[racket] Math library kudos

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
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