Infix Expressions for PLT Scheme

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})