# [oct-cvs] Oct commit: oct qd-methods.lisp qd-package.lisp

rtoy rtoy at common-lisp.net
Thu Sep 20 21:04:05 UTC 2007

```Update of /project/oct/cvsroot/oct
In directory clnet:/tmp/cvs-serv8223

Modified Files:
qd-methods.lisp qd-package.lisp
Log Message:
Add RATIONALIZE methods.  Algorithm graciously provided by Bruno
Haible.

qd-package.lisp:

qd-methods.lisp:
o Add RATIONALIZE methods for CL:REAL's and QD-REAL's..

--- /project/oct/cvsroot/oct/qd-methods.lisp	2007/09/19 17:30:04	1.59
+++ /project/oct/cvsroot/oct/qd-methods.lisp	2007/09/20 21:04:05	1.60
@@ -835,6 +835,102 @@
(cl:rational x2)
(cl:rational x3))))

+(defmethod rationalize ((x cl:real))
+  (cl:rationalize x))
+
+;;; The algorithm here is the method described in CLISP.  Bruno Haible has
+;;; graciously given permission to use this algorithm.  He says, "You can use
+;;; it, if you present the following explanation of the algorithm."
+;;;
+;;; Algorithm (recursively presented):
+;;;   If x is a rational number, return x.
+;;;   If x = 0.0, return 0.
+;;;   If x < 0.0, return (- (rationalize (- x))).
+;;;   If x > 0.0:
+;;;     Call (integer-decode-float x). It returns a m,e,s=1 (mantissa,
+;;;     exponent, sign).
+;;;     If m = 0 or e >= 0: return x = m*2^e.
+;;;     Search a rational number between a = (m-1/2)*2^e and b = (m+1/2)*2^e
+;;;     with smallest possible numerator and denominator.
+;;;     Note 1: If m is a power of 2, we ought to take a = (m-1/4)*2^e.
+;;;       But in this case the result will be x itself anyway, regardless of
+;;;       the choice of a. Therefore we can simply ignore this case.
+;;;     Note 2: At first, we need to consider the closed interval [a,b].
+;;;       but since a and b have the denominator 2^(|e|+1) whereas x itself
+;;;       has a denominator <= 2^|e|, we can restrict the seach to the open
+;;;       interval (a,b).
+;;;     So, for given a and b (0 < a < b) we are searching a rational number
+;;;     y with a <= y <= b.
+;;;     Recursive algorithm fraction_between(a,b):
+;;;       c := (ceiling a)
+;;;       if c < b
+;;;         then return c       ; because a <= c < b, c integer
+;;;         else
+;;;           ; a is not integer (otherwise we would have had c = a < b)
+;;;           k := c-1          ; k = floor(a), k < a < b <= k+1
+;;;           return y = k + 1/fraction_between(1/(b-k), 1/(a-k))
+;;;                             ; note 1 <= 1/(b-k) < 1/(a-k)
+;;;
+;;; You can see that we are actually computing a continued fraction expansion.
+;;;
+;;; Algorithm (iterative):
+;;;   If x is rational, return x.
+;;;   Call (integer-decode-float x). It returns a m,e,s (mantissa,
+;;;     exponent, sign).
+;;;   If m = 0 or e >= 0, return m*2^e*s. (This includes the case x = 0.0.)
+;;;   Create rational numbers a := (2*m-1)*2^(e-1) and b := (2*m+1)*2^(e-1)
+;;;   (positive and already in lowest terms because the denominator is a
+;;;   power of two and the numerator is odd).
+;;;   Start a continued fraction expansion
+;;;     p[-1] := 0, p[0] := 1, q[-1] := 1, q[0] := 0, i := 0.
+;;;   Loop
+;;;     c := (ceiling a)
+;;;     if c >= b
+;;;       then k := c-1, partial_quotient(k), (a,b) := (1/(b-k),1/(a-k)),
+;;;            goto Loop
+;;;   finally partial_quotient(c).
+;;;   Here partial_quotient(c) denotes the iteration
+;;;     i := i+1, p[i] := c*p[i-1]+p[i-2], q[i] := c*q[i-1]+q[i-2].
+;;;   At the end, return s * (p[i]/q[i]).
+;;;   This rational number is already in lowest terms because
+;;;   p[i]*q[i-1]-p[i-1]*q[i] = (-1)^i.
+;;;
+(defmethod rationalize ((x qd-real))
+  ;; This is a fairly straigtforward implementation of the iterative
+  ;; algorithm above.
+  (multiple-value-bind (frac expo sign)
+      (integer-decode-float x)
+    (cond ((or (zerop frac) (>= expo 0))
+	   (if (minusp sign)
+	       (- (ash frac expo))
+	       (ash frac expo)))
+	  (t
+	   ;; expo < 0 and (2*m-1) and (2*m+1) are coprime to 2^(1-e),
+	   ;; so build the fraction up immediately, without having to do
+	   ;; a gcd.
+	   (let ((a (/ (- (* 2 frac) 1) (ash 1 (- 1 expo))))
+		 (b (/ (+ (* 2 frac) 1) (ash 1 (- 1 expo))))
+		 (p0 0)
+		 (q0 1)
+		 (p1 1)
+		 (q1 0))
+	     (do ((c (ceiling a) (ceiling a)))
+		 ((< c b)
+		  (let ((top (+ (* c p1) p0))
+			(bot (+ (* c q1) q0)))
+		    (/ (if (minusp sign)
+			   (- top)
+			   top)
+		       bot)))
+	       (let* ((k (- c 1))
+		      (p2 (+ (* k p1) p0))
+		      (q2 (+ (* k q1) q0)))
+		 (psetf a (/ (- b k))
+			b (/ (- a k)))
+		 (setf p0 p1
+		       q0 q1
+		       p1 p2
+		       q1 q2))))))))

(define-compiler-macro + (&whole form &rest args)
(declare (ignore form))
--- /project/oct/cvsroot/oct/qd-package.lisp	2007/09/18 12:46:36	1.40
+++ /project/oct/cvsroot/oct/qd-package.lisp	2007/09/20 21:04:05	1.41
@@ -168,6 +168,7 @@
#:decf
#:float-digits
#:rational
+	   #:rationalize
)
;; Export types
(:export #:qd-real
@@ -237,6 +238,7 @@
#:decf
#:float-digits
#:rational
+	   #:rationalize
)
;; Constants
(:export #:+pi+

```