Continued fraction/Arithmetic/G(matrix ng, continued fraction n1, continued fraction n2)

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Task
Continued fraction/Arithmetic/G(matrix ng, continued fraction n1, continued fraction n2)
You are encouraged to solve this task according to the task description, using any language you may know.

This task performs the basic mathematical functions on 2 continued fractions. This requires the full version of matrix NG:

I may perform perform the following operations:

Input the next term of continued fraction N1
Input the next term of continued fraction N2
Output a term of the continued fraction resulting from the operation.

I output a term if the integer parts of and and and are equal. Otherwise I input a term from continued fraction N1 or continued fraction N2. If I need a term from N but N has no more terms I inject .

When I input a term t from continued fraction N1 I change my internal state:

is transposed thus

When I need a term from exhausted continued fraction N1 I change my internal state:

is transposed thus

When I input a term t from continued fraction N2 I change my internal state:

is transposed thus

When I need a term from exhausted continued fraction N2 I change my internal state:

is transposed thus

When I output a term t I change my internal state:

is transposed thus

When I need to choose to input from N1 or N2 I act:

if b and b2 are zero I choose N1
if b is zero I choose N2
if b2 is zero I choose N2
if abs( is greater than abs( I choose N1
otherwise I choose N2

When performing arithmetic operation on two potentially infinite continued fractions it is possible to generate a rational number. eg * should produce 2. This will require either that I determine that my internal state is approaching infinity, or limiting the number of terms I am willing to input without producing any output.

Ada

Translation of: Python
Works with: GCC version 12.2.1
pragma ada_2022;                -- When big_integers were introduced.

with ada.numerics.big_numbers.big_integers;
use ada.numerics.big_numbers.big_integers;

with ada.strings; use ada.strings;
with ada.strings.fixed; use ada.strings.fixed;
with ada.strings.unbounded; use ada.strings.unbounded;

with ada.text_io; use ada.text_io;

procedure BIVARIATE_CONTINUED_FRACTION_TASK is

  package CONTINUED_FRACTIONS is

    type memoization_storage is array (natural range <>) of big_integer;
    type memoization_access is access memoization_storage;

    type continued_fraction_record is abstract tagged
      record
        terminated : boolean := false;   -- Are there no more terms?
        memo_count : natural := 0;       -- How many terms are memoized?
        memo       : memoization_access  -- Memoized terms.
                       := new memoization_storage (0 .. 31);
      end record;

    procedure generate_term (cf : in out continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer) is abstract;

    type continued_fraction is access all
      continued_fraction_record'class; -- The 'class notation is important.

    function term_exists (cf : in continued_fraction;
                          i  : in natural)
                          return boolean;

    function get_term (cf : in continued_fraction;
                       i  : in natural)
                       return big_integer
      with pre => i < cf.memo_count;

    function cf2string (cf        : in continued_fraction;
                        max_terms : in positive := 20)
                        return unbounded_string;

  end CONTINUED_FRACTIONS;

  package body CONTINUED_FRACTIONS is

    function term_exists (cf : in continued_fraction;
                          i  : in natural)
                          return boolean is
      procedure resize_if_necessary is
        memo1 : memoization_access;
      begin
        if cf.memo'length <= i then
          memo1 := new memoization_storage(0 .. 2 * (i + 1));
          for i in 0 .. cf.memo_count - 1 loop
            memo1(i) := cf.memo(i);
          end loop;
          cf.memo := memo1;
        end if;
      end;
      exists : boolean;
      term   : big_integer;
    begin
      if i < cf.memo_count then
        exists := true;
      elsif cf.terminated then
        exists := false;
      else
        resize_if_necessary;
        while cf.memo_count <= i and not cf.terminated loop
          generate_term (cf.all, exists, term);
          if exists then
            cf.memo(cf.memo_count) := term;
            cf.memo_count := cf.memo_count + 1;
          else
            cf.terminated := true;
          end if;
        end loop;
        exists := term_exists (cf, i);
      end if;
      return exists;
    end;

    function get_term (cf : in continued_fraction;
                       i  : in natural)
                       return big_integer is
    begin
      return cf.memo(i);
    end;

    function cf2string (cf        : in continued_fraction;
                        max_terms : in positive := 20)
                        return unbounded_string is
      s    : unbounded_string := null_unbounded_string;
      done : boolean;
      i    : natural;
      term : big_integer;
    begin
      s := s & "[";
      i := 0;
      done := false;
      while not done loop
        if not term_exists (cf, i) then
          s := s & "]";
          done := true;
        elsif i = max_terms then
          s := s & ",...]";
          done := true;
        else
          if i = 1 then
            s := s & ";";
          elsif i /= 0 then
            s := s & ",";
          end if;
          term := get_term (cf, i);
          s := s & trim (term'image, left);
          i := i + 1;
        end if;
      end loop;
      return s;
    end;

  end CONTINUED_FRACTIONS;

  package CONSTANT_TERM_CONTINUED_FRACTIONS is

    use CONTINUED_FRACTIONS;

    type constant_term_continued_fraction_record is
         new continued_fraction_record with
      record
        term : big_integer;
      end record;

    type constant_term_continued_fraction is access all
      constant_term_continued_fraction_record;

    function constant_term_cf (term : in big_integer)
                              return continued_fraction;

    overriding
    procedure generate_term (cf : in out constant_term_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer);

  end CONSTANT_TERM_CONTINUED_FRACTIONS;

  package body CONSTANT_TERM_CONTINUED_FRACTIONS is

    function constant_term_cf (term : in big_integer)
                              return continued_fraction is
      cf : constant_term_continued_fraction;
    begin
      cf := new constant_term_continued_fraction_record;
      cf.term := term;
      return continued_fraction (cf);
    end;

    overriding
    procedure generate_term (cf : in out constant_term_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer) is
    begin
      output_exists := true;
      output := cf.term;
    end;

  end CONSTANT_TERM_CONTINUED_FRACTIONS;

  package INTEGER_CONTINUED_FRACTIONS is

    use CONTINUED_FRACTIONS;

    type integer_continued_fraction_record is
         new continued_fraction_record with
      record
        term : big_integer;
        done : boolean := false;
      end record;

    type integer_continued_fraction is access all
      integer_continued_fraction_record;

    function i2cf (term : in big_integer)
                   return continued_fraction;

    overriding
    procedure generate_term (cf : in out integer_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer);

  end INTEGER_CONTINUED_FRACTIONS;

  package body INTEGER_CONTINUED_FRACTIONS is

    function i2cf (term : in big_integer)
                   return continued_fraction is
      cf : integer_continued_fraction;
    begin
      cf := new integer_continued_fraction_record;
      cf.term := term;
      return continued_fraction (cf);
    end;

    overriding
    procedure generate_term (cf : in out integer_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer) is
    begin
      output_exists := not (cf.done);
      cf.done := true;
      if output_exists then
        output := cf.term;
      end if;
    end;

  end INTEGER_CONTINUED_FRACTIONS;

  package NG8_CONTINUED_FRACTIONS is

    use CONTINUED_FRACTIONS;

    stopping_processing_threshold : big_integer := 2 ** 512;
    infinitization_threshold      : big_integer := 2 ** 64;

    type ng8_continued_fraction_record is
         new continued_fraction_record with
      record
        a12, a1, a2, a : big_integer;
        b12, b1, b2, b : big_integer;
        x, y           : continued_fraction;
        ix, iy         : natural;
        xoverflow      : boolean;
        yoverflow      : boolean;
      end record;

    type ng8_continued_fraction is access all
      ng8_continued_fraction_record;

    function apply_ng8 (a12, a1, a2, a : in big_integer;
                        b12, b1, b2, b : in big_integer;
                        x, y           : in continued_fraction)
                        return continued_fraction;

    -- Addition.
    function "+" (x, y : in continued_fraction)
                  return continued_fraction;
    function "+" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction;
    function "+" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction;

    -- Keeping the same sign. (Effectively clones x as an
    -- ng8_continued_fraction.)
    function "+" (x : in continued_fraction)
                  return continued_fraction;

    -- Subtraction.
    function "-" (x, y : in continued_fraction)
                  return continued_fraction;
    function "-" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction;
    function "-" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction;

    -- Negation.
    function "-" (x : in continued_fraction)
                  return continued_fraction;

    -- Multiplication.
    function "*" (x, y : in continued_fraction)
                  return continued_fraction;
    function "*" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction;
    function "*" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction;

    -- Division.
    function "/" (x, y : in continued_fraction)
                  return continued_fraction;
    function "/" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction;
    function "/" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction;

    -- A rational number as a continued fraction. The terms are
    -- memoized, so this implementation will not be as inefficient as
    -- one might suppose.
    function r2cf (n, d : in big_integer)
                   return continued_fraction;

    overriding
    procedure generate_term (cf : in out ng8_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer);

  end NG8_CONTINUED_FRACTIONS;

  package body NG8_CONTINUED_FRACTIONS is

    use CONTINUED_FRACTIONS;
    use CONSTANT_TERM_CONTINUED_FRACTIONS;

    -- An arbitrary infinite source of non-zero finite terms.
    forever_cf : continued_fraction := constant_term_cf (1234);

    function apply_ng8 (a12, a1, a2, a : in big_integer;
                        b12, b1, b2, b : in big_integer;
                        x, y           : in continued_fraction)
                        return continued_fraction is
      cf : ng8_continued_fraction;
    begin
      cf := new ng8_continued_fraction_record;
      cf.a12 := a12;
      cf.a1  := a1;
      cf.a2  := a2;
      cf.a   := a;
      cf.b12 := b12;
      cf.b1  := b1;
      cf.b2  := b2;
      cf.b   := b;
      cf.x   := x;
      cf.y   := y;
      cf.ix  := 0;
      cf.iy  := 0;
      cf.xoverflow := false;
      cf.yoverflow := false;
      return continued_fraction (cf);
    end;

    function "+" (x, y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, 1, 0, 0, 0, 0, 1, x, y);
    end;

    function "+" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, 0, y, 0, 0, 0, 1, x, forever_cf);
    end;

    function "+" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, 1, x, 0, 0, 0, 1, forever_cf, y);
    end;

    function "+" (x : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, 1, 0, 0, 0, 0, 1, forever_cf, x);
    end;

    function "-" (x, y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, -1, 0, 0, 0, 0, 1, x, y);
    end;

    function "-" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, 0, -y, 0, 0, 0, 1, x, forever_cf);
    end;

    function "-" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, -1, x, 0, 0, 0, 1, forever_cf, y);
    end;

    function "-" (x : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, -1, 0, 0, 0, 0, 1, forever_cf, x);
    end;

    function "*" (x, y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (1, 0, 0, 0, 0, 0, 0, 1, x, y);
    end;

    function "*" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction is
    begin
      return apply_ng8 (0, y, 0, 0, 0, 0, 0, 1, x, forever_cf);
    end;

    function "*" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, x, 0, 0, 0, 0, 1, forever_cf, y);
    end;

    function "/" (x, y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, 0, 0, 0, 0, 1, 0, x, y);
    end;

    function "/" (x : in continued_fraction;
                  y : in big_integer)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 1, 0, 0, 0, 0, 0, y, x, forever_cf);
    end;

    function "/" (x : in big_integer;
                  y : in continued_fraction)
                  return continued_fraction is
    begin
      return apply_ng8 (0, 0, 0, x, 0, 0, 1, 0, forever_cf, y);
    end;

    function r2cf (n, d : in big_integer)
                   return continued_fraction is
    begin
      return apply_ng8 (0, 0, 0, n, 0, 0, 0, d, forever_cf, forever_cf);
    end;

    procedure possibly_infinitize_output (q             : in big_integer;
                                          output_exists : out boolean;
                                          output        : out big_integer) is
    begin
      output_exists := abs (q) < abs (infinitization_threshold);
      if output_exists then
        output := q;
      end if;
    end;

    procedure divide (a, b : in big_integer;
                      q, r : out big_integer) is
    begin
      if b /= 0 then
        q := a / b;
        r := a rem b;
      end if;
    end;

    function too_big (num : big_integer)
                      return boolean is
    begin
      return (abs (stopping_processing_threshold) <= abs (num));
    end;

    function any_too_big (a, b, c, d, e, f, g, h : in big_integer)
                          return boolean is
    begin
      return (too_big (a) or else
              too_big (b) or else
              too_big (c) or else
              too_big (d) or else
              too_big (e) or else
              too_big (f) or else
              too_big (g) or else
              too_big (h));
    end;

    overriding
    procedure generate_term (cf : in out ng8_continued_fraction_record;
                             output_exists : out boolean;
                             output        : out big_integer) is

      a12, a1, a2, a        : big_integer;
      b12, b1, b2, b        : big_integer;
      q12, q1, q2, q        : big_integer;
      r12, r1, r2, r        : big_integer;
      absorb_y_instead_of_x : boolean;
      done                  : boolean;

      function all_b_are_zero
               return boolean is
      begin
        return (b12 = 0 and b1 = 0 and b2 = 0 and b = 0);
      end;

      function all_q_are_equal
               return boolean is
      begin
        return (q = q1 and q = q2 and q = q12);
      end;

      procedure compare_fractions is
        n1, n2, n : big_integer;
      begin
        -- Rather than compare fractions, we will put the numerators over
        -- a common denominator of b*b1*b2, and then compare the new
        -- numerators.
        n1 := a1 * b2 * b;
        n2 := a2 * b1 * b;
        n  := a  * b1 * b2;
        absorb_y_instead_of_x := (abs (n1 - n) <= abs (n2 - n));
      end;

      procedure absorb_x_term is
        term                           : big_integer;
        new_a12, new_a1, new_a2, new_a : big_integer;
        new_b12, new_b1, new_b2, new_b : big_integer;
      begin
        new_a2 := a12;
        new_a  := a1;
        new_b2 := b12;
        new_b  := b1;
        if not cf.xoverflow and then term_exists (cf.x, cf.ix) then
          term := get_term (cf.x, cf.ix);
          new_a12 := a2 + (a12 * term);
          new_a1  := a  + (a1  * term);
          new_b12 := b2 + (b12 * term);
          new_b1  := b  + (b1  * term);
          if any_too_big (new_a12, new_a1, new_a2, new_a,
                          new_b12, new_b1, new_b2, new_b) then
            cf.xoverflow := true;
            new_a12 := a12;
            new_a1  := a1;
            new_b12 := b12;
            new_b1  := b1;
          else
            cf.ix := cf.ix + 1;
          end if;
        else
          new_a12 := a12;
          new_a1  := a1;
          new_b12 := b12;
          new_b1  := b1;
        end if;
        a12 := new_a12;
        a1  := new_a1;
        a2  := new_a2;
        a   := new_a;
        b12 := new_b12;
        b1  := new_b1;
        b2  := new_b2;
        b   := new_b;
      end;

      procedure absorb_y_term is
        term                           : big_integer;
        new_a12, new_a1, new_a2, new_a : big_integer;
        new_b12, new_b1, new_b2, new_b : big_integer;
      begin
        new_a1 := a12;
        new_a  := a2;
        new_b1 := b12;
        new_b  := b2;
        if not cf.yoverflow and then term_exists (cf.y, cf.iy) then
          term := get_term (cf.y, cf.iy);
          new_a12 := a1 + (a12 * term);
          new_a2  := a  + (a2  * term);
          new_b12 := b1 + (b12 * term);
          new_b2  := b  + (b2  * term);
          if any_too_big (new_a12, new_a1, new_a2, new_a,
                          new_b12, new_b1, new_b2, new_b) then
            cf.yoverflow := true;
            new_a12 := a12;
            new_a2  := a2;
            new_b12 := b12;
            new_b2  := b2;
          else
            cf.iy := cf.iy + 1;
          end if;
        else
          new_a12 := a12;
          new_a2  := a2;
          new_b12 := b12;
          new_b2  := b2;
        end if;
        a12 := new_a12;
        a1  := new_a1;
        a2  := new_a2;
        a   := new_a;
        b12 := new_b12;
        b1  := new_b1;
        b2  := new_b2;
        b   := new_b;
      end;

      procedure absorb_term is
      begin
        if absorb_y_instead_of_x then
          absorb_y_term;
        else
          absorb_x_term;
        end if;
      end;

    begin
      a12 := cf.a12;
      a1  := cf.a1;
      a2  := cf.a2;
      a   := cf.a;
      b12 := cf.b12;
      b1  := cf.b1;
      b2  := cf.b2;
      b   := cf.b;

      done := false;
      while not done loop
        absorb_y_instead_of_x := false;
        if all_b_are_zero then
          -- There are no more terms.
          output_exists := false;
          done := true;
        elsif b2 = 0 and b = 0 then
          null;
        elsif b2 = 0 or b = 0 then
          absorb_y_instead_of_x := true;
        elsif b1 = 0 then
          null;
        else
          divide (a12, b12, q12, r12);
          divide (a1, b1, q1, r1);
          divide (a2, b2, q2, r2);
          divide (a, b, q, r);
          if b12 /= 0 and then all_q_are_equal then
            -- Output a term.
            cf.a12 := b12;
            cf.a1  := b1;
            cf.a2  := b2;
            cf.a   := b;
            cf.b12 := r12;
            cf.b1  := r1;
            cf.b2  := r2;
            cf.b   := r;
            possibly_infinitize_output (q, output_exists, output);
            done := true;
          else
            compare_fractions;
          end if;
        end if;
        absorb_term;
      end loop;
    end;

  end NG8_CONTINUED_FRACTIONS;

  use CONTINUED_FRACTIONS;
  use CONSTANT_TERM_CONTINUED_FRACTIONS;
  use INTEGER_CONTINUED_FRACTIONS;
  use NG8_CONTINUED_FRACTIONS;

  procedure show (expression : in string;
                  cf         : in continued_fraction;
                  note       : in string := "") is
    expr     : string := 19 * ' ';
    contfrac : string := 48 * ' ';
  begin
    move (source => expression,
          target => expr,
          justify => right);
    put (expr);
    put (" =>  ");
    if note = "" then
      put_line (to_string (cf2string (cf)));
    else
      move (source => to_string (cf2string (cf)),
            target => contfrac,
            justify => left);
      put (contfrac);
      put_line (note);
    end if;
  end;

  golden_ratio : continued_fraction := constant_term_cf (1);
  silver_ratio : continued_fraction := constant_term_cf (2);
  one          : continued_fraction := i2cf (1);
  two          : continued_fraction := i2cf (2);
  three        : continued_fraction := i2cf (3);
  four         : continued_fraction := i2cf (4);
  sqrt2        : continued_fraction := silver_ratio - 1;

begin

  show ("golden ratio", golden_ratio, "(1 + sqrt(5))/2");
  show ("silver ratio", silver_ratio, "(1 + sqrt(2))");
  show ("sqrt(2)", sqrt2, "silver ratio minus 1");
  show ("13/11", r2cf (13, 11));
  show ("22/7", r2cf (22, 7), "approximately pi");
  show ("1", one);
  show ("2", two);
  show ("3", three);
  show ("4", four);
  show ("(1 + 1/sqrt(2))/2",
        apply_ng8 (0, 1, 0, 0, 0, 0, 2, 0, silver_ratio, sqrt2),
        "method 1");
  show ("(1 + 1/sqrt(2))/2",
        apply_ng8 (1, 0, 0, 1, 0, 0, 0, 8, silver_ratio, silver_ratio),
        "method 2");
  show ("(1 + 1/sqrt(2))/2",
        (one / 2) * (one + (1 / sqrt2)),
        "method 3");
  show ("sqrt(2) + sqrt(2)", sqrt2 + sqrt2);
  show ("sqrt(2) - sqrt(2)", sqrt2 - sqrt2);
  show ("sqrt(2) * sqrt(2)", sqrt2 * sqrt2);
  show ("sqrt(2) / sqrt(2)", sqrt2 / sqrt2);

end BIVARIATE_CONTINUED_FRACTION_TASK;

-- local variables:
-- mode: indented-text
-- tab-width: 2
-- end:
Output:
$ gnatmake -q -g bivariate_continued_fraction_task.adb && ./bivariate_continued_fraction_task
       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   (1 + sqrt(2))
            sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   silver ratio minus 1
              13/11 =>  [1;5,2]
               22/7 =>  [3;7]                                           approximately pi
                  1 =>  [1]
                  2 =>  [2]
                  3 =>  [3]
                  4 =>  [4]
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
  sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
  sqrt(2) - sqrt(2) =>  [0]
  sqrt(2) * sqrt(2) =>  [2]
  sqrt(2) / sqrt(2) =>  [1]

ATS

Translation of: Python

Using 128-bit integers

(Margin note: This program is a bug-fix of an ATS program on which I based Python code. It does not really matter, however, which program came first. So I am calling this a translation from Python.)

The following program uses GNU C extensions: 128-bit integers, and integer operations that detect overflow. I add the 128-bit integers as a new g0int(int128knd) type. The overflow-detecting operations I call +!, -!, and *!. There are no multiple-precision integers or rationals. Rather than detect numbers "getting too large", I catch integer overflows. I use the most negative 128-bit value to represent "no finite term".

I have not included any code to weed out large terms that show up where "infinities" belong.

(Margin note: Sometimes infinite sequences get truncated due to overflow, though I have not seen this happen very near the beginning of a continued fraction. You can give a "maximum number of terms to print" argument to this program, to see the phenomenon for yourself. It happens relatively quickly with 128-bit integers.)

