c_imp/rap/rap_c.c

/* $Archive:: /ACM Rational Approximation Paper And Algorithms/C-Language Implementation/rap_c.c      $ */
/* $Revision: 1.1 $ */
/* $Date: 2001/09/25 21:44:55 $ */
/* $Author: dtashley $ */
/* $Workfile:: rap_c.c                                                                                $ */
/* 
/*
//--------------------------------------------------------------------------------
//Copyright 2008 David T. Ashley
//-------------------------------------------------------------------------------------------------
//This source code and any program in which it is compiled/used is provided under the GNU GENERAL
//PUBLIC LICENSE, Version 3, full license text below.
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//THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
//PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
//IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
//ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
//
//  16. Limitation of Liability.
//
//  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
//WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
//THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
//GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
//USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
//DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
//PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
//EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
//SUCH DAMAGES.
//
//  17. Interpretation of Sections 15 and 16.
//
//  If the disclaimer of warranty and limitation of liability provided
//above cannot be given local legal effect according to their terms,
//reviewing courts shall apply local law that most closely approximates
//an absolute waiver of all civil liability in connection with the
//Program, unless a warranty or assumption of liability accompanies a
//copy of the Program in return for a fee.
//
//                     END OF TERMS AND CONDITIONS
//
//            How to Apply These Terms to Your New Programs
//
//  If you develop a new program, and you want it to be of the greatest
//possible use to the public, the best way to achieve this is to make it
//free software which everyone can redistribute and change under these terms.
//
//  To do so, attach the following notices to the program.  It is safest
//to attach them to the start of each source file to most effectively
//state the exclusion of warranty; and each file should have at least
//the "copyright" line and a pointer to where the full notice is found.
//
//    <one line to give the program's name and a brief idea of what it does.>
//    Copyright (C) <year>  <name of author>
//
//    This program is free software: you can redistribute it and/or modify
//    it under the terms of the GNU General Public License as published by
//    the Free Software Foundation, either version 3 of the License, or
//    (at your option) any later version.
//
//    This program is distributed in the hope that it will be useful,
//    but WITHOUT ANY WARRANTY; without even the implied warranty of
//    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
//    GNU General Public License for more details.
//
//    You should have received a copy of the GNU General Public License
//    along with this program.  If not, see <http://www.gnu.org/licenses/>.
//
//Also add information on how to contact you by electronic and paper mail.
//
//  If the program does terminal interaction, make it output a short
//notice like this when it starts in an interactive mode:
//
//    <program>  Copyright (C) <year>  <name of author>
//    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
//    This is free software, and you are welcome to redistribute it
//    under certain conditions; type `show c' for details.
//
//The hypothetical commands `show w' and `show c' should show the appropriate
//parts of the General Public License.  Of course, your program's commands
//might be different; for a GUI interface, you would use an "about box".
//
//  You should also get your employer (if you work as a programmer) or school,
//if any, to sign a "copyright disclaimer" for the program, if necessary.
//For more information on this, and how to apply and follow the GNU GPL, see
//<http://www.gnu.org/licenses/>.
//
//  The GNU General Public License does not permit incorporating your program
//into proprietary programs.  If your program is a subroutine library, you
//may consider it more useful to permit linking proprietary applications with
//the library.  If this is what you want to do, use the GNU Lesser General
//Public License instead of this License.  But first, please read
//<http://www.gnu.org/philosophy/why-not-lgpl.html>.
//-------------------------------------------------------------------------------------------------
//--------------------------------------------------------------------------------
** Rational Approximation Software Submitted To CALGO To Accompany TOMS Paper
** Entitled "On Best Rational Approximations Using Large Integers"
** --------------------------------------------------------------------------
** Please see accompanying user's guide for invocation instructions.
*/

#include <process.h>
   /* For the exit() function.                                              */

#include <stdio.h>
   /* printf() and similar functions require the stdio.h header file.       */

#include <string.h>
   /* String functions and memory manipulation functions defined there.     */

#include <malloc.h>
   /* Necessary for memory allocation functions.                            */


/* It wasn't straightforward to locate the file containing the constants TRUE
** and FALSE.  Must define them myself.
*/
#ifndef FALSE
   #define FALSE (0)
#endif
#ifndef TRUE
   #define TRUE  (1)
#endif


/* This source file is broken into segments, each 
** containing a specific family of functions
** that operate on a specific data type, or at a 
** specific level of abstraction.  In C++, these
** would often tend to be made into classes (and 
** this could be done here), but it is a more
** humane experience on anyone who would like to 
** recompile the program to keep everything in a
** single source file.
**
** Functions are arranged from the bottom of the calling tree upward.  This
** way, no function prototypes are needed to ensure the compiler will 
** detect definitions and invocations which are inconsistent.
*/

/****************************************************************************/
/****************************************************************************/
/*************   C O M P I L A T I O N    C O N S T A N T S   ***************/
/****************************************************************************/
/****************************************************************************/
/* Many aspects of the program's behavior can be changed by simply changing
** the constants below.  Many of the values chosen below are intended to 
** keep the software clear of platform-specific limits.  Changes to the
** constants may cause unforeseen difficulties.
*/
#define MAX_CMDLINE_PARS                                                (10)
   /* The maximum number of command-line parameters that will be accepted.
   ** This is ultimately dependent on what the subcommands take, but 
   ** 10 is a safe value, and let's me be lazy and not search to find the
   ** command that takes the most parameters.
   */

#define INPUT_INTEGER_MAX_DIGITS                                       (463)
   /* This value was chosen to allow operation in the Farey series of up to
   ** order 2**1536.
   */
#define INTERMEDIATE_CALC_MAX_DIGITS                                  (4000)
   /* This value wasn't chosen scientifically.  It is just known that
   ** multiplying a 463-digit integer by another will result in at most
   ** a 926-digit result ... it would be quite difficult in the calculations
   ** outlined in the TOMS paper to reach the 4,000 digit limit.  Note that
   ** this is also the limit for the size of the integer components of
   ** rational results.
   */
#define STDIN_MAX_CHARS                                              (32000)
   /* The maximum number of digits that may be read from the standard input
   ** if batch mode is selected.
   */
#define LINE_LEN                                                        (78)
   /* The number of columns assumed to be displayable on the output.
   ** because a standard monitor (legacy) was 80 columns wide, I've chosen
   ** a value just a tad less.
   */
#define NUMBER_DESC_WIDTH                                               (20)
   /* The number of characters at the left reserved for describing a particular
   ** integer.
   */
#define DIGITS_PER_LINE                                                 (27)
   /* The number of digits to be displayed per line for very long
   ** integers.
   */
#define HORIZONTAL_BAR_SEP_CHAR                                        ('-')
   /* The character used to form large horizontal separators.
   */

/****************************************************************************/
/****************************************************************************/
/*********   V E R S I O N    C O N T R O L    F U N C T I O N S  ***********/
/****************************************************************************/
/****************************************************************************/

/****************************************************************************/
/* vcGetVcData():                                                           */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns the embedded version control information for this program,     */
/*   arranged as a sequence of lines to be displayed on an 80-column        */
/*   monitor.                                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   n  :  Line number to return.  The first line is number 0.              */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--:  Pointer to a constant character string to display on that line,  */
/*         or NULL if there are no more lines of version control            */
/*         information to display.                                          */
/****************************************************************************/
const char *vcGetVcData(unsigned n)
   {
   const char *RAP_version_string_array[] 
      = {"$Revision: 1.1 $ $Date: 2001/09/25 21:44:55 $"};
      /* Note that the info above is filled in automatically by Visual
      ** Source Safe on every check-in--this is not manually maintained.
      */

   if (n < (sizeof(RAP_version_string_array)/sizeof(RAP_version_string_array[0])))
      {
      return(RAP_version_string_array[n]);
      }
   else
      {
      return(NULL);
      }
   }


/****************************************************************************/
/****************************************************************************/
/******   G E N E R A L    F O R M A T T I N G    F U N C T I O N S   *******/
/****************************************************************************/
/****************************************************************************/

/****************************************************************************/
/* gfRepChar():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Repeats a character to stdout the specified number of times.           */
/*                                                                          */
/* INPUTS                                                                   */
/*   c  :  The character to repeat.                                         */
/*                                                                          */
/*   n  :  The number of times to repeat it.                                */
/****************************************************************************/
void gfRepChar(char c, unsigned n)
   {
   while(n--)
           printf("%c", c);
   }


/****************************************************************************/
/* gfHline():                                                               */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Dumps a horizontal line of the standard length to the standard output. */
/****************************************************************************/
void gfHline(void)
   {
   gfRepChar(HORIZONTAL_BAR_SEP_CHAR, LINE_LEN);
   printf("\n");
   }


/****************************************************************************/
/* gfBannerHeading():                                                       */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Prints a banner heading bracketed by astreisks to the standard output. */
/*   This function is useful for separating different sections of output.   */
/****************************************************************************/
void gfBannerHeading(char *s, int n_extra_lines)
   {
   const int lr_padding = 3;
      /* The number of spaces on each side of what is printed.
      */
   int i;
      /* General iteration variable.
      */
   int n_asterisks;
   int input_arg_len;
   int n_left_spaces;
   int n_right_spaces;

   /* The order of the source deck prevents me from using the assertion
   ** function without function prototypes, and I don't like function
   ** prototypes, and I'm too lazy to move the functions, so I'll just
   ** protect against naughty parameters causing bizarre behavior.
   */
   if (!s)
      s = "";
   if (n_extra_lines < 0)
      n_extra_lines = 0;

   /* Print the right number of solid lines of asterisks to the
   ** standard output.
   */
   for (i=0; i<n_extra_lines; i++)
      {
      gfRepChar('*', LINE_LEN);
      printf("\n");
      }

   /* Figure out how many asterisks to print on each side of the
   ** argument, and how many spaces.  We also need to figure out
   ** how many characters of the input argument to print--if there
   ** are too many characters, we need to truncate.
   */
   input_arg_len = strlen(s);
   if(input_arg_len > (LINE_LEN - 2 * lr_padding - 2))
      input_arg_len = LINE_LEN - 2 * lr_padding - 2;

   n_asterisks = (LINE_LEN - 2*lr_padding - input_arg_len)/2;

   n_left_spaces = lr_padding;

   if ((LINE_LEN - 2*lr_padding - input_arg_len) % 2)
      {
      /* Odd, need to pad the right by one. */
      n_right_spaces = lr_padding+1;
      }
   else
      {
      n_right_spaces = lr_padding;
      }

   /* Print the text. */
   gfRepChar('*', n_asterisks);
   gfRepChar(' ', n_left_spaces);
   for (i=0; i<input_arg_len; i++)
      printf("%c", s[i]);
   gfRepChar(' ', n_right_spaces);
   gfRepChar('*', n_asterisks);
   printf("\n");

   /* Print the right number of solid lines of asterisks to the
   ** standard output.
   */
   for (i=0; i<n_extra_lines; i++)
      {
      gfRepChar('*', LINE_LEN);
      printf("\n");
      }
   }


/****************************************************************************/
/****************************************************************************/
/*******   L O G I C A L    A S S E R T I O N    F U N C T I O N S   ********/
/****************************************************************************/
/****************************************************************************/

/****************************************************************************/
/* asAssert():                                                              */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Terminates the program abnormally if the logical condition specified   */
/*   is false.  Roughly speaking, this fills the same role as the           */
/*   C-language assert() macro.                                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   c  :  The logical condition tested (or, more precisely, the result of  */
/*         the code inserted by the compiler to stuff this parameter).      */
/*                                                                          */
/*   l  :  The line number on which the call to this function occurs.       */
/*         This is filled in by using a special preprocessor symbol.        */
/****************************************************************************/
void asAssert(unsigned c, unsigned long l)
   {
   if (!c)
      {
      gfHline();
      printf("Abnormal termination; Assertion failed, line: %lu.\n", l);
      gfHline();

      /* Now need to convince the 'C' runtime environment to kill us.
      */
      exit(0);
      }
   }


/****************************************************************************/
/* asFatal():                                                               */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Terminates the program with a fatal error message.                     */
/*                                                                          */
/* INPUTS                                                                   */
/*   *msg:  The message to use.  Must be fairly short.                      */
/****************************************************************************/
void asFatal(const char *msg)
   {
   asAssert(msg != NULL, __LINE__);

   gfHline();
   printf("FATAL ERROR:  %s.\n", msg);
   gfHline();

   /* Now need to convince the 'C' runtime environment to kill us.
   */
   exit(0);
   }


/****************************************************************************/
/* asInfo():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Prints informational message to stdout and terminates the program.     */
/****************************************************************************/
void asInfo(void)           
   {
   static char *msglines[] =
      {
      "RAP.EXE:  rational approximation calculation program.",
      "!",
      "Copyright 2001, The Association For Computing Machinery (www.acm.org),",
      "David T. Ashley (dtashley@aol.com), Joseph P. DeVoe (jdevoe@visteon.com),",
      "Cory Pratt (cory_pratt@3com.com), Karl Perttunen (kperttun@visteon.com),",
      "and Anatoly Zhigljavsky (zhigljavskyaa@cardiff.ac.uk).",
      "!",
      "Please submit any bug reports or inquiries to all of the e-mail addresses",
      "listed above.",
      "!",
      "This program accompanies a paper entitled \"On Best Rational",
      "Approximations Using Large Integers\" submitted to ACM TOMS in late 2000/",
      "early 2001.  The paper fully documents the algorithms employed in this",
      "software.  In a nutshell, this program will execute algorithms on very",
      "large integers (up to 463 digits) and rational numbers with very large",
      "integer components which are helpful in finding best rational",
      "approximations in Farey series of very large order and in large rectangular",
      "regions of the integer lattice.  The source code for this program should",
      "be available from CALGO (collected algorithms of the ACM).",
      "!",
      "Format legend: MANDATORY_PARAMETER",
      "               [OPTIONAL_PARAMETER]",
      "               {a, b ...} = Exactly one must be chosen from set.",
      "               i...       = Integer",
      "               i+...      = Non-Negative Integer",
      "               i++...     = Positive Integer",
      "               r...       = Rational Number",
      "               r+...      = Non-Negative Rational Number",
      "               r++...     = Positive Rational Number",
      "!",
      "How to specify an integer:",
      "   [-]<digits>",
      "   <digits>.<more_digits>e[{-,+}]<exponent>",
      "      where the exponent creates an integer.  Example: 3.14e2 is an integer",
      "      but 3.14e1 is not (the former is 314, the latter is 31.4).  Note that",
      "      an integer can be substituted for a rational number, but a rational",
      "      number cannot always be substituted for an integer.",
      "!",
      "How to specify a rational number:",
      "   [-]<digits>/<more_digits>",
      "   [-]<digits>.<more_digits>[e[{-,+}]<exponent>]",
      "!",
      "RAP + i1 i2",
      "   Adds two integers to produce an integer result.  Result is i1 + i2.",
      "!",
      "RAP + r1 r2",
      "   Adds two rational numbers to produce a rational or integral result.",
      "   Result is r1 + r2.",
      "!",
      "RAP - i1 i2",
      "   Subtracts two integers to produce an integer result.  Result is i1 - i2.",
      "!",
      "RAP * i1 i2",
      "   Multiplies two integers to produce an integer result.  Result is i1 * i2.",
      "!",
      "RAP / i1 i2",
      "   Divides two integers to produce an integer result.  Result is i1 / i2.",
      "!",
      "RAP / r1 r2",
      "   Divides two rational numbers to produce a rational or integral result.",
      "   Result is r1 / r2.",
      "!",
      "RAP % i1 i2",
      "   Divides two integers and produces the remainder from the division.  The",
      "   result is the same as i1 % i2 as defined by the 'C' programming language",
      "   except that much longer integers are accomodated.",
      "!",
      "RAP ** i1 i2+",
      "   Exponentiates the integer i1 to the i2+'th power.",
      "!",
      "RAP ** r1 i1+",
      "   Exponentiates the rational number r1 to the i1+'th power.",
      "!",
      "RAP gcd i1++ i2++",
      "   Calculates the greatest common divisor of i1 and i2 using Euclid's",
      "   algorithm.",
      "!",
      "RAP dap r1 i1+",
      "   Given rational number r1, forms a rational approximation with a",
      "   different denominator i1.  This functionality is very useful",
      "   for converting a rational number to a more familiar decimal",
      "   approximation.  The accompanying user's manual gives full",
      "   details about the formula used by the DAP command.  DAP is also",
      "   used by several other commands to provide a more familiar",
      "   output form.",
      "!",
      "RAP cf r1+",
      "   Forms the continued fraction partial quotients and convergents",
      "   of a non-negative rational number r1.",
      "!",
      "RAP fn r1+ i1++",
      "   Finds the two best neighboring rational approximations to non-negative",
      "   rational number r1 in the Farey series of order i1.",
      "!",
      "RAP fn r1++ i1++ i2++ i3++",
      "   Same as form above except finds i2 neighbors on either side of r1",
      "   (rather than the default of 1), and uses i3 as the denominator for",
      "   presenting the approximations and errors in decimal form.",
      "!",
      "RAP mind r1+ r2+",
      "   Calculates a rational number in the interval [r1, r2] with the minimum",
      "   denominator.  If the rational number in the interval with the minimum",
      "   denominator is not unique (i.e. there is more than one with the same ",
      "   denominator), no assurances are made about which of the numbers with",
      "   minimum denominator will be returned (however, it is assured that the",
      "   result will be one of them).",
      "!",
      "RAP fab r1+ i1++ i2++",
      "   Finds the two neighbors to r1 in a rectangular area of the integer",
      "   lattice specified by h<=i1 and k<=i2.  This is a similar concept to",
      "   the \"fn\" command, but with numerator and denominator both constrained.",
      "   On output, 4 lines of digits after the decimal point are assumed by",
      "   default.",
      "!",
      "RAP fab r1+ i1++ i2++ i3++ i4++",
      "   Same as form above, except optional parameters i3 and i4 specify the",
      "   number of neighbors on both the left and right to generate, and the",
      "   denominator to use for display of DAP information.",
      "!",
      "RAP fndmax r1+ r2+ i++",
      "   Provides an upper bound on the distance between successive terms of",
      "   the Farey series of order i in the interval [r1, r2].  It is required",
      "   that r1 < r2 and that r1 and r2 are both in the Farey series of order",
      "   i.",
      "!",
      "RAP fndmax r1+ r2+ i++ j++",
      "   Same as form above except j may override the default denominator of",
      "   1e108 for DAP presentation.",
      "!",
      "RAP fabdmax r1+ r2+ i1+ i2+",
      "   Provides an upper bound on the distance between successive terms of",
      "   the \"rectangular\" Farey series (doubly-constrained) with h <= i1 and",
      "   j <= i2 in the interval [r1, r2].  It is required that r1 < r2 and that",
      "   both meet the constraints (i.e. are in the \"rectangular\" series).",
      "!",
      "RAP fabdmax r1+ r2+ i1+ i2+ j++",
      "   Same as form above except j may override the default denominator of 1e108",
      "   for DAP presentation.",
      "!",
      "If this information has scrolled off the screen and can't be read, try re-",
      "directing it to a file (for example, RAP >OUT.TXT) and then viewing the file",
      "with a text editor, or piping it through \"more\" (for example, RAP|MORE).",
      "!"
      };

