276 lines
6.7 KiB
Racket
276 lines
6.7 KiB
Racket
#lang scribble/doc
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@(require scribble/manual
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scribble/eval
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scribble/basic
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scribble/bnf
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(planet cce/scheme:4:1/planet)
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(for-label (planet dherman/pprint:4)
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scheme/base
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scheme/contract
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"../main.ss")
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"util.ss")
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@title[#:tag "top"]{@bold{Infix Expressions} for PLT Scheme}
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@author[(author+email "Jens Axel Søgaard" "jensaxel@soegaard.net")]
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This package provides infix notation for writing mathematical expressions.
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@section{Getting Started}
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A simple example, calculating 1+2*3.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(display (format "1+2*3 is ~a\n" @${1+2*3} )
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}|
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@subsection{Arithmetical Operations}
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The arithmetical operations +, -, *, / and ^ is written with standard
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mathematical notation. Normal parentheseses are used for grouping.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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@${2*(1+3^4)} ; evaluates to 164
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}|
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@subsection{Identifiers}
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Identifiers refer to the current lexical scope:
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(define x 41)
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@${ x+1 } ; evaluates to 42
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}|
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@subsection{Application}
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Function application use square brackets (as does Mathematica).
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Here @scheme[sqrt] is bound to the square root function defined
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in the language after at-exp, here the scheme language.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(display (format "The square root of 64 is ~a\n" @${sqrt[64]} ))
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@${ list[1,2,3] } evaluates to the list (1 2 3)
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}|
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@subsection{Lists}
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Lists are written with curly brackets {}.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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@${ {1,2,1+2} } ; evaluates to (1 2 3)
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}|
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@subsection{List Reference}
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List reference is written with double square brackets.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(define xs '(a b c))
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@${ xs[[1]] } ; evaluates to b
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}|
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@subsection{Anonymous Functions}
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The syntax (λ ids . expr) where ids are a space separated list
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of identifiers evaluates to function in which the ids are bound in
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body expressions.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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@${ (λ.1)[]} ; evaluates to 1
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@${ (λx.x+1)[2]} ; evaluates to 3
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@${ (λx y.x+y+1)[1,2]} ; evaluates to 4
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}|
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@subsection{Square Roots}
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Square roots can be written with a literal square root:
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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@${√4} ; evaluates to 2
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@${√(2+2)} ; evaluates to 2
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}|
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@subsection{Comparisons}
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The comparison operators <, =, >, <=, and >= are available.
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The syntaxes ≤ and ≥ for <= and >= respectively, works too.
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Inequality is tested with <>.
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@subsection{Logical Negation}
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Logical negations is written as ¬.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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@${ ¬true } ; evaluates to #f
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@${ ¬(1<2) } ; evaluates to #f
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}|
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@subsection{Assignment}
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Assignment is written with := .
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@subsection{Sequencing}
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A series of expresions can be evaluated by interspersing semi colons
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between the expressions.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(define x 0)
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@${ (x:=1); (x+3) } ; evaluates to 4
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}|
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@section{Examples}
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@subsection{Example: Fibonacci}
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This problem is from the Euler Project.
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Each new term in the Fibonacci sequence is generated by adding the
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previous two terms. By starting with 1 and 2, the first 10 terms will be:
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1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
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Find the sum of all the even-valued terms in the sequence which do not
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exceed four million.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require
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(planet soegaard/infix)
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(only-in (planet "while.scm" ("soegaard" "control.plt" 2 0))
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while))
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(define-values (f g t) (values 1 2 0))
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(define sum f)
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@${
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while[ g< 4000000,
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when[ even?[g], sum:=sum+g];
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t := f + g;
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f := g;
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g := t];
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sum}|
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@subsection{Example: Difference Between a Sum of Squares and the Square of a Sum}
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This problem is from the Euler Project.
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The sum of the squares of the first ten natural numbers is,
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1^2 + 2^2 + ... + 10^2 = 385
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The square of the sum of the first ten natural numbers is,
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(1 + 2 + ... + 10)^2 = 552 = 3025
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Hence the difference between the sum of the squares of the first ten natural
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numbers and the square of the sum is 3025 - 385 = 2640.
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Find the difference between the sum of the squares of the first one hundred
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natural numbers and the square of the sum.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(require (planet "while.scm" ("soegaard" "control.plt" 2 0))) ; while
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#|
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(define n 0)
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(define ns 0)
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(define squares 0)
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(define sum 0)
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@${
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sum:=0;
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while[ n<100,
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n := n+1;
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ns := ns+n;
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squares := squares + n^2];
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ns^2-squares
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}
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}|
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@subsection{Example: Pythagorean Triplets}
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This example is from the Euler Project.
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A Pythagorean triplet is a set of three natural numbers, a,b,c for which,
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a^2 + b^2 = c^2
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For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
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There exists exactly one Pythagorean triplet for which a + b + c = 1000.
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Find the product abc.
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(let-values ([(a b c) (values 0 0 0)])
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(let/cc return
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(for ([k (in-range 1 100)])
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(for ([m (in-range 2 1000)])
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(for ([n (in-range 1 m)])
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@${ a := k* 2*m*n;
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b := k* (m^2 - n^2);
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c := k* (m^2 + n^2);
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when[ a+b+c = 1000,
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display[{{k,m,n}, {a,b,c}}];
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newline[];
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return[a*b*c] ]})))))
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}|
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@subsection{Example: Miller Rabin Primality Test}
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This example was inspired by Programming Praxis:
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http://programmingpraxis.com/2009/05/01/primality-checking/
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@verbatim[#:indent 2]|{
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#lang at-exp scheme
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(require (planet soegaard/infix))
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(require srfi/27) ; random-integer
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(define (factor2 n)
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; return r and s, s.t n = 2^r * s where s odd
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; invariant: n = 2^r * s
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(let loop ([r 0] [s n])
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(let-values ([(q r) (quotient/remainder s 2)])
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(if (zero? r)
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(loop (+ r 1) q)
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(values r s)))))
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(define (miller-rabin n)
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; Input: n odd
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(define (mod x) (modulo x n))
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(define (expt x m)
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(cond [(zero? m) 1]
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[(even? m) @${mod[sqr[x^(m/2)] ]}]
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[(odd? m) @${mod[x*x^(m-1)]}]))
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(define (check? a)
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(let-values ([(r s) (factor2 (sub1 n))])
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; is a^s congruent to 1 or -1 modulo n ?
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(and @${member[a^s,{1,mod[-1]}]} #t)))
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(andmap check?
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(build-list 50 (λ (_) (+ 2 (random-integer (- n 3)))))))
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(define (prime? n)
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(cond [(< n 2) #f]
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[(= n 2) #t]
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[(even? n) #f]
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[else (miller-rabin n)]))
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(prime? @${2^89-1})
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