(*------------------------------------------------------------------*)

#include "share/atspre_staload.hats"
staload UN = "prelude/SATS/unsafe.sats"

%{#
#include <stdint.h>
#include <limits.h>
#include <float.h>
#include <math.h>
%}

#define NIL list_nil ()
#define ::  list_cons

exception gint_overflow of ()

(*------------------------------------------------------------------*)

extern fn {tk : tkind}
g0int_neginf :
  () -<> g0int tk

extern fn {tk : tkind}
g0int_add_overflow_exn :
  (g0int tk, g0int tk) -< !exn > g0int tk

extern fn {tk : tkind}
g0int_sub_overflow_exn :
  (g0int tk, g0int tk) -< !exn > g0int tk

extern fn {tk : tkind}
g0int_mul_overflow_exn :
  (g0int tk, g0int tk) -< !exn > g0int tk

infixl ( + ) +!
infixl ( - ) -!
infixl ( * ) *!

overload neginf with g0int_neginf
overload +! with g0int_add_overflow_exn
overload -! with g0int_sub_overflow_exn
overload *! with g0int_mul_overflow_exn

(*------------------------------------------------------------------*)
(* 128-bit integers. *)

%{#

/* The most negative int128 will be treated as "neginf" or "negative
   infinity". For our purposes the sign will not matter, though. */
#define neginf_int128() (((__int128) 1) << 127)

#define neg_c(x)    (-(x))
#define add_c(x, y) ((x) + (y))
#define sub_c(x, y) ((x) - (y))
#define mul_c(x, y) ((x) * (y))
#define div_c(x, y) ((x) / (y))
#define mod_c(x, y) ((x) % (y))
#define eq_c(x, y)  (((x) == (y)) ? atsbool_true : atsbool_false)
#define neq_c(x, y) (((x) != (y)) ? atsbool_true : atsbool_false)
#define lt_c(x, y)  (((x) < (y)) ? atsbool_true : atsbool_false)
#define lte_c(x, y) (((x) <= (y)) ? atsbool_true : atsbool_false)
#define gt_c(x, y)  (((x) > (y)) ? atsbool_true : atsbool_false)
#define gte_c(x, y) (((x) >= (y)) ? atsbool_true : atsbool_false)

/* GNU extensions for detection of integer overflow. */
#define add_overflow(x, y, pz) \
  (__builtin_add_overflow ((x), (y), (pz)) ? \
          atsbool_true : atsbool_false)
#define sub_overflow(x, y, pz) \
  (__builtin_sub_overflow ((x), (y), (pz)) ? \
          atsbool_true : atsbool_false)
#define mul_overflow(x, y, pz) \
  (__builtin_mul_overflow ((x), (y), (pz)) ? \
          atsbool_true : atsbool_false)

%}

tkindef int128_kind = "__int128" (* A GNU extension. *)
stadef int128knd = int128_kind
typedef int128_0 = g0int int128knd
typedef int128_1 (i : int) = g1int (int128knd, i)
stadef int128 = int128_1 // 2nd-select
stadef int128 = int128_0 // 1st-select
stadef Int128 = [i : int] int128_1 i

extern fn g0int_neginf_int128 : () -<> int128 = "mac#neginf_int128"
extern fn g0int_neg_int128 : int128 -<> int128 = "mac#neg_c"
extern fn g0int_add_int128 : (int128, int128) -<> int128 = "mac#add_c"
extern fn g0int_sub_int128 : (int128, int128) -<> int128 = "mac#sub_c"
extern fn g0int_mul_int128 : (int128, int128) -<> int128 = "mac#mul_c"
extern fn g0int_div_int128 : (int128, int128) -<> int128 = "mac#div_c"
extern fn g0int_mod_int128 : (int128, int128) -<> int128 = "mac#mod_c"
extern fn g0int_eq_int128 : (int128, int128) -<> bool = "mac#eq_c"
extern fn g0int_neq_int128 : (int128, int128) -<> bool = "mac#neq_c"
extern fn g0int_lt_int128 : (int128, int128) -<> bool = "mac#lt_c"
extern fn g0int_lte_int128 : (int128, int128) -<> bool = "mac#lte_c"
extern fn g0int_gt_int128 : (int128, int128) -<> bool = "mac#gt_c"
extern fn g0int_gte_int128 : (int128, int128) -<> bool = "mac#gte_c"

implement g0int_neginf<int128knd> () = g0int_neginf_int128 ()
implement g0int_neg<int128knd> x = g0int_neg_int128 x
implement g0int_add<int128knd> (x, y) = g0int_add_int128 (x, y)
implement g0int_sub<int128knd> (x, y) = g0int_sub_int128 (x, y)
implement g0int_mul<int128knd> (x, y) = g0int_mul_int128 (x, y)
implement g0int_div<int128knd> (x, y) = g0int_div_int128 (x, y)
implement g0int_mod<int128knd> (x, y) = g0int_mod_int128 (x, y)
implement g0int_eq<int128knd> (x, y) = g0int_eq_int128 (x, y)
implement g0int_neq<int128knd> (x, y) = g0int_neq_int128 (x, y)
implement g0int_lt<int128knd> (x, y) = g0int_lt_int128 (x, y)
implement g0int_lte<int128knd> (x, y) = g0int_lte_int128 (x, y)
implement g0int_gt<int128knd> (x, y) = g0int_gt_int128 (x, y)
implement g0int_gte<int128knd> (x, y) = g0int_gte_int128 (x, y)

implement g0int2int<intknd,int128knd> i = $UN.cast i
implement g0int2int<int128knd,intknd> i = $UN.cast i
implement g0int2float<int128knd,ldblknd> i = $UN.cast i

implement g0int_iseqz<int128knd> x = (x = g0i2i 0)
implement g0int_isneqz<int128knd> x = (x <> g0i2i 0)
implement g0int_isltz<int128knd> x = (x < g0i2i 0)
implement g0int_isltez<int128knd> x = (x <= g0i2i 0)
implement g0int_isgtz<int128knd> x = (x > g0i2i 0)
implement g0int_isgtez<int128knd> x = (x >= g0i2i 0)

implement g0int_abs<int128knd> x = (if isltz x then ~x else x)

local

  extern fn
  add_overflow_int128 :
    (int128, int128, &int128? >> int128) -< !wrt > bool
      = "mac#add_overflow"

  extern fn
  sub_overflow_int128 :
    (int128, int128, &int128? >> int128) -< !wrt > bool
      = "mac#sub_overflow"

  extern fn
  mul_overflow_int128 :
    (int128, int128, &int128? >> int128) -< !wrt > bool
      = "mac#mul_overflow"

in (* local *)

  fn
  g0int_add_overflow_exn_int128 (x : int128, y : int128)
      :<!exn> int128 =
    let
      var z : int128?
      val overflow = $effmask_wrt add_overflow_int128 (x, y, z)
    in
      if ~overflow then
        z
      else
        $raise gint_overflow ()
    end

  fn
  g0int_sub_overflow_exn_int128 (x : int128, y : int128)
      :<!exn> int128 =
    let
      var z : int128?
      val overflow = $effmask_wrt sub_overflow_int128 (x, y, z)
    in
      if ~overflow then
        z
      else
        $raise gint_overflow ()
    end

  fn
  g0int_mul_overflow_exn_int128 (x : int128, y : int128)
      :<!exn> int128 =
    let
      var z : int128?
      val overflow = $effmask_wrt mul_overflow_int128 (x, y, z)
    in
      if ~overflow then
        z
      else
        $raise gint_overflow ()
    end

end (* local *)

implement
g0int_add_overflow_exn<int128knd> (x, y) =
  g0int_add_overflow_exn_int128 (x, y)

implement
g0int_sub_overflow_exn<int128knd> (x, y) =
  g0int_sub_overflow_exn_int128 (x, y)

implement
g0int_mul_overflow_exn<int128knd> (x, y) =
  g0int_mul_overflow_exn_int128 (x, y)

(*------------------------------------------------------------------*)

(* We will truncate quotients towards zero. *)
infixl ( / ) div divrem
infixl ( mod ) rem
macdef div = g0int_div
macdef rem = g0int_mod
macdef divrem (a, b) =
  let
    val x = ,(a)
    and y = ,(b)
  in
    (* Optimizing C compilers will compute both quotient and remainder
       at the same time. *)
    @(x \g0int_div y, x \g0int_mod y)
  end

(*------------------------------------------------------------------*)
(* Continued fractions.

   cf_generator tk -- A closure that produces terms of type g0int tk,
                      sequentially.

   cf tk           -- A structure from which one can get the ith
                      term of a continued fraction. It gets terms
                      from a cf_generator tk.                       *)

typedef integer = int128
stadef integerknd = int128knd

typedef cf_generator = () -<cloref1> integer

local

  typedef _cf (terminated : bool,
               m          : int,
               n          : int) =
    [m <= n]
    @{
      terminated = bool terminated, (* No more terms? *)
      m = size_t m,         (* The number of terms computed so far. *)
      n = size_t n,         (* The size of memo storage.*)
      memo = arrayref (integer, n), (* Memoized terms. *)
      gen = cf_generator            (* A thunk to generate terms. *)
    }
  typedef _cf (m : int) =
    [terminated : bool]
    [n : int | m <= n]
    _cf (terminated, m, n)
  typedef _cf =
    [m : int]
    _cf m

  fn
  _cf_get_more_terms
            {terminated : bool}
            {m          : int}
            {n          : int}
            {needed     : int | m <= needed; needed <= n}
            (cf         : _cf (terminated, m, n),
             needed     : size_t needed)
      : [m1 : int | m <= m1; m1 <= needed]
        [n1 : int | m1 <= n1]
        _cf (m1 < needed, m1, n1) =
    let
      prval () = lemma_g1uint_param (cf.m)

      macdef memo = cf.memo

      fun
      loop {i : int | m <= i; i <= needed}
           .<needed - i>.
           (i : size_t i)
          : [m1 : int | m <= m1; m1 <= needed]
            [n1 : int | m1 <= n1]
            _cf (m1 < needed, m1, n1) =
        if i = needed then
          @{
            terminated = false,
            m = needed,
            n = (cf.n),
            memo = memo,
            gen = (cf.gen)
          }
        else
          let
            val term = (cf.gen) ()
          in
            if term <> neginf<integerknd> () then
              begin
                memo[i] := term;
                loop (succ i)
              end
            else
              @{
                terminated = true,
                m = i,
                n = (cf.n),
                memo = memo,
                gen = (cf.gen)
              }
          end
    in
      loop (cf.m)
    end

  fn
  _cf_update {terminated : bool}
             {m          : int}
             {n          : int | m <= n}
             {needed     : int}
             (cf         : _cf (terminated, m, n),
              needed     : size_t needed)
      : [terminated1 : bool]
        [m1 : int | m <= m1]
        [n1 : int | m1 <= n1]
        _cf (terminated1, m1, n1) =
    let
      prval () = lemma_g1uint_param (cf.m)
      macdef memo = cf.memo
      macdef gen = cf.gen
    in
      if (cf.terminated) then
        cf
      else if needed <= (cf.m) then
        cf
      else if needed <= (cf.n) then
        _cf_get_more_terms (cf, needed)
      else
        let                     (* Provides twice the room needed. *)
          val n1 = needed + needed
          val memo1 = arrayref_make_elt (n1, g0i2i 0)
          val () =
            let
              var i : [i : nat] size_t i
            in
              for (i := i2sz 0; i < (cf.m); i := succ i)
                memo1[i] := memo[i]
            end
          val cf1 =
            @{
              terminated = false,
              m = (cf.m),
              n = n1,
              memo = memo1,
              gen = (cf.gen)
            }
        in
          _cf_get_more_terms (cf1, needed)
        end
    end

in (* local *)

  typedef cf = ref _cf

  extern fn cf_make : cf_generator -> cf

  extern fn cf_get_at_size : {i : int} (cf, size_t i) -> integer
  extern fn cf_get_at_int : {i : nat} (cf, int i) -> integer

  extern fn cf2generator : cf -> cf_generator

  (* The precedence of the overloads has to exceed that of ref[] *)
  overload cf_get_at with cf_get_at_size of 1
  overload cf_get_at with cf_get_at_int of 1
  overload [] with cf_get_at of 1

  implement
  cf_make gen =
    let
      #ifndef CF_START_SIZE #then
        #define CF_START_SIZE 32
      #endif
    in
      ref
        @{
          terminated = false,
          m = i2sz 0,
          n = i2sz CF_START_SIZE,
          memo = arrayref_make_elt (i2sz CF_START_SIZE, g0i2i 0),
          gen = gen
        }
    end

  implement
  cf_get_at_size (cf, i) =
    let
      prval () = lemma_g1uint_param i
      val cf1 = _cf_update (!cf, succ i)
    in
      !cf := cf1;
      if i < (cf1.m) then
        arrayref_get_at<integer> (cf1.memo, i)
      else
        neginf<integerknd> ()
    end

  implement cf_get_at_int (cf, i) = cf_get_at_size (cf, g1i2u i)

  implement
  cf2generator cf =
    let
      val i : ref Size_t = ref (i2sz 0)
    in
      lam () =>
        let
          val term = cf[!i]
        in
          !i := succ !i;
          term
        end
    end

end (* local *)

(*------------------------------------------------------------------*)
(* Make a string from an int128. *)

fn
int128_2string (i : int128) : string =
  let
    fun
    loop (i : int128, accum : List0 charNZ) : List0 charNZ =
      if iseqz i then
        accum
      else
        let
          val @(i, digit) = i divrem (g0i2i 10)
          val digit = (g0i2i digit) : int
          val digit = g1ofg0 (int2char0 (digit + (char2int0 '0')))
          val () = assertloc (digit <> '\0')
        in
          loop (i, digit :: accum)
        end

    val minus_sign : Char = '-'
    val () = assertloc (minus_sign <> '\0')
  in
    if iseqz i then
      "0"
    else if i = neginf<integerknd> () then
      "neginf"
    else if isltz i then
      strnptr2string (string_implode (minus_sign :: (loop (~i, NIL))))
    else
      strnptr2string (string_implode (loop (i, NIL)))
  end

implement tostring_val<integer> i = $effmask_all int128_2string i

implement fprint_val<integer> (f, i) = fprint! (f, tostring_val<integer> i)
fn fprint_integer (f : FILEref, i : integer) = fprint_val<integer> (f, i)
fn print_integer (i : integer) = fprint_integer (stdout_ref, i)
fn prerr_integer (i : integer) = fprint_integer (stderr_ref, i)

overload fprint with fprint_integer
overload print with print_integer
overload prerr with prerr_integer

(*------------------------------------------------------------------*)
(* Converting a continued fraction to a string. *)

extern fn cf2string_with_default_max_terms : cf -> string
extern fn cf2string_given_max_terms : (cf, Size_t) -> string

overload cf2string with cf2string_with_default_max_terms
overload cf2string with cf2string_given_max_terms

val cf2string_default_max_terms : ref Size_t = ref (i2sz 20)

implement
cf2string_with_default_max_terms cf =
  cf2string_given_max_terms (cf, !cf2string_default_max_terms)
  
implement
cf2string_given_max_terms (cf, max_terms) =
  let
    fun
    loop (i     : Size_t,
          accum : List0 string)
        : List0 string =
      let
        val term = cf[i]
      in
        if i = max_terms then
          begin
            if term = neginf<integerknd> () then
              "]" :: accum
            else
              ",...]" :: accum
          end
        else if term = neginf<integerknd> () then
          "]" :: accum
        else
          let
            val separator =
              (if i = i2sz 0 then
                 ""
               else if i = i2sz 1 then
                 ";"
               else
                 ",")
            and term_str = tostring_val<integer> term
          in
            loop (succ i, term_str :: separator :: accum)
          end
        end

    val string_lst = loop (i2sz 0, "[" :: NIL)
  in
    strptr2string (stringlst_concat (list_vt2t (reverse string_lst)))
  end

(*------------------------------------------------------------------*)
(* The continued fraction for a rational number or integer. *)

typedef ratnum = @(integer, integer)

fn
r2cf (ratnum : ratnum) : cf =
  cf_make
    let
      val ratnum_ref : ref ratnum = ref ratnum
    in
      lam () =<cloref1>
        let
          val @(n, d) = !ratnum_ref
        in
          if iseqz d then
            neginf<integerknd> ()
          else
            let
              val @(q, r) = n divrem d
            in
              !ratnum_ref := @(d, r);
              q
            end
        end
    end

fn
i2cf (intnum : integer) : cf =
  r2cf @(intnum, g0i2i 1)

(*------------------------------------------------------------------*)
(* Application of a homographic function to a continued fraction. *)

typedef ng4 = @(integer, integer, integer, integer)

fn
apply_ng4 (ng4 : ng4, x : cf) : cf =
  cf_make
    let
      val state : ref @(ng4, Size_t) = ref @(ng4, i2sz 0)
    in
      lam () =<cloref1>
        let
          fnx
          loop (a1 : integer,
                a  : integer,
                b1 : integer,
                b  : integer,
                i  : Size_t)
              : integer =
            if (iseqz b1) * (iseqz b) then
              neginf<integerknd> ()
            else if (iseqz b1) + (iseqz b) then
              absorb_term (a1, a, b1, b, i)
            else
              let
                val @(q, r) = a divrem b
                and @(q1, r1) = a1 divrem b1
              in
                if q1 <> q then
                  absorb_term (a1, a, b1, b, i)
                else
                  begin
                    !state :=
                      @(@(b1, b, r1, r), i);
                    q
                  end
              end
          and
          absorb_term (a1 : integer,
                       a  : integer,
                       b1 : integer,
                       b  : integer,
                       i  : Size_t)
              : integer =
            let
              val term = x[i]
            in
              if term <> neginf<integerknd> () then
                loop (a + (a1 * term), a1,
                      b + (b1 * term), b1, succ i)
              else
                loop (a1, a1, b1, b1, succ i)
            end

          val @(@(a1, a, b1, b), i) = !state
        in
          loop (a1, a, b1, b, i)
        end
    end

(*------------------------------------------------------------------*)
(* Some basic operations involving only one continued fraction. *)

fn
cf_neg (x : cf) : cf =
  apply_ng4 (@(g0i2i ~1, g0i2i 0, g0i2i 0, g0i2i 1), x)

fn
cf_add_ratnum (x : cf, y : ratnum) : cf =
  let
    val @(n, d) = y
  in
    apply_ng4 (@(d, n, g0i2i 0, d), x)
  end

fn
ratnum_add_cf (x : ratnum, y : cf) : cf =
  cf_add_ratnum (y, x)

fn
cf_add_integer (x : cf, y : integer) : cf =
  cf_add_ratnum (x, @(y, g0i2i 1))

fn
integer_add_cf (x : integer, y : cf) : cf =
  cf_add_ratnum (y, @(x, g0i2i 1))

fn
cf_sub_ratnum (x : cf, y : ratnum) : cf =
  cf_add_ratnum (x, @(~(y.0), (y.1)))

fn
ratnum_sub_cf (x : ratnum, y : cf) : cf =
  let
    val @(n, d) = x
  in
    apply_ng4 (@(~d, n, g0i2i 0, d), y)
  end

fn
cf_sub_integer (x : cf, y : integer) : cf =
  cf_add_ratnum (x, @(~y, g0i2i 1))

fn
integer_sub_cf (x : integer, y : cf) : cf =
  ratnum_sub_cf (@(x, g0i2i 1), y)

fn
cf_mul_ratnum (x : cf, y : ratnum) : cf =
  let
    val @(n, d) = y
  in
    apply_ng4 (@(n, g0i2i 0, g0i2i 0, d), x)
  end

fn
ratnum_mul_cf (x : ratnum, y : cf) : cf =
  cf_mul_ratnum (y, x)

fn
cf_mul_integer (x : cf, y : integer) : cf =
  cf_mul_ratnum (x, @(y, g0i2i 1))

fn
integer_mul_cf (x : integer, y : cf) : cf =
  cf_mul_ratnum (y, @(x, g0i2i 1))

fn
cf_div_ratnum (x : cf, y : ratnum) : cf =
  cf_mul_ratnum (x, @(y.1, y.0))

fn
ratnum_div_cf (x : ratnum, y : cf) : cf =
  let
    val @(n, d) = x
  in
    apply_ng4 (@(g0i2i 0, n, d, g0i2i 0), y)
  end

fn
cf_div_integer (x : cf, y : integer) : cf =
  cf_mul_ratnum (x, @(g0i2i 1, y))

fn
integer_div_cf (x : integer, y : cf) : cf =
  ratnum_div_cf (@(x, g0i2i 1), y)

overload ~ with cf_neg

overload + with cf_add_ratnum
overload + with ratnum_add_cf
overload + with cf_add_integer
overload + with integer_add_cf

overload - with cf_sub_ratnum
overload - with ratnum_sub_cf
overload - with cf_sub_integer
overload - with integer_sub_cf

overload * with cf_mul_ratnum
overload * with ratnum_mul_cf
overload * with cf_mul_integer
overload * with integer_mul_cf

overload / with cf_div_ratnum
overload / with ratnum_div_cf
overload / with cf_div_integer
overload / with integer_div_cf

(*------------------------------------------------------------------*)
(* Application of a bihomographic function to a continued fraction. *)

typedef ng8 = @(integer, integer, integer, integer,
                integer, integer, integer, integer)

fn
apply_ng8 (ng8 : ng8, x : cf, y : cf) : cf =
  cf_make
    let
      val ng : ref ng8 = ref ng8
      and xsource : ref cf_generator = ref (cf2generator x)
      and ysource : ref cf_generator  = ref (cf2generator y)

      fn neginf_source () :<cloref1> integer = neginf<integerknd> ()
    in
      lam () =<cloref1>
        let
          fnx
          recurs () : integer =
            let
              val @(a12, a1, a2, a, b12, b1, b2, b) = !ng
              val bz = iseqz b
              and b1z = iseqz b1
              and b2z = iseqz b2
              and b12z = iseqz b12
            in
              if b12z * b1z * b2z * bz then
                neginf<integerknd> ()
              else if bz * b2z then
                absorb_x_term ()
              else if bz + b2z then
                absorb_y_term ()
              else if b1z then
                absorb_x_term ()
              else
                let
                  val @(q, r) = a divrem b
                  and @(q1, r1) = a1 divrem b1
                  and @(q2, r2) = a2 divrem b2
                  and @(q12, r12) =
                    (if ~b12z then
                       a12 divrem b12
                     else
                       @(neginf (), neginf ())) : @(integer, integer)
                in
                  if (~b12z) * (q = q1) * (q = q2) * (q = q12) then
                    begin       (* Output a term. *)
                      !ng := @(b12, b1, b2, b, r12, r1, r2, r);
                      q
                    end
                  else
                    let
                      (* Put numerators over a common denominator,
                         then compare the numerators. *)
                      val n = a *! b1 *! b2
                      and n1 = a1 *! b *! b2
                      and n2 = a2 *! b *! b1
                    in
                      if abs (n1 -! n) > abs (n2 -! n) then
                        absorb_x_term ()
                      else
                        absorb_y_term ()
                    end
                end
            end
          and
          absorb_x_term () : integer =
            let
              val @(a12, a1, a2, a, b12, b1, b2, b) = !ng
              val term = (!xsource) ()
            in
              if term <> neginf<integerknd> () then
                begin
                  try
                    let
                      val a12_ = a2 +! (a12 *! term)
                      and a1_ = a +! (a1 *! term)
                      and a2_ = a12
                      and a_ = a1
                      and b12_ = b2 +! (b12 *! term)
                      and b1_ = b +! (b1 *! term)
                      and b2_ = b12
                      and b_ = b1
                    in
                      !ng := @(a12_, a1_, a2_, a_,
                               b12_, b1_, b2_, b_);
                      (* Be aware: this is not a tail recursion! *)
                      recurs ()
                    end
                  with
                  | ~ gint_overflow () =>
                    begin
                      (* Replace the sources with ones that return no
                         terms. (You have to replace BOTH sources.) *)
                      !ng := @(a12, a1, a12, a1, b12, b1, b12, b1);
                      !xsource := neginf_source;
                      !ysource := neginf_source;
                      recurs ()
                    end
                end
              else
                begin
                  !ng := @(a12, a1, a12, a1, b12, b1, b12, b1);
                  recurs ()
                end
            end
          and
          absorb_y_term () : integer =
            let
              val @(a12, a1, a2, a, b12, b1, b2, b) = !ng
              val term = (!ysource) ()
            in
              if term <> neginf<integerknd> () then
                begin
                  try
                    let
                      val a12_ = a1 +! (a12 *! term)
                      and a1_ = a12
                      and a2_ = a +! (a2 *! term)
                      and a_ = a2
                      and b12_ = b1 +! (b12 *! term)
                      and b1_ = b12
                      and b2_ = b +! (b2 *! term)
                      and b_ = b2
                    in
                      !ng := @(a12_, a1_, a2_, a_,
                               b12_, b1_, b2_, b_);
                      (* Be aware: this is not a tail recursion! *)
                      recurs ()
                    end
                  with
                  | ~ gint_overflow () =>
                    begin
                      (* Replace the sources with ones that return no
                         terms. (You have to replace BOTH sources.) *)
                      !ng := @(a12, a12, a2, a2, b12, b12, b2, b2);
                      !xsource := neginf_source;
                      !ysource := neginf_source;
                      recurs ()
                    end
                end
              else
                begin
                  !ng := @(a12, a12, a2, a2, b12, b12, b2, b2);
                  recurs ()
                end
            end
        in
          recurs ()
        end
    end

(*------------------------------------------------------------------*)
(* Some basic operations on two continued fractions. *)

fn
cf_add_cf (x : cf, y : cf) : cf =
  apply_ng8 (@(g0i2i 0, g0i2i 1, g0i2i 1, g0i2i 0,
               g0i2i 0, g0i2i 0, g0i2i 0, g0i2i 1), x, y)

fn
cf_sub_cf (x : cf, y : cf) : cf =
  apply_ng8 (@(g0i2i 0, g0i2i 1, g0i2i ~1, g0i2i 0,
               g0i2i 0, g0i2i 0, g0i2i 0, g0i2i 1), x, y)

fn
cf_mul_cf (x : cf, y : cf) : cf =
  apply_ng8 (@(g0i2i 1, g0i2i 0, g0i2i 0, g0i2i 0,
               g0i2i 0, g0i2i 0, g0i2i 0, g0i2i 1), x, y)

fn
cf_div_cf (x : cf, y : cf) : cf =
  apply_ng8 (@(g0i2i 0, g0i2i 1, g0i2i 0, g0i2i 0,
               g0i2i 0, g0i2i 0, g0i2i 1, g0i2i 0), x, y)

overload + with cf_add_cf
overload - with cf_sub_cf
overload * with cf_mul_cf
overload / with cf_div_cf

(*------------------------------------------------------------------*)

typedef charptr = $extype"char *"

fn
show_with_note (expression : string,
                cf         : cf,
                note       : string)
    : void =
  if note = "" then
    ignoret ($extfcall (int, "printf", "%19s =>  %s\n",
                        $UN.cast{charptr} expression,
                        $UN.cast{charptr} (cf2string cf)))
  else
    ignoret ($extfcall (int, "printf", "%19s =>  %-46s  %s\n",
                        $UN.cast{charptr} expression,
                        $UN.cast{charptr} (cf2string cf),
                        $UN.cast{charptr} note))

fn
show_without_note (expression : string, cf : cf) : void =
  show_with_note (expression, cf, "")

overload show with show_with_note
overload show with show_without_note

(*------------------------------------------------------------------*)

val golden_ratio = cf_make (lam () =<cloref1> (g0i2i 1) : integer)
val silver_ratio = cf_make (lam () =<cloref1> (g0i2i 2) : integer)
val sqrt2 = silver_ratio - g0i2i 1

val frac_13_11 = r2cf @(g0i2i 13, g0i2i 11)
val frac_22_7 = r2cf @(g0i2i 22, g0i2i 7)

val one = i2cf (g0i2i 1)
val two = i2cf (g0i2i 2)
val three = i2cf (g0i2i 3)
val four = i2cf (g0i2i 4)

implement
main (argc, argv) =
  begin
    if 2 <= argc then
      let
        val n = g0string2uint (argv_get_at (argv, 1)) : uint
      in
        (* Set the maximum number of terms to print. *)
        !cf2string_default_max_terms := g1u2u (g1ofg0 n)
      end;

    show ("golden ratio", golden_ratio, "(1 + sqrt(5))/2");
    show ("silver ratio", silver_ratio, "1 + sqrt(2)");
    show ("sqrt(2)", sqrt2);
    show ("13/11", frac_13_11);
    show ("22/7", frac_22_7);
    show ("one", one);
    show ("two", two);
    show ("three", three);
    show ("four", four);
    show ("(1 + 1/sqrt(2))/2",
          apply_ng8 (@(g0i2i 0, g0i2i 1, g0i2i 0, g0i2i 0,
                       g0i2i 0, g0i2i 0, g0i2i 2, g0i2i 0),
                     silver_ratio, sqrt2),
          "method 1");
    show ("(1 + 1/sqrt(2))/2",
          apply_ng8 (@(g0i2i 1, g0i2i 0, g0i2i 0, g0i2i 1,
                       g0i2i 0, g0i2i 0, g0i2i 0, g0i2i 8),
                     silver_ratio, silver_ratio),
          "method 2");
    show ("(1 + 1/sqrt(2))/2", (g0i2i 1 + (one / sqrt2)) / two,
          "method 3");
    show ("sqrt(2) + sqrt(2)", sqrt2 + sqrt2);
    show ("sqrt(2) - sqrt(2)", sqrt2 - sqrt2);
    show ("sqrt(2) * sqrt(2)", sqrt2 * sqrt2);
    show ("sqrt(2) / sqrt(2)", sqrt2 / sqrt2);

    0
  end

(*------------------------------------------------------------------*)
Output:
$ patscc -g -O2 -std=gnu2x -DATS_MEMALLOC_GCBDW bivariate-continued-fraction-task-memoizing.dats -lgc && ./a.out
       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   1 + sqrt(2)
            sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
              13/11 =>  [1;5,2]
               22/7 =>  [3;7]
                one =>  [1]
                two =>  [2]
              three =>  [3]
               four =>  [4]
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
  sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
  sqrt(2) - sqrt(2) =>  [0]
  sqrt(2) * sqrt(2) =>  [2;7530688524100]
  sqrt(2) / sqrt(2) =>  [1]

Using multiple precision numbers

Translation of: Standard ML
Library: ats2-xprelude
Library: GMP

For this program you need ats2-xprelude.