   int i;

   /* Dump out the whole array and terminate the program.  Lines beginning
   ** with an exclamation point should be replaced with a horizontal line.
   */
   for (i=0; i<sizeof(msglines)/sizeof(msglines[0]); i++)
      {
      if ((msglines[i])[0] == '!')
         {
         gfHline();
         }
      else
         {
         printf("%s\n", msglines[i]);
         }
      }

   exit(0);
   }


/****************************************************************************/
/****************************************************************************/
/***************   D A T A - D R I V E N    F U N C T I O N S   *************/
/****************************************************************************/
/****************************************************************************/
/* This section is reserved for data-driven functions with no I/O 
** requirements and no side-effects.
*/
/****************************************************************************/
/* ddCharToLower():                                                         */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   If a character is upper-case, translates it to lower-case.             */
/****************************************************************************/
char ddCharToLower(char c)
   {
   switch(c)
      {
      case 'A' :  return ('a');
                  break;
      case 'B' :  return ('b');
                  break;
      case 'C' :  return ('c');
                  break;
      case 'D' :  return ('d');
                  break;
      case 'E' :  return ('e');
                  break;
      case 'F' :  return ('f');
                  break;
      case 'G' :  return ('g');
                  break;
      case 'H' :  return ('h');
                  break;
      case 'I' :  return ('i');
                  break;
      case 'J' :  return ('j');
                  break;
      case 'K' :  return ('k');
                  break;
      case 'L' :  return ('l');
                  break;
      case 'M' :  return ('m');
                  break;
      case 'N' :  return ('n');
                  break;
      case 'O' :  return ('o');
                  break;
      case 'P' :  return ('p');
                  break;
      case 'Q' :  return ('q');
                  break;
      case 'R' :  return ('r');
                  break;
      case 'S' :  return ('s');
                  break;
      case 'T' :  return ('t');
                  break;
      case 'U' :  return ('u');
                  break;
      case 'V' :  return ('v');
                  break;
      case 'W' :  return ('w');
                  break;
      case 'X' :  return ('x');
                  break;
      case 'Y' :  return ('y');
                  break;
      case 'Z' :  return ('z');
                  break;
      default:    return(c);
                  break;
      }
   }


/****************************************************************************/
/* ddIsInfoChar():                                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns TRUE if the passed character is a character that should be in  */
/*   a token, or FALSE otherwise.                                           */
/****************************************************************************/
unsigned ddIsInfoChar(char c)
   {
   int i;
   char d;
   const char CONST_info_characters[] = { 'a', 'b', 'c', 'd', 'e', 'f', 'g',
                                          'h', 'i', 'j', 'k', 'l', 'm', 'n',
                                          'o', 'p', 'q', 'r', 's', 't', 'u',
                                          'v', 'w', 'x', 'y', 'z', '0', '1',
                                          '2', '3', '4', '5', '6', '7', '8',
                                          '9', '+', '-', '*', '/', '_', '.',
                                          '\\','%' };
   
   d = ddCharToLower(c);

   for (i=0; i<sizeof(CONST_info_characters)/sizeof(CONST_info_characters[0]); i++)
      {
      if (CONST_info_characters[i] == d)
         return(TRUE);
      }

   return(FALSE);
   }


/****************************************************************************/
/* ddIsDiscardChar():                                                       */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns TRUE if the passed character is a character that should be     */
/*   discarded, or FALSE otherwise.                                         */
/****************************************************************************/
unsigned ddIsDiscardChar(char c)
   {
   int i;
   char d;
   const char CONST_discard_characters[] = {','};
   
   d = ddCharToLower(c);

   for (i=0; i<sizeof(CONST_discard_characters)/sizeof(CONST_discard_characters[0]); i++)
      {
      if (CONST_discard_characters[i] == d)
         return(TRUE);
      }

   return(FALSE);
   }


/****************************************************************************/
/* ddIsWhitespaceChar():                                                    */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns TRUE if the passed character is a character that should be     */
/*   treated as whitespace, or FALSE otherwise.                             */
/****************************************************************************/
unsigned ddIsWhitespaceChar(char c)
   {
   int i;
   char d;
   const char CONST_whitespace_chars[] = { ' ', '\t', '\n' };
   
   d = ddCharToLower(c);

   for (i=0; i<sizeof(CONST_whitespace_chars)/sizeof(CONST_whitespace_chars[0]); i++)
      {
      if (CONST_whitespace_chars[i] == d)
         return(TRUE);
      }

   return(FALSE);
   }


/****************************************************************************/
/* ddIsDigit():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns TRUE if the passed argument is a digit, or FALSE otherwise.    */
/*                                                                          */
/* INPUTS                                                                   */
/*   c  :  Character to evaluate for digitness (or is that digiality??).    */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--:  TRUE if a digit, FALSE otherwise.                                */
/****************************************************************************/
unsigned ddIsDigit(char c)
   {
   if ((c >= '0') && (c <= '9'))
      return(TRUE);
   else
      return(FALSE);
   }


/****************************************************************************/
/* ddDigitToValue():                                                        */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Converts from ASCII digit to unsigned integer value.  Fatal to call    */
/*   with non-digit.                                                        */
/*                                                                          */
/* INPUTS                                                                   */
/*   c  :  Character to convert to value.                                   */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--:  Value.                                                           */
/****************************************************************************/
unsigned ddDigitToValue(char c)
   {
   if ((c >= '0') && (c <= '9'))
      {
      return(c - '0');
      }
   else
      {
      asAssert(0, __LINE__);  /* Guaranteed fatal. */
      return(0);              /* To satisfy compiler warning about path not
                              ** returning a value.
                              */
      }
   }


/****************************************************************************/
/* ddValueToDigit():                                                        */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Converts from unsigned value to digit '0'..'9'.  Fatal to call with    */
/*   with value out of range.                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   i  :  Character to convert to value.                                   */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--:  Value.                                                           */
/****************************************************************************/
char ddValueToDigit(unsigned i)
   {
   if (i < 10)
      {
      return(i + '0');
      }
   else
      {
      asAssert(0, __LINE__);  /* Guaranteed fatal. */
      return('0');              /* To satisfy compiler warning about path not
                              ** returning a value.
                              */
      }
   }


/****************************************************************************/
/* ddStringContains():                                                      */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Determines whether STR1 contains any characters in STR2, and returns   */
/*   TRUE if so or FALSE otherwise.                                         */
/****************************************************************************/
int ddStringContains(const char *str1, const char *str2)
   {
   unsigned s1len, s2len;
   unsigned i, j;

   asAssert(str1 != NULL, __LINE__);
   asAssert(str2 != NULL, __LINE__);

   s1len = strlen(str1);
   s2len = strlen(str2);
   
   for (i=0; i<s1len; i++)
      {
      for (j=0; j<s2len; j++)
         {
         if (str1[i] == str2[j])
            return(TRUE);
         }
      }

   return(FALSE);
   }


/****************************************************************************/
/* ddStringContainsOnly():                                                  */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns TRUE if STR1 contains ONLY characters from STR2, or FALSE      */
/*   otherwise.                                                             */
/****************************************************************************/
int ddStringContainsOnly(const char *str1, const char *str2)
   {
   int i, j;
   int l1, l2;

   asAssert(str1 != NULL, __LINE__);
   asAssert(str2 != NULL, __LINE__);

   l1 = strlen(str1);
   l2 = strlen(str2);

   for (i=0; i<l1; i++)
      {
      for (j=0; j<l2; j++)
         {
         if (str1[i] == str2[j])
            break;
         }

      if (j == l2)
         return(FALSE);
      }

   return(TRUE);
   }


/****************************************************************************/
/* ddStringReverse():                                                       */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Reverses the order of characters in the string s.                      */
/****************************************************************************/
void ddStringReverse(char *s)                  
   {
   unsigned l;
   unsigned i;
   unsigned limit;
   char temp;

   asAssert(s != NULL, __LINE__);

   l = strlen(s);
   limit = l / 2;

   for (i=0; i<limit; i++)
      {
      temp = s[i];
      s[i] = s[l-i-1];
      s[l-i-1] = temp;
      }
   }


/****************************************************************************/
/* ddStringDeleteLeadingChar():                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Deletes the leading character of a string.  If there is no leading     */
/*   character, takes no action.                                            */
/****************************************************************************/
void ddStringDeleteLeadingChar(char *s)                  
   {
   int len;
   int i;

   asAssert(s != NULL, __LINE__);

   len = strlen(s);

   for (i=0; i<len; i++)
      s[i] = s[i+1];
   }


/****************************************************************************/
/* ddFundamentalAdditionCell():                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Fundamental cell of addition, used to add one ASCII digit to another   */
/*   ASCII digit, processing carries in and carries out.  Unexpected values */
/*   will generate a fatal error.                                           */
/*                                                                          */
/* INPUTS                                                                   */
/*   digit1:        Digit 1 in.                                             */
/*   digit2:        Digit 2 in.                                             */
/*   carry_in:      Carry in.  Must be '0' or '1'.                          */
/*                                                                          */
/* OUTPUTS                                                                  */
/*  *digit_out:     Digit result out.                                       */
/*  *carry_out:     Carry out.  Will be '0' or '1'.                         */
/****************************************************************************/
void ddFundamentalAdditionCell(char      digit1,
                               char      digit2,
                               char      carry_in,
                               char     *digit_out,
                               char     *carry_out)
   {
   unsigned vd1;
   unsigned vd2;
   unsigned vci;
   unsigned total;

   asAssert(ddIsDigit(digit1), __LINE__);
   asAssert(ddIsDigit(digit2), __LINE__);
   asAssert(ddDigitToValue(carry_in) <= 1, __LINE__);
   asAssert(digit_out != NULL, __LINE__);
   asAssert(carry_out != NULL, __LINE__);

   vd1 = ddDigitToValue(digit1);
   vd2 = ddDigitToValue(digit2);
   vci = ddDigitToValue(carry_in);

   total = vd1+vd2+vci;

   *digit_out = ddValueToDigit(total % 10);
   *carry_out = ddValueToDigit(total / 10);
   }


/****************************************************************************/
/* ddFundamentalMultiplicationCell():                                       */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Fundamental cell of multiplication, used to multiply one ASCII digit   */
/*   by another, processing carries in and out.  Unexpected inputs will     */
/*   generate a fatal error.                                                */
/*                                                                          */
/* INPUTS                                                                   */
/*   digit1:        Digit 1 in.                                             */
/*   digit2:        Digit 2 in.                                             */
/*   carry_in:      Carry in.  Because of the way multiplication works,     */
/*                  this must always be from '0' through '8'.               */
/*                                                                          */
/* OUTPUTS                                                                  */
/*  *digit_out:     Digit result out.                                       */
/*  *carry_out:     Carry out.  Because of the way multiplication works,    */
/*                  this must always be from '0' through '8'.               */
/****************************************************************************/
void ddFundamentalMultiplicationCell(char      digit1,
                                     char      digit2,
                                     char      carry_in,
                                     char     *digit_out,
                                     char     *carry_out)
   {
   unsigned vd1;
   unsigned vd2;
   unsigned vci;
   unsigned total;

   asAssert(ddIsDigit(digit1), __LINE__);
   asAssert(ddIsDigit(digit2), __LINE__);
   asAssert(ddDigitToValue(carry_in) <= 8, __LINE__);
   asAssert(digit_out != NULL, __LINE__);
   asAssert(carry_out != NULL, __LINE__);

   vd1 = ddDigitToValue(digit1);
   vd2 = ddDigitToValue(digit2);
   vci = ddDigitToValue(carry_in);

   total = (vd1 * vd2) + vci;

   *digit_out = ddValueToDigit(total % 10);
   *carry_out = ddValueToDigit(total / 10);
   }


/****************************************************************************/
/* ddFundamentalSubtractionCell():                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Fundamental cell of subtraction, used to subtract one ASCII digit from */
/*   another ASCII digit, processing borrows in and borrows out.            */
/*   Unexpected values will generate a fatal error.                         */
/*                                                                          */
/* INPUTS                                                                   */
/*   digit1:        Digit 1 in.                                             */
/*   digit2:        Digit 2 in.                                             */
/*   borrow_in:     Borrow in.  Must be '0' or '1'.                         */
/*                                                                          */
/* OUTPUTS                                                                  */
/*  *digit_out:     Digit result out, digit1-digit2-borrow_in.              */
/*  *borrow_out:    Borrow out.  Will be '0' or '1'.                        */
/****************************************************************************/
void ddFundamentalSubtractionCell(char      digit1,
                                  char      digit2,
                                  char      borrow_in,
                                  char     *digit_out,
                                  char     *borrow_out)
   {
   unsigned vd1;
   unsigned vd2;
   unsigned vbi;
   unsigned total;

   asAssert(ddIsDigit(digit1), __LINE__);
   asAssert(ddIsDigit(digit2), __LINE__);
   asAssert(ddDigitToValue(borrow_in) <= 1, __LINE__);
   asAssert(digit_out != NULL, __LINE__);
   asAssert(borrow_out != NULL, __LINE__);

   vd1 = ddDigitToValue(digit1);
   vd2 = ddDigitToValue(digit2);
   vbi = ddDigitToValue(borrow_in);

   total = 10+vd1-vd2-vbi;

