; sieve of sundaram
(define (primes n)
(let* ((m (quotient n 2)) (pv (make-vector (+ m 1) #t)))
(do ((i 1 (+ i 1))) ((< (quotient m 4) i))
(do ((j i (+ j 1))) ((< (quotient (- m i) (+ i i 1)) j))
(vector-set! pv (+ i j (* 2 i j)) #f)))
(let loop ((i 1) (ps (list 2)))
(cond ((= i m) (reverse ps))
((vector-ref pv i) (loop (+ i 1) (cons (+ i i 1) ps)))
(else (loop (+ i 1) ps))))))
(display "Sundaram: ") (time (display (length (primes 2000000))) (display " "))
(define (primes limit)
(define (exact x) (inexact->exact (floor x)))
(let ((sieve (make-vector (+ (quotient limit 2) (modulo limit 2)) #f))
(primes (list 3 2)))
(define (flip! m) (vector-set! sieve m (not (vector-ref sieve m))))
(let ((x-max (exact (sqrt (/ (- limit 1) 4)))) (x2 0))
(do ((xd 4 (+ xd 8))) ((<= (+ (* x-max 8) 2) xd))
(set! x2 (+ x2 xd))
(let* ((y-max (exact (sqrt (- limit x2))))
(n (+ x2 (* y-max y-max)))
(n-diff (+ y-max y-max -1)))
(when (even? n) (set! n (- n n-diff)) (set! n-diff (- n-diff 2)))
(do ((d (* (- n-diff 1) 2) (- d 8))) ((<= d -1))
(when (member (modulo n 12) (list 1 5)) (flip! (quotient n 2)))
(set! n (- n d))))))
(let ((x-max (exact (sqrt (/ (- limit 1) 3)))) (x2 0))
(do ((xd 3 (+ xd 6))) ((<= (+ (* x-max 6) 2) xd))
(set! x2 (+ x2 xd))
(let* ((y-max (exact (sqrt (- limit x2))))
(n (+ x2 (* y-max y-max)))
(n-diff (+ y-max y-max -1)))
(when (even? n) (set! n (- n n-diff)) (set! n-diff (- n-diff 2)))
(do ((d (* (- n-diff 1) 2) (- d 8))) ((<= d -1))
(when (= (modulo n 12) 7) (flip! (quotient n 2)))
(set! n (- n d))))))
(let ((x-max (exact (/ (+ (sqrt (- 4 (* (- 1 limit) 8))) 2) 4)))
(y-min -1) (x2 0) (xd 3))
(do ((x 1 (+ x 1))) ((<= (+ x-max 1) x))
(set! x2 (+ x2 xd)) (set! xd (+ xd 6))
(when (<= limit x2)
(set! y-min (* (- (* (- (inexact->exact (ceiling (sqrt (- x2 limit)))) 1) 2) 2) 2)))
(let ((n (- (* (+ (* x x) x) 2) 1))
(n-diff (* (- (* (- x 1) 2) 2) 2)))
(do ((d n-diff (- d 8))) ((<= d y-min))
(when (= (modulo n 12) 11) (flip! (quotient n 2)))
(set! n (+ n d))))))
(do ((n 2 (+ n 1))) ((<= (quotient (+ (exact (sqrt limit)) 1) 2) n))
(when (vector-ref sieve n)
(let* ((p (+ n n 1)) (p2 (* p p)))
(set! primes (cons p primes))
(do ((k p2 (+ k (+ p2 p2)))) ((<= limit k))
(vector-set! sieve (quotient k 2) #f)))))
(do ((p (+ (exact (sqrt limit)) 1) (+ p 2))) ((<= limit p))
(when (vector-ref sieve (quotient p 2))
(set! primes (cons p primes))))
(reverse primes)))
(display "Atkin: ") (time (display (length (primes 2000000))) (display " "))
(define (primes . args) ; (primes [lo] hi) inclusive at both ends
(let* ((lo (if (null? (cdr args)) 0 (car args)))
(hi (if (null? (cdr args)) (car args) (cadr args))))
(cond ((and (<= lo 100000) (<= hi 1000000)) ; simple sieve
(let* ((max-index (quotient (- hi 3) 2))
(v (make-vector (+ max-index 1) #t)))
(let loop ((i 0) (ps (list 2)))
(let ((p (+ i i 3)) (startj (+ (* 2 i i) (* 6 i) 3)))
(cond ((< hi (* p p))
(let loop ((j i) (ps ps))
(cond ((< max-index j)
(let loop ((ps (reverse ps)))
(if (<= lo (car ps)) ps
(loop (cdr ps)))))
((vector-ref v j)
(loop (+ j 1) (cons (+ j j 3) ps)))
(else (loop (+ j 1) ps)))))
((vector-ref v i)
(let loop ((j startj))
(when (<= j max-index)
(vector-set! v j #f) (loop (+ j p))))
(loop (+ i 1) (cons p ps)))
(else (loop (+ i 1) ps)))))))
((< lo (sqrt hi))
(let* ((r (inexact->exact (ceiling (sqrt hi))))
(r (if (even? r) r (+ r 1))))
(append (primes lo r) (primes r hi))))
(else ; segmented sieve
(let* ((lo (if (even? lo) lo (- lo 1)))
(b 50000) (bs (make-vector b #t))
(r (inexact->exact (ceiling (sqrt hi))))
(ps (cdr (primes r)))
(qs (map (lambda (p)
(modulo (* -1/2 (+ lo 1 p)) p)) ps))
(zs (list)) (z (lambda (p) (set! zs (cons p zs)))))
(do ((t lo (+ t b b))
(qs qs (map (lambda (p q) (modulo (- q b) p))
ps qs)))
((<= hi t)
(let loop ((zs zs))
(if (<= (car zs) hi) (reverse zs)
(loop (cdr zs)))))
(do ((i 0 (+ i 1))) ((= i b)) (vector-set! bs i #t))
(do ((ps ps (cdr ps)) (qs qs (cdr qs))) ((null? qs))
(do ((j (car qs) (+ j (car ps)))) ((<= b j))
(vector-set! bs j #f)))
(do ((j 0 (+ j 1))) ((= j b))
(if (vector-ref bs j) (z (+ t j j 1))))))))))
(display "Segmented: ") (time (display (length (primes 2000000))) (display " "))
(define (primes n)
(let* ((max-index (quotient (- n 3) 2))
(v (make-vector (+ 1 max-index) #t)))
(let loop ((i 0) (ps '(2)))
(let ((p (+ i i 3)) (startj (+ (* 2 i i) (* 6 i) 3)))
(cond ((>= (* p p) n)
(let loop ((j i) (ps ps))
(cond ((> j max-index) (reverse ps))
((vector-ref v j)
(loop (+ j 1) (cons (+ j j 3) ps)))
(else (loop (+ j 1) ps)))))
((vector-ref v i)
(let loop ((j startj))
(if (<= j max-index)
(begin (vector-set! v j #f)
(loop (+ j p)))))
(loop (+ 1 i) (cons p ps)))
(else (loop (+ 1 i) ps)))))))
(display "Eratosthenes: ") (time (display (length (primes 2000000))) (display " "))