The program closely follows the Standard ML code, so one can compare the two languages with each other. They have similar syntaxes, but are very different. Notice, for instance, that ATS has overloads, whereas SML does not. (SML has signatures with respective namespaces.) In ATS, a function is not a closure unless you explicitly make it one, whereas in SML no special notation is needed. And so on.

ATS is translated to C, and its functions (except closures) are essentially just C functions. One can write Arduino code and Linux kernel modules in ATS, because ATS is, in some sense, an elaborate way to write C. Nevertheless, there is enough similarity between ATS and Standard ML to easily translate the SML code for this Rosetta Code task to ATS.

I have broken the program into three files, to demonstrate what an ATS program might look like, if it were broken into separately compiled parts.

The first file is an "interface" specification for a continued_fraction data type. The file is called continued_fraction.sats:

(* "Static" file. (Exported declarations.) *)

(* To set up a predictable name-mangling scheme:  *)
#define ATS_PACKNAME "rosetta-code.continued_fraction"

(* Load declarations from ats2-xprelude: *)
#include "xprelude/HATS/xprelude_sats.hats"
staload "xprelude/SATS/exrat.sats"

(* A term_generator thunk generates terms, which a continued_fraction
   data structure memoizes. The internals of continued_fraction are
   not exposed here. It is an abstract type, the size of (but not the
   same type as) a pointer. SIDE NOTE: In ATS2, we get a conventional
   function, rather than a closure, unless we say explicitly that we
   want a closure; "cloref1" means a particular kind of closure one
   often uses when linking the program with Boehm GC. *)
typedef term_generator = () -<cloref1> Option exrat
abstype continued_fraction = ptr

(* Create a continued fraction. *)
fn continued_fraction_make : term_generator -> continued_fraction

(* Does the indexed term exist? *)
fn term_exists : (continued_fraction, intGte 0) -> bool

(* Retrieve the indexed term. Raise an exception if there is no such
   term. The precedence of the overload must exceed that of an
   overload that is in the prelude. (To see what I mean, try removing
   the "of 1".) *)
val get_term_exn : (continued_fraction, intGte 0) -> exrat
overload [] with get_term_exn of 1

(* Use a continued_fraction as a term_generator thunk. *)
fn continued_fraction_to_term_generator :
  continued_fraction -> term_generator

(* Get a human-readable string. *)
val default_max_terms : ref (intGte 1)
fn continued_fraction_to_string_given_max_terms :
  (continued_fraction, intGte 1) -> string
fn continued_fraction_to_string_default_max_terms :
  continued_fraction -> string
overload continued_fraction_to_string with 
  continued_fraction_to_string_given_max_terms
overload continued_fraction_to_string with 
  continued_fraction_to_string_default_max_terms
overload cf2string with continued_fraction_to_string

(* Make a continued_fraction for an integer. *)
fn int_to_continued_fraction : int -> continued_fraction
overload i2cf with int_to_continued_fraction

(* Make a continued_fraction for a rational number. *)
fn exrat_to_continued_fraction : exrat -> continued_fraction
fn rational_to_continued_fraction :
  (int, [d : int | d != 0] int d) -> continued_fraction
overload r2cf with exrat_to_continued_fraction
overload r2cf with rational_to_continued_fraction

(* Make a continued_fraction with one term repeated forever. *)
fn continued_fraction_make_constant_term : int -> continued_fraction
overload constant_term_cf with continued_fraction_make_constant_term

(* Make a continued fraction via binary arithmetic operations. (I have
   not bothered here to implement ng4, although one likely would wish
   to have ng4 as well.)  *)
(* The @() denotes an unboxed tuple. A boxed tuple is written '() and
   would be put in the heap. An unboxed tuple may also be written
   without the @-sign, but then the compiler might confuse it with,
   for instance, an argument list. (ATS2 has conventional argument
   lists that are distinct from tuples, and supports
   call-by-reference, where an argument is mutable.) *)
typedef ng8 = @(exrat, exrat, exrat, exrat,
                exrat, exrat, exrat, exrat)
typedef continued_fraction_binary_op_cloref =
  (continued_fraction, continued_fraction) -<cloref1> continued_fraction
(* ng8_make_int takes ONE argument, which is a tuple. *)
val ng8_make_int : @(int, int, int, int, int, int, int, int) -> ng8
val ng8_stopping_processing_threshold : ref exrat
val ng8_infinitization_threshold : ref exrat
val ng8_apply : ng8 -> continued_fraction_binary_op_cloref
val ng8_apply_add : continued_fraction_binary_op_cloref
val ng8_apply_sub : continued_fraction_binary_op_cloref
val ng8_apply_mul : continued_fraction_binary_op_cloref
val ng8_apply_div : continued_fraction_binary_op_cloref
(* The following two are regular functions, not closures. They are
   translated by the ATS compiler into ordinary C functions. *)
fn ng8_apply_neg : continued_fraction -> continued_fraction
fn ng8_apply_pow : (continued_fraction, int) -> continued_fraction
overload + with ng8_apply_add
overload - with ng8_apply_sub
overload * with ng8_apply_mul
overload / with ng8_apply_div
overload ~ with ng8_apply_neg
overload ** with ng8_apply_pow

(* Miscellanous continued fractions. *)
val zero : continued_fraction
val one : continued_fraction
val two : continued_fraction
val three : continued_fraction
val four : continued_fraction
//
val one_fourth : continued_fraction
val one_third : continued_fraction
val one_half : continued_fraction
val two_thirds : continued_fraction
val three_fourths : continued_fraction
//
val golden_ratio : continued_fraction
val silver_ratio : continued_fraction
val sqrt2 : continued_fraction
val sqrt5 : continued_fraction

The second file is an implementation of the stuff declared in the first file. The second file is called continued_fraction.dats:

(* "Dynamic" file. (Implementations.) *)

(* To set up a predictable name-mangling scheme:  *)
#define ATS_PACKNAME "rosetta-code.continued_fraction"

(* Load templates from the ATS2 prelude: *)
#include "share/atspre_staload.hats"

(* Load declarations and templates from ats2-xprelude: *)
#include "xprelude/HATS/xprelude.hats"
staload "xprelude/SATS/exrat.sats"
staload _ = "xprelude/DATS/exrat.dats"

(* Load the declarations for this package: *)
staload "continued_fraction.sats"

typedef cf_record =
  (* A cf_record is an unboxed record, denoted by @{}. A boxed record
     would be written '{} and would be placed in the heap. Either way,
     it is an immutable record. For a mutable record, we would have to
     use vtypedef to make it a LINEAR type. *)
  @{
    terminated = bool,          (* Is the generator exhausted? *)
    memo_count = size_t,        (* How many terms are memoized? *)
    
    (* An arrszref is an array with runtime bounds checking. An
       arrszref is less efficient than an arrayref, but will not force
       us to use dependent types for the indices. *)
    memo = arrszref exrat,      (* Memoized terms. *)

    generate = term_generator   (* The source of terms. *)
  }

(* The actual type of a continued_fraction is a MUTABLE reference to
   the (immutable) cf_record. Within this file, we may also call the
   type cf_t. *)
typedef cf_t = ref cf_record
assume continued_fraction = cf_t

implement
continued_fraction_make generator =
  let
    val record : cf_record =
      @{
        terminated = false,
        memo_count = i2sz 0,
        memo = arrszref_make_elt<exrat> (i2sz 32, exrat_make (0, 1)),
        generate = generator
      }
  in
    ref record
  end

(* "fn" means a non-recursive function. A function that might be
   recursive is written "fun" (or sometimes "fnx"). Incidentally: it
   is common to see the recursions put into nested functions, with the
   function a programmer is supposed to call being non-recursive. This
   is often a matter of style. (By the way, in a "*.sats" file there
   is no distinction between "fn" and "fun" that I know of.) *)
fn
resize_if_necessary (cf : cf_t, i : size_t) : void =
  let
    val @{
          terminated = terminated,
          memo_count = memo_count,
          memo = memo,
          generate = generate
        } = !cf
  in
    if size memo <= i then
      let
        val new_size = i2sz 2 * (succ i)
        val new_memo =
          arrszref_make_elt<exrat> (new_size, exrat_make (0, 1))
        val new_record : cf_record =
          @{
            terminated = terminated,
            memo_count = memo_count,
            memo = new_memo,
            generate = generate
          }

        var i : size_t          (* A C-style automatic variable. *)
      in
        (* A C-style for-loop. *)
        for (i := i2sz 0; i <> memo_count; i := succ i)
          new_memo[i] := memo[i];

        !cf := new_record
      end
  end

fn
update_terms (cf : cf_t, i : size_t) : void =
  let
    fun
    loop () : void =
      let
        val @{
              terminated = terminated,
              memo_count = memo_count,
              memo = memo,
              generate = generate
            } = !cf
      in
        if terminated then
          ()
        else if i < memo_count then
          ()
        else
          case generate () of
          | None () =>
            let
              val new_record =
                @{
                  terminated = true,
                  memo_count = memo_count,
                  memo = memo,
                  generate = generate
                }
            in
              !cf := new_record
            end
          | Some term =>
            (* "begin-end" is a synonym for "()". *)
            begin
              memo[memo_count] := term;
              let
                val new_record =
                  @{
                    terminated = false,
                    memo_count = succ memo_count,
                    memo = memo,
                    generate = generate
                  }
              in
                !cf := new_record;
                loop ()
              end
            end
      end
  in
    loop ()
  end

implement
term_exists (cf, i) =
  let
    val i = i2sz i

    fun
    loop () =
      let
        val @{
              terminated = terminated,
              memo_count = memo_count,
              memo = memo,
              generate = generate
            } = !cf
      in
        if i < memo_count then
          true
        else if terminated then
          false
        else
          begin
            resize_if_necessary (cf, i);
            update_terms (cf, i);
            loop ()
          end
      end
  in
    loop ()
  end

implement
get_term_exn (cf, i) =
  if i2sz i < (!cf).memo_count then
    let
      val memo = (!cf).memo
    in
      memo[i]
    end
  else
    $raise IllegalArgExn "get_term_exn:out_of_bounds"

implement
continued_fraction_to_term_generator cf =
  let
    val i : ref (intGte 0) = ref 0
  in
    lam () =<cloref1>
      let
        val j = !i
      in
        if term_exists (cf, j) then
          begin
            !i := succ j;
            Some (cf[j])
          end
        else
          None ()
      end
  end

implement default_max_terms = ref 20

implement
continued_fraction_to_string_given_max_terms (cf, max_terms) =
  let
    fun
    loop (i : intGte 0, accum : string) : string =
      if ~term_exists (cf, i) then
        (* The return value of string_append is a LINEAR, MUTABLE
           strptr, which we cast to a nonlinear, immutable string.
           (One could introduce one's own shorthands, though.) *)
        strptr2string (string_append (accum, "]"))
      else if i = max_terms then
        strptr2string (string_append (accum, ",...]"))
      else
        let
          val separator =
            if i = 0 then
              ""
            else if i = 1 then
              ";"
            else
              ","
          and term_string = tostring_val<exrat> cf[i]
        in
          loop (succ i,
                strptr2string (string_append (accum, separator,
                                              term_string)))
        end
  in
    loop (0, "[")
  end

implement
continued_fraction_to_string_default_max_terms cf =
  let
    val max_terms = !default_max_terms
  in
    continued_fraction_to_string_given_max_terms (cf, max_terms)
  end

implement
int_to_continued_fraction i =
  let
    val done : ref bool = ref false
    val i = (g0i2f i) : exrat
  in
    continued_fraction_make
      (lam () =<cloref1>
        if !done then
          None ()
        else
          begin
            !done := true;
            Some i
          end)
  end

implement
exrat_to_continued_fraction num =
  let
    val done : ref bool = ref false
    val num : ref exrat = ref num
  in
    continued_fraction_make
      (lam () =<cloref1>
        if !done then
          None ()
        else
          let
            val q = floor !num
            val r = !num - q
          in
            if iseqz r then
              !done := true
            else
              !num := reciprocal r;
            Some q
          end)
  end

implement
rational_to_continued_fraction (numer, denom) =
  exrat_to_continued_fraction (exrat_make (numer, denom))

implement
continued_fraction_make_constant_term i =
  let
    val i = (g0i2f i) : exrat
  in
    continued_fraction_make (lam () =<cloref1> Some i)
  end

implement
ng8_make_int tuple =
  let
    val @(a12, a1, a2, a, b12, b1, b2, b) = tuple
    fn f (i : int) : exrat = exrat_make (i, 1)
  in
    @(f a12, f a1, f a2, f a, f b12, f b1, f b2, f b)
  end

implement ng8_stopping_processing_threshold =
  ref (exrat_make (2, 1) ** 512)
implement ng8_infinitization_threshold =
  ref (exrat_make (2, 1) ** 64)

fn
too_big (term : exrat) : bool =
  abs (term) >= abs (!ng8_stopping_processing_threshold)

fn
any_too_big (ng : ng8) : bool =
  (* "orelse" may also be (and usually is) written "||", as in C.
     The "orelse" notation resembles that of Standard ML.
     Non-shortcircuiting OR also exists, and can be written "+". *)
  case+ ng of  (* <-- the + sign means all cases must have a match. *)
  | @(a, b, c, d, e, f, g, h) =>
    too_big (a) orelse too_big (b) orelse
    too_big (c) orelse too_big (d) orelse
    too_big (e) orelse too_big (f) orelse
    too_big (g) orelse too_big (h)

fn
infinitize (term : exrat) : Option exrat =
  if abs (term) >= abs (!ng8_infinitization_threshold) then
    None ()
  else
    Some term

val no_terms_source : term_generator =
  lam () =<cloref1> None ()

fn
divide (a : exrat, b : exrat) : @(exrat, exrat) =
  if iseqz b then
    @(exrat_make (0, 1), exrat_make (0, 1))
  else
    (* Do integer division of the numerators of a and b. The following
       particular function does floor division if the divisor is
       positive, ceiling division if the divisor is negative. Thus the
       remainder is never negative. *)
    exrat_numerator_euclid_division (a, b)

implement
ng8_apply ng =
  lam (x, y) =>
    let
      val ng : ref ng8 = ref ng
      and xsource : ref term_generator =
        ref (continued_fraction_to_term_generator x)
      and ysource : ref term_generator =
        ref (continued_fraction_to_term_generator y)

      fn
      all_b_are_zero () : bool =
        let
          val @(_, _, _, _, b12, b1, b2, b) = !ng
        in
          (* Instead of the Standard ML-like notation "andalso", one
             may (and usually does) use the C-like notation
             "&&". There is also non-shortcircuiting AND, written
             "*". *)
          iseqz b andalso
          iseqz b2 andalso
          iseqz b1 andalso
          iseqz b12
        end

      fn
      all_four_equal (a : exrat, b : exrat,
                      c : exrat, d : exrat) : bool =
        a = b && a = c && a = d

      fn
      absorb_x_term () =
        let
          val @(a12, a1, a2, a, b12, b1, b2, b) = !ng
        in
          case (!xsource) () of
          | Some term =>
            let
              val new_ng = (a2 + (a12 * term),
                            a + (a1 * term), a12, a1,
                            b2 + (b12 * term),
                            b + (b1 * term), b12, b1)
            in
              if any_too_big new_ng then
                (* Pretend all further x terms are infinite. *)
                (!ng := @(a12, a1, a12, a1, b12, b1, b12, b1);
                 !xsource := no_terms_source)
              else
                !ng := new_ng
            end
          | None () =>
              !ng := @(a12, a1, a12, a1, b12, b1, b12, b1)
        end

      fn
      absorb_y_term () =
        let
          val @(a12, a1, a2, a, b12, b1, b2, b) = !ng
        in
          case (!ysource) () of
          | Some term =>
            let
              val new_ng = (a1 + (a12 * term), a12,
                            a + (a2 * term), a2,
                            b1 + (b12 * term), b12,
                            b + (b2 * term), b2)
            in
              if any_too_big new_ng then
                (* Pretend all further y terms are infinite. *)
                (!ng := @(a12, a12, a2, a2, b12, b12, b2, b2);
                 !ysource := no_terms_source)
              else
                !ng := new_ng
            end
          | None () =>
              !ng := @(a12, a12, a2, a2, b12, b12, b2, b2)
        end

      fun
      loop () =
        (* ATS2 can do mutual recursion with proper tail calls, but,
           to stay closer to the Standard ML code, here I use only
           single tail recursion. To do mutual recursion with proper
           tail calls, one says "fnx" instead of "fun". *)
        if all_b_are_zero () then
          None ()             (* There are no more terms to output. *)
        else
          let
            val @(_, _, _, _, b12, b1, b2, b) = !ng
          in
            if iseqz b andalso iseqz b2 then
              (absorb_x_term (); loop ())
            else if iseqz b orelse iseqz b2 then
              (absorb_y_term (); loop ())
            else if iseqz b1 then
              (absorb_x_term (); loop ())
            else
              let
                val @(a12, a1, a2, a, _, _, _, _) = !ng
                val @(q12, r12) = divide (a12, b12)
                and @(q1, r1) = divide (a1, b1)
                and @(q2, r2) = divide (a2, b2)
                and @(q, r) = divide (a, b)
              in
                if isneqz b12 andalso
                      all_four_equal (q12, q1, q2, q) then
                  (!ng := (b12, b1, b2, b, r12, r1, r2, r);
                   (* Return a term--or, if a magnitude threshold is
                      reached, return no more terms . *)
                   infinitize q)
                else
                  let
                    (* Put a1, a2, and a over a common denominator and
                       compare some magnitudes. (SIDE NOTE: We are
                       representing big integers as EXACT rationals
                       with denominator one, so in fact could have put
                       a1, a2, and a over their respective
                       denominators and compared the
                       fractions. However, I have retained the
                       phrasing of the Standard ML program.) *)
                    val n1 = a1 * b2 * b
                    and n2 = a2 * b1 * b
                    and n = a * b1 * b2
                  in
                    if abs (n1 - n) > abs (n2 - n) then
                      (absorb_x_term (); loop ())
                    else
                      (absorb_y_term (); loop ())
                    end
                  end
              end
    in
      continued_fraction_make (lam () =<cloref1> loop ())
    end

(* A macro definition: *)
macdef make_op (tuple) = ng8_apply (ng8_make_int ,(tuple))

implement ng8_apply_add = make_op @(0, 1, 1, 0, 0, 0, 0, 1)
implement ng8_apply_sub = make_op @(0, 1, ~1, 0, 0, 0, 0, 1)
implement ng8_apply_mul = make_op @(1, 0, 0, 0, 0, 0, 0, 1)
implement ng8_apply_div = make_op @(0, 1, 0, 0, 0, 0, 1, 0)

(* Here the closure is "wrapped" in an ordinary function. *)
val _ng8_apply_neg = make_op @(0, 0, ~1, 0, 0, 0, 0, 1)
implement ng8_apply_neg cf = _ng8_apply_neg (cf, cf)


val _reciprocal = make_op @(0, 0, 0, 1, 0, 1, 0, 0)

implement
ng8_apply_pow (cf, i) =
  let
    macdef reciprocal cf = _reciprocal (,(cf), ,(cf))

    fun
    loop (x     : continued_fraction,
          n     : int,
          accum : continued_fraction) : continued_fraction =
      if 1 < n then
        let
          val nhalf = n / 2
          and xsquare = x * x
        in
          if nhalf + nhalf <> n then
            loop (xsquare, nhalf, accum * x)
          else
            loop (xsquare, nhalf, accum)
        end
      else if n = 1 then
        accum * x
      else
        accum
  in
    if 0 <= i then
      loop (cf, i, one)
    else
      reciprocal (loop (cf, ~i, one))
  end

implement zero = i2cf 0
implement one = i2cf 1
implement two = i2cf 2
implement three = i2cf 3
implement four = i2cf 4

implement one_fourth = r2cf (1, 4)
implement one_third = r2cf (1, 3)
implement one_half = r2cf (1, 2)
implement two_thirds = r2cf (2, 3)
implement three_fourths = r2cf (3, 4)

implement golden_ratio = constant_term_cf 1
implement silver_ratio = constant_term_cf 2
implement sqrt2 = silver_ratio - one
implement sqrt5 = (two * golden_ratio) - one

The third file is the main program. It is called continued-fraction-task.dats:

(* Main program. *)
(*

  Install ats2-xprelude, being sure to enable GMP support:
  https://sourceforge.net/p/chemoelectric/ats2-xprelude
  
  If you have it installed already, there might have been bugfixes
  since. So try updating.
  
