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#lang racket
(provide
iterative-improve
close-enough?
repeated
fixed-point
average
average-damp
smoother
square
cube
power
compose
double
deriver
golden-ratio
miller-raban-test
all-miller-raban
coprimer
filtered-accumulate
product
sum
simpson
fermat?
all-fermat
expmoder
divides?
find-divisor
smallest-divisor)
(require sicp)
;;abstract procedures
(define (iterative-improve good-enuf? improve)
(lambda (guess)
(define (try g)
(let ((next (improve g)))
(if (good-enuf? next g)
next
(try next))))
(try guess)))
(define (filtered-accumulate pred combiner null-value term a next b)
(define (iter a result)
(cond
((> a b) result)
((pred a)
(iter (next a) (combiner (term a) result)))
(else (iter (next a) result))))
(iter a null-value))
;; basic
(define (cube x) (* x x x))
(define (square x) (* x x))
(define (power x n)
(cond
((= n 0) 1)
((= n 1) x)
(else (* x (power x (dec n))))))
(define (close-enough? tolerance)
(lambda (x y)
(< (abs (- x y)) tolerance)))
(define (average a b)
(/ (+ a b) 2))
(define (divides? a b)
(= (remainder b a) 0))
(define (find-divisor n test-divisor)
(cond
((> (square test-divisor) n) n)
((divides? test-divisor n) test-divisor)
(else (find-divisor
n
((lambda (td) (if (= td 2) 3 (+ td 2)))
test-divisor)))))
(define (smallest-divisor n)
(find-divisor n 2))
;; functions
(define (double f)
(lambda (x)
(f (f x))))
(define (compose f g)
(lambda (x) (f (g x))))
(define (repeated f n)
(if (= n 0)
(lambda (x) x)
(lambda (x)
(define (iter result i)
(if (> i n)
result
(iter (f result) (inc i))))
(iter x 1))))
(define (average-damp f n)
(lambda (guess)
(let ((next (f guess)))
((repeated
(lambda (g) (average guess g))
n) next))))
;; math
(define (sum term a next b)
(define (iter a result)
(if (> a b)
result
(iter (next a) (+ result (term a)))))
(iter a 0))
(define (product term a next b)
(define (iter result a)
(if (> a b)
result
(iter (* (term a) result) (next a))))
(iter 1 a))
(define (smoother dx)
(lambda (f)
(lambda (x)
(/ (+ (f (- x dx)) (f x) (f (+ x dx))) 3))))
(define (deriver dx)
(lambda (f)
(lambda (x)
(/ (- (f (+ x dx)) (f x))
dx))))
(define (coprimer n)
(lambda (a)
(if (= 1 (gcd a n))
((lambda ()
(println a)
true))
false)))
(define (fixed-point f first-guess)
((iterative-improve
(close-enough? 0.0001)
f)
first-guess))
(define (golden-ratio)
(fixed-point
(lambda (x) (+ 1 (/ 1 x)))
1.0))
(define (simpson f lower upper n)
(define h (/ (- upper lower) n))
(define (nth-term k)
(f (+ lower (* k h))))
(define (term k)
(cond
((= k 0) (f lower))
((= k upper) (f upper))
((even? k) (* 2 (nth-term k)))
(else (* 4 (nth-term k)))))
(* (/ h 3.0)
(sum term lower inc n)))
(define (expmoder signal)
(define (expmod base e m)
(cond
((= e 0) 1)
((even? e)
(signal base e m (remainder (square (expmod base (/ e 2) m)) m)))
(else
(remainder (* base (expmod base (- e 1) m)) m))))
expmod)
(define (fermat? a n)
(= ((expmoder (lambda (b e m x) x)) a n n) a))
(define (all-fermat n)
(define (f a n)
(cond
((= a 0) true)
((fermat? (- a 1) n) (f (- a 1) n))
(else false)))
(f n n))
(define (miller-raban-test a n)
(define (signal-mr b e m x)
(cond
((= e (- m 1)) x) ;; end result
((= b (- m 1)) x) ;; base squared wil result in trivial root
((= x 1) 0) ;; non-trivial root
(else x))) ;; no root found
(= ((expmoder signal-mr) a (- n 1) n) 1))
(define (all-miller-raban n)
(define (iter a n)
(cond
((<= a 2) true)
((miller-raban-test (- a 1) n) (iter (- a 1) n))
(else false)))
(iter n n))
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