# [racket] Math library kudos

 From: Luke Vilnis (lvilnis at gmail.com) Date: Tue Feb 19 23:26:12 EST 2013 Previous message: [racket] Math library kudos Next message: [racket] Math library kudos Messages sorted by: [date] [thread] [subject] [author]

```FYI, log gamma is another fast way to calculate the number of combinations
if you want to deal with really big numbers.

On Tue, Feb 19, 2013 at 7:28 PM, Joe Gilray <jgilray at gmail.com> wrote:

> Racketeers,
>
> Thanks for putting together the fantastic math library.  It will be a
> wonderful resource.  Here are some quick impressions (after playing mostly
> with math/number-theory)
>
> 1) The functions passed all my tests and were very fast.  If you need even
> more speed you can keep a list of primes around and write functions to use
> that, but that should be rarely necessary
>
> 2) I have a couple of functions to donate if you want them:
>
> 2a) Probablistic primality test:
>
> ; function that performs a Miller-Rabin probabalistic primality test k
> times, returns #t if n is probably prime
> ; algorithm from http://rosettacode.org/wiki/Miller-Rabin_primality_test,
> code adapted from Lisp example
> ; (module+ test (check-equal? (is-mr-prime? 1000000000000037 8) #t))
> (define (is-mr-prime? n k)
>   ; function that returns two values r and e such that number = divisor^e
> * r, and r is not divisible by divisor
>   (define (factor-out number divisor)
>     (do ([e 0 (add1 e)] [r number (/ r divisor)])
>       ((not (zero? (remainder r divisor))) (values r e))))
>
>   ; function that performs fast modular exponentiation by repeated squaring
>   (define (expt-mod base exponent modulus)
>     (let expt-mod-iter ([b base] [e exponent] [p 1])
>       (cond
>         [(zero? e) p]
>         [(even? e) (expt-mod-iter (modulo (* b b) modulus) (/ e 2) p)]
>         [else (expt-mod-iter b (sub1 e) (modulo (* b p) modulus))])))
>
>   ; function to return a random, exact number in the passed range
> (inclusive)
>   (define (shifted-rand lower upper)
>     (+ lower (random (add1 (- (modulo upper 4294967088) (modulo lower
> 4294967088))))))
>
>   (cond
>     [(= n 1) #f]
>     [(< n 4) #t]
>     [(even? n) #f]
>     [else
>      (let-values ([(d s) (factor-out (- n 1) 2)]) ; represent n-1 as 2^s-d
>        (let lp ([a (shifted-rand 2 (- n 2))] [cnt k])
>          (if (zero? cnt) #t
>              (let ([x (expt-mod a d n)])
>                (if (or (= x 1) (= x (sub1 n))) (lp (shifted-rand 2 (- n
> 2)) (sub1 cnt))
>                    (let ctestlp ([r 1] [ctest (modulo (* x x) n)])
>                      (cond
>                        [(>= r s) #f]
>                        [(= ctest 1) #f]
>                        [(= ctest (sub1 n)) (lp (shifted-rand 2 (- n 2))
> (sub1 cnt))]
>                        [else (ctestlp (add1 r) (modulo (* ctest ctest)
> n))])))))))]))
>
> 2b) combinations calculator
>
> ; function that returns the number of combinations, not the combinations
> themselves
> ; faster than using n! / (r! (n-r)!)
> (define (combinations n r)
>   (cond
>     [(or (< n 0) (< r 0)) (error "combinations: illegal arguments, n and r
> must be >= 0")]
>     [(> r n) 0]
>     [else
>      (let lp ([mord n] [total 1] [mult #t])
>        (cond
>          [(or (= 0 mord) (= 1 mord)) total]
>          [(and mult (= mord (- n r))) (lp r total #f)]
>          [(and mult (= mord r)) (lp (- n r) total #f)]
>          [mult (lp (sub1 mord) (* total mord) #t)]
>          [else (lp (sub1 mord) (/ total mord) #f)]))]))
>
> Thanks again!
> -Joe
>
> ____________________
>   Racket Users list:
>   http://lists.racket-lang.org/users
>
>
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```

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