  Then, to compile the program:
  
    patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW \
      $(pkg-config --cflags ats2-xprelude) \
      $(pkg-config --variable=PATSCCFLAGS ats2-xprelude) \
      continued-fraction-task.dats continued_fraction.{s,d}ats \
      $(pkg-config --libs ats2-xprelude) -lgc -lm

*)

(* Load templates from the ATS2 prelude: *)
#include "share/atspre_staload.hats"

staload "continued_fraction.sats" (* Programmer access to exported stuff. *)
dynload "continued_fraction.dats" (* Initialize the "val". *)

(* Load declarations and templates from ats2-xprelude: *)
#include "xprelude/HATS/xprelude.hats"
staload "xprelude/SATS/exrat.sats"
staload _ = "xprelude/DATS/exrat.dats"

fn
make_pad (n : size_t) : string =
  let
    val n = g1ofg0 n
    prval () = lemma_g1uint_param n
    implement string_tabulate$fopr<> _ = ' '
  in
    strnptr2string (string_tabulate<> n)
  end

fn
show_with_note (expression : string,
                cf         : continued_fraction,
                note       : string) : void =
  let
    val cf_str = cf2string cf

    val expr_sz = strlen expression
    and cf_sz = strlen cf_str
    and note_sz = strlen note

    val expr_pad_sz = max (i2sz 19 - expr_sz, i2sz 0)
    and cf_pad_sz =
      if iseqz note_sz then
        i2sz 0
      else
        max (i2sz 48 - cf_sz, i2sz 0)

    val expr_pad = make_pad expr_pad_sz
    and cf_pad = make_pad cf_pad_sz
  in
    println! (expr_pad, expression, " =>  ",
              cf_str, cf_pad, note)
  end

fn
show_without_note (expression : string,
                   cf         : continued_fraction) : void =
  show_with_note (expression, cf, "")

overload show with show_with_note
overload show with show_without_note

implement
main0 () =         (* A main that takes no arguments and returns 0. *)
  begin
    show ("golden ratio", golden_ratio, "(1 + sqrt(5))/2");
    show ("silver ratio", silver_ratio, "(1 + sqrt(2))");
    show ("sqrt2", sqrt2);
    show ("sqrt5", sqrt5);

    show ("1/4", one_fourth);
    show ("1/3", one_third);
    show ("1/2", one_half);
    show ("2/3", two_thirds);
    show ("3/4", three_fourths);

    show ("13/11", r2cf (13, 11));
    show ("22/7", r2cf (22, 7), "approximately pi");

    show ("0", zero);
    show ("1", one);
    show ("2", two);
    show ("3", three);
    show ("4", four);

    show ("4 + 3", four + three);
    show ("4 - 3", four - three);
    show ("4 * 3", four * three);
    show ("4 / 3", four / three);
    show ("4 ** 3", four ** 3);
    show ("4 ** (-3)", four ** (~3));
    show ("negative 4", ~four);

    show ("(1 + 1/sqrt(2))/2",
          (one + (one / sqrt2)) / two, "method 1");
    show ("(1 + 1/sqrt(2))/2",
          silver_ratio * (sqrt2 ** (~3)), "method 2");
    show ("(1 + 1/sqrt(2))/2",
          ((silver_ratio ** 2) + one) / (four * two), "method 3");

    show ("sqrt2 + sqrt2", sqrt2 + sqrt2);
    show ("sqrt2 - sqrt2", sqrt2 - sqrt2);
    show ("sqrt2 * sqrt2", sqrt2 * sqrt2);
    show ("sqrt2 / sqrt2", sqrt2 / sqrt2);
  end
Output:

To compile the program, you might try something like the following (assuming you have Boehm GC):

patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW $(pkg-config --cflags ats2-xprelude) $(pkg-config --variable=PATSCCFLAGS ats2-xprelude) continued-fraction-task.dats continued_fraction.{s,d}ats $(pkg-config --libs ats2-xprelude) -lgc -lm

You have to specify some C language standard, because patscc defaults to C99.

Then run the program by typing

./a.out

The output should resemble that of the Standard ML program from which the ATS was translated. Minus signs might look different:

       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   (1 + sqrt(2))
              sqrt2 =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
              sqrt5 =>  [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]
                1/4 =>  [0;4]
                1/3 =>  [0;3]
                1/2 =>  [0;2]
                2/3 =>  [0;1,2]
                3/4 =>  [0;1,3]
              13/11 =>  [1;5,2]
               22/7 =>  [3;7]                                           approximately pi
                  0 =>  [0]
                  1 =>  [1]
                  2 =>  [2]
                  3 =>  [3]
                  4 =>  [4]
              4 + 3 =>  [7]
              4 - 3 =>  [1]
              4 * 3 =>  [12]
              4 / 3 =>  [1;3]
             4 ** 3 =>  [64]
          4 ** (-3) =>  [0;64]
         negative 4 =>  [-4]
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
      sqrt2 + sqrt2 =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
      sqrt2 - sqrt2 =>  [0]
      sqrt2 * sqrt2 =>  [2]
      sqrt2 / sqrt2 =>  [1]

C

Translation of: Python
Translation of: Scheme

You will need Boehm GC and the GNU Multiple Precision Library.

(Actually you can leave out the Boehm GC parts. The program leaks memory, but harmlessly. Also, you can leave out the C23 attribute specifiers.)

/*------------------------------------------------------------------*/

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <string.h>
#include <limits.h>
#include <float.h>
#include <math.h>
#include <gc/gc.h>              /* Boehm GC. */
#include <gmp.h>                /* GNU Multiple Precision. */

void *
my_malloc (size_t size)
{
  void *p = GC_MALLOC (size);
  if (p == NULL)
    {
      fprintf (stderr, "Memory allocation failed.\n");
      exit (1);
    }
  return p;
}

void *
my_realloc (void *p, size_t size)
{
  void *q = GC_REALLOC (p, size);
  if (q == NULL)
    {
      fprintf (stderr, "Memory allocation failed.\n");
      exit (1);
    }
  return q;
}

void
my_free (void *p)
{
  GC_FREE (p);
}

/*------------------------------------------------------------------*/
/* Some helper functions. */

#define MIN(x, y) (((x) < (y)) ? (x) : (y))
#define MAX(x, y) (((x) > (y)) ? (x) : (y))

char *
string_append1 (const char *s1)
{
  size_t n1 = strlen (s1);
  char *s = my_malloc ((n1 + 1) * sizeof (char));
  s[n1] = '\0';
  memcpy (s, s1, n1);
  return s;
}

char *
string_append2 (const char *s1, const char *s2)
{
  size_t n1 = strlen (s1);
  size_t n2 = strlen (s2);
  char *s = my_malloc ((n1 + n2 + 1) * sizeof (char));
  s[n1 + n2] = '\0';
  memcpy (s, s1, n1);
  memcpy (s + n1, s2, n2);
  return s;
}

char *
string_append3 (const char *s1, const char *s2, const char *s3)
{
  size_t n1 = strlen (s1);
  size_t n2 = strlen (s2);
  size_t n3 = strlen (s3);
  char *s = my_malloc ((n1 + n2 + n3 + 1) * sizeof (char));
  s[n1 + n2 + n3] = '\0';
  memcpy (s, s1, n1);
  memcpy (s + n1, s2, n2);
  memcpy (s + n1 + n2, s3, n3);
  return s;
}

char *
string_repeat (size_t n, const char *s)
{
  /* This brute force implementation will suffice. */
  char *t = "";
  for (size_t i = 0; i != n; i += 1)
    t = string_append2 (t, s);
  return t;
}

/*------------------------------------------------------------------*/

typedef mpz_t *cf_func (size_t i, void *env);

struct cf
{
  bool terminated;              /* Are there no more terms? */
  size_t m;                     /* The number of terms memoized. */
  size_t n;                     /* The size of memoization storage. */
  mpz_t **memo;                 /* Memoization storage. */
  cf_func *func;                /* A function that produces terms. */
  void *env;                    /* An environment for func. */
};

typedef struct cf *cf_t;

cf_t
make_cf (cf_func *func, void *env)
{
  cf_t cf = my_malloc (sizeof (struct cf));
  cf->terminated = false;
  cf->m = 0;
  cf->n = 32;
  cf->memo = my_malloc (cf->n * sizeof (mpz_t *));
  cf->func = func;
  cf->env = env;
  return cf;
}

void
resize_cf (cf_t cf, size_t minimum)
{
  /* Ensure there is at least twice the minimum storage. */
  size_t size = 2 * minimum;
  if (cf->n < size)
    {
      cf->memo = my_realloc (cf->memo, size * sizeof (mpz_t *));
      cf->n = size;
    }
}

void
update_cf (cf_t cf, size_t needed)
{
  /* Ensure there are at least a certain number of finite terms
     memoized (or else that all of them are memoized). */
  if (!cf->terminated && cf->m < needed)
    {
      if (cf->n < needed)
        resize_cf (cf, needed);
      while (!cf->terminated && cf->m != needed)
        {
          cf->memo[cf->m] = cf->func (cf->m, cf->env);
          cf->m += 1;
        }
    }
}

mpz_t *
cf_ref (cf_t cf, size_t i)
{
  /* Get the ith term, or a NULL pointer if there is no finite ith
     term. */
  update_cf (cf, i + 1);
  return (i < cf->m) ? cf->memo[i] : NULL;
}

size_t default_max_terms = 20;

char *
cf2string (cf_t cf, size_t max_terms)
{
  if (max_terms == 0)
    max_terms = default_max_terms;

  size_t i = 0;
  char *s = string_append1 ("[");
  bool done = false;
  while (!done)
    {
      mpz_t *term = cf_ref (cf, i);
      if (term == NULL)
        {
          s = string_append2 (s, "]");
          done = true;
        }
      else if (i == max_terms)
        {
          s = string_append2 (s, ",...]");
          done = true;
        }
      else
        {
          static const char *separators[3] = { "", ";", "," };
          const char *separator = separators[(i <= 1) ? i : 2];
          const char *term_str = mpz_get_str (NULL, 10, *term);
          s = string_append3 (s, separator, term_str);
          i += 1;
        }
    }
  return s;
}

/*------------------------------------------------------------------*/

cf_t golden_ratio;
cf_t silver_ratio;

mpz_t *
return_constant ([[maybe_unused]] size_t i, void *env)
{
  mpz_t *term = my_malloc (sizeof (mpz_t));
  mpz_init_set (*term, *((mpz_t *) env));
  return term;
}

cf_t
make_cf_with_constant_terms (int term_si)
{
  mpz_t *env = my_malloc (sizeof (mpz_t));
  mpz_init_set_si (*env, term_si);
  return make_cf (return_constant, env);
}

/*------------------------------------------------------------------*/

cf_t sqrt2;

mpz_t *
return_sqrt2_term (size_t i, [[maybe_unused]] void *env)
{
  mpz_t *term = my_malloc (sizeof (mpz_t));
  mpz_init_set_si (*term, (i == 0) ? 1 : 2);
  return term;
}

cf_t
make_cf_sqrt2 (void)
{
  return make_cf (return_sqrt2_term, NULL);
}

/*------------------------------------------------------------------*/

cf_t frac_13_11;
cf_t frac_22_7;
cf_t one;
cf_t two;
cf_t three;
cf_t four;

mpz_t *
return_rational_term ([[maybe_unused]] size_t i, void *env)
{
  mpz_t *frac = env;
  mpz_t *term = NULL;
  if (mpz_sgn (frac[1]) != 0)
    {
      term = my_malloc (sizeof (mpz_t));
      mpz_init (*term);
      mpz_t r;
      mpz_init (r);
      mpz_fdiv_qr (*term, r, frac[0], frac[1]);
      mpz_set (frac[0], frac[1]);
      mpz_set (frac[1], r);
    }
  return term;
}

cf_t
make_cf_rational (int numerator_si, int denominator_si)
{
  mpz_t *env = my_malloc (2 * sizeof (mpz_t));
  mpz_init_set_si (env[0], numerator_si);
  mpz_init_set_si (env[1], denominator_si);
  return make_cf (return_rational_term, env);
}

cf_t
make_cf_integer (int integer_si)
{
  return make_cf_rational (integer_si, 1);
}

/*------------------------------------------------------------------*/

/* Thresholds. */
mpz_t number_that_is_too_big;
mpz_t practically_infinite;

struct ng8_env
{
  mpz_t ng[8];
  cf_t x;
  cf_t y;
  size_t ix;
  size_t iy;
  bool xoverflow;
  bool yoverflow;
};

typedef struct ng8_env *ng8_env_t;

enum ng8_index
{
  ng8a12 = 0,
  ng8a1  = 1,
  ng8a2  = 2,
  ng8a   = 3,
  ng8b12 = 4,
  ng8b1  = 5,
  ng8b2  = 6,
  ng8b   = 7
};

static bool
ng8_too_big (const mpz_t ng[8])
{
  bool too_big = false;
  int i = 0;
  while (!too_big && i != 8)
    {
      too_big = (0 <= mpz_cmpabs (ng[i], number_that_is_too_big));
      i += 1;
    }
  return too_big;
}

static bool
treat_ng8_term_as_infinite (const mpz_t term)
{
  return (0 <= mpz_cmpabs (term, practically_infinite));
}

static void
a_plus_bc (mpz_t result, const mpz_t a,  const mpz_t b,
           const mpz_t c)
{
  mpz_set (result, a);
  mpz_addmul (result, b, c);
}

static void
abc (mpz_t result, const mpz_t a,  const mpz_t b, const mpz_t c)
{
  mpz_mul (result, a, b);
  mpz_mul (result, result, c);
}

static void
absorb_x_term (ng8_env_t env)
{
  mpz_t tmp[8];
  for (int i = 0; i != 8; i += 1)
    mpz_init_set (tmp[i], env->ng[i]);
  mpz_t *term = (!env->xoverflow) ? cf_ref (env->x, env->ix) : NULL;
  env->ix += 1;
  mpz_set (env->ng[ng8a2], tmp[ng8a12]);
  mpz_set (env->ng[ng8a], tmp[ng8a1]);
  mpz_set (env->ng[ng8b2], tmp[ng8b12]);
  mpz_set (env->ng[ng8b], tmp[ng8b1]);
  if (term != NULL)
    {
      a_plus_bc (env->ng[ng8a12], tmp[ng8a2], tmp[ng8a12], *term);
      a_plus_bc (env->ng[ng8a1], tmp[ng8a], tmp[ng8a1], *term);
      a_plus_bc (env->ng[ng8b12], tmp[ng8b2], tmp[ng8b12], *term);
      a_plus_bc (env->ng[ng8b1], tmp[ng8b], tmp[ng8b1], *term);
      if (ng8_too_big (env->ng))
        {
          env->xoverflow = true;
          mpz_set (env->ng[ng8a12], tmp[ng8a12]);
          mpz_set (env->ng[ng8a1], tmp[ng8a1]);
          mpz_set (env->ng[ng8b12], tmp[ng8b12]);
          mpz_set (env->ng[ng8b1], tmp[ng8b1]);
        }
    }
}

static void
absorb_y_term (ng8_env_t env)
{
  mpz_t tmp[8];
  for (int i = 0; i != 8; i += 1)
    mpz_init_set (tmp[i], env->ng[i]);
  mpz_t *term = (!env->yoverflow) ? cf_ref (env->y, env->iy) : NULL;
  env->iy += 1;
  mpz_set (env->ng[ng8a1], tmp[ng8a12]);
  mpz_set (env->ng[ng8a], tmp[ng8a2]);
  mpz_set (env->ng[ng8b1], tmp[ng8b12]);
  mpz_set (env->ng[ng8b], tmp[ng8b2]);
  if (term != NULL)
    {
      a_plus_bc (env->ng[ng8a12], tmp[ng8a1], tmp[ng8a12], *term);
      a_plus_bc (env->ng[ng8a2], tmp[ng8a], tmp[ng8a2], *term);
      a_plus_bc (env->ng[ng8b12], tmp[ng8b1], tmp[ng8b12], *term);
      a_plus_bc (env->ng[ng8b2], tmp[ng8b], tmp[ng8b2], *term);
      if (ng8_too_big (env->ng))
        {
          env->yoverflow = true;
          mpz_set (env->ng[ng8a12], tmp[ng8a12]);
          mpz_set (env->ng[ng8a2], tmp[ng8a2]);
          mpz_set (env->ng[ng8b12], tmp[ng8b12]);
          mpz_set (env->ng[ng8b2], tmp[ng8b2]);
        }
    }
}

mpz_t *
return_ng8_term ([[maybe_unused]] size_t i, void *env)
{
  /* The McCabe complexity of this function is high. Please be careful
     if modifying the code. */

  ng8_env_t p = env;

  mpz_t *term = NULL;

  bool done = false;
  while (!done)
    {
      const bool b12_zero = (mpz_sgn (p->ng[ng8b12]) == 0);
      const bool b1_zero = (mpz_sgn (p->ng[ng8b1]) == 0);
      const bool b2_zero = (mpz_sgn (p->ng[ng8b2]) == 0);
      const bool b_zero = (mpz_sgn (p->ng[ng8b]) == 0);

      if (b_zero && b1_zero && b2_zero && b12_zero)
        done = true;            /* There are no more terms. */
      else if (b_zero && b2_zero)
        absorb_x_term (p);
      else if (b_zero || b2_zero)
        absorb_y_term (p);
      else if (b1_zero)
        absorb_x_term (p);
      else
        {
          mpz_t q, r;
          mpz_inits (q, r, NULL);
          mpz_t q1, r1;
          mpz_inits (q1, r1, NULL);
          mpz_t q2, r2;
          mpz_inits (q2, r2, NULL);
          mpz_t q12, r12;
          mpz_inits (q12, r12, NULL);

          mpz_fdiv_qr (q, r, p->ng[ng8a], p->ng[ng8b]);
          mpz_fdiv_qr (q1, r1, p->ng[ng8a1], p->ng[ng8b1]);
          mpz_fdiv_qr (q2, r2, p->ng[ng8a2], p->ng[ng8b2]);
          if (!b12_zero)
            mpz_fdiv_qr (q12, r12, p->ng[ng8a12], p->ng[ng8b12]);

          if (!b12_zero
              && mpz_cmp (q, q1) == 0
              && mpz_cmp (q, q2) == 0
              && mpz_cmp (q, q12) == 0)
            {
              // Output a term.
              mpz_set (p->ng[ng8a12], p->ng[ng8b12]);
              mpz_set (p->ng[ng8a1],  p->ng[ng8b1]);
              mpz_set (p->ng[ng8a2],  p->ng[ng8b2]);
              mpz_set (p->ng[ng8a],   p->ng[ng8b]);
              mpz_set (p->ng[ng8b12], r12);
              mpz_set (p->ng[ng8b1],  r1);
              mpz_set (p->ng[ng8b2],  r2);
              mpz_set (p->ng[ng8b],   r);
              if (!treat_ng8_term_as_infinite (q))
                {
                  term = my_malloc (sizeof (mpz_t));
                  mpz_init_set (*term, q);
                }
              done = true;
            }
          else
            {
              /* Rather than compare fractions, we will put the
                 numerators over a common denominator of b*b1*b2, and
                 then compare the new numerators. */
              mpz_t n, n1, n2, n1_diff, n2_diff;
              mpz_inits (n, n1, n2, n1_diff, n2_diff, NULL);
              abc (n, p->ng[ng8a], p->ng[ng8b1], p->ng[ng8b2]);
              abc (n1, p->ng[ng8a1], p->ng[ng8b], p->ng[ng8b2]);
              abc (n2, p->ng[ng8a2], p->ng[ng8b], p->ng[ng8b1]);
              mpz_sub (n1_diff, n1, n);
              mpz_sub (n2_diff, n2, n);
              if (mpz_cmpabs (n1_diff, n2_diff) > 0)
                absorb_x_term (p);
              else
                absorb_y_term (p);
            }
        }
    }

  return term;
}

cf_t
make_cf_ng8 (int ng[8], cf_t x, cf_t y)
{
  ng8_env_t env = my_malloc (sizeof (struct ng8_env));
  for (int i = 0; i != 8; i += 1)
    mpz_init_set_si (env->ng[i], ng[i]);
  env->x = x;
  env->y = y;
  env->ix = 0;
  env->iy = 0;
  env->xoverflow = false;
  env->yoverflow = false;
  return make_cf (return_ng8_term, env);
}

/*------------------------------------------------------------------*/

static void *
gmp_malloc (size_t alloc_size)
{
  return my_malloc (alloc_size);
}

static void *
gmp_realloc (void *p,
             [[maybe_unused]] size_t old_size,
             size_t new_size)
{
  return my_realloc (p, new_size);
}

static void
gmp_free (void *p, [[maybe_unused]] size_t size)
{
  /* There is no need for us to explicitly free memory, and
     performance might even suffer if we do. On the other hand, maybe
     GMP will free memory that otherwise would have been passed over
     for collection. */
  my_free (p);                  /* <-- optional */
}

void
show (const char *expression, cf_t cf, const char *note)
{
  size_t nexpr = strlen (expression);
  char *padding = string_repeat (MAX (19, nexpr + 1) - nexpr, " ");
  char *line = string_append3 (padding, expression, " =>  ");
  char *cfstr = cf2string (cf, 0);
  line = string_append2 (line, cfstr);
  if (note != NULL)
    {
      size_t ncfstr = strlen (cfstr);
      padding = string_repeat (MAX (48, ncfstr + 1) - ncfstr, " ");
      line = string_append3 (line, padding, note);
    }
  puts (line);
}

int ng8_add[8] = { 0, 1, 1, 0, 0, 0, 0, 1 };
int ng8_sub[8] = { 0, 1, -1, 0, 0, 0, 0, 1 };
int ng8_mul[8] = { 1, 0, 0, 0, 0, 0, 0, 1 };
int ng8_div[8] = { 0, 1, 0, 0, 0, 0, 1, 0 };

int
main (void)
{
  GC_INIT ();

  /* GMP has to be told to use Boehm GC as its allocator. */
  mp_set_memory_functions (gmp_malloc, gmp_realloc, gmp_free);

  /* Initialize thresholds, to values chosen merely for
     demonstration. */
  mpz_init_set_si (number_that_is_too_big, 1);
  mpz_mul_2exp (number_that_is_too_big, number_that_is_too_big,
                512);           /* 2**512 */
  mpz_init_set_si (practically_infinite, 1);
  mpz_mul_2exp (practically_infinite, practically_infinite,
                64);            /* 2**64 */

  /* Initialize global continued fractions. */
  golden_ratio = make_cf_with_constant_terms (1);
  silver_ratio = make_cf_with_constant_terms (2);
  sqrt2 = make_cf_sqrt2 ();
  frac_13_11 = make_cf_rational (13, 11);
  frac_22_7 = make_cf_rational (22, 7);
  one = make_cf_integer (1);
  two = make_cf_integer (2);
  three = make_cf_integer (3);
  four = make_cf_integer (4);

  /* Divide the silver ratio by 2 times the square root of 2. */
  int ng8_method1[8] = { 0, 1, 0, 0, 0, 0, 2, 0 };
  cf_t method1 = make_cf_ng8 (ng8_method1, silver_ratio, sqrt2);

  /* Add 1/8 to 1/8th of the square of the silver ratio. */
  int ng8_method2[8] = { 1, 0, 0, 1, 0, 0, 0, 8 };
  cf_t method2 = make_cf_ng8 (ng8_method2, silver_ratio,
                              silver_ratio);