   *digit_out = ddValueToDigit(total % 10);
   if (total < 10)
      {
      *borrow_out = '1';
        /* If we went neggy neggy on the subtraction, we need a borrow.
        */
      }
   else
      {
      *borrow_out = '0';  /* No borrow. */
      }
   }


/****************************************************************************/
/* ddUmin():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns the minimum of two unsigned integers.                          */
/*                                                                          */
/* INPUTS                                                                   */
/*   arg1, arg2  :  Unsigned integers to compare.                           */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--         :  Samller of arg1, arg2.                                  */
/****************************************************************************/
unsigned ddUmin(unsigned arg1, unsigned arg2)
   {
   if (arg1 < arg2)
      return(arg1);
   else
      return(arg2);
   }


/****************************************************************************/
/* ddUmax():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns the maximum of two unsigned integers.                          */
/*                                                                          */
/* INPUTS                                                                   */
/*   arg1, arg2  :  Unsigned integers to compare.                           */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--         :  Larger of arg1, arg2.                                   */
/****************************************************************************/
unsigned ddUmax(unsigned arg1, unsigned arg2)
   {
   if (arg1 > arg2)
      return(arg1);
   else
      return(arg2);
   }


/****************************************************************************/
/* ddSmin():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns the minimum of two signed integers.                            */
/*                                                                          */
/* INPUTS                                                                   */
/*   arg1, arg2  :  Signed integers to compare.                             */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--         :  Samller of arg1, arg2.                                  */
/****************************************************************************/
int ddSmin(int arg1, int arg2)
   {
   if (arg1 < arg2)
      return(arg1);
   else
      return(arg2);
   }


/****************************************************************************/
/* ddSmax():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Returns the maximum of two signed integer.                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   arg1, arg2  :  Signed integers to compare.                             */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   <--         :  Larger of arg1, arg2.                                   */
/****************************************************************************/
int ddSmax(int arg1, int arg2)
   {
   if (arg1 > arg2)
      return(arg1);
   else
      return(arg2);
   }


/****************************************************************************/
/****************************************************************************/
/*******   M E M O R Y    A L L O C A T I O N    F U N C T I O N S   ********/
/****************************************************************************/
/****************************************************************************/
/* These functions are primarily wrappers for the standard 'C' library
** functions.  They will trap "out of memory" errors, and also provide a
** place to insert debugging code if it becomes necessary.
*/
/****************************************************************************/
/* maMalloc():                                                              */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Wrapper for malloc().  See malloc() documentation.                     */
/****************************************************************************/
void *maMalloc(size_t n)
   {
   void *rv;

   rv = malloc(n);

   asAssert(rv != NULL, __LINE__);

   return(rv);
   }


/****************************************************************************/
/* maFree():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Wrapper for free().  See free() documentation.                         */
/****************************************************************************/
void maFree(void *p)
   {
   asAssert(p != NULL, __LINE__);

   free(p);
   }


/****************************************************************************/
/* maRealloc():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Wrapper for realloc().  See realloc() documentation.                   */
/****************************************************************************/
void *maRealloc(void *ptr, size_t n)
   {
   void *rv;

   rv = realloc(ptr, n);

   asAssert(rv != NULL, __LINE__);

   return(rv);
   }


/****************************************************************************/
/****************************************************************************/
/*******   S Y N T H E T I C    I N T E G E R    F U N C T I O N S   ********/
/****************************************************************************/
/****************************************************************************/
/* Definition of a synthetic integer.
*/
struct synthetic_integer_struct
   {
   unsigned nan;
      /* Boolean variable used to remember errors that render the integer
      ** as "non a number", so that its value is invalid or unknown.
      ** NAN's propagate so that performing any operating involving a NAN
      ** also yields a NAN.  A non-zero value here means that an error has
      ** been thrown and the synthetic integer is invalid.
      */
   unsigned neg;
      /* Only the absolute value of the integer is stored as a character
      ** string.  This boolean remembers if the number is negative.
      ** A non-zero value here means the integer is negative.  This may have
      ** either value when the value of zero is represented by the 
      ** synthetic integer.
      */
   unsigned len;
      /* The number of characters in the string representing the integer.
      ** This is the same as would be returned by strlen().  This field
      ** is maintained to avoid the expense of counting characters before
      ** the terminator each time the length is needed.
      */
   unsigned char digits[INTERMEDIATE_CALC_MAX_DIGITS + 1];
      /* The string representing the synthetic integer.  To avoid the need
      ** to move large blocks of characters, the integer is stored with the
      ** least significant digits first.  The string must be \0 terminated--
      ** this is done primarily to make printing things for debugging
      ** easier, although the terminator is redundant with the "len" field
      ** above.  The valid representation of zero is a "len" field of 
      ** zero and a terminating \0 as element [0] here.  No commas or other
      ** characters besides digits are allowed.
      */
   };


typedef struct synthetic_integer_struct SYNTHETIC_INTEGER;
   /* Typedef for more compact reference to data type.
   */


/****************************************************************************/
/* siCreate():                                                              */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Creates a synthetic integer (allocates the memory and initializes) and */
/*   sets value to zero.                                                    */
/*                                                                          */
/* INPUTS                                                                   */
/*   *i :  Pointer to pointer to synthetic integer.  The caller's pointer   */
/*         will be filled in to point to the allocated object.              */
/****************************************************************************/
void siCreate(SYNTHETIC_INTEGER **i)
   {
   asAssert(i != NULL, __LINE__);
      /* Input pointer may not be NULL.
      */

   *i = maMalloc(sizeof(SYNTHETIC_INTEGER));
      /* Allocate the memory, stuff the pointer in the caller's
      ** area.
      */

   /* Initialize the synthetic integer to what is required for
   ** the number zero.
   */
   (*i)->nan = 0;
   (*i)->neg = 0;
   (*i)->len = 0;
   (*i)->digits[0] = 0;
   }


/****************************************************************************/
/* siDestroy():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Destroys a synthetic integer (deallocates the memory and sets the      */
/*   caller's pointer to NULL.                                              */
/*                                                                          */
/* INPUTS                                                                   */
/*   *i :  Pointer to pointer to synthetic integer.  The synthetic integer  */
/*         will be destroyed and the caller's pointer will be filled to     */
/*         NULL.                                                            */
/****************************************************************************/
void siDestroy(SYNTHETIC_INTEGER **i)
   {
   asAssert(i != NULL, __LINE__);
      /* Input pointer to pointer may not be NULL.
      */
   asAssert(*i != NULL, __LINE__);
      /* Input pointer to block of memory may not be NULL.
      */

   maFree(*i);
      /* Deallocate block of memory.
      */

   *i = NULL;
      /* Fill in the caller's variable to NULL.
      */
   }


/****************************************************************************/
/* siCopy():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Copies a synthetic integer.  The integer must already be created.      */
/*                                                                          */
/* INPUTS                                                                   */
/*   *src, *dst  :  Source and destination synthetic integers.              */
/****************************************************************************/
void siCopy(SYNTHETIC_INTEGER **src, SYNTHETIC_INTEGER **dst)
   {
   asAssert(src != NULL, __LINE__);
   asAssert(*src != NULL, __LINE__);
   asAssert(dst != NULL, __LINE__);
   asAssert(*dst != NULL, __LINE__);

   /* There are a lot of ways to copy, and pros and cons.
   ** I'll choose the one that gives the least typing
   ** here in the source code, but maybe I'll need to
   ** optimize it later.
   */
   memcpy(*dst, *src, sizeof(SYNTHETIC_INTEGER));
   }


/****************************************************************************/
/* siMulByTen():                                                            */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Multiplies a synthetic integer by 10, producing a NAN result if there  */
/*   is an overflow in the number of digits.  This is a primitive operation */
/*   used in multiplication of arbitrary integers.                          */
/****************************************************************************/
void siMulByTen(SYNTHETIC_INTEGER **arg)
   {
   asAssert(arg != NULL, __LINE__);
   asAssert(*arg != NULL, __LINE__);

   /* If the integer is already NAN, NAN it stays.
   */
   if (!((*arg)->nan))
      {
      /* If the integer is 0, 0 it stays. */
      if ((*arg)->len)
         {
         /* The result will overflow iff the string is already
         ** at its length limit.  If so, must declare a NAN.  Multiplication
         ** by 10 is guaranteed to add exactly one digit.
         */
         asAssert((*arg)->len <= INTERMEDIATE_CALC_MAX_DIGITS, __LINE__);
         if ((*arg)->len == INTERMEDIATE_CALC_MAX_DIGITS)
            {
            /* Overflow.  Must declare a NAN. */
            (*arg)->digits[0] = '\0';
            (*arg)->len       = 0;
            (*arg)->nan       = TRUE;
            (*arg)->neg       = FALSE;
            }
         else
            {
            /* The digit shift will come off alright.  Carry it out.
            */
            unsigned tgt_idx;

            for (tgt_idx = (*arg)->len; tgt_idx >= 1; tgt_idx--)
               {
               (*arg)->digits[tgt_idx] = (*arg)->digits[tgt_idx-1];
               }
            (*arg)->digits[0] = '0';
            (*arg)->digits[(*arg)->len + 1] = '\0';
            ((*arg)->len)++;
            }
         }
      }
   }


/****************************************************************************/
/* siDivByTen():                                                            */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Divides a synthetic integer by 10 using digit-shifting (remainders     */
/*   are discarded.  This function is used in division of arbitrary         */
/*   integers.                                                              */
/****************************************************************************/
void siDivByTen(SYNTHETIC_INTEGER **arg)
   {
   asAssert(arg != NULL, __LINE__);
   asAssert(*arg != NULL, __LINE__);

   /* If the integer is already NAN, NAN it stays.
   */
   if (!((*arg)->nan))
      {
      /* If the integer is a single digit, it must go to zero.
      */
      if ((*arg)->len == 1)
         {
         (*arg)->len       = 0;
         (*arg)->digits[0] = '\0';
         (*arg)->neg       = FALSE;
         }
      else
         {
         unsigned i;

         asAssert((*arg)->len <= INTERMEDIATE_CALC_MAX_DIGITS, __LINE__);

         /* This is just a normal digit shift. */
         for (i=0; i<((*arg)->len); i++)
            {
            (*arg)->digits[i] = (*arg)->digits[i+1];
            }

         ((*arg)->len)--;
         }
      }
   }


/****************************************************************************/
/* siMulByDigit():                                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Multiplies a synthetic integer by an arbitrary digit, producing a      */
/*   result that may be zero, the argument, or an integer of larger mag-    */
/*   nitude.  A NAN is declared on overflow.  This is a primitive operation */
/*   used in multiplication of arbitrary integers.                          */
/****************************************************************************/
void siMulByDigit(SYNTHETIC_INTEGER **arg, char digit)
   {
   asAssert(arg != NULL, __LINE__);
   asAssert(*arg != NULL, __LINE__);
   asAssert(ddIsDigit(digit), __LINE__);

   /* If the integer is already NAN, NAN it stays.
   */
   if (!((*arg)->nan))
      {
      /* If the integer is 0 or the digit is '0', force the result
      ** to 0.
      */
      if (!((*arg)->len) || (digit == '0'))
         {
         (*arg)->digits[0] = '\0';
         (*arg)->len       = 0;
         (*arg)->nan       = FALSE;
         (*arg)->neg       = FALSE;
         }
      else
         {
         /* This is now a valid multiplication.  Regrettably, we can't
         ** determine an overflow or lack of decisively except by carrying 
         ** the multiplication out to its end.
         */
         char carry;
         char new_digit;
         unsigned cur_digit;

         carry = '0';

         asAssert((*arg)->len <= INTERMEDIATE_CALC_MAX_DIGITS, __LINE__);

         for (cur_digit=0; cur_digit < (*arg)->len; cur_digit++)
            {
            ddFundamentalMultiplicationCell((*arg)->digits[cur_digit],
                                            digit,
                                            carry,
                                            &new_digit,
                                            &carry);

            if (cur_digit == ((*arg)->len - 1)) /* If last digit in loop. */
               {
               if (cur_digit == (INTERMEDIATE_CALC_MAX_DIGITS-1)) /* arg was max len. */
                  {
                  if (carry != '0')
                     {
                     /* We have an SI that was at the length limit, we've processed.
                     ** the last digit, and there is a carry out.  Must declare a NAN.
                     */
                     (*arg)->digits[0] = '\0';
                     (*arg)->len       = 0;
                     (*arg)->nan       = TRUE;
                     (*arg)->neg       = FALSE;
                     }
                  else
                     {
                     /* We have an SI that was max length, but did not overflow.
                     ** Close it up normally.
                     */
                     (*arg)->digits[cur_digit] = new_digit;
                     /* Nothing else to do.  Everything is preset. */
                     }
                  }
               else
                  {
                  /* We are at the last digit of multiplication, but not at the 
                  ** maximum length of a string of digits.  Cleanup without fear
                  ** of overflow.
                  */
                  if (carry == '0')
                     {
                     /* Number stayed same length.  Just assign digit.  Terminator 
                     ** and other info is still valid.
                     */
                     (*arg)->digits[cur_digit] = new_digit;
                     }
                  else
                     {
                     /* At last digit of multiplication.  Number grew by one digit.
                     ** Clean up.
                     */
                     (*arg)->digits[cur_digit]   = new_digit;
                     (*arg)->digits[cur_digit+1] = carry;
                     (*arg)->digits[cur_digit+2] = '\0';
                     ((*arg)->len)++;

                     /* Must break out of the for() loop.  Otherwise, modifying
                     ** the length above will keep us going.
                     */
                     break;
                     }
                  }
               }
            else
               {
               /* Not the last digit in the loop.  Just assign digit.
               */
               (*arg)->digits[cur_digit] = new_digit;
               }
            } /* End for() each digit. */
         }
      }
   }


/****************************************************************************/
/* siCompareAbs():                                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Compares the absolute value of two synthetic integers (this involves   */
/*   comparing only the string representation without considering the       */
/*   sign).  This function is useful from within the full compare function  */
/*   to avoid duplication of code--it isn't very useful in any other        */
/*   context.                                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   *arg1, *arg2  :  Two synthetic integers whose absolute values should   */
/*                    be compared.                                          */
/*                                                                          */
/* OUTPUT                                                                   */
/*   <--   :   (-1)  : ABS(*arg1) < ABS(*arg2)                              */
/*              (0)  : ABS(*arg1) = ABS(*arg2)                              */
/*              (1)  : ABS(*arg1) > ABS(*arg2)                              */
/****************************************************************************/
int siCompareAbs(SYNTHETIC_INTEGER **arg1, SYNTHETIC_INTEGER **arg2)
   {
   int i;

   asAssert(arg1 != NULL, __LINE__);
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__);
   asAssert(*arg2 != NULL, __LINE__);

   if ((*arg1)->len < (*arg2)->len)
      {
      return(-1);
      }
   else if ((*arg1)->len > (*arg2)->len)
      {
      return(1);
      }
   else if (!((*arg1)->len) && !((*arg2)->len))
      {
      /* Both are zero, return equal. */
      return(0);
      }
   else
      {
      /* Lengths of strings are equal--must look at the strings in
      ** more detail.
      */
      for (i = (*arg1)->len; i >= 0; i--)
         {
         if ((*arg1)->digits[i] < (*arg2)->digits[i])
            {
            return(-1);
            }
         else if ((*arg1)->digits[i] > (*arg2)->digits[i])
            {
            return(1);
            }
         }

      /* Have iterated through the strings--couldn't find any differences.
      ** They are equal.  This also covers the zero case.
      */
      return(0);
      }
   }


/****************************************************************************/
/* siCompare():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Establishes the logical ordering of two synthetic integers.            */
/*                                                                          */
/* INPUTS                                                                   */
/*   *arg1, *arg2  :  Two synthetic integers to compare.                    */
/*                                                                          */
/* OUTPUT                                                                   */
/*   <--   :   (-1)  : *arg1 < *arg2.                                       */
/*              (0)  : *arg1 = *arg2                                        */
/*              (1)  : *arg1 > *arg2                                        */
/****************************************************************************/
int siCompare(SYNTHETIC_INTEGER **arg1, SYNTHETIC_INTEGER **arg2)
   {
   asAssert(arg1 != NULL, __LINE__);
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__);
   asAssert(*arg2 != NULL, __LINE__);

