That's also not accounting for the fact that you're running a multi-process operating system and time-sharing the CPU, so the number of actual cycles consumed will be less than expected given the amount of real-time duration measured.

All of that said: don't worry about the frequency; instead just print the output of import platform; platform.processor(). The cycle estimate is unfortunately baloney.

## Formatting standards

PEP8 linters will tell you that:

• segmentedSieve should be segmented_sieve (and so on for statsOnly, etc.)
• there should only be one blank line before # get sieving primes recursively,
• m30_wheel should be M30Wheel due to being a class
• etc.

## Reduce print calls

print("sieve size:", end=' ')
ss = len(memoryview(sieve))
print(ss//1024, "KB") if ss > 1024 else print(ss, "bytes")


can be

ss = len(memoryview(sieve))
size = f'{ss//1024} KiB' if ss > 1024 else f'{ss} bytes'
print(f'sieve size: {size}')


Also note that KB is not a unit. kB is 1000 bytes, and KiB is 1024 bytes.

## Don't exponentiate needlessly

I don't trust Python to convert n ** 0.5 to a more efficient sqrt automatically. Just call sqrt.

## Use Numpy

Operations like this:

       sieve[i * i // 2::i] = [False] * ((n - i * i - 1) // (2 * i) + 1)


where array segments are copied over - can be made much more efficient through the use of Numpy. Numpy is built exactly for this kind of thing - fast array operations for numerical work.

## Type hints

You're concerned about performance, and that's fine - type hints do not incur a performance hit. So something like this:

def update_new_mult(self, multiple, wheel_index, wheel):


def update_new_mult(self, multiple: int, wheel_index: int, wheel: M30Wheel) -> None:


## Immutability

Something like

    self.gaps             = [4,2,4,2,4,6,2,6, 4,2,4,2,4,6,2,6]  # 2 loops for overflow


is written once and read many times, so use a tuple, not a list. Past that: since it's only calculated during initialization, you really shouldn't be hard-coding these values. Calculate them in a simple loop based on your wheel_primes. This will improve maintainability if ever you change your modulus.

## In-place subtraction

n = n - 7  # adjust for wheel starting at 7 vs. 0


should be

n -= 7  # adjust for wheel starting at 7 vs. 0


## Combined division and modulation

    return (n//30 << 3) + self.wheel_indices[n % 30]


should use divmod(n, 30) to get both the quotient and remainder at the same time.

## Magic numbers

30 should be stored in a constant, for the same reasons that you should be calculating gaps - what if it changes? And for third parties, or you in three years, it isn't immediately evident what 30 means.

The same goes for basically every number in these lines:

    n = n - 7  # adjust for wheel starting at 7 vs. 0
return (n//30 << 3) + self.wheel_indices[n % 30]

return 30 * (i >> 3) + self.wheel_primes[i & 7]


I don't know where 7 comes from, but I suspect that it should be calculated from (1 << 3) - 1 based on its usage as a mask.

## Name collisions

Don't call a variable sqrt. It's common enough that there's a bare import of that symbol from math.

• @bullseye Seems I trampled your edit - sorry - feel free to suggest it again and I'll accept it. – Reinderien Oct 8 '19 at 4:23
• Time tests with divmod(n, d) have shown that separate expressions actually perform better, probably due to the return tuple creation & unpacking that is needed to use the result. (I was shocked.) – AJNeufeld Oct 8 '19 at 5:01
• Expanding a little on "just call sqrt", Python 3.8 will have math.isqrt. Look for it, starting October 14th, 2019 (fingers crossed). – AJNeufeld Oct 8 '19 at 5:10
• @Reinderien done – bullseye Oct 8 '19 at 6:27
• The likely case is that it's "not just sqrt" accounting for the 4.1% difference, and that there's something else in play as well - maybe module lookup. – Reinderien Oct 9 '19 at 0:56