  /* Thrice divide the silver ratio by the square root of 2. */
  cf_t method3 = make_cf_ng8 (ng8_div, silver_ratio, sqrt2);
  method3 = make_cf_ng8 (ng8_div, method3, sqrt2);
  method3 = make_cf_ng8 (ng8_div, method3, sqrt2);

  show ("golden ratio", golden_ratio, "(1 + sqrt(5))/2");
  show ("silver ratio", silver_ratio, "1 + sqrt(2)");
  show ("sqrt(2)", sqrt2, NULL);
  show ("13/11", frac_13_11, NULL);
  show ("22/7", frac_22_7, NULL);
  show ("one", one, NULL);
  show ("two", two, NULL);
  show ("three", three, NULL);
  show ("four", four, NULL);
  show ("(1 + 1/sqrt(2))/2", method1, "method 1");
  show ("(1 + 1/sqrt(2))/2", method2, "method 2");
  show ("(1 + 1/sqrt(2))/2", method3, "method 3");
  show ("sqrt(2) + sqrt(2)", make_cf_ng8 (ng8_add, sqrt2, sqrt2),
        NULL);
  show ("sqrt(2) - sqrt(2)", make_cf_ng8 (ng8_sub, sqrt2, sqrt2),
        NULL);
  show ("sqrt(2) * sqrt(2)", make_cf_ng8 (ng8_mul, sqrt2, sqrt2),
        NULL);
  show ("sqrt(2) / sqrt(2)", make_cf_ng8 (ng8_div, sqrt2, sqrt2),
        NULL);

  return 0;
}

/*------------------------------------------------------------------*/
Output:
$ gcc -std=gnu2x -Wall -Wextra -g bivariate-continued-fraction-task-gmp.c -lgmp -lgc && ./a.out
       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   1 + sqrt(2)
            sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
              13/11 =>  [1;5,2]
               22/7 =>  [3;7]
                one =>  [1]
                two =>  [2]
              three =>  [3]
               four =>  [4]
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
  sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
  sqrt(2) - sqrt(2) =>  [0]
  sqrt(2) * sqrt(2) =>  [2]
  sqrt(2) / sqrt(2) =>  [1]

C++

Uses matrixNG, NG_4 and NG from Continued_fraction/Arithmetic/G(matrix_NG,_Contined_Fraction_N)#C++, and r2cf from Continued_fraction/Arithmetic/Construct_from_rational_number#C++

/* Implement matrix NG
   Nigel Galloway, February 12., 2013
*/
class NG_8 : public matrixNG {
  private: int a12, a1, a2, a, b12, b1, b2, b, t;
           double ab, a1b1, a2b2, a12b12;
  const int chooseCFN(){return fabs(a1b1-ab) > fabs(a2b2-ab)? 0 : 1;}
  const bool needTerm() {
    if (b1==0 and b==0 and b2==0 and b12==0) return false;
    if (b==0){cfn = b2==0? 0:1; return true;} else ab = ((double)a)/b;
    if (b2==0){cfn = 1; return true;} else a2b2 = ((double)a2)/b2;
    if (b1==0){cfn = 0; return true;} else a1b1 = ((double)a1)/b1;
    if (b12==0){cfn = chooseCFN(); return true;} else a12b12 = ((double)a12)/b12;
    thisTerm = (int)ab;
    if (thisTerm==(int)a1b1 and thisTerm==(int)a2b2 and thisTerm==(int)a12b12){
      t=a; a=b; b=t-b*thisTerm; t=a1; a1=b1; b1=t-b1*thisTerm; t=a2; a2=b2; b2=t-b2*thisTerm; t=a12; a12=b12; b12=t-b12*thisTerm;
      haveTerm = true; return false;
    }
    cfn = chooseCFN();
    return true;
  }
  void consumeTerm(){if(cfn==0){a=a1; a2=a12; b=b1; b2=b12;} else{a=a2; a1=a12; b=b2; b1=b12;}}
  void consumeTerm(int n){
    if(cfn==0){t=a; a=a1; a1=t+a1*n; t=a2; a2=a12; a12=t+a12*n; t=b; b=b1; b1=t+b1*n; t=b2; b2=b12; b12=t+b12*n;}
    else{t=a; a=a2; a2=t+a2*n; t=a1; a1=a12; a12=t+a12*n; t=b; b=b2; b2=t+b2*n; t=b1; b1=b12; b12=t+b12*n;}
  }
  public:
  NG_8(int a12, int a1, int a2, int a, int b12, int b1, int b2, int b): a12(a12), a1(a1), a2(a2), a(a), b12(b12), b1(b1), b2(b2), b(b){
}};

Testing

[3;7] + [0;2]

int main() {
  NG_8 a(0,1,1,0,0,0,0,1);
  r2cf n2(22,7);
  r2cf n1(1,2);
  for(NG n(&a, &n1, &n2); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;

  NG_4 a3(2,1,0,2);
  r2cf n3(22,7);
  for(NG n(&a3, &n3); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  return 0;
}
Output:
3 1 1 1 4
3 1 1 1 4

[1:5,2] * [3;7]

int main() {
  NG_8 b(1,0,0,0,0,0,0,1);
  r2cf b1(13,11);
  r2cf b2(22,7);
  for(NG n(&b, &b1, &b2); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  for(NG n(&a, &b2, &b1); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  for(r2cf cf(286,77); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
  std::cout << std::endl;
  return 0;
}
Output:
3 1 2 2
3 1 2 2

[1:5,2] - [3;7]

int main() {
  NG_8 c(0,1,-1,0,0,0,0,1);
  r2cf c1(13,11);
  r2cf c2(22,7);
  for(NG n(&c, &c1, &c2); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  for(r2cf cf(-151,77); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
  std::cout << std::endl;
  return 0;
}
Output:
-1 -1 -24 -1 -2
-1 -1 -24 -1 -2

Divide [] by [3;7]

int main() {
  NG_8 d(0,1,0,0,0,0,1,0);
  r2cf d1(22*22,7*7);
  r2cf d2(22,7);
  for(NG n(&d, &d1, &d2); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  return 0;
}
Output:
3 7

([0;3,2] + [1;5,2]) * ([0;3,2] - [1;5,2])

int main() {
  r2cf a1(2,7);
  r2cf a2(13,11);
  NG_8 na(0,1,1,0,0,0,0,1);
  NG A(&na, &a1, &a2); //[0;3,2] + [1;5,2]
  r2cf b1(2,7);
  r2cf b2(13,11);
  NG_8 nb(0,1,-1,0,0,0,0,1);
  NG B(&nb, &b1, &b2); //[0;3,2] - [1;5,2]
  NG_8 nc(1,0,0,0,0,0,0,1); //A*B
  for(NG n(&nc, &A, &B); n.moreTerms(); std::cout << n.nextTerm() << " ");
  std::cout << std::endl;
  for(r2cf cf(2,7); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
  std::cout << std::endl;
  for(r2cf cf(13,11); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
  std::cout << std::endl;
  for(r2cf cf(-7797,5929); cf.moreTerms(); std::cout << cf.nextTerm() << " ");
  std::cout << std::endl;
  return 0;
}

Common Lisp

Translation of: Python
(defstruct (cf (:conc-name %%cf-)
               (:constructor make-cf (generator)))
  "continued fraction"
  (generator            nil :type function)
  (terminated-p         nil :type boolean)
  (memo  (make-array '(32)) :type (array integer))
  (memo-count             0 :type fixnum))

(defstruct (ng8 (:constructor ng8 (a12 a1 a2 a
                                   b12 b1 b2 b)))
  "coefficients of a bihomographic function"
  (a12 0 :type integer)
  (a1  0 :type integer)
  (a2  0 :type integer)
  (a   0 :type integer)
  (b12 0 :type integer)
  (b1  0 :type integer)
  (b2  0 :type integer)
  (b   0 :type integer))

(defun cf-ref (cf i)
  "Return the ith term, or nil if there is none."
  (declare (cf cf) (fixnum i))
  (defun get-more-terms (needed)
    (declare (fixnum needed))
    (loop while (not (%%cf-terminated-p cf))
          while (< (%%cf-memo-count cf) needed)
          do (let ((term (funcall (%%cf-generator cf))))
               (cond (term (let ((memo (%%cf-memo cf))
                                 (m (%%cf-memo-count cf)))
                             (setf (aref memo m) term)
                             (setf (%%cf-memo-count cf) (1+ m))))
                     (t (setf (%%cf-terminated-p cf) t))))))
  (defun update (needed)
    (declare (fixnum needed))
    (cond ((%%cf-terminated-p cf) (progn))
          ((<= needed (%%cf-memo-count cf)) (progn))
          ((<= needed (array-dimension (%%cf-memo cf) 0))
           (get-more-terms needed))
          (t (let* ((n1 (+ needed needed))
                    (memo1 (make-array (list n1))))
               (loop for i from 0 upto (1- (%%cf-memo-count cf))
                     do (setf (aref memo1 i) (aref (%%cf-memo cf) i)))
               (setf (%%cf-memo cf) memo1)
               (get-more-terms needed)))))
  (update (1+ i))
  (the (or integer null) (and (< i (%%cf-memo-count cf))
                              (aref (%%cf-memo cf) i))))

(defparameter *cf-max-terms* 20
  "Default term-count limit for cf->string.")

(defun cf->string (cf &optional (max-terms *cf-max-terms*))
  "Make a readable string from a continued fraction."
  (declare (cf cf))
  (loop with i = 0
        with s = "["
        do (let ((term (cf-ref cf i)))
             (cond ((not term)
                    (return (concatenate 'string s "]")))
                   ((= i max-terms)
                    (return (concatenate 'string s ",...]")))
                   (t (let ((separator (case i
                                         ((0) "")
                                         ((1) ";")
                                         (t ",")))
                            (term-str (format nil "~A" term)))
                        (setf i (1+ i))
                        (setf s (concatenate 'string s separator
                                             term-str))))))))

(defun integer->cf (num)
  "Transform an integer to a continued fraction."
  (declare (integer num))
  (let ((terminated-p nil))
    (declare (boolean terminated-p))
    (make-cf #'(lambda ()
                 (and (not terminated-p)
                      (progn (setf terminated-p t)
                             num))))))

(defun ratio->cf (num)
  "Transform a ratio to a continued fraction."
  (declare (ratio num))
  (let ((n (the integer (numerator num)))
        (d (the integer (denominator num))))
    (make-cf #'(lambda ()
                 (and (not (zerop d))
                      (multiple-value-bind (q r) (floor n d)
                        (setf n d)
                        (setf d r)
                        q))))))

;; Thresholds chosen merely for demonstration.
(defparameter number-that-is-too-big (expt 2 512))
(defparameter practically-infinite (expt 2 64))

(defun num-too-big-p (num)
  (declare (integer num))
  (>= (abs num) (abs number-that-is-too-big)))

(defun ng8-too-big-p (ng)
  (declare (ng8 ng))
  (or (num-too-big-p (ng8-a12 ng))
      (num-too-big-p (ng8-a1 ng))
      (num-too-big-p (ng8-a2 ng))
      (num-too-big-p (ng8-a ng))
      (num-too-big-p (ng8-b12 ng))
      (num-too-big-p (ng8-b1 ng))
      (num-too-big-p (ng8-b2 ng))
      (num-too-big-p (ng8-b ng))))

(defun treat-as-infinite-p (term)
  (declare (integer term))
  (>= (abs term) (abs practically-infinite)))

(defun quotient (u v)
  (declare (integer u v))
  (if (zerop v)
      (list nil nil)
      (multiple-value-list (floor u v))))

(defmacro absorb-x-term (ng xsource)
  `(let ((a12 (ng8-a12 ,ng))
         (a1 (ng8-a1 ,ng))
         (a2 (ng8-a2 ,ng))
         (a (ng8-a ,ng))
         (b12 (ng8-b12 ,ng))
         (b1 (ng8-b1 ,ng))
         (b2 (ng8-b2 ,ng))
         (b (ng8-b ,ng))
         (term (funcall ,xsource)))
     (if term
         (let ((ng^ (ng8 (+ a2 (* a12 term))
                         (+ a  (* a1  term)) a12 a1
                         (+ b2 (* b12 term))
                         (+ b  (* b1  term)) b12 b1)))
           (if (not (ng8-too-big-p ng^))
               (setf ,ng ng^)
               (progn (setf ,ng (ng8 a12 a1 a12 a1 b12 b1 b12 b1))
                      ;; Replace the x source with one that never
                      ;; returns a term.
                      (setf ,xsource #'no-terms-thunk))))
         (setf ,ng (ng8 a12 a1 a12 a1 b12 b1 b12 b1)))))

(defmacro absorb-y-term (ng ysource)
  `(let ((a12 (ng8-a12 ,ng))
         (a1 (ng8-a1 ,ng))
         (a2 (ng8-a2 ,ng))
         (a (ng8-a ,ng))
         (b12 (ng8-b12 ,ng))
         (b1 (ng8-b1 ,ng))
         (b2 (ng8-b2 ,ng))
         (b (ng8-b ,ng))
         (term (funcall ,ysource)))
     (if term
         (let ((ng^ (ng8 (+ a1 (* a12 term)) a12
                         (+ a  (* a2  term)) a2
                         (+ b1 (* b12 term)) b12
                         (+ b  (* b2  term)) b2)))
           (if (not (ng8-too-big-p ng^))
               (setf ,ng ng^)
               (progn (setf ,ng (ng8 a12 a12 a2 a2 b12 b12 b2 b2))
                      ;; Replace the y source with one that never
                      ;; returns a term.
                      (setf ysource #'no-terms-thunk))))
         (setf ,ng (ng8 a12 a12 a2 a2 b12 b12 b2 b2)))))

(defun cf->thunk (cf)
  (let ((i 0))
    #'(lambda ()
        (let ((term (cf-ref cf i)))
          (setf i (1+ i))
          term))))

(defun no-terms-thunk ()
  nil)

(defun apply-ng8 (ng8 x y)
  (declare (ng8 ng8))
  (let ((ng ng8)
        (xsource (cf->thunk x))
        (ysource (cf->thunk y)))
    (flet
        ((main ()
           (loop
             with absorb

             for bzero = (zerop (ng8-b ng))
             for b1zero = (zerop (ng8-b1 ng))
             for b2zero = (zerop (ng8-b2 ng))
             for b12zero = (zerop (ng8-b12 ng))

             do (multiple-value-bind (q r q1 r1 q2 r2 q12 r12)
                    (values-list
                     `(,@(quotient (ng8-a ng) (ng8-b ng))
                       ,@(quotient (ng8-a1 ng) (ng8-b1 ng))
                       ,@(quotient (ng8-a2 ng) (ng8-b2 ng))
                       ,@(quotient (ng8-a12 ng) (ng8-b12 ng))))
                  (cond
                    ((and bzero b1zero b2zero b12zero) (return nil))
                    ((and bzero b2zero) (setf absorb 'x))
                    ((or bzero b2zero) (setf absorb 'y))
                    (b1zero (setf absorb 'x))
                    ((and (not b12zero) (= q q1 q2 q12))
                     ;;
                     ;; Output a term.
                     ;;
                     (setf ng (ng8 (ng8-b12 ng) (ng8-b1 ng)
                                   (ng8-b2 ng) (ng8-b ng)
                                   r12 r1 r2 r))
                     (return (and (not (treat-as-infinite-p q)) q)))
                    (t
                     ;;
                     ;; Rather than compare fractions, we will put the
                     ;; numerators over a common denominator of
                     ;; b*b1*b2, and then compare the new numerators.
                     ;;
                     (let ((n  (* (ng8-a ng) (ng8-b1 ng) (ng8-b2 ng)))
                           (n1 (* (ng8-a1 ng) (ng8-b ng) (ng8-b2 ng)))
                           (n2 (* (ng8-a2 ng) (ng8-b ng) (ng8-b1 ng))))
                       (if (> (abs (- n1 n)) (abs (- n2 n)))
                           (setf absorb 'x)
                           (setf absorb 'y))))))

             when (eq absorb 'x)
               do (absorb-x-term ng xsource)

             when (eq absorb 'y)
               do (absorb-y-term ng ysource))))

      (make-cf #'main))))

(defun show (expression cf &optional (note ""))
  (format t "~A =>  ~A~A~%" expression (cf->string cf) note))

(defvar golden-ratio (make-cf #'(lambda () 1)))
(defvar silver-ratio (make-cf #'(lambda () 2)))
(defvar sqrt2 (make-cf (let ((next-term 1))
                  #'(lambda ()
                      (let ((term next-term))
                        (setf next-term 2)
                        term)))))
(defvar frac13/11 (ratio->cf 13/11))
(defvar frac22/7 (ratio->cf 22/7))
(defvar one (integer->cf 1))
(defvar two (integer->cf 2))
(defvar three (integer->cf 3))
(defvar four (integer->cf 4))

(defun cf+ (x y) (apply-ng8 (ng8 0 1 1 0 0 0 0 1) x y))
(defun cf- (x y) (apply-ng8 (ng8 0 1 -1 0 0 0 0 1) x y))
(defun cf* (x y) (apply-ng8 (ng8 1 0 0 0 0 0 0 1) x y))
(defun cf/ (x y) (apply-ng8 (ng8 0 1 0 0 0 0 1 0) x y))

(show "      golden ratio" golden-ratio)
(show "      silver ratio" silver-ratio)
(show "           sqrt(2)" sqrt2)
(show "             13/11" frac13/11)
(show "              22/7" frac22/7)
(show "                 1" one)
(show "                 2" two)
(show "                 3" three)
(show "                 4" four)
(show " (1 + 1/sqrt(2))/2" (cf/ silver-ratio
                                (cf* sqrt2 (cf* sqrt2 sqrt2)))
      "  method 1")
(show " (1 + 1/sqrt(2))/2" (apply-ng8 (ng8 1 0 0 1 0 0 0 8)
                                      silver-ratio
                                      silver-ratio)
      "  method 2")
(show " (1 + 1/sqrt(2))/2" (cf/ (cf/ (cf/ silver-ratio sqrt2)
                                     sqrt2)
                                sqrt2)
      "  method 3")
(show " sqrt(2) + sqrt(2)" (cf+ sqrt2 sqrt2))
(show " sqrt(2) - sqrt(2)" (cf- sqrt2 sqrt2))
(show " sqrt(2) * sqrt(2)" (cf* sqrt2 sqrt2))
(show " sqrt(2) / sqrt(2)" (cf/ sqrt2 sqrt2))
Output:
$ sbcl --script bivariate-continued-fraction-task.lisp
      golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]
      silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
           sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
             13/11 =>  [1;5,2]
              22/7 =>  [3;7]
                 1 =>  [1]
                 2 =>  [2]
                 3 =>  [3]
                 4 =>  [4]
 (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]  method 1
 (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]  method 2
 (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]  method 3
 sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
 sqrt(2) - sqrt(2) =>  [0]
 sqrt(2) * sqrt(2) =>  [2]
 sqrt(2) / sqrt(2) =>  [1]

D

Translation of: Python
Works with: Digital Mars D version 2.099.1
Works with: GCC version 12.2.1
//--------------------------------------------------------------------

import std.algorithm;
import std.bigint;
import std.conv;
import std.math;
import std.stdio;
import std.string;
import std.typecons;

//--------------------------------------------------------------------

class CF                        // Continued fraction.
{
  alias Term = BigInt;          // The type for terms.
  alias Index = size_t;         // The type for indexing terms.

  protected bool terminated;    // Are there no more terms?
  protected size_t m;           // The number of terms memoized.
  private Term[] memo;          // Memoization storage.

  static Index maxTerms = 20;   // Maximum number of terms in the
                                // string representation.

  this ()
  {
    terminated = false;
    m = 0;
    memo.length = 32;
  }

  protected Nullable!Term generate ()
  {
    // Return terms for zero. To get different terms, override this
    // method.
    auto retval = (Term(0)).nullable;
    if (m != 0)
      retval.nullify();
    return retval;
  }

  public Nullable!Term opIndex (Index i)
  {
    void update (size_t needed)
    {
      // Ensure all finite terms with indices 0 <= i < needed are
      // memoized.
      if (!terminated && m < needed)
        {
          if (memo.length < needed)
            // To reduce the frequency of reallocation, increase the
            // space to twice what might be needed right now.
            memo.length = 2 * needed;

          while (m != needed && !terminated)
            {
              auto term = generate ();
              if (!term.isNull())
                {
                  memo[m] = term.get();
                  m += 1;
                }
              else
                terminated = true;
            }
        }
    }

    update (i + 1);

    Nullable!Term retval;
    if (i < m)
      retval = memo[i].nullable;
    return retval;
  }

  public override string toString ()
  {
    static string[3] separators = ["", ";", ","];

    string s = "[";
    Index i = 0;
    bool done = false;
    while (!done)
      {
        auto term = this[i];
        if (term.isNull())
          {
            s ~= "]";
            done = true;
          }
        else if (i == maxTerms)
          {
            s ~= ",...]";
            done = true;
          }
        else
          {
            s ~= separators[(i <= 1) ? i : 2];
            s ~= to!string (term.get());
            i += 1;
          }
      }
    return s;
  }

  public CF opBinary(string op : "+") (CF other)
  {
    return new cfNG8 (ng8_add, this, other);
  }

  public CF opBinary(string op : "-") (CF other)
  {
    return new cfNG8 (ng8_sub, this, other);
  }

  public CF opBinary(string op : "*") (CF other)
  {
    return new cfNG8 (ng8_mul, this, other);
  }

  public CF opBinary(string op : "/") (CF other)
  {
    return new cfNG8 (ng8_div, this, other);
  }

};

//--------------------------------------------------------------------

class cfIndexed : CF  // Continued fraction with an index-to-term map.
{
  alias Mapper = Nullable!Term delegate (Index);
  
  protected Mapper map;

  this (Mapper map)
  {
    this.map = map;
  }

  protected override Nullable!Term generate ()
  {
    return map (m);
  }
}

__gshared goldenRatio =
  new cfIndexed ((i) => CF.Term(1).nullable);

__gshared silverRatio =
  new cfIndexed ((i) => CF.Term(2).nullable);

__gshared sqrt2 =
  new cfIndexed ((i) => CF.Term(min (i + 1, 2)).nullable);

//--------------------------------------------------------------------

class cfRational : CF           // CF for a rational number.
{
  private Term n;
  private Term d;

  this (Term numer, Term denom = Term(1))
  {
    n = numer;
    d = denom;
  }

  protected override Nullable!Term generate ()
  {
    Nullable!Term term;
    if (d != 0)
      {
        auto q = n / d;
        auto r = n % d;
        n = d;
        d = r;
        term = q.nullable;
      }
    return term;
  }
}

__gshared frac_13_11 = new cfRational (CF.Term(13), CF.Term(11));
__gshared frac_22_7 = new cfRational (CF.Term(22), CF.Term(7));
__gshared one = new cfRational (CF.Term(1));
__gshared two = new cfRational (CF.Term(2));
__gshared three = new cfRational (CF.Term(3));
__gshared four = new cfRational (CF.Term(4));

//--------------------------------------------------------------------

class NG8                       // Bihomographic function.
{
  public CF.Term a12, a1, a2, a;
  public CF.Term b12, b1, b2, b;

  this (CF.Term a12, CF.Term a1, CF.Term a2, CF.Term a,
        CF.Term b12, CF.Term b1, CF.Term b2, CF.Term b)
  {
    this.a12 = a12;
    this.a1 = a1;
    this.a2 = a2;
    this.a = a;
    this.b12 = b12;
    this.b1 = b1;
    this.b2 = b2;
    this.b = b;
  }

  this (long a12, long a1, long a2, long a,
        long b12, long b1, long b2, long b)
  {
    this.a12 = a12;
    this.a1 = a1;
    this.a2 = a2;
    this.a = a;
    this.b12 = b12;
    this.b1 = b1;
    this.b2 = b2;
    this.b = b;
  }

  this (NG8 other)
  {
    this.a12 = other.a12;
    this.a1 = other.a1;
    this.a2 = other.a2;
    this.a = other.a;
    this.b12 = other.b12;
    this.b1 = other.b1;
    this.b2 = other.b2;
    this.b = other.b;
  }
}

class cfNG8 : CF   // CF that is a bihomographic function of other CF.
{
  private NG8 ng;
  private CF x;
  private CF y;
  private Index ix;
  private Index iy;
  private bool xoverflow;
  private bool yoverflow;

  //
  // Thresholds chosen merely for demonstration.
  //
  static number_that_is_too_big = // 2 ** 512
    BigInt ("13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096");
  static practically_infinite = // 2 ** 64
    BigInt ("18446744073709551616");

  this (NG8 ng, CF x, CF y)
  {
    this.ng = new NG8 (ng);
    this.x = x;
    this.y = y;
    ix = 0;
    iy = 0;
    xoverflow = false;
    yoverflow = false;
  }

  protected override Nullable!Term generate ()
  {
    // The McCabe complexity of this function is high. Please be
    // careful if modifying the code.