   /* If either integer has been marked NAN, we shouldn't be
   ** comparing, and there is no reason to handle this case,
   ** as it is a minority case and harmless whatever happens.
   */
   /* Cover the two easiest cases, which are where one operand is
   ** positive and the other is negative.
   */
   if (((*arg1)->neg) && (!((*arg2)->neg)))
      {
      return(-1);
      }
   else if ((!((*arg1)->neg)) && ((*arg2)->neg))
      {
      return(1);
      }
   /* We know both integers are of the same sign.  We can call the
   ** absolute value compare function to get their ordering.
   */
   else if ((*arg1)->neg)
      {
      return(siCompareAbs(arg2, arg1));  /* Reverse the order to negate the
                                         ** results provided by
                                         ** siCompareAbs().
                                         */
      }
   else if (!((*arg1)->neg))
      {
      return(siCompareAbs(arg1, arg2));  /* Normal non-neg compare. 
                                         */
      }
   else
      {
      /* It should be impossible to reach this case.
      */
      asAssert(0, __LINE__);  /* Fatal. */
      return(0);  /* Suppress compiler warning. */
      }
   }


/****************************************************************************/
/* siDump():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Outputs a synthetic integer to the standard output stream.  The output */
/*   includes a title, and provisions are made for NAN, for a description,  */
/*   and prints commas.  It is the caller's responsibility to provide lines */
/*   before and lines after if desired.  Ths function assumes that it       */
/*   starts with the cursor in column 1.                                    */
/*                                                                          */
/* INPUTS                                                                   */
/*   **si          :  The synthetic integer which is being output.          */
/*                                                                          */
/*   *desc         :  The description to use for the integer.  It will be   */
/*                    right-justified up against the start of the integer,  */
/*                    and this function will automatically include a        */
/*                    colon.  If the description is too long, this function */
/*                    will truncate it to fit the space available.          */
/****************************************************************************/
void siDump(SYNTHETIC_INTEGER **si, const char *desc)
   {
   unsigned cur_line;
   unsigned nlines;
   unsigned digits_per_line;

   /* Make sure the caller isn't doing something bad for the program's health.
   */
   asAssert(si != NULL, __LINE__); 
   asAssert(*si != NULL, __LINE__);
   asAssert(desc != NULL, __LINE__);

   /* The number of digits per line that we assume must be a multiple of
   ** three.  Round this up in case the preprocessor constant was set
   ** dubiously.
   */
   digits_per_line = ddUmax(3, ((DIGITS_PER_LINE + 2) /3) * 3);

   /* As the first order of business, figure out how many lines the beast
   ** will require.
   */
   if ((*si)->nan)
      {
      nlines = 1;  /* Only one line required for NAN verbeage. */
      }
   else if (!((*si)->len))
      {
      nlines = 1;  /* The zero value requires one line. */
      }
   else
      {
      /* In any other case, have a formula.
      */
      nlines = 1 + ((*si)->len - 1) / digits_per_line;
      }

   /* Iterate through each line, spitting out whatever is appropriate. */
   for (cur_line = 0; cur_line < nlines; cur_line++)
      {
      unsigned cur_digit_on_line;

      /* If this is the first line, spit out the description, right-aligned.
      ** Otherwise, spit spaces.
      */
      if (!cur_line)
         {
         /* First line. */
         unsigned len;

         len = strlen(desc);

         if (len <= NUMBER_DESC_WIDTH)
            {
            /* Description is shorter or equal, pad on left. */
            gfRepChar(' ', NUMBER_DESC_WIDTH - len);
            printf("%s", desc);
            }
         else
            {
            /* Description is too long, truncate. */
            unsigned i;

            for (i=0; i<len; i++)
               printf("%c", desc[i]);
            }

         printf(": ");

         /* If the number is negative, throw in a minus sign. */
         if ((*si)->neg && !(*si)->nan)
            {
            printf("- ");
            }
         else
            {
            printf("  ");
            }
         }
      else
         {
         /* Every line but first line. */
         gfRepChar(' ', NUMBER_DESC_WIDTH+4);
         }

      for(cur_digit_on_line=0; cur_digit_on_line < digits_per_line; cur_digit_on_line++)
         {
         unsigned idx_into_string;
            /* Index into the string which is our digit of interest.
            */

         /* Compute the index.  The equation is based on the ordering
         ** of presentation, for example,
         **
         **            7 6
         **          5 4 3
         **          2 1 0.
         **
         ** With a little thought, the equation should make sense.
         ** The index won't always be used to index into the string.
         */
         idx_into_string =
           (((nlines-1) - cur_line) * digits_per_line)
           +
           (digits_per_line - 1 - cur_digit_on_line);

         /* Print the appropriate digit or a space.  The NAN case and the 
         ** zero case need to be treated specially.
         */
         if ((*si)->nan)
            {
            /* Not a number.  Everything is blank, except spell out NAN
            ** at the end of the string of digits.
            */
            if (cur_digit_on_line == (digits_per_line - 3))
               {
               printf("N");
               }
            else if (cur_digit_on_line == (digits_per_line - 2))
               {
               printf("A");
               }
            else if (cur_digit_on_line == (digits_per_line - 1))
               {
               printf("N");
               }
            else
               {
               printf(" ");
               }
            }
         else if (!(*si)->len)
            {
            /* This is the zero case.  For zero, there is only one line,
            ** and every character except the last one is a blank.
            */
            if (cur_digit_on_line == (digits_per_line - 1))
               {
               printf("0");
               }
            else
               {
               printf(" ");
               }
            }
         else
            {
            /* This is a valid number which is not zero.  Need to print
            ** the digits.
            */

            if (idx_into_string < (*si)->len)
               {
               printf("%c", (*si)->digits[idx_into_string]);
               }
            else
               {
               printf(" ");
               }
            } /* End of digit case.

         /* Now handle the commas.  The rules for commas are straightforward.
         **    a)NAN never has a comma.
         **    b)Zeros never have a comma.
         **    c)The final line, last digit never has a comma.
         **    d)Everything else in multiples of three ...
         */
         if (!(idx_into_string % 3) && (idx_into_string))
            {
            if ((*si)->nan)
               {
               printf(" ");
               }
            else if (!(*si)->len)
               {
               printf(" ");
               }
            else
               {
               if (idx_into_string < (*si)->len)
                  {
                  printf(",");
                  }
               else
                  {
                  printf(" ");
                  }
               }
            }
         } /* End for each digit on the current line. */
      
      /* For the first line, print out an informative message
      ** advising of the number of digits.  For all other lines
      ** print nothing.
      */
      if (!cur_line && !((*si)->nan))
         {
         if (nlines == 1)
            printf(" ");

         if ((*si)->len <= 1)
            {
            printf("  (    1 digit)\n");
            }
         else if ((*si)->len < 1000)
            {
            printf("  (%5u digits)\n", (*si)->len);
            }
         else
            {
            printf("  (%u,%03u digits)\n", (*si)->len / 1000, (*si)->len % 1000);
            }
         }
      else
         {
         printf("\n");
         }
      } /* End for each line. */
   }


/****************************************************************************/
/* siSetToLong():                                                           */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Sets a synthetic integer (which must already be created) to a long     */
/*   value.  This is used primarily in preliminary unit testing.            */
/*                                                                          */
/* INPUTS                                                                   */
/*   **si          :  The synthetic integer to operate on.                  */
/*                                                                          */
/*   val           :  The value to set the synthetic integer to.            */
/****************************************************************************/
void siSetToLong(SYNTHETIC_INTEGER **si, long val)
   {
   char buf[100];
   char *p;
   unsigned len;
   unsigned idx;

   asAssert(si != NULL, __LINE__); 
   asAssert(*si != NULL, __LINE__);

   sprintf(buf, "%ld", val);

   if (buf[0] == '-')
      p = buf+1;
   else
      p = buf;

   len = strlen(p);

   (*si)->nan = FALSE;

   if (val == 0)
      {
      (*si)->nan = FALSE;
      (*si)->neg = FALSE;
      (*si)->len = 0;
      (*si)->digits[0] = '\0';
      }
   else 
      {
      if (val < 0)
         {
         (*si)->neg = TRUE;
         }
      else
         {
         (*si)->neg = FALSE;
         }
        
      (*si)->len = len;

      for (idx=0; idx<len; idx++)
         {
         (*si)->digits[idx] = p[len - idx - 1];
         }

      (*si)->digits[len] = '\0';
      }
   }


/****************************************************************************/
/* siSetToNan():                                                            */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Sets a synthetic integer (which must already be created) to NAN (not   */
/*   a number).                                                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   **si          :  The synthetic integer to set to NAN.                  */
/*                                                                          */
/* NOTES                                                                    */
/*   This program "evolved", and this function evolved after many of the    */
/*   functions that set a number to NAN.  There are may places in the code  */
/*   where an integer is set to NAN without the use of this function.       */
/****************************************************************************/
void siSetToNan(SYNTHETIC_INTEGER **si)
   {
   asAssert(si != NULL, __LINE__); 
   asAssert(*si != NULL, __LINE__);

   (*si)->digits[0] = '\0';
   (*si)->len       = 0;
   (*si)->nan       = TRUE;
   (*si)->neg       = FALSE;
   }


/****************************************************************************/
/* siSetToPowerOfTen():                                                     */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Sets a synthetic integer (which must already be created) to an         */
/*   integral power of ten.  This function is helpful for math in many      */
/*   cases.                                                                 */
/*                                                                          */
/* INPUTS                                                                   */
/*   **si          :  The synthetic integer to operate on.                  */
/*                                                                          */
/*   exp           :  The integral exponent value (non-negative).  For      */
/*                    example, a value of 0 here will result in the         */
/*                    assignment of 1, 1 here will result in the assignment */
/*                    of 10, 2 here will result in the assignment of 100.   */
/*                                                                          */
/* ERRORS                                                                   */
/*   A negative exponent will result in an assertion failure.  An exponent  */
/*   too large will result in the assignment of NAN.                        */
/*                                                                          */
/* NOTE                                                                     */
/*   This program "evolved", and this function evolved after many of the    */
/*   functions that set a number to a power of 10 "directly".  There may be */
/*   many places in the code where an integer is set to a power of 10       */
/*   without the use of this function.                                      */
/****************************************************************************/
void siSetToPowerOfTen(SYNTHETIC_INTEGER **si, int exp)
   {
   int i;

   asAssert(si != NULL, __LINE__); 
   asAssert(*si != NULL, __LINE__);
   asAssert(exp >= 0, __LINE__);

   if (exp < INTERMEDIATE_CALC_MAX_DIGITS)
      {
      (*si)->len = exp+1;
      (*si)->nan = FALSE;
      (*si)->neg = FALSE;
      (*si)->digits[exp+1] = '\0';
      (*si)->digits[exp] = '1';
      for (i=0; i<exp; i++)
         (*si)->digits[i] = '0';
      }
   else
      {
      siSetToNan(si);
      }
   }


/****************************************************************************/
/* siAddTwoNonnegative():                                                   */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Adds two synthetic integers which are guaranteed to be non-negative.   */
/*   The result may be set NAN if there is an overflow.  This is a low-     */
/*   level function called by other higher-level functions to do more       */
/*   complex arithmetic.                                                    */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, **arg2  :  The two operands to add.  Both operands must be     */
/*                      non-negative.  It is legal for both SI's to         */
/*                      use the same storage or to be the same variable.    */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  The synthetic integer where the output result is    */
/*                      placed.  This may be the same SI as either or both  */
/*                      of the inputs.                                      */
/****************************************************************************/
void siAddTwoNonnegative(SYNTHETIC_INTEGER **arg1,
                         SYNTHETIC_INTEGER **arg2,
                         SYNTHETIC_INTEGER **result)
   {
   SYNTHETIC_INTEGER *callers_orig_value = NULL;
   unsigned i;
   unsigned len1, len2, lenmax;
   char carry;
   char operand1, operand2, digitresult;

   /* Be sure the caller isn't doing anything silly. */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(!((*arg1)->neg), __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(!((*arg2)->neg), __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg1 != arg2) && (arg2 != result) && (result != arg1), __LINE__);

   /* If either input is NAN, the output is NAN by definition.
   */
   if (((*arg1)->nan) || ((*arg2)->nan))
      {
      (*result)->nan = TRUE;
      }
   else
      {
      /* The two input arguments don't matter because they aren't written,
      ** but if the caller has allowed the result to be the same as the 
      ** either or both of the two inputs, must use a different memory
      ** location.  However, only do this if need to, as will slow things
      ** down to create and destroy.
      */
      if ((*arg1 == *result) || (*arg2 == *result))
         {
         /* Caller has allowed result to be the same as either
         ** or both inputs.  Must replace caller's pointer temporarily.
         */
         callers_orig_value = *result;
         siCreate(result);
         }

      /* Zero out the potential result.
      */
      (*result)->neg = FALSE;
      (*result)->nan = FALSE;
      (*result)->len = 0;
      (*result)->digits[0] = '\0';

      /* Buffer important handy values.
      */
      len1 = (*arg1)->len;
      len2 = (*arg2)->len;
      lenmax = ddUmax(len1, len2);

      /* Be absolutely sure that both lengths are permissible values.  Memory
      ** reference errors can be hard to track down.
      */
      asAssert(lenmax <= INTERMEDIATE_CALC_MAX_DIGITS, __LINE__);

      /* Start off with an LSD carry input of '0', as there is no carry into the
      ** first digit.
      */
      carry = '0';

      /* Iterate through, performing the addition, character by character.
      ** characters that are unpopulated are treated as zero.
      */
      for (i=0; i<lenmax; i++)
         {
         /* Grab the right character from the arguments. */
         if (i < len1)
            operand1 = (*arg1)->digits[i];
         else
            operand1 = '0';
         if (i < len2)
            operand2 = (*arg2)->digits[i];
         else
            operand2 = '0';

         /* Perform the actual addition.  Note that the carry value is copied
         ** copied onto the stack for the function call, so carry can double
         ** as both input and output.
         */
         ddFundamentalAdditionCell(operand1, operand2, carry, &digitresult, &carry);         

         /* Stuff the result digit in the right place. */
         (*result)->digits[i] = digitresult;
         }

      /* After we're out of the loop, there are a few possible outcomes.  If the
      ** carry is zero, then there is no risk of any kind of overflow.
      */
      if (carry == '0')
         {
         /* No risk of overflow. */
         (*result)->digits[i] = '\0';
         (*result)->len = i;
         }
      else
         {
         /* carry == '1' */
         /* There are two possibilities here.  Either the result has grown one
         ** digit more than the longest input operand, and this is legal, or the
         ** result has grown one digit more and there is an overflow,
         ** forcing declaration of a NAN.
         */
         if (i == INTERMEDIATE_CALC_MAX_DIGITS)
            {
            (*result)->nan = TRUE;
            (*result)->len = 0;
            (*result)->digits[0] = '\0';
            }
         else
            {
            (*result)->len = i+1;
            (*result)->digits[i] = carry;
            (*result)->digits[i+1] = '\0';
            }
         }

      /* If it was necessary to evacuate the caller's result area, swap it back
      ** and deallocate memory.
      */
      if (callers_orig_value)
         {
         siCopy(result, &callers_orig_value);
         siDestroy(result);
         *result = callers_orig_value;
         }
      }
   }


/****************************************************************************/
/* siSubtractToProduceNonnegativeResult():                                  */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Subtracts two synthetic integers, with the understanding that the      */
/*   result will be non-negative.  This function is used as a building      */
/*   block for subtraction which can span zero.                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, **arg2  :  The two operands to subtract.  Both must be non-    */
/*                      negative.                                           */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  **arg1 - **arg2, or a fatal error if the result     */
/*                      would be negative.                                  */
/****************************************************************************/
void siSubtractToProduceNonnegativeResult(SYNTHETIC_INTEGER **arg1,
                                          SYNTHETIC_INTEGER **arg2,
                                          SYNTHETIC_INTEGER **result)
   {
   SYNTHETIC_INTEGER *callers_orig_value = NULL;
   unsigned i;
   unsigned len;
   char borrow;
   char operand1, operand2, digitresult;

   /* Be sure the caller isn't doing anything silly. */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(!((*arg1)->neg), __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(!((*arg2)->neg), __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg1 != arg2) && (arg2 != result) && (result != arg1), __LINE__);

