Up to 128 precision is obtainable if we approximate a real   
   number with a fraction  n/d where n d is single precision 64 bit.   
   What is needed is an "oracle" that determines whether   
   n/d is too small or too large.   
   In this example the oracle say that p/q is larger than x,   
   but this oracle can be replaced to whatever you like.   
      
   `find-part finds the n-th convergent. If n is to large to accommodate   
   the largest ratio is returned.   
      
   U> is used to indicate overflow in adding positive number,   
   a technique acceptable in Forth.   
      
   \ -------------------- 8< --------------------------------------   
   WANT U>   
   VARIABLE x   
   \ For  p1  p2  leave  p2 . (pairs)   
   : PNIP  >R >R 2DROP R> R> ;   
      
   \ Multiply  double  single : result  double .   
   : DS*D      >R  R@ M* SWAP  ROT R> UM* D+ ;   
      
   : D< DNEGATE D+ 0< NIP ;   
      
      
   \ For P Q :" The point is above the line"  Q^2>RATIO*P^2   
   : oracle   DUP 0< 1010 AND THROW   
       >R DUP M* x @ DS*D R> DUP M* D< ;   
      
   \ Leave START value for convergents.   
   : \\:   0 1 1 0 ;   
      
   \ For CONV1 CONV2 incorporate K (cf term), return CONV2 CONV3   
   : \\   >R 2SWAP 2OVER R@ * SWAP R> * SWAP  D+ ;   
      
   \ For consecutive convergents C1 and C2 find next convergent.   
   \ Leave next C2 and C3.   
   VARIABLE CFI   
   : next-convergent   2SWAP 2OVER D+ 2DUP oracle >R   0 CFI !   
      BEGIN  1 CFI +! 2OVER D+ 2DUP oracle R@ <> UNTIL RDROP   
      2OVER DNEGATE D+  ;   
      
   : MAX-INT  -1 1 RSHIFT ;   
      
   \ continued-fraction   
   \ For current values of 'a 'b 'c whatnot find   
   \ the  nth  convergent of as  numerator  denominator .   
   : find-part >R \\:  2SWAP   
       R> 0 DO   
           'next-convergent CATCH IF &* EMIT 2SWAP LEAVE THEN   
           DUP MAX-INT U> IF && EMIT 2SWAP LEAVE THEN   
       LOOP   
       PNIP ;   
      
   \ For A B print its continued fraction and the GCD.   
   : .CF SWAP BEGIN OVER /MOD . DUP WHILE SWAP REPEAT DROP &| EMIT . ;   
      
   \ ----------------------------------------------------------------   
      
   Example. Printing the continued fraction proves that the result   
   is correct.   
      
   \ -------------------- 8< --------------------------------------   
   INCLUDE cfsqrtd.frt   
   S[ ] OK 2 x !   
      
   S[ ] OK 3 find-part   
      
   S[ 5 7 ] OK .CF   
   0 1 2 2 |1   
   S[ ] OK 100 find-part   
   S[ 2015874949414289041 2850877693509864481 ] OK .CF   
   0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   
   2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 |1   
      
   29 x !   
   100 find-part .CF   
   *0 5 2 1 1 2 10 2 1 1 2 10 2 1 1 2 10 2 1 1 2 10 2 1 1 2 10 2 1 1   
   2 10 2 1 1 2 10 2 1 1 2 10 2 2 |1   
   \ -------------------- 8< --------------------------------------   
      
   Groetjes Albert   
   --   
   The Chinese government is satisfied with its military superiority over USA.   
   The next 5 year plan has as primary goal to advance life expectancy   
   over 80 years, like Western Europe.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)