1 Getting Started
1.1 Arithmetical Operations
1.2 Identifiers
1.3 Application
1.4 Lists
1.5 List Reference
1.6 Anonymous Functions
1.7 Square Roots
1.8 Comparisons
1.9 Logical Negation
1.10 Assignment
1.11 Sequencing
2 Examples
2.1 Example: Fibonacci
2.2 Example: Difference Between a Sum of Squares and the Square of a Sum
2.3 Example: Pythagorean Triplets
2.4 Example: Miller Rabin Primality Test

Infix Expressions for PLT Scheme

Jens Axel Søgaard <jensaxel at soegaard dot net>

This package provides infix notation for writing mathematical expressions.

1 Getting Started

A simple example, calculating 1+2*3.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (display (format "1+2*3 is ~a\n" @${1+2*3} )

1.1 Arithmetical Operations

The arithmetical operations +, -, *, / and ^ is written with standard mathematical notation. Normal parentheseses are used for grouping.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  @${2*(1+3^4)}   ; evaluates to 164

1.2 Identifiers

Identifiers refer to the current lexical scope:

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (define x 41)

  @${ x+1 }   ; evaluates to 42

1.3 Application

Function application use square brackets (as does Mathematica). Here sqrt is bound to the square root function defined in the language after at-exp, here the scheme language.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (display (format "The square root of 64 is ~a\n" @${sqrt[64]} ))

  @${ list[1,2,3] }  evaluates to the list (1 2 3)

1.4 Lists

Lists are written with curly brackets {}.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  @${ {1,2,1+2} }  ; evaluates to (1 2 3)

1.5 List Reference

List reference is written with double square brackets.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (define xs '(a b c))

  @${ xs[[1]] }  ; evaluates to b

Note: Since ]] denotes "closing double square brackets", one currently needs to insert a space in nested function applications:

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (define xs '(a b c))

  @${ exp[log[10] ] }

1.6 Anonymous Functions

The syntax (λ ids . expr) where ids are a space separated list of identifiers evaluates to function in which the ids are bound in body expressions.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  

  @${ (λ.1)[]}           ; evaluates to 1

  @${ (λx.x+1)[2]}       ; evaluates to 3

  @${ (λx y.x+y+1)[1,2]} ; evaluates to 4

1.7 Square Roots

Square roots can be written with a literal square root:

  #lang at-exp scheme

  (require (planet soegaard/infix))

  @${√4}     ; evaluates to 2

  @${√(2+2)} ; evaluates to 2

1.8 Comparisons

The comparison operators <, =, >, <=, and >= are available. The syntaxes ≤ and ≥ for <= and >= respectively, works too. Inequality is tested with <>.

1.9 Logical Negation

Logical negations is written as ¬.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  @${ ¬true }      ; evaluates to #f

  @${ ¬(1<2) }   ; evaluates to #f

1.10 Assignment

Assignment is written with := .

1.11 Sequencing

A series of expresions can be evaluated by interspersing semi colons between the expressions.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (define x 0)

  @${ (x:=1); (x+3) }  ; evaluates to 4

2 Examples

2.1 Example: Fibonacci

This problem is from the Euler Project.

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Find the sum of all the even-valued terms in the sequence which do not exceed four million.

  #lang at-exp scheme

  (require

    (planet soegaard/infix)

    (only-in (planet "while.scm" ("soegaard" "control.plt" 2 0))

             while))

  

  (define-values (f g t) (values 1 2 0))

  (define sum f)

  @${

  while[ g< 4000000,

    when[ even?[g], sum:=sum+g];

    t := f + g;

    f := g;

    g := t];

  sum

2.2 Example: Difference Between a Sum of Squares and the Square of a Sum

This problem is from the Euler Project.

The sum of the squares of the first ten natural numbers is, 1^2 + 2^2 + ... + 10^2 = 385 The square of the sum of the first ten natural numbers is, (1 + 2 + ... + 10)^2 = 552 = 3025 Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (require (planet "while.scm" ("soegaard" "control.plt" 2 0))) ; while

  #|

  

  (define n 0)

  (define ns 0)

  (define squares 0)

  (define sum 0)

  @${

  sum:=0;

  while[ n<100,

    n := n+1;

    ns := ns+n;

    squares := squares + n^2];

  ns^2-squares

  }

2.3 Example: Pythagorean Triplets

This example is from the Euler Project.

A Pythagorean triplet is a set of three natural numbers, a,b,c for which, a^2 + b^2 = c^2 For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.

  #lang at-exp scheme

  (require (planet soegaard/infix))

  

  (let-values ([(a b c) (values 0 0 0)])

    (let/cc return

      (for ([k (in-range 1 100)])

        (for ([m (in-range 2 1000)])

          (for ([n (in-range 1 m)])

            @${ a := k* 2*m*n;

                b := k* (m^2 - n^2);

                c := k* (m^2 + n^2);

                when[ a+b+c = 1000,

                   display[{{k,m,n}, {a,b,c}}];

                   newline[];

                   return[a*b*c] ]})))))

2.4 Example: Miller Rabin Primality Test

This example was inspired by Programming Praxis:

http://programmingpraxis.com/2009/05/01/primality-checking/

  #lang at-exp scheme

  (require (planet soegaard/infix))

  (require srfi/27) ; random-integer

  

  (define (factor2 n)

    ; return r and s, s.t n = 2^r * s where s odd

    ; invariant: n = 2^r * s

    (let loop ([r 0] [s n])

      (let-values ([(q r) (quotient/remainder s 2)])

        (if (zero? r)

            (loop (+ r 1) q)

            (values r s)))))

  

  (define (miller-rabin n)

    ; Input: n odd

    (define (mod x) (modulo x n))

    (define (expt x m)

      (cond [(zero? m) 1]

            [(even? m) @${mod[sqr[x^(m/2)] ]}]

            [(odd? m)  @${mod[x*x^(m-1)]}]))

    (define (check? a)

      (let-values ([(r s) (factor2 (sub1 n))])

        ; is a^s congruent to 1 or -1 modulo n ?

        (and @${member[a^s,{1,mod[-1]}]} #t)))

    (andmap check?

            (build-list 50 (λ (_) (+ 2 (random-integer (- n 3)))))))

  

  (define (prime? n)

    (cond [(< n 2) #f]

          [(= n 2) #t]

          [(even? n) #f]

          [else (miller-rabin n)]))

  

  (prime? @${2^89-1})