    Nullable!Term term;

    bool done = false;
    while (!done)
      {
        bool bz = (ng.b == 0);
        bool b1z = (ng.b1 == 0);
        bool b2z = (ng.b2 == 0);
        bool b12z = (ng.b12 == 0);
        if (bz && b1z && b2z && b12z)
          done = true;          // There are no more terms.
        else if (bz && b2z)
          absorb_x_term ();
        else if (bz || b2z)
          absorb_y_term ();
        else if (b1z)
          absorb_x_term ();
        else
          {
            Term q, r;
            Term q1, r1;
            Term q2, r2;
            Term q12, r12;
            divMod (ng.a, ng.b, q, r);
            divMod (ng.a1, ng.b1, q1, r1);
            divMod (ng.a2, ng.b2, q2, r2);
            if (ng.b12 != 0)
              divMod (ng.a12, ng.b12, q12, r12);
            if (!b12z && q == q1 && q == q2 && q == q12)
              {
                // Output a term.
                ng = new NG8 (ng.b12, ng.b1, ng.b2, ng.b,
                              r12, r1, r2, r);
                if (!treat_as_infinite (q))
                  term = q.nullable;
                done = true;
              }
            else
              {
                //
                // Rather than compare fractions, we will put the
                // numerators over a common denominator of b*b1*b2,
                // and then compare the new numerators.
                //
                Term n = ng.a * ng.b1 * ng.b2;
                Term n1 = ng.a1 * ng.b * ng.b2;
                Term n2 = ng.a2 * ng.b * ng.b1;
                if (abs (n1 - n) > abs (n2 - n))
                  absorb_x_term ();
                else
                  absorb_y_term ();
              }
          }
      }

    return term;
  }

  private void absorb_x_term ()
  {
    Nullable!Term term;
    if (!xoverflow)
      term = x[ix];
    ix += 1;
    if (!term.isNull())
      {
        auto t = term.get();
        auto new_ng = new NG8 (ng.a2 + (ng.a12 * t),
                               ng.a + (ng.a1 * t),
                               ng.a12, ng.a1,
                               ng.b2 + (ng.b12 * t),
                               ng.b + (ng.b1 * t),
                               ng.b12, ng.b1);
        if (!too_big (new_ng))
          ng = new_ng;
        else
          {
            ng = new NG8 (ng.a12, ng.a1, ng.a12, ng.a1,
                          ng.b12, ng.b1, ng.b12, ng.b1);
            xoverflow = true;
          }
      }
    else
      ng = new NG8 (ng.a12, ng.a1, ng.a12, ng.a1,
                    ng.b12, ng.b1, ng.b12, ng.b1);
  }

  private void absorb_y_term ()
  {
    Nullable!Term term;
    if (!yoverflow)
      term = y[iy];
    iy += 1;
    if (!term.isNull())
      {
        auto t = term.get();
        auto new_ng = new NG8 (ng.a1 + (ng.a12 * t), ng.a12,
                               ng.a + (ng.a2 * t), ng.a2,
                               ng.b1 + (ng.b12 * t), ng.b12,
                               ng.b + (ng.b2 * t), ng.b2);
        if (!too_big (new_ng))
          ng = new_ng;
        else
          {
            ng = new NG8 (ng.a12, ng.a12, ng.a2, ng.a2,
                          ng.b12, ng.b12, ng.b2, ng.b2);
            yoverflow = true;
          }
      }
    else
      ng = new NG8 (ng.a12, ng.a12, ng.a2, ng.a2,
                    ng.b12, ng.b12, ng.b2, ng.b2);
  }

  private bool too_big (NG8 ng)
  {
    // Stop computing if a number reaches the threshold.
    return (too_big (ng.a12) || too_big (ng.a1) ||
            too_big (ng.a2) || too_big (ng.a) ||
            too_big (ng.b12) || too_big (ng.b1) ||
            too_big (ng.b2) || too_big (ng.b));
  }

  private bool too_big (Term u)
  {
    return (abs (u) >= abs (number_that_is_too_big));
  }

  private bool treat_as_infinite (Term u)
  {
    return (abs(u) >= abs (practically_infinite));
  }
}

__gshared NG8 ng8_add = new NG8 (0, 1, 1, 0, 0, 0, 0, 1);
__gshared NG8 ng8_sub = new NG8 (0, 1, -1, 0, 0, 0, 0, 1);
__gshared NG8 ng8_mul = new NG8 (1, 0, 0, 0, 0, 0, 0, 1 );
__gshared NG8 ng8_div = new NG8 (0, 1, 0, 0, 0, 0, 1, 0);

//--------------------------------------------------------------------

void
show (string expression, CF cf, string note = "")
{
  auto line = rightJustify (expression, 19) ~ " =>  ";
  auto cf_str = to!string (cf);
  if (note == "")
    line ~= cf_str;
  else
    line ~= leftJustify (cf_str, 48) ~ note;
  writeln (line);
}

int
main (char[][] args)
{
  show ("golden ratio", goldenRatio, "(1 + sqrt(5))/2");
  show ("silver ratio", silverRatio, "1 + sqrt(2)");
  show ("sqrt(2)", sqrt2);
  show ("13/11", frac_13_11);
  show ("22/7", frac_22_7);
  show ("one", one);
  show ("two", two);
  show ("three", three);
  show ("four", four);
  show ("(1 + 1/sqrt(2))/2",
        new cfNG8 (new NG8 (0, 1, 0, 0, 0, 0, 2, 0),
                   silverRatio, sqrt2),
        "method 1");
  show ("(1 + 1/sqrt(2))/2",
        new cfNG8 (new NG8 (1, 0, 0, 1, 0, 0, 0, 8),
                   silverRatio, silverRatio),
        "method 2");
  show ("(1 + 1/sqrt(2))/2", (one + (one / sqrt2)) / two,
        "method 3");
  show ("sqrt(2) + sqrt(2)", sqrt2 + sqrt2);
  show ("sqrt(2) - sqrt(2)", sqrt2 - sqrt2);
  show ("sqrt(2) * sqrt(2)", sqrt2 * sqrt2);
  show ("sqrt(2) / sqrt(2)", sqrt2 / sqrt2);

  return 0;
}

//--------------------------------------------------------------------
Output:
$ gdc -g -Wall -Wextra bivariate_continued_fraction_task_dlang.d && ./a.out
       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   1 + sqrt(2)
            sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
              13/11 =>  [1;5,2]
               22/7 =>  [3;7]
                one =>  [1]
                two =>  [2]
              three =>  [3]
               four =>  [4]
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
  sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]
  sqrt(2) - sqrt(2) =>  [0]
  sqrt(2) * sqrt(2) =>  [2]
  sqrt(2) / sqrt(2) =>  [1]

Fortran

Translation of: Python
Translation of: ObjectIcon

This program includes a primitive module for multiple-precision integer arithmetic. It is adequate for the task.

Parts of the program might assume two's-complement representation of signed integers. The requirement that integers be two's-complement seems to me unlikely ever to become a part of Fortran standards (even though it will be required in future C standards).

!---------------------------------------------------------------------

module big_integers             ! Big (but not very big) integers.

  ! NOTE: I assume that iachar and achar do not alter the most
  !       significant bit.

  use, intrinsic :: iso_fortran_env, only: int16
  implicit none
  private

  public :: big_integer
  public :: integer2big
  public :: string2big
  public :: big2string
  public :: big_sgn
  public :: big_cmp, big_cmpabs
  public :: big_neg, big_abs
  public :: big_addabs, big_add
  public :: big_subabs, big_sub
  public :: big_mul      ! One might also include a big_muladd.
  public :: big_divrem   ! One could also include big_div and big_rem.
  public :: operator(+)
  public :: operator(-)
  public :: operator(*)

  type :: big_integer
     ! The representation is sign-magnitude. The radix is 256, which
     ! is not speed-efficient, but which seemed relatively easy to
     ! work with if one were writing in standard Fortran (and assuming
     ! iachar and achar were "8-bit clean").
     logical :: sign = .false.  ! .false. => +sign, .true. => -sign.
     character, allocatable :: bytes(:)
  end type big_integer

  character, parameter :: zero = achar (0)
  character, parameter :: one = achar (1)

  ! An integer type capable of holding an unsigned 8-bit value.
  integer, parameter :: bytekind = int16

  interface operator(+)
     module procedure big_add
  end interface

  interface operator(-)
     module procedure big_neg
     module procedure big_sub
  end interface

  interface operator(*)
     module procedure big_mul
  end interface

contains

  elemental function logical2byte (bool) result (byte)
    logical, intent(in) :: bool
    character :: byte
    if (bool) then
       byte = one
    else
       byte = zero
    end if
  end function logical2byte

  elemental function logical2i (bool) result (i)
    logical, intent(in) :: bool
    integer :: i
    if (bool) then
       i = 1
    else
       i = 0
    end if
  end function logical2i

  elemental function byte2i (c) result (i)
    character, intent(in) :: c
    integer :: i
    i = iachar (c)
  end function byte2i

  elemental function i2byte (i) result (c)
    integer, intent(in) :: i
    character :: c
    c = achar (i)
  end function i2byte

  elemental function byte2bk (c) result (i)
    character, intent(in) :: c
    integer(bytekind) :: i
    i = iachar (c, kind = bytekind)
  end function byte2bk

  elemental function bk2byte (i) result (c)
    integer(bytekind), intent(in) :: i
    character :: c
    c = achar (i)
  end function bk2byte

  elemental function bk2i (i) result (j)
    integer(bytekind), intent(in) :: i
    integer :: j
    j = int (i)
  end function bk2i

  elemental function i2bk (i) result (j)
    integer, intent(in) :: i
    integer(bytekind) :: j
    j = int (iand (i, 255), kind = bytekind)
  end function i2bk

  ! Left shift of the least significant 8 bits of a bytekind integer.
  elemental function lshftbk (a, i) result (c)
    integer(bytekind), intent(in) :: a
    integer, intent(in) :: i
    integer(bytekind) :: c
    c = ishft (ibits (a, 0, 8 - i), i)
  end function lshftbk

  ! Right shift of the least significant 8 bits of a bytekind integer.
  elemental function rshftbk (a, i) result (c)
    integer(bytekind), intent(in) :: a
    integer, intent(in) :: i
    integer(bytekind) :: c
    c = ibits (a, i, 8 - i)
  end function rshftbk

  ! Left shift an integer.
  elemental function lshfti (a, i) result (c)
    integer, intent(in) :: a
    integer, intent(in) :: i
    integer :: c
    c = ishft (a, i)
  end function lshfti

  ! Right shift an integer.
  elemental function rshfti (a, i) result (c)
    integer, intent(in) :: a
    integer, intent(in) :: i
    integer :: c
    c = ishft (a, -i)
  end function rshfti

  function integer2big (i) result (a)
    integer, intent(in) :: i
    type(big_integer), allocatable :: a

    !
    ! To write a more efficient implementation of this procedure is
    ! left as an exercise for the reader.
    !

    character(len = 100) :: buffer

    write (buffer, '(I0)') i
    a = string2big (trim (buffer))
  end function integer2big

  function string2big (s) result (a)
    character(len = *), intent(in) :: s
    type(big_integer), allocatable :: a

    integer :: n, i, istart, iend
    integer :: digit

    if ((s(1:1) == '-') .or. s(1:1) == '+') then
       istart = 2
    else
       istart = 1
    end if

    iend = len (s)

    n = (iend - istart + 2) / 2

    allocate (a)
    allocate (a%bytes(n))

    a%bytes = zero
    do i = istart, iend
       digit = ichar (s(i:i)) - ichar ('0')
       if (digit < 0 .or. 9 < digit) error stop
       a = short_multiplication (a, 10)
       a = short_addition (a, digit)
    end do
    a%sign = (s(1:1) == '-')
    call normalize (a)
  end function string2big

  function big2string (a) result (s)
    type(big_integer), intent(in) :: a
    character(len = :), allocatable :: s

    type(big_integer), allocatable :: q
    integer :: r
    integer :: sgn

    sgn = big_sgn (a)
    if (sgn == 0) then
       s = '0'
    else
       q = a
       s = ''
       do while (big_sgn (q) /= 0)
          call short_division (q, 10, q, r)
          s = achar (r + ichar ('0')) // s
       end do
       if (sgn < 0) s = '-' // s
    end if
  end function big2string

  function big_sgn (a) result (sgn)
    type(big_integer), intent(in) :: a
    integer :: sgn

    integer :: n, i

    n = size (a%bytes)
    i = 1
    sgn = 1234
    do while (sgn == 1234)
       if (i == n + 1) then
          sgn = 0
       else if (a%bytes(i) /= zero) then
          if (a%sign) then
             sgn = -1
          else
             sgn = 1
          end if
       else
          i = i + 1
       end if
    end do
  end function big_sgn

  function big_cmp (a, b) result (cmp)
    type(big_integer(*)), intent(in) :: a, b
    integer :: cmp

    if (a%sign) then
       if (b%sign) then
          cmp = -big_cmpabs (a, b)
       else
          cmp = -1
       end if
    else
       if (b%sign) then
          cmp = 1
       else
          cmp = big_cmpabs (a, b)
       end if
    end if
  end function big_cmp

  function big_cmpabs (a, b) result (cmp)
    type(big_integer(*)), intent(in) :: a, b
    integer :: cmp

    integer :: n, i
    integer :: ia, ib

    cmp = 1234
    n = max (size (a%bytes), size (b%bytes))
    i = n
    do while (cmp == 1234)
       if (i == 0) then
          cmp = 0
       else
          ia = byteval (a, i)
          ib = byteval (b, i)
          if (ia < ib) then
             cmp = -1
          else if (ia > ib) then
             cmp = 1
          else
             i = i - 1
          end if
       end if
    end do
  end function big_cmpabs

  function big_neg (a) result (c)
    type(big_integer), intent(in) :: a
    type(big_integer), allocatable :: c
    c = a
    c%sign = .not. c%sign
  end function big_neg

  function big_abs (a) result (c)
    type(big_integer), intent(in) :: a
    type(big_integer), allocatable :: c
    c = a
    c%sign = .false.
  end function big_abs

  function big_add (a, b) result (c)
    type(big_integer), intent(in) :: a
    type(big_integer), intent(in) :: b
    type(big_integer), allocatable :: c

    logical :: sign

    if (a%sign) then
       if (b%sign) then      ! a <= 0, b <= 0
          c = big_addabs (a, b)
          sign = .true.
       else                  ! a <= 0, b >= 0
          c = big_subabs (a, b)
          sign = .not. c%sign
       end if
    else
       if (b%sign) then      ! a >= 0, b <= 0
          c = big_subabs (a, b)
          sign = c%sign
       else                  ! a >= 0, b >= 0
          c = big_addabs (a, b)
          sign = .false.
       end if
    end if
    c%sign = sign
  end function big_add

  function big_sub (a, b) result (c)
    type(big_integer), intent(in) :: a
    type(big_integer), intent(in) :: b
    type(big_integer), allocatable :: c

    logical :: sign

    if (a%sign) then
       if (b%sign) then      ! a <= 0, b <= 0
          c = big_subabs (a, b)
          sign = .not. c%sign
       else                  ! a <= 0, b >= 0
          c = big_addabs (a, b)
          sign = .true.
       end if
    else
       if (b%sign) then      ! a >= 0, b <= 0
          c = big_addabs (a, b)
          sign = .false.
       else                  ! a >= 0, b >= 0
          c = big_subabs (a, b)
          sign = c%sign
       end if
    end if
    c%sign = sign
  end function big_sub

  function big_addabs (a, b) result (c)
    type(big_integer), intent(in) :: a, b
    type(big_integer), allocatable :: c

    ! Compute abs(a) + abs(b).

    integer :: n, nc, i
    logical :: carry
    type(big_integer), allocatable :: tmp

    n = max (size (a%bytes), size (b%bytes))
    nc = n + 1

    allocate(tmp)
    allocate(tmp%bytes(nc))

    call add_bytes (get_byte (a, 1), get_byte (b, 1), .false., tmp%bytes(1), carry)
    do i = 2, n
       call add_bytes (get_byte (a, i), get_byte (b, i), carry, tmp%bytes(i), carry)
    end do
    tmp%bytes(nc) = logical2byte (carry)
    call normalize (tmp)
    c = tmp
  end function big_addabs

  function big_subabs (a, b) result (c)
    type(big_integer), intent(in) :: a, b
    type(big_integer), allocatable :: c

    ! Compute abs(a) - abs(b). The result is signed.

    integer :: n, i
    logical :: carry
    type(big_integer), allocatable :: tmp

    n = max (size (a%bytes), size (b%bytes))
    allocate(tmp)
    allocate(tmp%bytes(n))

    if (big_cmpabs (a, b) >= 0) then
       tmp%sign = .false.
       call sub_bytes (get_byte (a, 1), get_byte (b, 1), .false., tmp%bytes(1), carry)
       do i = 2, n
          call sub_bytes (get_byte (a, i), get_byte (b, i), carry, tmp%bytes(i), carry)
       end do
    else
       tmp%sign = .true.
       call sub_bytes (get_byte (b, 1), get_byte (a, 1), .false., tmp%bytes(1), carry)
       do i = 2, n
          call sub_bytes (get_byte (b, i), get_byte (a, i), carry, tmp%bytes(i), carry)
       end do
    end if
    call normalize (tmp)
    c = tmp
  end function big_subabs

  function big_mul (a, b) result (c)
    type(big_integer), intent(in) :: a, b
    type(big_integer), allocatable :: c

    !
    ! This is Knuth, Volume 2, Algorithm 4.3.1M.
    !

    integer :: na, nb, nc
    integer :: i, j
    integer :: ia, ib, ic
    integer :: carry
    type(big_integer), allocatable :: tmp

    na = size (a%bytes)
    nb = size (b%bytes)
    nc = na + nb + 1

    allocate (tmp)
    allocate (tmp%bytes(nc))

    tmp%bytes = zero
    j = 1
    do j = 1, nb
       ib = byte2i (b%bytes(j))
       if (ib /= 0) then
          carry = 0
          do i = 1, na
             ia = byte2i (a%bytes(i))
             ic = byte2i (tmp%bytes(i + j - 1))
             ic = (ia * ib) + ic + carry
             tmp%bytes(i + j - 1) = i2byte (iand (ic, 255))
             carry = ishft (ic, -8)
          end do
          tmp%bytes(na + j) = i2byte (carry)
       end if
    end do
    tmp%sign = (a%sign .neqv. b%sign)
    call normalize (tmp)
    c = tmp
  end function big_mul

  subroutine big_divrem (a, b, q, r)
    type(big_integer), intent(in) :: a, b
    type(big_integer), allocatable, intent(inout) :: q, r

    !
    ! Division with a remainder that is never negative. Equivalently,
    ! this is floor division if the divisor is positive, and ceiling
    ! division if the divisor is negative.
    !
    ! See Raymond T. Boute, "The Euclidean definition of the functions
    ! div and mod", ACM Transactions on Programming Languages and
    ! Systems, Volume 14, Issue 2, pp. 127-144.
    ! https://doi.org/10.1145/128861.128862
    !

    call nonnegative_division (a, b, .true., .true., q, r)
    if (a%sign) then
       if (big_sgn (r) /= 0) then
          q = short_addition (q, 1)
          r = big_sub (big_abs (b), r)
       end if
       q%sign = .not. b%sign
    else
       q%sign = b%sign
    end if
  end subroutine big_divrem

  function short_addition (a, b) result (c)
    type(big_integer), intent(in) :: a
    integer, intent(in) :: b
    type(big_integer), allocatable :: c

    ! Compute abs(a) + b.

    integer :: na, nc, i
    logical :: carry
    type(big_integer), allocatable :: tmp

    na = size (a%bytes)
    nc = na + 1

    allocate(tmp)
    allocate(tmp%bytes(nc))

    call add_bytes (a%bytes(1), i2byte (b), .false., tmp%bytes(1), carry)
    do i = 2, na
       call add_bytes (a%bytes(i), zero, carry, tmp%bytes(i), carry)
    end do
    tmp%bytes(nc) = logical2byte (carry)
    call normalize (tmp)
    c = tmp
  end function short_addition

  function short_multiplication (a, b) result (c)
    type(big_integer), intent(in) :: a
    integer, intent(in) :: b
    type(big_integer), allocatable :: c

    integer :: i, na, nc
    integer :: ia, ic
    integer :: carry
    type(big_integer), allocatable :: tmp

    na = size (a%bytes)
    nc = na + 1

    allocate (tmp)
    allocate (tmp%bytes(nc))

    tmp%sign = a%sign
    carry = 0
    do i = 1, na
       ia = byte2i (a%bytes(i))
       ic = (ia * b) + carry
       tmp%bytes(i) = i2byte (iand (ic, 255))
       carry = ishft (ic, -8)
    end do
    tmp%bytes(nc) = i2byte (carry)
    call normalize (tmp)
    c = tmp
  end function short_multiplication

  ! Division without regard to signs.
  subroutine nonnegative_division (a, b, want_q, want_r, q, r)
    type(big_integer), intent(in) :: a, b
    logical, intent(in) :: want_q, want_r
    type(big_integer), intent(inout), allocatable :: q, r

    integer :: na, nb
    integer :: remainder

    na = size (a%bytes)
    nb = size (b%bytes)

    ! It is an error if b has "significant" zero-bytes or is equal to
    ! zero.
    if (b%bytes(nb) == zero) error stop

    if (nb == 1) then
       if (want_q) then
          call short_division (a, byte2i (b%bytes(1)), q, remainder)
       else
          block
            type(big_integer), allocatable :: bit_bucket
            call short_division (a, byte2i (b%bytes(1)), bit_bucket, remainder)
          end block
       end if
       if (want_r) then
          if (allocated (r)) deallocate (r)
          allocate (r)
          allocate (r%bytes(1))
          r%bytes(1) = i2byte (remainder)
       end if
    else
       if (na >= nb) then
          call long_division (a, b, want_q, want_r, q, r)
       else
          if (want_q) q = string2big ("0")
          if (want_r) r = a
       end if
    end if
  end subroutine nonnegative_division

  subroutine short_division (a, b, q, r)
    type(big_integer), intent(in) :: a
    integer, intent(in) :: b
    type(big_integer), intent(inout), allocatable :: q
    integer, intent(inout) :: r

    !
    ! This is Knuth, Volume 2, Exercise 4.3.1.16.
    !
    ! The divisor is assumed to be positive.
    !

    integer :: n, i
    integer :: ia, ib, iq
    type(big_integer), allocatable :: tmp

    ib = b
    n = size (a%bytes)

    allocate (tmp)
    allocate (tmp%bytes(n))

    r = 0
    do i = n, 1, -1
       ia = (256 * r) + byte2i (a%bytes(i))
       iq = ia / ib
       r = mod (ia, ib)
       tmp%bytes(i) = i2byte (iq)
    end do
    tmp%sign = a%sign
    call normalize (tmp)
    q = tmp
  end subroutine short_division

  subroutine long_division (a, b, want_quotient, want_remainder, quotient, remainder)
    type(big_integer), intent(in) :: a, b
    logical, intent(in) :: want_quotient, want_remainder
    type(big_integer), intent(inout), allocatable :: quotient
    type(big_integer), intent(inout), allocatable :: remainder

    !
    ! This is Knuth, Volume 2, Algorithm 4.3.1D.
    !
    ! We do not deal here with the signs of the inputs and outputs.
    !
    ! It is assumed size(a%bytes) >= size(b%bytes), and that b has no
    ! leading zero-bytes and is at least two bytes long. If b is one
    ! byte long and nonzero, use short division.
    !

    integer :: na, nb, m, n
    integer :: num_lz, num_nonlz
    integer :: j
    integer :: qhat
    logical :: carry

    !
    ! We will NOT be working with VERY large numbers, and so it will
    ! be safe to put temporary storage on the stack. (Note: your
    ! Fortran might put this storage in a heap instead of the stack.)
    !
    !    v = b, normalized to put its most significant 1-bit all the
    !           way left.
    !
    !    u = a, shifted left by the same amount as b.
    !
    !    q = the quotient.
    !
    ! The remainder, although shifted left, will end up in u.
    !
    integer(bytekind) :: u(0:size (a%bytes) + size (b%bytes))
    integer(bytekind) :: v(0:size (b%bytes) - 1)
    integer(bytekind) :: q(0:size (a%bytes) - size (b%bytes))

    na = size (a%bytes)
    nb = size (b%bytes)

    n = nb
    m = na - nb

    ! In the most significant byte of the divisor, find the number of
    ! leading zero bits, and the number of bits after that.
    block
      integer(bytekind) :: tmp
      tmp = byte2bk (b%bytes(n))
      num_nonlz = bit_size (tmp) - leadz (tmp)
      num_lz = 8 - num_nonlz
    end block

    call normalize_v (b%bytes) ! Make the most significant bit of v be one.
    call normalize_u (a%bytes) ! Shifted by the same amount as v.

