/usr/share/zenlisp/rmath.l is in zenlisp 2013.11.22-2.
This file is owned by root:root, with mode 0o644.
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; By Nils M Holm, 2007, 2008
; Feel free to copy, share, and modify this code.
; See the file LICENSE for details.
; would use REQUIRE, but REQUIRE is in BASE
(cond ((defined 'base) :f)
(t (load base)))
(define rmath :t)
(require 'imath)
(define (numerator x)
(reverse (cdr (memq '/ (reverse x)))))
(define (denominator x) (cdr (memq '/ x)))
(define (rational-p x)
(and (listp x)
(memq '/ x)
(integer-p (numerator x))
(integer-p (denominator x))))
(define (r-number-p x)
(or (integer-p x)
(rational-p x)))
(define (make-rational num den)
(append num '#/ den))
(define (rational x)
(cond ((rational-p x) x)
(t (make-rational x '#1))))
(define (r-zero x)
(cond ((rational-p x) (r= x '#0))
(t (i-zero x))))
(define (r-one x)
(cond ((rational-p x) (r= x '#1))
(t (i-one x))))
(define (%least-terms x)
(let ((cd (gcd (numerator x) (denominator x))))
(cond ((r-one cd) x)
(t (make-rational (quotient (numerator x) cd)
(quotient (denominator x) cd))))))
(define (%decay x)
(cond ((r-one (denominator x))
(numerator x))
(t x)))
(define r-normalize
(let ()
(lambda (x)
(letrec
((norm-sign (lambda (x)
(let ((num (numerator x))
(den (denominator x)))
(let ((pos (eq (i-negative num)
(i-negative den))))
(make-rational (cond (pos (i-abs num))
(t (cons '- (i-abs num))))
(i-abs den)))))))
(cond ((rational-p x)
(%decay (%least-terms (norm-sign x))))
(t (i-normalize x)))))))
(define (r-integer x)
(let ((xlt (+ '#0 x)))
(cond ((rational-p xlt)
(bottom (list 'r-integer x)))
(t xlt))))
(define (r-natural x)
(i-natural (r-integer x)))
(define (r-abs x)
(cond ((rational-p x)
(make-rational (i-abs (numerator x))
(i-abs (denominator x))))
(t (i-abs x))))
(define (%equalize a b)
(let ((num-a (numerator a))
(num-b (numerator b))
(den-a (denominator a))
(den-b (denominator b)))
(let ((cd (gcd den-a den-b)))
(cond
((r-one cd)
(list (make-rational (i* num-a den-b)
(i* den-a den-b))
(make-rational (i* num-b den-a)
(i* den-b den-a))))
(t (list (make-rational (quotient (i* num-a den-b) cd)
(quotient (i* den-a den-b) cd))
(make-rational (quotient (i* num-b den-a) cd)
(quotient (i* den-b den-a) cd))))))))
(define r+
(let ()
(lambda (a b)
(let ((factors (%equalize (rational a) (rational b)))
(radd
(lambda (a b)
(r-normalize
(make-rational (i+ (numerator a) (numerator b))
(denominator a))))))
(radd (car factors) (cadr factors))))))
(define r-
(let ()
(lambda (a b)
(let ((factors (%equalize (rational a) (rational b)))
(rsub
(lambda (a b)
(r-normalize
(make-rational (i- (numerator a) (numerator b))
(denominator a))))))
(rsub (car factors) (cadr factors))))))
(define (r* a b)
(let ((rmul
(lambda (a b)
(r-normalize
(make-rational (i* (numerator a) (numerator b))
(i* (denominator a) (denominator b)))))))
(rmul (rational a) (rational b))))
(define (r/ a b)
(let ((rdiv
(lambda (a b)
(r-normalize
(make-rational (i* (numerator a) (denominator b))
(i* (denominator a) (numerator b)))))))
(cond ((r-zero b) (bottom (list 'r/ a b)))
(t (rdiv (rational a) (rational b))))))
(define r<
(let ()
(lambda (a b)
(let ((factors (%equalize (rational a) (rational b))))
(i< (numerator (car factors))
(numerator (cadr factors)))))))
(define (r> a b) (r< b a))
(define (r<= a b) (eq (r> a b) :f))
(define (r>= a b) (eq (r< a b) :f))
(define r=
(let ()
(lambda (a b)
(cond ((or (rational-p a) (rational-p b))
(equal (%least-terms (rational a))
(%least-terms (rational b))))
(t (i= a b))))))
(define (r-expt x y)
(letrec
((rx (cond ((i-negative (r-integer y))
(r/ '#1 (rational x)))
(t (rational x))))
(square
(lambda (x)
(r* x x)))
(exp
(lambda (x y)
(cond ((r-zero y) '#1)
((even y)
(square (exp x (quotient y '#2))))
(t (r* x (square (exp x (quotient y '#2)))))))))
(exp rx (i-abs (r-integer y)))))
(define (r-negative x)
(cond ((rational-p x)
(i-negative (numerator (r-normalize x))))
(t (i-negative x))))
(define (r-negate x)
(cond ((rational-p x)
(let ((nx (r-normalize x)))
(make-rational (i-negate (numerator nx))
(denominator nx))))
(t (i-negate x))))
(define (r-max . a) (apply limit r> a))
(define (r-min . a) (apply limit r< a))
(define (r-sqrt square precision)
(let ((e (make-rational '#1 (r-expt '#10 (r-natural precision)))))
(letrec
((sqr (lambda (x)
(cond ((r< (r-abs (r- (r* x x) square))
e)
x)
(t (sqr (r/ (r+ x (r/ square x))
'#2)))))))
(sqr (n-sqrt (r-natural square))))))
(require 'iter)
(define * (arithmetic-iterator rational r* '#1))
(define + (arithmetic-iterator rational r+ '#0))
(define (- . x)
(cond ((null x)
(bottom '(too few arguments to rational -)))
((eq (cdr x) ()) (r-negate (car x)))
(t (fold (lambda (a b)
(r- (rational a) (rational b)))
(car x)
(cdr x)))))
(define (/ . x)
(cond ((null x)
(bottom '(too few arguments to rational /)))
((eq (cdr x) ())
(/ '#1 (car x)))
(t (fold (lambda (a b)
(r/ (rational a) (rational b)))
(car x)
(cdr x)))))
(define < (predicate-iterator rational r<))
(define <= (predicate-iterator rational r<=))
(define = (predicate-iterator rational r=))
(define > (predicate-iterator rational r>))
(define >= (predicate-iterator rational r>=))
(define abs r-abs)
(define *epsilon* '#10)
(define expt r-expt)
(define integer r-integer)
(define max r-max)
(define min r-min)
(define natural r-natural)
(define negate r-negate)
(define negative r-negative)
(define number-p r-number-p)
(define one r-one)
(define (sqrt x) (r-sqrt x *epsilon*))
(define zero r-zero)
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