   /* If either input is NAN, the output is NAN by definition.
   */
   if (((*arg1)->nan) || ((*arg2)->nan))
      {
      (*result)->nan = TRUE;
      }
   else
      {
      /* The two input arguments don't matter because they aren't written,
      ** but if the caller has allowed the result to be the same as the 
      ** either or both of the two inputs, must use a different memory
      ** location.  However, only do this if need to, as will slow things
      ** down to create and destroy.
      */
      if ((*arg1 == *result) || (*arg2 == *result))
         {
         /* Caller has allowed result to be the same as either
         ** or both inputs.  Must replace caller's pointer temporarily.
         */
         callers_orig_value = *result;
         siCreate(result);
         }

      /* Zero out the potential result.
      */
      (*result)->neg = FALSE;
      (*result)->nan = FALSE;
      (*result)->len = 0;
      (*result)->digits[0] = '\0';

      /* Buffer important handy values.
      */
      len = (*arg1)->len;  /* By definition, this one is the max. */

      /* Be absolutely sure that the length used for iteration is a 
      ** permissible value.
      */
      asAssert(len <= INTERMEDIATE_CALC_MAX_DIGITS, __LINE__);

      /* Start off with an LSD borrow input of '0', as there is no borrow into the
      ** first digit.
      */
      borrow = '0';

      /* Iterate through, performing the subtraction, character by character.
      ** Characters that are unpopulated are treated as zero.
      */
      for (i=0; i<len; i++)
         {
         /* Grab the right character from the arguments. */
         operand1 = (*arg1)->digits[i];
         if (i < (*arg2)->len)
            operand2 = (*arg2)->digits[i];
         else
            operand2 = '0';

         /* Perform the actual subtraction.  Note that the borrow value is
         ** copied onto the stack for the function call, so double can double
         ** as both input and output.
         */
         /* printf("Idx: %u, borrow in: %c, ", i, borrow); */

         ddFundamentalSubtractionCell(operand1, operand2, borrow, &digitresult, &borrow);         

         /* printf("op1: %c, op2: %c, borrow out: %c, result out: %c.\n", 
                operand1,
                operand2,
                borrow,
                digitresult); */

         /* Stuff the result digit in the right place. */
         (*result)->digits[i] = digitresult;
         }

      /* Trim the leading zeros from the subtraction result.  It would be more
      ** elegant to just identify the ripstop during the subtraction, but
      ** this will work, too.
      */
      if (!i)
         {
         /* We had a 0-0 subtraction. */
         (*result)->digits[0] = '\0';
         (*result)->len = 0;
         }
      else
         {
         (*result)->digits[i] = '\0';
         (*result)->len = i;
         while ((i >= 1) && ((*result)->digits[i-1] == '0'))
            {
            (*result)->digits[i-1] = '\0';
            (*result)->len = i-1;
            i--;
            }
         }

      /* If it was necessary to evacuate the caller's result area, swap it back
      ** and deallocate memory.
      */
      if (callers_orig_value)
         {
         siCopy(result, &callers_orig_value);
         siDestroy(result);
         *result = callers_orig_value;
         }
      }
   }


/****************************************************************************/
/* siUnrestrictedAddition():                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Adds two integers of any sign.  NAN is produced if anything overflows. */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, **arg2  :  The two synthetic integer operand to add.  It is    */
/*                      permitted for both to point to the same place.      */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  The result.  This may point to the same place as    */
/*                      either or both of arg1, arg2.                       */
/****************************************************************************/
void siUnrestrictedAddition(SYNTHETIC_INTEGER **arg1,
                            SYNTHETIC_INTEGER **arg2,
                            SYNTHETIC_INTEGER **result)
   {
   SYNTHETIC_INTEGER *callers_orig_value = NULL;
   int arg1_rel_arg2;
   unsigned arg1_is_neg;
   unsigned arg2_is_neg;
   unsigned case_number;

   /* Be sure the caller isn't doing anything silly. Because double-
   ** indirection is involved, another word of explanation is needed.
   ** The caller is passing a pointer to his pointer.  This isn't necessary
   ** in this circumstance (we'll never need to reallocate his object, for
   ** example), but it is just simpler to keep everthing consistent 
   ** throughout the program.  When I say that the caller can have pointers
   ** the same for the args and the result, I mean that the caller can have
   ** separate physical pointers which point to the same thing.  However, the
   ** caller may not have one physical pointer only and pass us three identical
   ** pointers to the same physical pointer.  This is illegal.  The caller must
   ** have three seperate pointers, even if they point to the same place.
   */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg1 != arg2) && (arg2 != result) && (result != arg1), __LINE__);

   /* The two input arguments don't matter because they aren't written,
   ** but if the caller has allowed the result to be the same as the 
   ** either or both of the two inputs, must use a different memory
   ** location.  However, only do this if need to, as will slow things
   ** down to create and destroy.
   */
   if ((*arg1 == *result) || (*arg2 == *result))
      {
      /* Caller has allowed result to be the same as either
      ** or both inputs.  Must replace caller's pointer temporarily.
      */
      callers_orig_value = *result;
      siCreate(result);
      }

   /* Zero out the potential result.
   */
   (*result)->neg = FALSE;
   (*result)->nan = FALSE;
   (*result)->len = 0;
   (*result)->digits[0] = '\0';

   /* If either operand is NAN, the result is NAN.  That is all the
   ** needs to be said.
   */
   if (((*arg1)->nan) || ((*arg2)->nan))
      {
      (*result)->nan = TRUE;
      }
   else
      {
      /* Gather some vital statistics about the two numbers passed.  These
      ** will be used to break into cases.
      */
      arg1_rel_arg2 = siCompareAbs(arg1, arg2) + 1;
      arg1_is_neg   = (*arg1)->neg;
      arg2_is_neg   = (*arg2)->neg;

      /* Zero out the signs of the two numbers.  They will be restored
      ** later.
      */
      (*arg1)->neg = FALSE;
      (*arg2)->neg = FALSE;
   
      /* Create a case number based on the above examination.  It is easier
      ** to do it this way so there is only one switch statement.
      */
      case_number =   ((unsigned)arg1_rel_arg2 * 4)
                  + arg1_is_neg * 2
                  + arg2_is_neg;
   
      /* printf("Case number: %u.\n", case_number); */

      switch(case_number)
         {
         /*********************************************************************/
         case 0:
            /* abs(arg1) < abs(arg2)  */
            /* arg1 >= 0              */
            /* arg2 >= 0              */
            /* Both arguments were non-negative.  The result will be non-
            ** negative, perhaps NAN.  Just add the two numbers, perhaps
            ** resulting in NAN.
            */
            siAddTwoNonnegative(arg1, arg2, result);
            break;
         /*********************************************************************/
         case 1:
            /* abs(arg1) < abs(arg2)  */
            /* arg1 >= 0              */
            /* arg2 < 0               */
            /* arg2 is negative, and will overwhelm arg1 and produce a negative
            ** result.  Get the positive result and flip the sign.
            */
            siSubtractToProduceNonnegativeResult(arg2, arg1, result);
            if (!((*result)->nan))
               (*result)->neg = TRUE;
            break;
         /*********************************************************************/
         case 2:
            /* abs(arg1) < abs(arg2)  */
            /* arg1 < 0               */
            /* arg2 >= 0              */
            /* Result will be positive, dominated by arg2. */
            siSubtractToProduceNonnegativeResult(arg2, arg1, result);
            break;
         /*********************************************************************/
         case 3:
            /* abs(arg1) < abs(arg2)  */
            /* arg1 < 0               */
            /* arg2 < 0               */
            /* arg1 and arg2 both negative.  Add them in a positive sense and
            ** negate the result.
            */
            siAddTwoNonnegative(arg1, arg2, result);
            if (!((*result)->nan))
               (*result)->neg = TRUE;
            break;
         /*********************************************************************/
         case 4:
            /* abs(arg1) == abs(arg2) */
            /* arg1 >= 0              */
            /* arg2 >= 0              */
            /* Two are equal and non-negative.  Just add. 
            */
            siAddTwoNonnegative(arg1, arg2, result);
            break;
         /*********************************************************************/
         case 5:
            /* abs(arg1) == abs(arg2) */
            /* arg1 >= 0              */
            /* arg2 < 0               */
            /* Result is zero.  Just leave the result the way it is, as we
            ** set the result to zero higher up in the code.
            */
            break;
         /*********************************************************************/
         case 6:
            /* abs(arg1) == abs(arg2) */
            /* arg1 < 0               */
            /* arg2 >= 0              */
            /* Result is zero.  Just leave the result the way it is, as we
            ** set the result to zero higher up in the code.
            */
            break;
         /*********************************************************************/
         case 7:
            /* abs(arg1) == abs(arg2) */
            /* arg1 < 0               */
            /* arg2 < 0               */
            /* Two are equal and negative.  Add and negate.
            */
            siAddTwoNonnegative(arg1, arg2, result);
            if (!((*result)->nan))
               (*result)->neg = TRUE;
            break;
         /*********************************************************************/
         case 8:
            /* abs(arg1) > abs(arg2)  */
            /* arg1 >= 0              */
            /* arg2 >= 0              */
            /* Both non-negative.  Just add.
            */
            siAddTwoNonnegative(arg1, arg2, result);
            break;
         /*********************************************************************/
         case 9:
            /* abs(arg1) > abs(arg2)  */
            /* arg1 >= 0              */
            /* arg2 < 0               */
            /* Result is positive and driven by arg1.  Subtract. */
            siSubtractToProduceNonnegativeResult(arg1, arg2, result);
            break;
         /*********************************************************************/
         case 10:
            /* abs(arg1) > abs(arg2)  */
            /* arg1 < 0               */
            /* arg2 >= 0              */
            /* Result is negative and driven by arg1.  Subtract and negate.
            */
            siSubtractToProduceNonnegativeResult(arg1, arg2, result);
            if (!((*result)->nan))
               (*result)->neg = TRUE;
            break;
         /*********************************************************************/
         case 11:
            /* abs(arg1) > abs(arg2)  */
            /* arg1 < 0               */
            /* arg2 < 0               */
            /* Result is negative.  Add and negate.
            */
            siAddTwoNonnegative(arg1, arg2, result);
            if (!((*result)->nan))
               (*result)->neg = TRUE;
            break;
         /*********************************************************************/
         default:
            /* Should never be here.  Fatal out. */
            asAssert(0, __LINE__); 
            break;
         /*********************************************************************/
         }
      
      /* Restore the results of the input arguments.  We will really mess up
      ** the caller if we don't.
      */
      (*arg1)->neg = arg1_is_neg;
      (*arg2)->neg = arg2_is_neg;
      }


   /* If it was necessary to evacuate the caller's result area, swap it back
   ** and deallocate memory.
   */
   if (callers_orig_value)
      {
      siCopy(result, &callers_orig_value);
      siDestroy(result);
      *result = callers_orig_value;
      }
   }

/****************************************************************************/
/* siUnrestrictedSubtraction():                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Subtracts two integers of any sign.  NAN is produced if anything       */
/*   overflows.                                                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, **arg2  :  The two synthetic integer operand to subtract.      */
/*                      Both may point to the same place.                   */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  The result, arg1-arg2.  May point to the same       */
/*                      place as either or both of arg1, arg2.              */
/****************************************************************************/
void siUnrestrictedSubtraction(SYNTHETIC_INTEGER **arg1,
                               SYNTHETIC_INTEGER **arg2,
                               SYNTHETIC_INTEGER **result)
   {
   unsigned neg;

   /* Be sure the caller is doing nothing silly. */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg1 != arg2) && (arg2 != result) && (result != arg1), __LINE__);

   /* If either operand is NAN, the result is NAN. */
   if ((*arg1)->nan || (*arg2)->nan)
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   else
      {
      /* Subtraction is easy to express in terms of addition, since the
      ** addition function already handles both operands of both signs.
      */
      if ((*arg2)->len)  /* Can't negate zero in this way. */
         {
         neg = (*arg2)->neg;
         (*arg2)->neg = !((*arg2)->neg);
         }

      siUnrestrictedAddition(arg1, arg2, result);

      if ((*arg2)->len)
         (*arg2)->neg = neg;
      }
   }


/****************************************************************************/
/* siUnrestrictedMultiplication():                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Muiltiplies two integers of any sign.  NAN is produced if anything     */
/*   overflows.                                                             */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, **arg2  :  The two synthetic integer operands to multiply.     */
/*                      Both may point to the same place.                   */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  The result, arg1*arg2.  May point to the same       */
/*                      place as either or both of arg1, arg2.              */
/****************************************************************************/
void siUnrestrictedMultiplication(SYNTHETIC_INTEGER **arg1,
                                  SYNTHETIC_INTEGER **arg2,
                                  SYNTHETIC_INTEGER **result)
   {
   /* Be sure the caller is doing nothing silly. */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg1 != arg2) && (arg2 != result) && (result != arg1), __LINE__);

   /* If either argument is NAN, the result is NAN. 
   */
   if ((*arg1)->nan || (*arg2)->nan)
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   /* If either argument is zero, result is zero. */
   else if (!((*arg1)->len) || !((*arg2)->len))
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = FALSE;
      (*result)->neg       = FALSE;
      }
   else
      {
      /* We have two non-zero operands, fit to be multiplied.
      */
      SYNTHETIC_INTEGER *rt, *temp, *rt2;
      int cidx, len;
      char     cur_digit;
      unsigned negarg1, negarg2;

      /* Must save the signs of the arguments and restore. */
      negarg1 = (*arg1)->neg;
      negarg2 = (*arg2)->neg;
      (*arg1)->neg = FALSE;
      (*arg2)->neg = FALSE;

      /* Create (allocate) a synthetic integer to contain the result
      ** that will be returned to the caller.
      */
      siCreate(&rt);

      /* Must create an alias.  The reasons for this are complex.  I coded
      ** myself into a corner and have to live with it.
      */
      rt2 = rt;

      /* Create (allocate) a synthetic integer that will be used to hold
      ** the result of multiplying arg2 by a digit.
      */
      siCreate(&temp);

      /* Go through, multiplying in much the same way as one would by hand.
      */
      len = (*arg1)->len;

      for (cidx = len-1; cidx >= 0; cidx--)
         {
         #if 0
         printf("cidx: %d\n", cidx);
         printf("rt before * 10 multiply\n");
         siDump(&rt, "rt before");
         /* Multiply the pending result by 10 */
         #endif
         siMulByTen(&rt);
         /* siDump(&rt, "rt after"); */

         /* Grab the current digit of interest and multiply arg2 by it. */
         cur_digit = (*arg1)->digits[cidx];
         /* printf("cur digit: %c\n", cur_digit); */

         siCopy(arg2, &temp);
         /* siDump(&temp, "copy of arg2"); */
         siMulByDigit(&temp, cur_digit);
         /* siDump(&temp, "arg2 * digit"); */

         /* Add the result of the multiplication to the pending
         ** result.
         */
         siUnrestrictedAddition(&rt2, &temp, &rt);

         #if 0
         siDump(&rt2, "rt2");
         siDump(&rt, "rt");
         siDump(&temp, "temp");
         #endif
         }

      /* We're done with the mathematical operation.  Signs should be a self-
      ** healing matter.
      */
      /* Copy the result back to caller's area. */
      siCopy(&rt, result);

      /* Destroy temporary areas. */
      siDestroy(&rt);
      siDestroy(&temp);

      /* Restore the signs of the integers that were arguments, then the
      ** sign of the result.
      */
      (*arg1)->neg = negarg1;
      (*arg2)->neg = negarg2;

      if ((negarg1 && negarg2) || (!negarg1 && !negarg2))
         (*result)->neg = FALSE;
      else
         (*result)->neg = TRUE;

