Can the execution time of this prime number generator be improved?

Question

My initial goal when writing this was to leave the smallest footprint possible. I can say with confidence that this goal has been met. Unfortunately, this leaves me with a r

Answer 1

Just for funsies, let's take a look at this page.

pi(x) is the prime counting function, it returns the number of primes below x. You can approximate pi(x) using the following inequalities:

(x/log x)(1 + 0.992/log x) < pi(x) < (x/log x)(1 + 1.2762/log x) 
// The upper bound holds for all x > 1

p(x) is the nth's prime function, which can be approximated using:

n ln n + n ln ln n - n < p(n) < n ln n + n ln ln n
// for n >= 6

With that in mind, here is a very fast generator which computes the first n primes, where each element at index i equal p(i). So, if we want to cap our array at primes below 2,000,000, we use the upperbound inequality for the prime counting function:

let rec is_prime (primes : int[]) i testPrime maxFactor =
    if primes.[i] > maxFactor then true
    else
        if testPrime % primes.[i] = 0 then false
        else is_prime primes (i + 1) testPrime maxFactor

let rec prime_n (primes : int[]) primeCount testPrime tog =
    if primeCount < primes.Length then
        let primeCount' =
            if is_prime primes 2 testPrime (float testPrime |> sqrt |> int) then
                primes.[primeCount] <- testPrime
                primeCount + 1
            else
                primeCount
        prime_n primes primeCount' (testPrime + tog) (6 - tog)

let getPrimes upTo =
    let x = float upTo
    let arraySize = int <| (x / log x) * (1.0 + 1.2762 / log x)
    let primes = Array.zeroCreate (max arraySize 3)
    primes.[0] <- 2
    primes.[1] <- 3
    primes.[2] <- 5

    prime_n primes 3 7 4
    primes

Cool! So how fast is it? On my 3.2ghz quad-core, I get the following in fsi:

> let primes = getPrimes 2000000;;
Real: 00:00:00.534, CPU: 00:00:00.546, GC gen0: 0, gen1: 0, gen2: 0

val primes : int [] =
  [|2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
    73; 79; 83; 89; 97; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
    157; 163; 167; 173; 179; 181; 191; 193; 197; 199; 211; 223; 227; 229; 233;
    239; 241; 251; 257; 263; 269; 271; 277; 281; 283; 293; 307; 311; 313; 317;
    331; 337; 347; 349; 353; 359; 367; 373; 379; 383; 389; 397; 401; 409; 419;
    421; 431; 433; 439; 443; 449; 457; 461; 463; 467; 479; 487; 491; 499; 503;
    509; 521; 523; 541; ...|]

> printfn "total primes: %i. Last prime: %i" (primes.Length - 1) primes.[primes.Length - 1];;
total primes: 149973. Last prime: 2014853

So that's all the primes around 2 million in less than half a second :)