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; 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 " "))
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