      /* siDump(result, "result"); */
      }
   }


/****************************************************************************/
/* siUnrestrictedDivision():                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Divides two synthetic integers of any sign, producing an integer       */
/*   quotient and integer remainder.  NAN is assigned to both the quotient  */
/*   and remainder on invalid inputs.  NAN due to overflow is not possible  */
/*   in this function, as any integer divided by 1 or -1 yields an integer  */
/*   of the same magnitude.                                                 */
/*                                                                          */
/* INPUTS                                                                   */
/*   **dividend      :  The dividend of the division.  May be any value.    */
/*                                                                          */
/*   **divisor       :  The divisor of the division.  May be any value      */
/*                      except zero.                                        */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **quotient      :  The integer quotient of the division.  This should  */
/*                      behave identically to the C "/" operator when       */
/*                      applied to integers.                                */
/*                                                                          */
/*   **remainder     :  The remainder of the division.  This should behave  */
/*                      identically to the C "%" operator.  There was some  */
/*                      question in my mind about the best way to define a  */
/*                      remainder with a negative dividend or divisor, but  */
/*                      I simply used the behavior of the "%" operator in   */
/*                      C as a guide.                                       */
/*                                                                          */
/* USAGE NOTES                                                              */
/*   The functions in this program are inconsistent about whether they      */
/*   allow the inputs and the outputs to be pointers to the same memory     */
/*   locations, etc.  This program was constructed impromptu, and I made    */
/*   up the rules as I went along.  I've now decided that it isn't that     */
/*   useful to allow this, so for this function I'll require uniqueness     */
/*   between the four parameters.  This doesn't affect the correctness      */
/*   of the program--just its internal elegance.  These inconsistent        */
/*   decisions are not visible by using the program.                        */
/****************************************************************************/
void siUnrestrictedDivision(SYNTHETIC_INTEGER **dividend,
                            SYNTHETIC_INTEGER **divisor,
                            SYNTHETIC_INTEGER **quotient,
                            SYNTHETIC_INTEGER **remainder)
   {
   /* Be sure the caller is doing nothing silly. */
   /* No first- or second-level pointers may be NULL. */
   asAssert(dividend != NULL, __LINE__); 
   asAssert(*dividend != NULL, __LINE__);
   asAssert(divisor != NULL, __LINE__); 
   asAssert(*divisor != NULL, __LINE__);
   asAssert(quotient != NULL, __LINE__); 
   asAssert(*quotient != NULL, __LINE__);
   asAssert(remainder != NULL, __LINE__); 
   asAssert(*remainder != NULL, __LINE__);
   /* All pointers must be mutually exclusive. */
   asAssert((dividend != divisor) && (dividend != quotient) && (dividend != remainder), __LINE__);
   asAssert((*dividend != *divisor) && (*dividend != *quotient) && (*dividend != *remainder), __LINE__);
   asAssert((divisor != quotient) && (divisor != remainder), __LINE__);
   asAssert((*divisor != *quotient) && (*divisor != *remainder), __LINE__);
   asAssert(quotient != remainder, __LINE__);
   asAssert(*quotient != *remainder, __LINE__);

   /* If either argument is NAN, the results are NAN. Also, if the divisor is zero,
   ** that causes two NAN's, as well.
   */
   if ((*dividend)->nan || (*divisor)->nan || !((*divisor)->len))
      {
      (*quotient)->digits[0]  = '\0';
      (*quotient)->len        = 0;
      (*quotient)->nan        = TRUE;
      (*quotient)->neg        = FALSE;
      (*remainder)->digits[0] = '\0';
      (*remainder)->len       = 0;
      (*remainder)->nan       = TRUE;
      (*remainder)->neg       = FALSE;
      }
   /* Zero in the numerator always gets us zero on both outputs. 
   */
   else if (!((*dividend)->len))  
      {
      siSetToLong(quotient, 0);
      siSetToLong(remainder, 0);
      }
   /* Must do a legitimate long division.  We know for sure that
   ** both dividend and divisor are non-zero.
   */
   else
      {
      unsigned dividend_is_neg;
      unsigned divisor_is_neg;
         /* Signs of dividend and divisor, saved so we only
         ** operate on positive numbers.
         */
      int quotient_ndigits;
         /* The number of digits in the quotient.
         */
      int i;
         /* General-purpose iteration variable.
         */
      char j;
         /* Trial digit for trial multiplication.
         */
      SYNTHETIC_INTEGER *si_temp1 = NULL,
                        *si_temp2 = NULL,
                        *si_temp3 = NULL;
         /* Temporary large integers.  These need to be set
         ** to NULL here, because in the event of an
         ** error or unexpected condition, need to know
         ** which ones to release.
         */

      /* Remember and reset the signs of the passed dividend and 
      ** divisor.  It makes it simpler just to consider positive
      ** numbers.
      */
      dividend_is_neg  = (*dividend)->neg;
      divisor_is_neg   = (*divisor)->neg;
      (*dividend)->neg = 0;
      (*divisor)->neg  = 0;

      /* If the dividend is less (in magnitude) than the divisor,
      ** the quotient is zero and the remainder is the
      ** dividend.
      */
      if (siCompare(dividend, divisor) == -1)
         {
         siSetToLong(quotient, 0);
         siCopy(dividend, remainder);
         (*remainder)->neg = dividend_is_neg;
         }
      else
         {
         /* Allocate all of the temporary integers.
         ** Done all in one place to eliminate paths
         ** through the code where may allocate something
         ** already allocated and cause a memory leak.
         */
         siCreate(&si_temp1);
         siCreate(&si_temp2);
         siCreate(&si_temp3);

         /* Establish the number of digits in the
         ** quotient.  There is a window near the maximum
         ** number of digits where we won't be able to 
         ** successfully start the division because we
         ** can't form an upper bound which is a power
         ** of 10 times the divisor.  In this case,
         ** just return NAN.  This window won't be a problem
         ** in practice.  It could be worked around, but
         ** no need to close it.
         */
         siCopy(divisor, &si_temp1);
         siMulByTen(&si_temp1);
         quotient_ndigits = 1;
         while(siCompare(&si_temp1, dividend) <= 0)
            {
            siMulByTen(&si_temp1);
            if (si_temp1->nan)
               goto nan_return;
            quotient_ndigits++;
            }

         siDivByTen(&si_temp1);

         /* printf("quotient number of digits: %d\n",
                quotient_ndigits);
         siDump(&si_temp1, "trial divisor"); */

         /* We now know the number of digits that the
         ** quotient will contain.  We also know the
         ** starting point to use for trial subtraction.
         ** Count down from the highest order digit to the
         ** lowest-order digit, subtracting the maximum
         ** amount each time.
         */
         /* si_temp1 is the trial divisor reference
         ** copy. 
         */
         /* si_temp2 is a copy of the trial divisor,
         ** used to see what happens as we try multiplying
         ** by various trial digits.
         */
         /* remainder (in the caller's area) is used for
         ** staging the dividend after subtractions made.
         ** This will end up with the remainder in the
         ** end.
         */
         /* quotient (in the caller's area) contains the
         ** quotient.  We'll stage it directly there.
         */
         (*quotient)->len = (unsigned)quotient_ndigits;
         (*quotient)->digits[(*quotient)->len] = '\0';
         siCopy(dividend, remainder);

         /* For each digit in the result, going from most
         ** significant to least significant.
         */
         for (i=quotient_ndigits-1; i>=0; i--)
            {
            /* Try to find the largest digit for the spot
            ** that will not overflow the remainder.
            ** This is the right digit.
            */
            /* For each digit we might try.
            */
            for (j='0'; j<='9'; j++)
               {
               siCopy(&si_temp1, &si_temp2);
                  /* Buffer the value so we can do a trial
                  ** multiplication.
                  */
               siMulByDigit(&si_temp2, j);
                  /* Perform the trial multiplication
                  ** by a digit.
                  */
               siUnrestrictedSubtraction(remainder,
                                         &si_temp2,
                                         &si_temp3);
                  /* Perform a trial subtraction involving the trial
                  ** digit.
                  */

               /* We have success if what is left over after
               ** the trial subtraction is less than the
               ** trial divisor.
               */
               if (siCompare(&si_temp3, &si_temp1) < 0)
                  break;

               /* It is a fatal internal software error if
               ** we are at the digit '9' and the constraint
               ** still is not met.
               */
               asAssert(j != '9', __LINE__);
               } /* End for each possible digit. */

            /* OK, we have the digit, and it is in the 
            ** variable 'j'.  We also have the value
            ** we should subtract and the result.
            ** Stuff the digit, and the result.
            */
            (*quotient)->digits[i] = j;
            siCopy(&si_temp3, remainder);

            /* Divide the trial divisor by 10 for the
            ** next iteration.
            */
            siDivByTen(&si_temp1);
            } /* End for each digit position. */

         /* We now need to figure out what the sign
         ** of the quotient and the remainder are.
         */
         if ((dividend_is_neg && !divisor_is_neg) ||
            (!dividend_is_neg && divisor_is_neg))
            (*quotient)->neg = TRUE;
         else
            (*quotient)->neg = FALSE;

         if (dividend_is_neg && (*remainder)->len)
            (*remainder)->neg = TRUE;
         else
            (*remainder)->neg = FALSE;
         }

      goto normal_return;
      nan_return: ;
      /* This is the return point where the quotient
      ** and remainder are marked NAN.  Thank goodness
      ** they didn't forget about the "goto" in 'C'.
      */

      normal_return:
      /* Restore the signs of the input arguments.
      */
      (*dividend)->neg = dividend_is_neg;
      (*divisor)->neg  = divisor_is_neg;

      /* Deallocate all allocated memory.
      */
      if (si_temp1)
         siDestroy(&si_temp1);
      if (si_temp2)
         siDestroy(&si_temp2);
      if (si_temp3)
         siDestroy(&si_temp3);
      }
   }


/****************************************************************************/
/* siGcd():                                                                 */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Applies Euclid's algorithm to obtain the GCD of two positive integers. */
/*   Any unexpected input operands will generate a NAN for the GCD.         */
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg1, arg2    :  The two positive integers whose GCD is to be found. */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  gcd(**arg1, **arg2).                                */
/****************************************************************************/
void siGcd(SYNTHETIC_INTEGER **arg1,
           SYNTHETIC_INTEGER **arg2,
           SYNTHETIC_INTEGER **result)
   {

   /* Be sure the caller isn't doing anything silly. */
   asAssert(arg1 != NULL, __LINE__); 
   asAssert(*arg1 != NULL, __LINE__);
   asAssert(arg2 != NULL, __LINE__); 
   asAssert(*arg2 != NULL, __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);

   /* A NAN input or zero or negative integers gets a NAN result.
   */
   if ((*arg1)->nan || (*arg2)->nan || !(*arg1)->len || !(*arg2)->len
      || (*arg1)->neg || (*arg2)->neg)
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   else
      {
      /* This algorithm is so well known and documented that I won't insult
      ** the reader with comments.
      */
      SYNTHETIC_INTEGER *h, *k, *q, *r;

      siCreate(&h);
      siCreate(&k);
      siCreate(&q);
      siCreate(&r);

      if (siCompare(arg1, arg2) == -1)
         {
         siCopy(arg1, &h);
         siCopy(arg2, &k);
         }
      else
         {
         siCopy(arg1, &k);
         siCopy(arg2, &h);
         }

      siCopy(&h, &r);

      do 
         {
         siCopy(&r, result);
         siUnrestrictedDivision(&h,&k,&q,&r);
         siCopy(&k, &h);
         siCopy(&r, &k);
         } while (r->len);

      siDestroy(&h);
      siDestroy(&k);
      siDestroy(&q);
      siDestroy(&r);
      }
   }

                             
/****************************************************************************/
/* siIntegerExponentiation():                                               */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Exponentiates one integer to the integral power of another.  The       */
/*   integer to be exponentiated may be of any sign, but the exponent must  */
/*   be non-negative.  Invalid or overflow conditions result in the         */
/*   assignment of NAN.                                                     */                       
/*                                                                          */
/* INPUTS                                                                   */
/*   **arg           :  Integer to be exponentiated.                        */
/*                                                                          */
/*   **exponent      :  Exponent to use.  Must be non-negative.             */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **result        :  The result, arg1**arg2.                             */
/*                                                                          */
/* NOTE                                                                     */
/*   In other functions (written chronologically earlier), it was allowed   */
/*   for the arguments and the result to be the same physical object.  This */
/*   didn't prove to be a useful feature, so it is disallowed in this       */
/*   function.                                                              */
/****************************************************************************/
void siIntegerExponentiation(SYNTHETIC_INTEGER **arg,
                             SYNTHETIC_INTEGER **exponent,
                             SYNTHETIC_INTEGER **result)
   {
   /* Be sure the caller is doing nothing silly. */
   asAssert(arg != NULL, __LINE__); 
   asAssert(*arg != NULL, __LINE__);
   asAssert(exponent != NULL, __LINE__); 
   asAssert(*exponent != NULL, __LINE__);
   asAssert(result != NULL, __LINE__); 
   asAssert(*result != NULL, __LINE__);
   asAssert((arg != exponent) && (exponent != result) && (result != arg), __LINE__);
   asAssert((*arg != *exponent) && (*exponent != *result) && (*result != *arg), __LINE__);

   /* If either argument is NAN, the result is NAN. */
   if (((*arg)->nan) || ((*exponent)->nan))
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   /* If we're trying to raise 0 to the 0th power, this is a no-no,
   ** as well.
   */
   else if (!((*arg)->len) && !((*exponent)->len))
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   /* A negative exponent is a no-no, as well.
   */
   else if ((*exponent)->neg)
      {
      (*result)->digits[0] = '\0';
      (*result)->len       = 0;
      (*result)->nan       = TRUE;
      (*result)->neg       = FALSE;
      }
   /* Zero to any power but zero is zero.
   */
   else if (!((*arg)->len))
      {
      siSetToLong(result, 0);
      }
   /* Any number but zero to the zeroth power is one. */
   else if (!((*exponent)->len))
      {
      siSetToLong(result, 1);
      }
   else
      {
      unsigned binary_exponent;
      SYNTHETIC_INTEGER *temp_si1, *result_copy1, *temp_si2, *temp_si3;

      /* Here we have a non-zero integer raised to a non-zero power.
      ** The best algorithm to do this is to examine the bit pattern
      ** corresponding to the exponent--we can generate arg**2,
      ** arg**4, arg**8, etc., just by repeatedly squaring it.  NAN's
      ** are picked up automatically.
      */
      /* First, it is convenient convert the exponent to an 
      ** unsigned integer, so we can examine the
      ** bit pattern directly.  Will agree that anything over
      ** 32000 is treated as 32000, to avoid platform issues.
      */
      siCreate(&temp_si1);
      siCreate(&temp_si2);
      siCreate(&temp_si3);
      siCreate(&result_copy1);

      siSetToLong(&temp_si1, 32000);

      if (siCompare(exponent, &temp_si1) > 0)
         {
         asFatal("exponent too large (max is 32,000)");
         }
      else
         {
         /* Must convert from the string representation
         ** buried in the synthetic integer to a binary
         ** unsigned integer.  This is a bit tricky because the
         ** string is in reverse order from the traditional.
         */
            {
            int sidx;

            sidx = (*exponent)->len - 1;
            binary_exponent = 0;

            while (sidx >= 0)
               {
               binary_exponent *= 10;
               binary_exponent += ddDigitToValue((*exponent)->digits[sidx]);
               sidx--;
               }
            }
         }

      /* Start off with a result of 1 */
      siSetToLong(result, 1);
      siSetToLong(&result_copy1, 1);

      /* Start off with arg**1 */
      siCopy(arg, &temp_si1);
      siCopy(arg, &temp_si2);
      siCopy(arg, &temp_si3);

      while (binary_exponent)
         {
         /* Multiply result by the working register iff we have a 1 as the
         ** LSB of the exponent. 
         */
         if (binary_exponent & 1)
            {
            siUnrestrictedMultiplication(
               &result_copy1, 
               &temp_si1,
               result);

            /* Clone out the result, because we're keeping an additional copy. */
            siCopy(result, &result_copy1);
            }


         /* Square the quantity cloned from the argument, and clone the copies
         ** back. 
         */
         siUnrestrictedMultiplication(
            &temp_si1, 
            &temp_si2,
            &temp_si3);
         siCopy(&temp_si3, &temp_si1);
         siCopy(&temp_si3, &temp_si2);

         /* Shift the exponent down one bit. */
         binary_exponent >>= 1;
         }

      /* Destroy all of the data structures created on the fly. */
      siDestroy(&temp_si1);
      siDestroy(&temp_si2);
      siDestroy(&temp_si3);
      siDestroy(&result_copy1);

      /* At this point the result should contain the right value and
      ** we should be ready to return.
      */
      }
   }


/****************************************************************************/
/****************************************************************************/
/**********   R A T I O N A L    N U M B E R    F U N C T I O N S   *********/
/****************************************************************************/
/****************************************************************************/
/* This section is reserved for functions which operate on rational numbers.
** For directness, rational numbers are represented as integer pairs (no
** sense adding more data structures to this program.
*/

