;;;; "ratize.scm" Find simplest number ratios ;;@code{(require 'rationalize)} ;;@ftindex rationalize ;;The procedure @dfn{rationalize} is interesting because most programming ;;languages do not provide anything analogous to it. Thanks to Alan ;;Bawden for contributing this algorithm. ;;@body ;;Computes the correct result for exact arguments (provided the ;;implementation supports exact rational numbers of unlimited precision); ;;and produces a reasonable answer for inexact arguments when inexact ;;arithmetic is implemented using floating-point. ;; (define (rationalize x e) (apply / (find-ratio x e))) ;;@code{Rationalize} has limited use in implementations lacking exact ;;(non-integer) rational numbers. The following procedures return a list ;;of the numerator and denominator. ;;@body ;;@0 returns the list of the @emph{simplest} ;;numerator and denominator whose quotient differs from @1 by no more ;;than @2. ;; ;;@format ;;@t{(find-ratio 3/97 .0001) @result{} (3 97) ;;(find-ratio 3/97 .001) @result{} (1 32) ;;} ;;@end format (define (find-ratio x e) (find-ratio-between (- x e) (+ x e))) ;;@body ;;@0 returns the list of the @emph{simplest} ;;numerator and denominator between @1 and @2. ;; ;;@format ;;@t{(find-ratio-between 2/7 3/5) @result{} (1 2) ;;(find-ratio-between -3/5 -2/7) @result{} (-1 2) ;;} ;;@end format (define (find-ratio-between x y) (define (sr x y) (let ((fx (inexact->exact (floor x))) (fy (inexact->exact (floor y)))) (cond ((>= fx x) (list fx 1)) ((= fx fy) (let ((rat (sr (/ (- y fy)) (/ (- x fx))))) (list (+ (cadr rat) (* fx (car rat))) (car rat)))) (else (list (+ 1 fx) 1))))) (cond ((< y x) (find-ratio-between y x)) ((>= x y) (list x 1)) ((positive? x) (sr x y)) ((negative? y) (let ((rat (sr (- y) (- x)))) (list (- (car rat)) (cadr rat)))) (else '(0 1))))