    ! Assure ourselves that the most significant bit of v is a one.
    if (.not. btest (v(n - 1), 7)) error stop

    do j = m, 0, -1
       call calculate_qhat (qhat)
       call multiply_and_subtract (carry)
       q(j) = i2bk (qhat)
       if (carry) call add_back
    end do

    if (want_quotient) then
       if (allocated (quotient)) deallocate (quotient)
       allocate (quotient)
       allocate (quotient%bytes(m + 1))
       quotient%bytes = bk2byte (q)
       call normalize (quotient)
    end if

    if (want_remainder) then
       if (allocated (remainder)) deallocate (remainder)
       allocate (remainder)
       allocate (remainder%bytes(n))
       call unnormalize_u (remainder%bytes)
       call normalize (remainder)
    end if

  contains

    subroutine normalize_v (b_bytes)
      character, intent(in) :: b_bytes(n)

      !
      ! Normalize v so its most significant bit is a one. Any
      ! normalization factor that achieves this goal will suffice; we
      ! choose 2**num_lz. (Knuth uses (2**32) div (y[n-1] + 1).)
      !
      ! Strictly for readability, we use linear stack space for an
      ! intermediate result.
      !

      integer :: i
      integer(bytekind) :: btmp(0:n - 1)

      btmp = byte2bk (b_bytes)

      v(0) = lshftbk (btmp(0), num_lz)
      do i = 1, n - 1
         v(i) = ior (lshftbk (btmp(i), num_lz), &
              &      rshftbk (btmp(i - 1), num_nonlz))
      end do
    end subroutine normalize_v

    subroutine normalize_u (a_bytes)
      character, intent(in) :: a_bytes(m + n)

      !
      ! Shift a leftwards to get u. Shift by as much as b was shifted
      ! to get v.
      !
      ! Strictly for readability, we use linear stack space for an
      ! intermediate result.
      !

      integer :: i
      integer(bytekind) :: atmp(0:m + n - 1)

      atmp = byte2bk (a_bytes)

      u(0) = lshftbk (atmp(0), num_lz)
      do i = 1, m + n - 1
         u(i) = ior (lshftbk (atmp(i), num_lz), &
              &      rshftbk (atmp(i - 1), num_nonlz))
      end do
      u(m + n) = rshftbk (atmp(m + n - 1), num_nonlz)
    end subroutine normalize_u

    subroutine unnormalize_u (r_bytes)
      character, intent(out) :: r_bytes(n)

      !
      ! Strictly for readability, we use linear stack space for an
      ! intermediate result.
      !

      integer :: i
      integer(bytekind) :: rtmp(0:n - 1)

      do i = 0, n - 1
         rtmp(i) = ior (rshftbk (u(i), num_lz), &
              &         lshftbk (u(i + 1), num_nonlz))
      end do
      rtmp(n - 1) = rshftbk (u(n - 1), num_lz)

      r_bytes = bk2byte (rtmp)
    end subroutine unnormalize_u

    subroutine calculate_qhat (qhat)
      integer, intent(out) :: qhat

      integer :: itmp, rhat
      logical :: adjust

      itmp = ior (lshfti (bk2i (u(j + n)), 8), &
           &      bk2i (u(j + n - 1)))
      qhat = itmp / bk2i (v(n - 1))
      rhat = mod (itmp, bk2i (v(n - 1)))
      adjust = .true.
      do while (adjust)
         if (rshfti (qhat, 8) /= 0) then
            continue
         else if (qhat * bk2i (v(n - 2)) &
              &     > ior (lshfti (rhat, 8), &
              &            bk2i (u(j + n - 2)))) then
            continue
         else
            adjust = .false.
         end if
         if (adjust) then
            qhat = qhat - 1
            rhat = rhat + bk2i (v(n - 1))
            if (rshfti (rhat, 8) == 0) then
               adjust = .false.
            end if
         end if
      end do
    end subroutine calculate_qhat

    subroutine multiply_and_subtract (carry)
      logical, intent(out) :: carry

      integer :: i
      integer :: qhat_v
      integer :: mul_carry, sub_carry
      integer :: diff

      mul_carry = 0
      sub_carry = 0
      do i = 0, n
         ! Multiplication.
         qhat_v = mul_carry
         if (i /= n) qhat_v = qhat_v + (qhat * bk2i (v(i)))
         mul_carry = rshfti (qhat_v, 8)
         qhat_v = iand (qhat_v, 255)

         ! Subtraction.
         diff = bk2i (u(j + i)) - qhat_v + sub_carry
         sub_carry = -(logical2i (diff < 0)) ! Carry 0 or -1.
         u(j + i) = i2bk (diff)
      end do
      carry = (sub_carry /= 0)
    end subroutine multiply_and_subtract

    subroutine add_back
      integer :: i, carry, sum

      q(j) = q(j) - 1_bytekind
      carry = 0
      do i = 0, n - 1
         sum = bk2i (u(j + i)) + bk2i (v(i)) + carry
         carry = ishft (sum, -8)
         u(j + i) = i2bk (sum)
      end do
    end subroutine add_back

  end subroutine long_division

  subroutine add_bytes (a, b, carry_in, c, carry_out)
    character, intent(in) :: a, b
    logical, value :: carry_in
    character, intent(inout) :: c
    logical, intent(inout) :: carry_out

    integer :: ia, ib, ic

    ia = byte2i (a)
    if (carry_in) ia = ia + 1
    ib = byte2i (b)
    ic = ia + ib
    c = i2byte (iand (ic, 255))
    carry_out = (ic >= 256)
  end subroutine add_bytes

  subroutine sub_bytes (a, b, carry_in, c, carry_out)
    character, intent(in) :: a, b
    logical, value :: carry_in
    character, intent(inout) :: c
    logical, intent(inout) :: carry_out

    integer :: ia, ib, ic

    ia = byte2i (a)
    ib = byte2i (b)
    if (carry_in) ib = ib + 1
    ic = ia - ib
    carry_out = (ic < 0)
    if (carry_out) ic = ic + 256
    c = i2byte (iand (ic, 255))
  end subroutine sub_bytes

  function get_byte (a, i) result (byte)
    type(big_integer), intent(in) :: a
    integer, intent(in) :: i
    character :: byte

    if (size (a%bytes) < i) then
       byte = zero
    else
       byte = a%bytes(i)
    end if
  end function get_byte

  function byteval (a, i) result (v)
    type(big_integer), intent(in) :: a
    integer, intent(in) :: i
    integer :: v

    if (size (a%bytes) < i) then
       v = 0
    else
       v = byte2i (a%bytes(i))
    end if
  end function byteval

  subroutine normalize (a)
    type(big_integer), intent(inout) :: a

    logical :: done
    integer :: i
    character, allocatable :: fewer_bytes(:)

    ! Shorten to the minimum number of bytes.
    i = size (a%bytes)
    done = .false.
    do while (.not. done)
       if (i == 1) then
          done = .true.
       else if (a%bytes(i) /= zero) then
          done = .true.
       else
          i = i - 1
       end if
    end do
    if (i /= size (a%bytes)) then
       allocate (fewer_bytes (i))
       fewer_bytes = a%bytes(1:i)
       call move_alloc (fewer_bytes, a%bytes)
    end if

    ! If the magnitude is zero, then clear the sign bit.
    if (size (a%bytes) == 1) then
       if (a%bytes(1) == zero) then
          a%sign = .false.
       end if
    end if
  end subroutine normalize

end module big_integers

!---------------------------------------------------------------------

module continued_fractions

  use, non_intrinsic :: big_integers
  implicit none
  private

  public :: continued_fraction

  public :: term_generator
  public :: term_generator_procedure

  public :: make_continued_fraction

  public :: i2cf
  public :: make_integer_continued_fraction
  public :: make_integer_continued_fraction_from_integer

  public :: constant_term_cf
  public :: make_constant_term_continued_fraction
  public :: make_constant_term_continued_fraction_from_integer

  public :: apply_ng8
  public :: apply_ng8_big_integers
  public :: apply_ng8_integers
  public :: ng8_coefficient_threshold
  public :: ng8_term_threshold

  public :: add_continued_fractions
  public :: subtract_continued_fractions
  public :: multiply_continued_fractions
  public :: divide_continued_fractions

  public :: cf2string
  public :: continued_fraction_to_string_given_max_terms
  public :: continued_fraction_to_string_with_default_max_terms
  public :: default_continued_fraction_max_terms

  type :: continued_fraction

     class(continued_fraction_record), pointer, private :: p => null ()

   contains

     procedure, pass :: get_term => get_continued_fraction_term
     procedure, pass :: term_exists => continued_fraction_term_exists
     procedure, pass :: term => continued_fraction_term

     procedure, pass :: to_string => continued_fraction_to_string_with_default_max_terms

     procedure, pass :: add => add_continued_fractions
     generic :: operator(+) => add

     procedure, pass :: subtract => subtract_continued_fractions
     generic :: operator(-) => subtract

     procedure, pass :: multiply => multiply_continued_fractions
     generic :: operator(*) => multiply

     procedure, pass :: divide => divide_continued_fractions
     generic :: operator(/) => divide

     procedure, pass, private :: continued_fraction_make_new_ref
     generic :: assignment(=) => continued_fraction_make_new_ref

     final :: continued_fraction_final

  end type continued_fraction

  type :: continued_fraction_record
     logical, private :: terminated = .false. ! No more terms?
     integer, private :: m = 0                ! No. of terms memoized.
     type(big_integer), private, allocatable :: memo(:) ! Memoized terms.
     class(term_generator), pointer :: gen ! Where terms come from.
     integer :: refcount = 0
   contains
     procedure, pass :: get_term => get_continued_fraction_record_term
     procedure, pass :: term_exists => continued_fraction_record_term_exists
     procedure, pass :: term => continued_fraction_record_term
     final :: continued_fraction_record_final
  end type continued_fraction_record

  type, abstract :: term_generator
   contains
     procedure(term_generator_procedure), pass, deferred :: generate
  end type term_generator

  interface
     subroutine term_generator_procedure (gen, term_exists, term)
       import term_generator
       import big_integer
       class(term_generator), intent(inout) :: gen
       logical, intent(out) :: term_exists
       type(big_integer), allocatable, intent(out) :: term
     end subroutine term_generator_procedure
  end interface

  type, extends (term_generator) :: integer_term_generator
     type(big_integer), allocatable :: term
     logical :: no_more_terms = .false.
   contains
     procedure, pass :: generate => integer_term_generator_generate
  end type integer_term_generator

  type, extends (term_generator) :: constant_term_generator
     type(big_integer), allocatable :: term
   contains
     procedure, pass :: generate => constant_term_generator_generate
  end type constant_term_generator

  type, extends (term_generator) :: ng8_term_generator
     type(big_integer), allocatable :: a12, a1, a2, a
     type(big_integer), allocatable :: b12, b1, b2, b
     type(continued_fraction) :: x, y
     integer :: ix = 0
     integer :: iy = 0
     logical :: x_overflow = .false.
     logical :: y_overflow = .false.
   contains
     procedure, pass :: generate => ng8_term_generator_generate
  end type ng8_term_generator

  interface i2cf
     module procedure make_integer_continued_fraction
     module procedure make_integer_continued_fraction_from_integer
  end interface i2cf

  interface constant_term_cf
     module procedure make_constant_term_continued_fraction
     module procedure make_constant_term_continued_fraction_from_integer
  end interface constant_term_cf

  interface apply_ng8
     module procedure apply_ng8_big_integers
     module procedure apply_ng8_integers
  end interface apply_ng8

  interface cf2string
    module procedure continued_fraction_to_string_given_max_terms
    module procedure continued_fraction_to_string_with_default_max_terms
  end interface cf2string

  integer :: default_continued_fraction_max_terms = 20

  type(big_integer), allocatable :: ng8_coefficient_threshold
  type(big_integer), allocatable :: ng8_term_threshold

contains

  subroutine continued_fraction_make_new_ref (dst, src)
    class(continued_fraction), intent(inout) :: dst
    class(continued_fraction), intent(in) :: src

    if (associated (dst%p)) deallocate (dst%p)
    dst%p => src%p
    dst%p%refcount = dst%p%refcount + 1
  end subroutine continued_fraction_make_new_ref

  subroutine continued_fraction_final (cf)
    type(continued_fraction), intent(inout) :: cf
    cf%p%refcount = cf%p%refcount - 1
    if (cf%p%refcount == 0) deallocate (cf%p)
  end subroutine continued_fraction_final

  function make_continued_fraction (gen) result (cf)
    class(term_generator), pointer, intent(in) :: gen
    type(continued_fraction) :: cf

    allocate (cf%p)
    allocate (cf%p%memo(0:31))  ! The starting size is arbitrary.
    cf%p%gen => gen
    cf%p%refcount = cf%p%refcount + 1
  end function make_continued_fraction

  subroutine continued_fraction_record_final (cfrec)
    type(continued_fraction_record), intent(inout) :: cfrec
    deallocate (cfrec%gen)
  end subroutine continued_fraction_record_final

  function make_integer_continued_fraction (bigint) result (cf)
    type(big_integer), intent(in) :: bigint
    type(continued_fraction) :: cf

    class(integer_term_generator), pointer :: gen

    allocate (gen)
    gen%term = bigint
    cf = make_continued_fraction (gen)
  end function make_integer_continued_fraction

  function make_integer_continued_fraction_from_integer (i) result (cf)
    integer, intent(in) :: i
    type(continued_fraction) :: cf
    cf = make_integer_continued_fraction (integer2big (i))
  end function make_integer_continued_fraction_from_integer

  subroutine integer_term_generator_generate (gen, term_exists, term)
    class(integer_term_generator), intent(inout) :: gen
    logical, intent(out) :: term_exists
    type(big_integer), allocatable, intent(out) :: term

    term_exists = (.not. gen%no_more_terms)
    if (term_exists) term = gen%term
    gen%no_more_terms = .true.
  end subroutine integer_term_generator_generate

  function make_constant_term_continued_fraction (bigint) result (cf)
    type(big_integer), intent(in) :: bigint
    type(continued_fraction) :: cf

    class(constant_term_generator), pointer :: gen

    allocate (gen)
    gen%term = bigint
    cf = make_continued_fraction (gen)
  end function make_constant_term_continued_fraction

  function make_constant_term_continued_fraction_from_integer (i) result (cf)
    integer, intent(in) :: i
    type(continued_fraction) :: cf
    cf = make_constant_term_continued_fraction (integer2big (i))
  end function make_constant_term_continued_fraction_from_integer

  subroutine constant_term_generator_generate (gen, term_exists, term)
    class(constant_term_generator), intent(inout) :: gen
    logical, intent(out) :: term_exists
    type(big_integer), allocatable, intent(out) :: term

    term_exists = .true.
    if (term_exists) term = gen%term
  end subroutine constant_term_generator_generate

  function apply_ng8_big_integers (a12, a1, a2, a, &
       &                           b12, b1, b2, b, x, y) result (cf)
    type(big_integer), intent(in) :: a12, a1, a2, a
    type(big_integer), intent(in) :: b12, b1, b2, b
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf

    class(ng8_term_generator), pointer :: gen

    allocate (gen)
    gen%a12 = a12;  gen%a1 = a1;  gen%a2 = a2;  gen%a = a
    gen%b12 = b12;  gen%b1 = b1;  gen%b2 = b2;  gen%b = b
    gen%x = x
    gen%y = y
    cf = make_continued_fraction (gen)
  end function apply_ng8_big_integers

  function apply_ng8_integers (a12, a1, a2, a, &
       &                       b12, b1, b2, b, x, y) result (cf)
    integer, intent(in) :: a12, a1, a2, a
    integer, intent(in) :: b12, b1, b2, b
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf

    cf = apply_ng8_big_integers (integer2big (a12), &
         &                       integer2big (a1),  &
         &                       integer2big (a2),  &
         &                       integer2big (a),   &
         &                       integer2big (b12), &
         &                       integer2big (b1),  &
         &                       integer2big (b2),  &
         &                       integer2big (b), x, y)
  end function apply_ng8_integers

  subroutine ng8_term_generator_generate (gen, term_exists, term)
    class(ng8_term_generator), intent(inout) :: gen
    logical, intent(out) :: term_exists
    type(big_integer), allocatable, intent(out) :: term

    logical :: done
    logical :: b12z, b1z, b2z, bz
    type(big_integer), allocatable :: q12, r12
    type(big_integer), allocatable :: q1, r1
    type(big_integer), allocatable :: q2, r2
    type(big_integer), allocatable :: q, r
    logical :: absorb_x, absorb_y, compare_fracs

    done = .false.
    do while (.not. done)
       absorb_x = .false.
       absorb_y = .false.
       compare_fracs = .false.

       b12z = (big_sgn (gen%b12) == 0)
       b1z = (big_sgn (gen%b1) == 0)
       b2z = (big_sgn (gen%b2) == 0)
       bz = (big_sgn (gen%b) == 0)

       if (b12z .and. b1z .and. b2z .and. bz) then
          ! There are no more terms.
          term_exists = .false.
          done = .true.
       else if (b2z .and. bz) then
          absorb_x = .true.
       else if (b2z .or. bz) then
          absorb_y = .true.
       else if (b1z) then
          absorb_x = .true.
       else
          call big_divrem (gen%a1, gen%b1, q1, r1)
          call big_divrem (gen%a2, gen%b2, q2, r2)
          call big_divrem (gen%a, gen%b, q, r)
          if (.not. b12z) then
             call big_divrem (gen%a12, gen%b12, q12, r12)
             if (big_cmp (q, q1) /= 0) then
                compare_fracs = .true.
             else if (big_cmp (q, q2) /= 0) then
                compare_fracs = .true.
             else if (big_cmp (q, q12) /= 0) then
                compare_fracs = .true.
             else
                call output_term
                done = .true.
             end if
          end if
       end if

       if (compare_fracs) call compare_fractions (absorb_x, absorb_y)
       if (absorb_x) call absorb_x_term
       if (absorb_y) call absorb_y_term
    end do

  contains

    subroutine output_term
      gen%a12 = gen%b12;  gen%a1 = gen%b1;  gen%a2 = gen%b2;  gen%a = gen%b
      gen%b12 = r12;      gen%b1 = r1;      gen%b2 = r2;      gen%b = r
      term_exists = (.not. treat_as_infinite (q))
      if (term_exists) term = q
    end subroutine output_term

    subroutine compare_fractions (absorb_x, absorb_y)
      logical, intent(inout) :: absorb_x, absorb_y

      ! Rather than compare fractions, we will put the numerators over
      ! a common denominator of b1*b2*b, and then compare the new
      ! numerators.

      type(big_integer), allocatable :: n1, n2, n

      n1 = gen%a1 * gen%b2 * gen%b
      n2 = gen%a2 * gen%b1 * gen%b
      n  = gen%a  * gen%b1 * gen%b2
      if (big_cmpabs (n1 - n, n2 - n) > 0) then
         absorb_x = .true.
      else
         absorb_y = .true.
      end if
    end subroutine compare_fractions

    subroutine absorb_x_term
      logical :: term_exists
      type(big_integer), allocatable :: term
      type(big_integer), allocatable :: new_a12, new_a1, new_a2, new_a
      type(big_integer), allocatable :: new_b12, new_b1, new_b2, new_b

      if (gen%x_overflow) then
         term_exists = .false.
      else
         term_exists = gen%x%term_exists(gen%ix)
      end if
      new_a2 = gen%a12
      new_a  = gen%a1
      new_b2 = gen%b12
      new_b  = gen%b1
      if (term_exists) then
         term = gen%x%term(gen%ix)
         new_a12 = gen%a2 + (gen%a12 * term)
         new_a1  = gen%a  + (gen%a1  * term)
         new_b12 = gen%b2 + (gen%b12 * term)
         new_b1  = gen%b  + (gen%b1  * term)
         if (any_too_big (new_a12, new_a1, new_a2, new_a, &
              &           new_b12, new_b1, new_b2, new_b)) then
            gen%x_overflow = .true.
            new_a12 = gen%a12
            new_a1  = gen%a1
            new_b12 = gen%b12
            new_b1  = gen%b1
         end if
      else
         new_a12 = gen%a12
         new_a1  = gen%a1
         new_b12 = gen%b12
         new_b1  = gen%b1
      end if
      gen%a12 = new_a12;  gen%a1 = new_a1;  gen%a2 = new_a2;  gen%a = new_a
      gen%b12 = new_b12;  gen%b1 = new_b1;  gen%b2 = new_b2;  gen%b = new_b
      gen%ix = gen%ix + 1
    end subroutine absorb_x_term

    subroutine absorb_y_term
      logical :: term_exists
      type(big_integer), allocatable :: term
      type(big_integer), allocatable :: new_a12, new_a1, new_a2, new_a
      type(big_integer), allocatable :: new_b12, new_b1, new_b2, new_b

      if (gen%y_overflow) then
         term_exists = .false.
      else
         term_exists = gen%y%term_exists(gen%iy)
      end if
      new_a1 = gen%a12
      new_a  = gen%a2
      new_b1 = gen%b12
      new_b  = gen%b2
      if (term_exists) then
         term = gen%y%term(gen%iy)
         new_a12 = gen%a1 + (gen%a12 * term)
         new_a2  = gen%a  + (gen%a2  * term)
         new_b12 = gen%b1 + (gen%b12 * term)
         new_b2  = gen%b  + (gen%b2  * term)
         if (any_too_big (new_a12, new_a1, new_a2, new_a, &
              &           new_b12, new_b1, new_b2, new_b)) then
            gen%y_overflow = .true.
            new_a12 = gen%a12
            new_a2  = gen%a2
            new_b12 = gen%b12
            new_b2  = gen%b2
         end if
      else
         new_a12 = gen%a12
         new_a2  = gen%a2
         new_b12 = gen%b12
         new_b2  = gen%b2
      end if
      gen%a12 = new_a12;  gen%a1 = new_a1;  gen%a2 = new_a2;  gen%a = new_a
      gen%b12 = new_b12;  gen%b1 = new_b1;  gen%b2 = new_b2;  gen%b = new_b
      gen%iy = gen%iy + 1
    end subroutine absorb_y_term

    function any_too_big (a, b, c, d, e, f, g, h) result (yes)
      type(big_integer), intent(in) :: a, b, c, d, e, f, g, h
      logical :: yes

      if (too_big (a)) then
         yes = .true.
      else if (too_big (b)) then
         yes = .true.
      else if (too_big (c)) then
         yes = .true.
      else if (too_big (d)) then
         yes = .true.
      else if (too_big (e)) then
         yes = .true.
      else if (too_big (f)) then
         yes = .true.
      else if (too_big (g)) then
         yes = .true.
      else if (too_big (h)) then
         yes = .true.
      else
         yes = .false.
      end if
    end function any_too_big

    function too_big (coef) result (yes)
      type(big_integer), intent(in) :: coef
      logical :: yes

      if (.not. allocated (ng8_coefficient_threshold)) then
         ! 2**512
         ng8_coefficient_threshold = string2big ('1340780792994259709957&
              &402499820584612747936582059239337772356144372176403007354&
              &697680187429816690342769003185818648605085375388281194656&
              &9946433649006084096')
      end if

      yes = (big_cmpabs (coef, ng8_coefficient_threshold) >= 0)
    end function too_big

    function treat_as_infinite (term) result (yes)
      type(big_integer), intent(in) :: term
      logical :: yes

      if (.not. allocated (ng8_term_threshold)) then
         ! 2**64
         ng8_term_threshold = string2big ('18446744073709551616')
      end if