/****************************************************************************/
/* rnSum():                                                                 */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Calculates the sum of two rational numbers, supplying the results in   */
/*   lowest terms.                                                          */
/*                                                                          */
/* INPUTS                                                                   */
/*   **h1_formal_par_in,                                                    */
/*   **k1_formal_par_in,                                                    */
/*   **h2_formal_par_in,                                                    */
/*   **k2_formal_par_in :  The numerators and denominators of the two       */
/*                         rational numbers to add.  Zero denominators      */
/*                         are not allowed, but duplicate pointers of all   */
/*                         types are.                                       */
/*   **h_result,                                                            */
/*   **k_result         :  The result, in lowest terms.  Zero is  repre-    */
/*                         sented canonically as 0/1.  The memory must be   */
/*                         allocated in the caller's area, and the two      */
/*                         data items must be distinct.                     */
/****************************************************************************/
void rnSum(SYNTHETIC_INTEGER **h1_formal_par_in,
           SYNTHETIC_INTEGER **k1_formal_par_in,
           SYNTHETIC_INTEGER **h2_formal_par_in,
           SYNTHETIC_INTEGER **k2_formal_par_in,
           SYNTHETIC_INTEGER **h_result,
           SYNTHETIC_INTEGER **k_result)
   {
   SYNTHETIC_INTEGER *arg1_h,
                     *arg1_k,
                     *arg2_h,
                     *arg2_k,
                     *common_denominator,
                     *arg1_numerator,
                     *arg2_numerator,
                     *cumulative_numerator,
                     *result_gcd,
                     *trash_remainder;

   unsigned is_neg = FALSE;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h1_formal_par_in != NULL, __LINE__);
   asAssert(*h1_formal_par_in != NULL, __LINE__);
   asAssert(k1_formal_par_in != NULL, __LINE__);
   asAssert(*k1_formal_par_in != NULL, __LINE__);
   asAssert(h2_formal_par_in != NULL, __LINE__);
   asAssert(*h2_formal_par_in != NULL, __LINE__);
   asAssert(k2_formal_par_in != NULL, __LINE__);
   asAssert(*k2_formal_par_in != NULL, __LINE__);
   asAssert(h_result != NULL, __LINE__);
   asAssert(*h_result != NULL, __LINE__);
   asAssert(k_result != NULL, __LINE__);
   asAssert(*k_result != NULL, __LINE__);
   asAssert(h_result != k_result, __LINE__);
   asAssert(*h_result != *k_result, __LINE__);

   /* Allocate space for temporary variables.
   */
   siCreate(&arg1_h);
   siCreate(&arg1_k);
   siCreate(&arg2_h);
   siCreate(&arg2_k);
   siCreate(&common_denominator);
   siCreate(&arg1_numerator);
   siCreate(&arg2_numerator);
   siCreate(&cumulative_numerator);
   siCreate(&result_gcd);
   siCreate(&trash_remainder);

   /* Copy over the formal parameters to local variables.  This
   ** means we can modify the inputs without worry.
   */
   siCopy(h1_formal_par_in, &arg1_h);
   siCopy(k1_formal_par_in, &arg1_k);
   siCopy(h2_formal_par_in, &arg2_h);
   siCopy(k2_formal_par_in, &arg2_k);

   /* If any of the four inputs are NAN, the results must be
   ** NAN.
   */
   if (arg1_h->nan || arg1_k->nan || arg2_h->nan || arg2_k->nan)
      {
      goto nan_return;
      }
   
   /* If either denominator is zero, the result is NAN.
   */
   if (!(arg1_k->len) || !(arg2_k->len))
      {
      goto nan_return;
      }

   /* Adjust the signs of the the arguments so that the denominators are
   ** always positive.
   */
   if (arg1_h->neg && arg1_k->neg)
      {
      arg1_h->neg = FALSE;
      arg1_k->neg = FALSE;
      }
   else if (!(arg1_h->neg) && arg1_k->neg)
      {
      arg1_h->neg = TRUE;
      arg1_k->neg = FALSE;
      }

   if (arg2_h->neg && arg2_k->neg)
      {
      arg2_h->neg = FALSE;
      arg2_k->neg = FALSE;
      }
   else if (!(arg2_h->neg) && arg2_k->neg)
      {
      arg2_h->neg = TRUE;
      arg2_k->neg = FALSE;
      }

   /* Calculate the common denominator.
   */
   siUnrestrictedMultiplication(&arg1_k, &arg2_k, &common_denominator);

   /* Calculate the arg1 numerator.
   */
   siUnrestrictedMultiplication(&arg1_h, &arg2_k, &arg1_numerator);

   /* Calculate the arg2 numerator.
   */
   siUnrestrictedMultiplication(&arg2_h, &arg1_k, &arg2_numerator);

   /* Add the numerators to get the additive result. */
   siUnrestrictedAddition(&arg1_numerator,
                          &arg2_numerator,
                          &cumulative_numerator);

   /* If we NAN'd out ... 
   */
   if (cumulative_numerator->nan)
      goto nan_return;

   /* If we've reached a result of zero, it won't be legal to form
   ** a gcd().  Assign the canonical form of zero.
   */
   if (!(cumulative_numerator->len))
      {
      siSetToLong(h_result, 0);
      siSetToLong(k_result, 1);
      goto normal_return;
      }

   /* Snatch the sign temporarily.
   */
   if (cumulative_numerator->neg)
      {
      is_neg = TRUE;
      cumulative_numerator->neg = FALSE;
      }

   /* Form the gcd. */
   siGcd(&cumulative_numerator, 
         &common_denominator,
         &result_gcd);

   /* Set the sign back. */
   cumulative_numerator->neg = is_neg;

   /* Divide both numerator and denominator by
   ** the gcd().
   */
   siUnrestrictedDivision(&cumulative_numerator,
                          &result_gcd, 
                          h_result,
                          &trash_remainder);
   siUnrestrictedDivision(&common_denominator,
                          &result_gcd, 
                          k_result,
                          &trash_remainder);

   goto normal_return;
   nan_return: ;
   siSetToNan(h_result);
   siSetToNan(k_result);
   normal_return: ;

   /* Destroy temporary variables.
   */
   siDestroy(&arg1_h);
   siDestroy(&arg1_k);
   siDestroy(&arg2_h);
   siDestroy(&arg2_k);
   siDestroy(&common_denominator);
   siDestroy(&arg1_numerator);
   siDestroy(&arg2_numerator);
   siDestroy(&cumulative_numerator);
   siDestroy(&result_gcd);
   siDestroy(&trash_remainder);
   }


/****************************************************************************/
/* rnDifference():                                                          */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Calculates the difference of two rational numbers, returning the       */
/*   results in lowest terms.                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   **h1_formal_par_in,                                                    */
/*   **k1_formal_par_in,                                                    */
/*   **h2_formal_par_in,                                                    */
/*   **k2_formal_par_in :  The numerators and denominators of the two       */
/*                         rational numbers to subtract.  Zero denominators */
/*                         are not allowed, but duplicate pointers of all   */
/*                         types are.                                       */
/*   **h_result,                                                            */
/*   **k_result         :  The result, in lowest terms.  Zero is  repre-    */
/*                         sented canonically as 0/1.  The memory must be   */
/*                         allocated in the caller's area, and the two      */
/*                         data items must be distinct.                     */
/****************************************************************************/
void rnDifference(SYNTHETIC_INTEGER **h1_formal_par_in,
                  SYNTHETIC_INTEGER **k1_formal_par_in,
                  SYNTHETIC_INTEGER **h2_formal_par_in,
                  SYNTHETIC_INTEGER **k2_formal_par_in,
                  SYNTHETIC_INTEGER **h_result,
                  SYNTHETIC_INTEGER **k_result)
   {
   SYNTHETIC_INTEGER *arg1_h,
                     *arg1_k,
                     *arg2_h,
                     *arg2_k,
                     *common_denominator,
                     *arg1_numerator,
                     *arg2_numerator,
                     *cumulative_numerator,
                     *result_gcd,
                     *trash_remainder;

   unsigned is_neg = FALSE;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h1_formal_par_in != NULL, __LINE__);
   asAssert(*h1_formal_par_in != NULL, __LINE__);
   asAssert(k1_formal_par_in != NULL, __LINE__);
   asAssert(*k1_formal_par_in != NULL, __LINE__);
   asAssert(h2_formal_par_in != NULL, __LINE__);
   asAssert(*h2_formal_par_in != NULL, __LINE__);
   asAssert(k2_formal_par_in != NULL, __LINE__);
   asAssert(*k2_formal_par_in != NULL, __LINE__);
   asAssert(h_result != NULL, __LINE__);
   asAssert(*h_result != NULL, __LINE__);
   asAssert(k_result != NULL, __LINE__);
   asAssert(*k_result != NULL, __LINE__);
   asAssert(h_result != k_result, __LINE__);
   asAssert(*h_result != *k_result, __LINE__);

   /* Allocate space for temporary variables.
   */
   siCreate(&arg1_h);
   siCreate(&arg1_k);
   siCreate(&arg2_h);
   siCreate(&arg2_k);
   siCreate(&common_denominator);
   siCreate(&arg1_numerator);
   siCreate(&arg2_numerator);
   siCreate(&cumulative_numerator);
   siCreate(&result_gcd);
   siCreate(&trash_remainder);

   /* Copy over the formal parameters to local variables.  This
   ** means we can modify the inputs without worry.
   */
   siCopy(h1_formal_par_in, &arg1_h);
   siCopy(k1_formal_par_in, &arg1_k);
   siCopy(h2_formal_par_in, &arg2_h);
   siCopy(k2_formal_par_in, &arg2_k);

   /* If any of the four inputs are NAN, the results must be
   ** NAN.
   */
   if (arg1_h->nan || arg1_k->nan || arg2_h->nan || arg2_k->nan)
      {
      goto nan_return;
      }

   /* If either denominator is zero, the result is NAN.
   */
   if (!(arg1_k->len) || !(arg2_k->len))
      {
      goto nan_return;
      }

   /* Adjust the signs of the the arguments so that the denominators are
   ** always positive.
   */
   if (arg1_h->neg && arg1_k->neg)
      {
      arg1_h->neg = FALSE;
      arg1_k->neg = FALSE;
      }
   else if (!(arg1_h->neg) && arg1_k->neg)
      {
      arg1_h->neg = TRUE;
      arg1_k->neg = FALSE;
      }

   if (arg2_h->neg && arg2_k->neg)
      {
      arg2_h->neg = FALSE;
      arg2_k->neg = FALSE;
      }
   else if (!(arg2_h->neg) && arg2_k->neg)
      {
      arg2_h->neg = TRUE;
      arg2_k->neg = FALSE;
      }

   /* Calculate the common denominator.
   */
   siUnrestrictedMultiplication(&arg1_k, &arg2_k, &common_denominator);

   /* Calculate the arg1 numerator.
   */
   siUnrestrictedMultiplication(&arg1_h, &arg2_k, &arg1_numerator);

   /* Calculate the arg2 numerator.
   */
   siUnrestrictedMultiplication(&arg2_h, &arg1_k, &arg2_numerator);

   /* Subtract the numerators to get the result. */
   siUnrestrictedSubtraction(&arg1_numerator,
                             &arg2_numerator,
                             &cumulative_numerator);

   /* If we NAN'd out ... 
   */
   if (cumulative_numerator->nan)
      goto nan_return;

   /* If we've reached a result of zero, it won't be legal to form
   ** a gcd().  Assign the canonical form of zero.
   */
   if (!(cumulative_numerator->len))
      {
      siSetToLong(h_result, 0);
      siSetToLong(k_result, 1);
      goto normal_return;
      }

   /* Snatch the sign temporarily.
   */
   if (cumulative_numerator->neg)
      {
      is_neg = TRUE;
      cumulative_numerator->neg = FALSE;
      }

   /* Form the gcd. */
   siGcd(&cumulative_numerator, 
         &common_denominator,
         &result_gcd);

   /* Set the sign back. */
   cumulative_numerator->neg = is_neg;

   /* Divide both numerator and denominator by
   ** the gcd().
   */
   siUnrestrictedDivision(&cumulative_numerator,
                          &result_gcd, 
                          h_result,
                          &trash_remainder);
   siUnrestrictedDivision(&common_denominator,
                          &result_gcd, 
                          k_result,
                          &trash_remainder);

   goto normal_return;
   nan_return: ;
   siSetToNan(h_result);
   siSetToNan(k_result);
   normal_return: ;

   /* Destroy temporary variables.
   */
   siDestroy(&arg1_h);
   siDestroy(&arg1_k);
   siDestroy(&arg2_h);
   siDestroy(&arg2_k);
   siDestroy(&common_denominator);
   siDestroy(&arg1_numerator);
   siDestroy(&arg2_numerator);
   siDestroy(&cumulative_numerator);
   siDestroy(&result_gcd);
   siDestroy(&trash_remainder);
   }


/****************************************************************************/
/* rnProduct():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Calculates the product of two rational numbers, returning the          */
/*   results in lowest terms.                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   **h1_formal_par_in,                                                    */
/*   **k1_formal_par_in,                                                    */
/*   **h2_formal_par_in,                                                    */
/*   **k2_formal_par_in :  The numerators and denominators of the two       */
/*                         rational numbers to multiply.  Zero denominators */
/*                         are not allowed, but duplicate pointers of all   */
/*                         types are.                                       */
/*   **h_result,                                                            */
/*   **k_result         :  The result, in lowest terms.  Zero is  repre-    */
/*                         sented canonically as 0/1.  The memory must be   */
/*                         allocated in the caller's area, and the two      */
/*                         data items must be distinct.                     */
/****************************************************************************/
void rnProduct(SYNTHETIC_INTEGER **h1_formal_par_in,
               SYNTHETIC_INTEGER **k1_formal_par_in,
               SYNTHETIC_INTEGER **h2_formal_par_in,
               SYNTHETIC_INTEGER **k2_formal_par_in,
               SYNTHETIC_INTEGER **h_result,
               SYNTHETIC_INTEGER **k_result)
   {
   SYNTHETIC_INTEGER *arg1_h,
                     *arg1_k,
                     *arg2_h,
                     *arg2_k,
                     *result_numerator,
                     *result_denominator,
                     *gcd,
                     *trash_remainder;

   unsigned is_neg = FALSE;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h1_formal_par_in != NULL, __LINE__);
   asAssert(*h1_formal_par_in != NULL, __LINE__);
   asAssert(k1_formal_par_in != NULL, __LINE__);
   asAssert(*k1_formal_par_in != NULL, __LINE__);
   asAssert(h2_formal_par_in != NULL, __LINE__);
   asAssert(*h2_formal_par_in != NULL, __LINE__);
   asAssert(k2_formal_par_in != NULL, __LINE__);
   asAssert(*k2_formal_par_in != NULL, __LINE__);
   asAssert(h_result != NULL, __LINE__);
   asAssert(*h_result != NULL, __LINE__);
   asAssert(k_result != NULL, __LINE__);
   asAssert(*k_result != NULL, __LINE__);
   asAssert(h_result != k_result, __LINE__);
   asAssert(*h_result != *k_result, __LINE__);

   /* Allocate space for temporary variables.
   */
   siCreate(&arg1_h);
   siCreate(&arg1_k);
   siCreate(&arg2_h);
   siCreate(&arg2_k);
   siCreate(&result_numerator);
   siCreate(&result_denominator);
   siCreate(&gcd);
   siCreate(&trash_remainder);

   /* Copy over the formal parameters to local variables.  This
   ** means we can modify the inputs without worry.
   */
   siCopy(h1_formal_par_in, &arg1_h);
   siCopy(k1_formal_par_in, &arg1_k);
   siCopy(h2_formal_par_in, &arg2_h);
   siCopy(k2_formal_par_in, &arg2_k);

   /* If any of the four inputs are NAN, the results must be
   ** NAN.
   */
   if (arg1_h->nan || arg1_k->nan || arg2_h->nan || arg2_k->nan)
      {
      goto nan_return;
      }

   /* If either denominator is zero, the result is NAN.
   */
   if (!(arg1_k->len) || !(arg2_k->len))
      {
      goto nan_return;
      }