      yes = (big_cmpabs (term, ng8_term_threshold) >= 0)
    end function treat_as_infinite

  end subroutine ng8_term_generator_generate

  function add_continued_fractions (x, y) result (cf)
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf
    cf = apply_ng8 (0, 1, 1, 0, 0, 0, 0, 1, x, y)
  end function add_continued_fractions

  function subtract_continued_fractions (x, y) result (cf)
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf
    cf = apply_ng8 (0, 1, -1, 0, 0, 0, 0, 1, x, y)
  end function subtract_continued_fractions

  function multiply_continued_fractions (x, y) result (cf)
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf
    cf = apply_ng8 (1, 0, 0, 0, 0, 0, 0, 1, x, y)
  end function multiply_continued_fractions

  function divide_continued_fractions (x, y) result (cf)
    class(continued_fraction), intent(in) :: x, y
    type(continued_fraction) :: cf
    cf = apply_ng8 (0, 1, 0, 0, 0, 0, 1, 0, x, y)
  end function divide_continued_fractions

  subroutine get_continued_fraction_term (cf, i, term_exists, term)
    class(continued_fraction), intent(in) :: cf
    integer, intent(in) :: i
    logical, intent(out) :: term_exists
    type(big_integer), allocatable, intent(out) :: term

    call get_continued_fraction_record_term (cf%p, i, term_exists, term)
  end subroutine get_continued_fraction_term

  subroutine get_continued_fraction_record_term (cfrec, i, term_exists, term)
    class(continued_fraction_record), intent(inout) :: cfrec
    integer, intent(in) :: i
    logical, intent(out) :: term_exists
    type(big_integer), allocatable, intent(out) :: term

    if (i < 0) error stop

    call update (i + 1)
    term_exists = (i < cfrec%m)
    if (term_exists) term = cfrec%memo(i)

  contains

    subroutine update (needed)
      integer :: needed
      logical :: term_exists1
      type(big_integer), allocatable :: term1

      if (.not. cfrec%terminated .and. cfrec%m < needed) then
         if (size (cfrec%memo) < needed) then
            block               ! Allocate more storage.
              type(big_integer), allocatable :: memo1(:)
              allocate (memo1(0 : (2 * needed) - 1))
              memo1(0:cfrec%m - 1) = cfrec%memo(0:cfrec%m - 1)
              call move_alloc (memo1, cfrec%memo)
            end block
         end if
         do while (.not. cfrec%terminated .and. cfrec%m < needed)
            call cfrec%gen%generate (term_exists1, term1)
            if (term_exists1) then
               cfrec%memo(cfrec%m) = term1
               cfrec%m = cfrec%m + 1
            else
               cfrec%terminated = .true.
            end if
         end do
      end if
    end subroutine update

  end subroutine get_continued_fraction_record_term

  function continued_fraction_term_exists (cf, i) result (term_exists)
    class(continued_fraction), intent(in) :: cf
    integer, intent(in) :: i
    logical :: term_exists
    term_exists = continued_fraction_record_term_exists (cf%p, i)
  end function continued_fraction_term_exists

  function continued_fraction_record_term_exists (cfrec, i) result (term_exists)
    class(continued_fraction_record), intent(inout) :: cfrec
    integer, intent(in) :: i
    logical :: term_exists

    type(big_integer), allocatable :: term

    call get_continued_fraction_record_term (cfrec, i, term_exists, term)
  end function continued_fraction_record_term_exists

  function continued_fraction_term (cf, i) result (term)
    class(continued_fraction), intent(in) :: cf
    integer, intent(in) :: i
    type(big_integer), allocatable :: term
    term = continued_fraction_record_term (cf%p, i)
  end function continued_fraction_term

  function continued_fraction_record_term (cfrec, i) result (term)
    class(continued_fraction_record), intent(inout) :: cfrec
    integer, intent(in) :: i
    type(big_integer), allocatable :: term

    logical :: term_exists

    call get_continued_fraction_record_term (cfrec, i, term_exists, term)
    if (.not. term_exists) error stop
  end function continued_fraction_record_term

  function continued_fraction_to_string_given_max_terms (cf, max_terms) result (s)
    class(continued_fraction), intent(in) :: cf
    integer, intent(in) :: max_terms
    character(len = :), allocatable :: s
    s = continued_fraction_record_to_string_given_max_terms (cf%p, max_terms)
  end function continued_fraction_to_string_given_max_terms

  function continued_fraction_record_to_string_given_max_terms (cfrec, max_terms) result (s)
    class(continued_fraction_record), intent(inout) :: cfrec
    integer, intent(in) :: max_terms
    character(len = :), allocatable :: s

    integer :: i
    logical :: done

    i = 0
    s = '['
    done = .false.
    do while (.not. done)
       if (.not. cfrec%term_exists(i)) then
          s = s // "]"
          done = .true.
       else if (i == max_terms) then
          s = s // ",...]"
          done = .true.
       else
          select case (i)
          case (0);      continue
          case (1);      s = s // ";"
          case default;  s = s // ","
          end select
          s = s // big2string (cfrec%term(i))
          i = i + 1
       end if
    end do
  end function continued_fraction_record_to_string_given_max_terms

  function continued_fraction_to_string_with_default_max_terms (cf) result (s)
    class(continued_fraction), intent(in) :: cf
    character(len = :), allocatable :: s
    s = continued_fraction_record_to_string_with_default_max_terms (cf%p)
  end function continued_fraction_to_string_with_default_max_terms

  function continued_fraction_record_to_string_with_default_max_terms (cfrec) result (s)
    class(continued_fraction_record), intent(inout) :: cfrec
    character(len = :), allocatable :: s

    integer :: max_terms

    max_terms = max (default_continued_fraction_max_terms, 1)
    s = continued_fraction_record_to_string_given_max_terms (cfrec, max_terms)
  end function continued_fraction_record_to_string_with_default_max_terms

end module continued_fractions

!---------------------------------------------------------------------

program bivariate_continued_fraction_task

  use, non_intrinsic :: big_integers
  use, non_intrinsic :: continued_fractions
  implicit none

  type(continued_fraction) :: golden_ratio
  type(continued_fraction) :: silver_ratio
  type(continued_fraction) :: sqrt2
  type(continued_fraction) :: one
  type(continued_fraction) :: two
  type(continued_fraction) :: three
  type(continued_fraction) :: four
  type(continued_fraction) :: method1
  type(continued_fraction) :: method2
  type(continued_fraction) :: method3

  block

    !
    ! Let us start with a test of the long division routine, with some
    ! numbers known to trigger a bug in the first version I
    ! posted. That version had a buggy "add_back" routine.
    !
    ! (How I found such numbers is easy: I used random search.)
    !

    type(big_integer), allocatable :: a, b, q, r

    a = string2big ("95292350834616415707142739736356097545484658215015733475&
         &690528634954280668802285176284181482202905629004984123564335019024318905")
    b = string2big ("63677949970178275389170357551071104191609722674550547056511830750")
    call big_divrem (a, b, q, r)
    if (big_sgn (((b * q) + r) - a) /= 0) error stop

    a = string2big ("5286200801181288750950358142425694618335361315503743069535407838&
         &1104411448878793976933118436177295215313131557463887718741957154")
    b = string2big ("54401097470188014066128968444633185848791550678521")
    call big_divrem (a, b, q, r)
    if (big_sgn (((b * q) + r) - a) /= 0) error stop

    a = string2big ("74352827755975214935544861176217106447734695144315262422&
         &288346418457330023596489437154599028318030933202302606937951415862696330")
    b = string2big ("291979433784649910583546698460221489986784915256036707914578&
         &892106828527219639012712")
    call big_divrem (a, b, q, r)
    if (big_sgn (((b * q) + r) - a) /= 0) error stop

    a = string2big ("7515839498480018152556264500298705705667515770181724145893&
         &9544448656273749453586884931339958104923411455488844854494605760712247")
    b = string2big ("8600698996698965932302079501896131441135807557744554902970070&
         &402964318496325075877027770784963001")
    call big_divrem (a, b, q, r)
    if (big_sgn (((b * q) + r) - a) /= 0) error stop

    a = string2big ("13370595927769020368832742717678609885835798503146654175875&
         &149837801359758893206045930442389897206420586502996797614097489470778")
    b = string2big ("871343613388")
    call big_divrem (a, b, q, r)
    if (big_sgn (((b * q) + r) - a) /= 0) error stop

  end block

  golden_ratio = constant_term_cf (1)
  silver_ratio = constant_term_cf (2)  
  one = i2cf (1)
  two = i2cf (2)
  three = i2cf (3)
  four = i2cf (4)
  sqrt2 = silver_ratio - one

  method1 = apply_ng8 (0, 1, 0, 0, 0, 0, 2, 0, silver_ratio, sqrt2)
  method2 = apply_ng8 (1, 0, 0, 1, 0, 0, 0, 8, silver_ratio, silver_ratio)
  method3 = (one / two) * (one + (one / sqrt2))

  call show ("golden ratio", golden_ratio, "(1 + sqrt(5))/2")
  call show ("silver ratio", silver_ratio, "(1 + sqrt(2))")
  call show ("sqrt(2)", sqrt2, "silver ratio minus 1")
  call show ("13/11", i2cf (13) / i2cf (11), "")
  call show ("22/7", i2cf (22) / i2cf (7), "")
  call show ("1", one, "")
  call show ("2", two, "")
  call show ("3", three, "")
  call show ("4", four, "")
  call show ("(1 + 1/sqrt(2))/2", method1, "method 1")
  call show ("(1 + 1/sqrt(2))/2", method2, "method 2")
  call show ("(1 + 1/sqrt(2))/2", method3, "method 3")
  call show ("sqrt(2) + sqrt(2)", sqrt2 + sqrt2, "")
  call show ("sqrt(2) - sqrt(2)", sqrt2 - sqrt2, "")
  call show ("sqrt(2) * sqrt(2)", sqrt2 * sqrt2, "")
  call show ("sqrt(2) / sqrt(2)", sqrt2 / sqrt2, "")

contains

  subroutine show (expression, cf, note)
    character(len = *), intent(in) :: expression
    class(continued_fraction), intent(in) :: cf
    character(len = *), intent(in) :: note

    write (*, '(A19, " =>  ", A, T73, A)') &
         & expression, cf%to_string(), note
  end subroutine show

end program bivariate_continued_fraction_task

!---------------------------------------------------------------------
Output:
$ gfortran -O2 -g -fbounds-check -Wall -Wextra bivariate_continued_fraction_task.f90 && ./a.out
       golden ratio =>  [1;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]   (1 + sqrt(5))/2
       silver ratio =>  [2;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   (1 + sqrt(2))
            sqrt(2) =>  [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]   silver ratio minus 1
              13/11 =>  [1;5,2]                                         
               22/7 =>  [3;7]                                           
                  1 =>  [1]                                             
                  2 =>  [2]                                             
                  3 =>  [3]                                             
                  4 =>  [4]                                             
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 1
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 2
  (1 + 1/sqrt(2))/2 =>  [0;1,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   method 3
  sqrt(2) + sqrt(2) =>  [2;1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,...]   
  sqrt(2) - sqrt(2) =>  [0]                                             
  sqrt(2) * sqrt(2) =>  [2]                                             
  sqrt(2) / sqrt(2) =>  [1]                                             

Go

Adding to the existing package from the Continued_fraction/Arithmetic/Construct_from_rational_number#Go task, re-uses cf.go and rat.go as given in that task.

File ng8.go:

package cf

import "math"

// A 2×4 matix:
//     [ a₁₂   a₁   a₂   a ]
//     [ b₁₂   b₁   b₂   b ]
//
// which when "applied" to two continued fractions N1 and N2
// gives a new continued fraction z such that:
//
//         a₁₂ * N1 * N2  +  a₁ * N1  +  a₂ * N2  +  a
//     z = -------------------------------------------
//         b₁₂ * N1 * N2  +  b₁ * N1  +  b₂ * N2  +  b
//
// Examples:
//      NG8{0,1,1,0,  0,0,0,1} gives N1 + N2
//      NG8{0,1,-1,0, 0,0,0,1} gives N1 - N2
//      NG8{1,0,0,0,  0,0,0,1} gives N1 * N2
//      NG8{0,1,0,0,  0,0,1,0} gives N1 / N2
//      NG8{21,-15,28,-20, 0,0,0,1} gives 21*N1*N2 -15*N1 +28*N2 -20
//                               which is (3*N1 + 4) * (7*N2 - 5)
type NG8 struct {
	A12, A1, A2, A int64
	B12, B1, B2, B int64
}

// Basic identities as NG8 matrices.
var (
	NG8Add = NG8{0, 1, 1, 0, 0, 0, 0, 1}
	NG8Sub = NG8{0, 1, -1, 0, 0, 0, 0, 1}
	NG8Mul = NG8{1, 0, 0, 0, 0, 0, 0, 1}
	NG8Div = NG8{0, 1, 0, 0, 0, 0, 1, 0}
)

func (ng *NG8) needsIngest() bool {
	if ng.B12 == 0 || ng.B1 == 0 || ng.B2 == 0 || ng.B == 0 {
		return true
	}
	x := ng.A / ng.B
	return ng.A1/ng.B1 != x || ng.A2/ng.B2 != x && ng.A12/ng.B12 != x
}

func (ng *NG8) isDone() bool {
	return ng.B12 == 0 && ng.B1 == 0 && ng.B2 == 0 && ng.B == 0
}

func (ng *NG8) ingestWhich() bool { // true for N1, false for N2
	if ng.B == 0 && ng.B2 == 0 {
		return true
	}
	if ng.B == 0 || ng.B2 == 0 {
		return false
	}
	x1 := float64(ng.A1) / float64(ng.B1)
	x2 := float64(ng.A2) / float64(ng.B2)
	x := float64(ng.A) / float64(ng.B)
	return math.Abs(x1-x) > math.Abs(x2-x)
}

func (ng *NG8) ingest(isN1 bool, t int64) {
	if isN1 {
		// [ a₁₂   a₁   a₂   a ] becomes [ a₂+a₁₂*t  a+a₁*t  a₁₂  a₁]
		// [ b₁₂   b₁   b₂   b ]         [ b₂+b₁₂*t  b+b₁*t  b₁₂  b₁]
		ng.A12, ng.A1, ng.A2, ng.A,
			ng.B12, ng.B1, ng.B2, ng.B =
			ng.A2+ng.A12*t, ng.A+ng.A1*t, ng.A12, ng.A1,
			ng.B2+ng.B12*t, ng.B+ng.B1*t, ng.B12, ng.B1
	} else {
		// [ a₁₂   a₁   a₂   a ] becomes [ a₁+a₁₂*t  a₁₂  a+a₂*t  a₂]
		// [ b₁₂   b₁   b₂   b ]         [ b₁+b₁₂*t  b₁₂  b+b₂*t  b₂]
		ng.A12, ng.A1, ng.A2, ng.A,
			ng.B12, ng.B1, ng.B2, ng.B =
			ng.A1+ng.A12*t, ng.A12, ng.A+ng.A2*t, ng.A2,
			ng.B1+ng.B12*t, ng.B12, ng.B+ng.B2*t, ng.B2
	}
}

func (ng *NG8) ingestInfinite(isN1 bool) {
	if isN1 {
		// [ a₁₂   a₁   a₂   a ] becomes [ a₁₂  a₁  a₁₂  a₁ ]
		// [ b₁₂   b₁   b₂   b ]         [ b₁₂  b₁  b₁₂  b₁ ]
		ng.A2, ng.A, ng.B2, ng.B =
			ng.A12, ng.A1,
			ng.B12, ng.B1
	} else {
		// [ a₁₂   a₁   a₂   a ] becomes [ a₁₂  a₁₂  a₂  a₂ ]
		// [ b₁₂   b₁   b₂   b ]         [ b₁₂  b₁₂  b₂  b₂ ]
		ng.A1, ng.A, ng.B1, ng.B =
			ng.A12, ng.A2,
			ng.B12, ng.B2
	}
}

func (ng *NG8) egest(t int64) {
	// [ a₁₂   a₁   a₂   a ] becomes [     b₁₂       b₁       b₂      b   ]
	// [ b₁₂   b₁   b₂   b ]         [ a₁₂-b₁₂*t  a₁-b₁*t  a₂-b₂*t  a-b*t ]
	ng.A12, ng.A1, ng.A2, ng.A,
		ng.B12, ng.B1, ng.B2, ng.B =
		ng.B12, ng.B1, ng.B2, ng.B,
		ng.A12-ng.B12*t, ng.A1-ng.B1*t, ng.A2-ng.B2*t, ng.A-ng.B*t
}

// ApplyTo "applies" the matrix `ng` to the continued fractions
// `N1` and `N2`, returning the resulting continued fraction.
// After ingesting `limit` terms without any output terms the resulting
// continued fraction is terminated.
func (ng NG8) ApplyTo(N1, N2 ContinuedFraction, limit int) ContinuedFraction {
	return func() NextFn {
		next1, next2 := N1(), N2()
		done := false
		sinceEgest := 0
		return func() (int64, bool) {
			if done {
				return 0, false
			}
			for ng.needsIngest() {
				sinceEgest++
				if sinceEgest > limit {
					done = true
					return 0, false
				}
				isN1 := ng.ingestWhich()
				next := next2
				if isN1 {
					next = next1
				}
				if t, ok := next(); ok {
					ng.ingest(isN1, t)
				} else {
					ng.ingestInfinite(isN1)
				}
			}
			sinceEgest = 0
			t := ng.A / ng.B
			ng.egest(t)
			done = ng.isDone()
			return t, true
		}
	}
}

File ng8_test.go:

package cf

import "fmt"

func ExampleNG8() {
	cases := [...]struct {
		op     string
		r1, r2 Rat
		ng     NG8
	}{
		{"+", Rat{22, 7}, Rat{1, 2}, NG8Add},
		{"*", Rat{13, 11}, Rat{22, 7}, NG8Mul},
		{"-", Rat{13, 11}, Rat{22, 7}, NG8Sub},
		{"/", Rat{22 * 22, 7 * 7}, Rat{22, 7}, NG8Div},
	}
	for _, tc := range cases {
		n1 := tc.r1.AsContinuedFraction()
		n2 := tc.r2.AsContinuedFraction()
		z := tc.ng.ApplyTo(n1, n2, 1000)
		fmt.Printf("%v %s %v is %v %v %v gives %v\n",
			tc.r1, tc.op, tc.r2,
			tc.ng, n1, n2, z,
		)
	}

	z := NG8Mul.ApplyTo(Sqrt2, Sqrt2, 1000)
	fmt.Println("√2 * √2 =", z)

	// Output:
	// 22/7 + 1/2 is {0 1 1 0 0 0 0 1} [3; 7] [0; 2] gives [3; 1, 1, 1, 4]
	// 13/11 * 22/7 is {1 0 0 0 0 0 0 1} [1; 5, 2] [3; 7] gives [3; 1, 2, 2]
	// 13/11 - 22/7 is {0 1 -1 0 0 0 0 1} [1; 5, 2] [3; 7] gives [-1; -1, -24, -1, -2]
	// 484/49 / 22/7 is {0 1 0 0 0 0 1 0} [9; 1, 7, 6] [3; 7] gives [3; 7]
	// √2 * √2 = [1; 0, 1]
}
Output:

(Note, [1; 0, 1] = 1 + 1 / (0 + 1/1 ) = 2, so the answer is correct, it should however be normalised to the more reasonable form of [2].)

22/7 + 1/2 is {0 1 1 0 0 0 0 1} [3; 7] [0; 2] gives [3; 1, 1, 1, 4]
13/11 * 22/7 is {1 0 0 0 0 0 0 1} [1; 5, 2] [3; 7] gives [3; 1, 2, 2]
13/11 - 22/7 is {0 1 -1 0 0 0 0 1} [1; 5, 2] [3; 7] gives [-1; -1, -24, -1, -2]
484/49 / 22/7 is {0 1 0 0 0 0 1 0} [9; 1, 7, 6] [3; 7] gives [3; 7]
√2 * √2 = [1; 0, 1]

Haskell

Translation of: Mercury

This Haskell follows the Mercury, in using infinitely long lazy lists to represent continued fractions. There are two kinds of terms: "infinite" and "finite integer".

----------------------------------------------------------------------

data Term = InfiniteTerm | IntegerTerm Integer
type ContinuedFraction = [Term] -- The list should be infinitely long.

type NG8 = (Integer, Integer, Integer, Integer,
            Integer, Integer, Integer, Integer)

----------------------------------------------------------------------

cf2string (cf :: ContinuedFraction) = 
  loop 0 "[" cf
  where loop i s lst =
          case lst of {
            (InfiniteTerm : _) -> s ++ "]" ;
            (IntegerTerm value : tail) ->
              (if i == 20 then
                 s ++ ",...]"
               else
                 let {
                   sepStr =
                       case i of {
                         0 -> "";
                         1 -> ";";
                         _ -> ","
                         };
                   termStr = show value;
                   s1 = s ++ sepStr ++ termStr
                   }
                 in loop (i + 1) s1 tail)
            }

----------------------------------------------------------------------

repeatingTerm (term :: Term) =
  term : repeatingTerm term

infiniteContinuedFraction = repeatingTerm InfiniteTerm

i2cf (i :: Integer) =
  -- Continued fraction representing an integer.
  IntegerTerm i : infiniteContinuedFraction

r2cf (n :: Integer) (d :: Integer) =
  -- Continued fraction representing a rational number.
  let (q, r) = divMod n d in
    (if r == 0 then
        (IntegerTerm q : infiniteContinuedFraction)
     else
        (IntegerTerm q : r2cf d r))

----------------------------------------------------------------------

add_cf = apply_ng8 (0, 1, 1, 0, 0, 0, 0, 1)
sub_cf = apply_ng8 (0, 1, -1, 0, 0, 0, 0, 1)
mul_cf = apply_ng8 (1, 0, 0, 0, 0, 0, 0, 1)
div_cf = apply_ng8 (0, 1, 0, 0, 0, 0, 1, 0)

apply_ng8
  (ng :: NG8)
  (x :: ContinuedFraction)
  (y :: ContinuedFraction) =
  --
  let (a12, a1, a2, a, b12, b1, b2, b) = ng in
    if iseqz [b12, b1, b2, b] then
      infiniteContinuedFraction -- No more finite terms to output.
    else if iseqz [b2, b] then
      let (ng1, x1, y1) = absorb_x_term ng x y in
        apply_ng8 ng1 x1 y1
    else if atLeastOne_iseqz [b2, b] then
      let (ng1, x1, y1) = absorb_y_term ng x y in
        apply_ng8 ng1 x1 y1
    else if iseqz [b1] then
      let (ng1, x1, y1) = absorb_x_term ng x y in
        apply_ng8 ng1 x1 y1
    else
      let {
        (q12, r12) = maybeDivide a12 b12;
        (q1, r1) = maybeDivide a1 b1;
        (q2, r2) = maybeDivide a2 b2;
        (q, r) = maybeDivide a b
        }
      in
        if not (iseqz [b12]) && q == q12 && q == q1 && q == q2 then
          -- Output a term.
          (if integerExceedsInfinitizingThreshold q then
             infiniteContinuedFraction
           else
             let new_ng = (b12, b1, b2, b, r12, r1, r2, r) in
               (IntegerTerm q : apply_ng8 new_ng x y))
        else
          -- Put a1, a2, and a over a common denominator and compare
          -- some magnitudes.
          let {
            n1 = a1 * b2 * b;
            n2 = a2 * b1 * b;
            n = a * b1 * b2
            }
          in
            (if abs (n1 - n) > abs (n2 - n) then
               let (ng1, x1, y1) = absorb_x_term ng x y in
                 apply_ng8 ng1 x1 y1
             else
               let (ng1, x1, y1) = absorb_y_term ng x y in
                 apply_ng8 ng1 x1 y1)

absorb_x_term
  ((a12, a1, a2, a, b12, b1, b2, b) :: NG8)
  (x :: ContinuedFraction)
  (y :: ContinuedFraction) =
  --
  case x of {
    (IntegerTerm n : xtail) -> (
        let new_ng = (a2 + (a12 * n), a + (a1 * n), a12, a1,
                      b2 + (b12 * n), b