   /* Carry out the multiplication of numerators and denominators to get
   ** the two products.
   */
   siUnrestrictedMultiplication(&arg1_h, &arg2_h, &result_numerator);
   siUnrestrictedMultiplication(&arg1_k, &arg2_k, &result_denominator);

   /* If either result is NAN, the results are NAN.
   */
   if (result_numerator->nan || result_denominator->nan)
      {
      goto nan_return;
      }

   /* If the numerator is zero, we must return canonical zero.
   */
   if (!(result_numerator->len))
      {
      siSetToLong(h_result, 0);
      siSetToLong(k_result, 1);
      goto normal_return;
      }

   /* We need to acquire the sign and make both results positive otherwise
   ** can't obtain the GCD.
   */
   if (result_numerator->neg)
      {
      is_neg = TRUE;
      result_numerator->neg = FALSE;
      }
   if (result_denominator->neg)
      {
      is_neg = !is_neg;
      result_denominator->neg = FALSE;
      }

   /* Form the gcd. */
   siGcd(&result_numerator, 
         &result_denominator,
         &gcd);

   /* Set the sign back. */
   result_numerator->neg = is_neg;

   /* Divide both numerator and denominator by
   ** the gcd().
   */
   siUnrestrictedDivision(&result_numerator,
                          &gcd, 
                          h_result,
                          &trash_remainder);
   siUnrestrictedDivision(&result_denominator,
                          &gcd, 
                          k_result,
                          &trash_remainder);

   goto normal_return;
   nan_return: ;
   siSetToNan(h_result);
   siSetToNan(k_result);
   normal_return: ;

   /* Destroy temporary variables.
   */
   siDestroy(&arg1_h);
   siDestroy(&arg1_k);
   siDestroy(&arg2_h);
   siDestroy(&arg2_k);
   siDestroy(&result_numerator);
   siDestroy(&result_denominator);
   siDestroy(&gcd);
   siDestroy(&trash_remainder);
   }


/****************************************************************************/
/* rnQuotient():                                                            */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Calculates the quotient of two rational numbers, returning the         */
/*   results in lowest terms.                                               */
/*                                                                          */
/* INPUTS                                                                   */
/*   **h1_formal_par_in,                                                    */
/*   **k1_formal_par_in,                                                    */
/*   **h2_formal_par_in,                                                    */
/*   **k2_formal_par_in :  The numerators and denominators of the two       */
/*                         rational numbers to divide.  Zero denominators   */
/*                         and zero divisors are not allowed, but duplicate */
/*                         pointers of all types are.                       */
/*   **h_result,                                                            */
/*   **k_result         :  The result, in lowest terms.  Zero is  repre-    */
/*                         sented canonically as 0/1.  The memory must be   */
/*                         allocated in the caller's area, and the two      */
/*                         data items must be distinct.                     */
/****************************************************************************/
void rnQuotient(SYNTHETIC_INTEGER **h1_formal_par_in,
                SYNTHETIC_INTEGER **k1_formal_par_in,
                SYNTHETIC_INTEGER **h2_formal_par_in,
                SYNTHETIC_INTEGER **k2_formal_par_in,
                SYNTHETIC_INTEGER **h_result,
                SYNTHETIC_INTEGER **k_result)
   {
   SYNTHETIC_INTEGER *arg1_h,
                     *arg1_k,
                     *arg2_h,
                     *arg2_k,
                     *result_numerator,
                     *result_denominator,
                     *gcd,
                     *trash_remainder;

   unsigned is_neg = FALSE;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h1_formal_par_in != NULL, __LINE__);
   asAssert(*h1_formal_par_in != NULL, __LINE__);
   asAssert(k1_formal_par_in != NULL, __LINE__);
   asAssert(*k1_formal_par_in != NULL, __LINE__);
   asAssert(h2_formal_par_in != NULL, __LINE__);
   asAssert(*h2_formal_par_in != NULL, __LINE__);
   asAssert(k2_formal_par_in != NULL, __LINE__);
   asAssert(*k2_formal_par_in != NULL, __LINE__);
   asAssert(h_result != NULL, __LINE__);
   asAssert(*h_result != NULL, __LINE__);
   asAssert(k_result != NULL, __LINE__);
   asAssert(*k_result != NULL, __LINE__);
   asAssert(h_result != k_result, __LINE__);
   asAssert(*h_result != *k_result, __LINE__);

   /* Allocate space for temporary variables.
   */
   siCreate(&arg1_h);
   siCreate(&arg1_k);
   siCreate(&arg2_h);
   siCreate(&arg2_k);
   siCreate(&result_numerator);
   siCreate(&result_denominator);
   siCreate(&gcd);
   siCreate(&trash_remainder);

   /* Copy over the formal parameters to local variables.  This
   ** means we can modify the inputs without worry.
   */
   siCopy(h1_formal_par_in, &arg1_h);
   siCopy(k1_formal_par_in, &arg1_k);
   siCopy(h2_formal_par_in, &arg2_h);
   siCopy(k2_formal_par_in, &arg2_k);

   /* If any of the four inputs are NAN, the results must be
   ** NAN.
   */
   if (arg1_h->nan || arg1_k->nan || arg2_h->nan || arg2_k->nan)
      {
      goto nan_return;
      }

   /* If either denominator is zero, the result is NAN.
   */
   if (!(arg1_k->len) || !(arg2_k->len))
      {
      goto nan_return;
      }

   /* Carry out the multiplication of numerators and denominators to get
   ** the two products.
   */
   siUnrestrictedMultiplication(&arg1_h, &arg2_k, &result_numerator);
   siUnrestrictedMultiplication(&arg1_k, &arg2_h, &result_denominator);

   /* If either result is NAN, the results are NAN.
   */
   if (result_numerator->nan || result_denominator->nan)
      {
      goto nan_return;
      }

   /* If the denominator of the result is zero, this is a no-no-NAN, too.
   */
   if (!(result_denominator->len))
      {
      goto nan_return;
      }

   /* If the numerator is zero, we must return canonical zero.
   */
   if (!(result_numerator->len))
      {
      siSetToLong(h_result, 0);
      siSetToLong(k_result, 1);
      goto normal_return;
      }

   /* We need to acquire the sign and make both results positive otherwise
   ** can't obtain the GCD.
   */
   if (result_numerator->neg)
      {
      is_neg = TRUE;
      result_numerator->neg = FALSE;
      }
   if (result_denominator->neg)
      {
      is_neg = !is_neg;
      result_denominator->neg = FALSE;
      }

   /* Form the gcd. */
   siGcd(&result_numerator, 
         &result_denominator,
         &gcd);

   /* Set the sign back. */
   result_numerator->neg = is_neg;

   /* Divide both numerator and denominator by
   ** the gcd().
   */
   siUnrestrictedDivision(&result_numerator,
                          &gcd, 
                          h_result,
                          &trash_remainder);
   siUnrestrictedDivision(&result_denominator,
                          &gcd, 
                          k_result,
                          &trash_remainder);

   goto normal_return;
   nan_return: ;
   siSetToNan(h_result);
   siSetToNan(k_result);
   normal_return: ;

   /* Destroy temporary variables.
   */
   siDestroy(&arg1_h);
   siDestroy(&arg1_k);
   siDestroy(&arg2_h);
   siDestroy(&arg2_k);
   siDestroy(&result_numerator);
   siDestroy(&result_denominator);
   siDestroy(&gcd);
   siDestroy(&trash_remainder);
   }


/****************************************************************************/
/* rnCanonize():                                                            */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Puts the rational number passed into a canonical form, which means:    */
/*      a)If either synthetic integer is NAN, both are marked NAN.          */
/*      b)If the denominator is zero, both synthetic integers are marked    */
/*        NAN.                                                              */
/*      c)If the number has a value of zero, it is set to the canonical     */
/*        form of 0/1.                                                      */
/*      d)If the number represents a positive value, the signs of both      */
/*        synthetic integers are set +.  If the number represents a         */
/*        negative value, the numerator will be set - and the denominator   */
/*        -.                                                                */
/*      e)Any g.c.d. will be removed so that the number is in lowest        */
/*        terms.                                                            */
/*                                                                          */
/* INPUTS (AND OUTPUTS)                                                     */
/*   **h_formal_par_in,                                                     */
/*   **k_formal_par_in  :  The numerator and denominator of the number to   */
/*                         operate on.                                      */
/****************************************************************************/
void rnCanonize(SYNTHETIC_INTEGER **h_formal_par_in,
                SYNTHETIC_INTEGER **k_formal_par_in)
   {
   int is_neg = FALSE;
   SYNTHETIC_INTEGER *gcd, *trash_remainder, *si_temp;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h_formal_par_in != NULL, __LINE__);
   asAssert(*h_formal_par_in != NULL, __LINE__);
   asAssert(k_formal_par_in != NULL, __LINE__);
   asAssert(*k_formal_par_in != NULL, __LINE__);
   asAssert(h_formal_par_in != k_formal_par_in, __LINE__);
   asAssert(*h_formal_par_in != *k_formal_par_in, __LINE__);

   /* Create the local integers.
   */
   siCreate(&gcd);
   siCreate(&trash_remainder);
   siCreate(&si_temp);

   /* If either input is NAN, must mark both NAN.
   */
   if ((*h_formal_par_in)->nan || (*k_formal_par_in)->nan)
      goto nan_return;

   /* If the denominator is zero, must also mark both NAN.
   */
   if (!((*k_formal_par_in)->len))
      goto nan_return;

   /* If the numerator is zero, the right answer is canonical zero, and
   ** must return canonical 0/1.
   */
   if (!((*h_formal_par_in)->len))
      goto zero_return;

   /* If we made it to this point, we have two non-zero integers, and the
   ** result will be non-zero.  Must canonize the sign and factor out
   ** any g.c.d.
   */
   if ((((*h_formal_par_in)->neg) && !((*k_formal_par_in)->neg)) 
      || 
      (!((*h_formal_par_in)->neg) && ((*k_formal_par_in)->neg)))
      {
      is_neg = TRUE;
      }

   (*h_formal_par_in)->neg = FALSE;
   (*k_formal_par_in)->neg = FALSE;

   siGcd(h_formal_par_in, k_formal_par_in, &gcd);

   siUnrestrictedDivision(h_formal_par_in, &gcd,
                          &si_temp, &trash_remainder);
   siCopy(&si_temp, h_formal_par_in);
   siUnrestrictedDivision(k_formal_par_in, &gcd,
                          &si_temp, &trash_remainder);
   siCopy(&si_temp, k_formal_par_in);

   if (is_neg)
      (*h_formal_par_in)->neg = TRUE;

   goto normal_return;
   nan_return: ;
      siSetToNan(h_formal_par_in);
      siSetToNan(k_formal_par_in);
      goto normal_return;
   zero_return:
      siSetToLong(h_formal_par_in, 0);
      siSetToLong(k_formal_par_in, 1);
      goto normal_return;
   normal_return: ;

   /* Destroy the local integers.
   */
   siDestroy(&gcd);
   siDestroy(&trash_remainder);
   siDestroy(&si_temp);
   }


/****************************************************************************/
/* rnCompare():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Compares two rational numbers to determine thier ranking on the number */
/*   line.                                                                  */
/*                                                                          */
/* INPUTS    .                                                              */
/*   **h_1, **k_1,                                                          */
/*   **h_2, **k_2  :  The two rational numbers to compare.  They are not    */
/*                    required to be in canonical form, but the             */
/*                    denominators may not be zero.                         */
/*                                                                          */
/* OUTPUT                                                                   */
/*   <--           :  -1 if h1/k1 < h2/k2                                   */
/*                     0 if h1/k1 = h2/k2                                   */
/*                     1 if h1/k1 > h2/k2                                   */
/****************************************************************************/
int rnCompare(SYNTHETIC_INTEGER **h_1,
              SYNTHETIC_INTEGER **k_1,
              SYNTHETIC_INTEGER **h_2,
              SYNTHETIC_INTEGER **k_2)
   {
   int rv=0;

   SYNTHETIC_INTEGER *h1, 
                     *k1, 
                     *h2, 
                     *k2, 
                     *left_cross_product, 
                     *right_cross_product;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h_1 != NULL, __LINE__);
   asAssert(*h_1 != NULL, __LINE__);
   asAssert(k_1 != NULL, __LINE__);
   asAssert(*k_1 != NULL, __LINE__);
   asAssert(h_2 != NULL, __LINE__);
   asAssert(*h_2 != NULL, __LINE__);
   asAssert(k_2 != NULL, __LINE__);
   asAssert(*k_2 != NULL, __LINE__);

   /* Allocate all of our temps.
   */
   siCreate(&h1);
   siCreate(&k1);
   siCreate(&h2);
   siCreate(&k2);
   siCreate(&left_cross_product);
   siCreate(&right_cross_product);

   /* Copy the formal parameters to the temps so we can mess
   ** around with them.
   */
   siCopy(h_1, &h1);
   siCopy(k_1, &k1);
   siCopy(h_2, &h2);
   siCopy(k_2, &k2);

   /* The denominators may not be zero.
   */
   asAssert((k1->len != 0) && (k2->len != 0) ,__LINE__);

   /* We need to normalize the signs on the two fractions.
   ** The cross-product inequality utilized breaks down
   ** if either denominator is negative.
   */
   if ((!(h1->neg) && (k1->neg)) || ((h1->neg) && !(k1->neg)))
      {
      h1->neg = TRUE;
      k1->neg = FALSE;
      }
   else
      {
      h1->neg = FALSE;
      k1->neg = FALSE;
      }

   if ((!(h2->neg) && (k2->neg)) || ((h2->neg) && !(k2->neg)))
      {
      h2->neg = TRUE;
      k2->neg = FALSE;
      }
   else
      {
      h2->neg = FALSE;
      k2->neg = FALSE;
      }

   /* Form the left cross-product and right cross-product.  If we get
   ** a NAN, we have to kill this function, as it would compromise
   ** the integrity of the program (there is no notion of NAN for a
   ** comparison result).
   */
   siUnrestrictedMultiplication(&h1, &k2, &left_cross_product);
   siUnrestrictedMultiplication(&h2, &k1, &right_cross_product);
   asAssert(!(left_cross_product->nan) && !(right_cross_product->nan), __LINE__);

   /* The comparison result is the same as the relationship between the
   ** two cross-products.  A few words should be said about this.
   **
   ** Let a/b and c/d be the two rational numbers.
   ** Assume that b>0 and d>0.
   ** ad < bc
   ** -> ad/b < c 
   ** ->a/b < c/d.
   ** and the same logic can be applied to the other relational
   ** operators.
   */
   rv = siCompare(&left_cross_product, &right_cross_product);

#if 0
   siDump(&left_cross_product, "left");
   siDump(&right_cross_product, "right");
#endif

   /* Destroy all of our temps.
   */
   siDestroy(&h1);
   siDestroy(&k1);
   siDestroy(&h2);
   siDestroy(&k2);
   siDestroy(&left_cross_product);
   siDestroy(&right_cross_product);

   /* Return the return value.
   */
   return(rv);
   }


/****************************************************************************/
/* rnDap():                                                                 */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Converts a rational number to an approximately equivalent value        */
/*   with a different denominator.  This functionality is the same as       */
/*   implemented by the DAP command.                                        */
/*                                                                          */
/* INPUTS    .                                                              */
/*   **h, **k      :  The rational number to represent differently.         */
/*                    The denominator may not be zero.                      */
/*                                                                          */
/*   **D           :  The new denominator.  May not be zero.                */
/*                                                                          */
/* OUTPUTS                                                                  */
/*   **N           :  The new numerator.                                    */
/****************************************************************************/
void rnDap(SYNTHETIC_INTEGER **h,
           SYNTHETIC_INTEGER **k,
           SYNTHETIC_INTEGER **N,
           SYNTHETIC_INTEGER **D)
   {
   SYNTHETIC_INTEGER *numerator_product,
                     *quotient,
                     *trash_remainder;

   /* Be sure the caller is doing nothing silly with pointers.
   */
   asAssert(h != NULL, __LINE__);
   asAssert(*h != NULL, __LINE__);
   asAssert(k != NULL, __LINE__);
   asAssert(*k != NULL, __LINE__);
   asAssert(N != NULL, __LINE__);
   asAssert(*N != NULL, __LINE__);
   asAssert(D != NULL, __LINE__);
   asAssert(*D != NULL, __LINE__);

   /* Allocate our locals.
   */
   siCreate(&numerator_product);
   siCreate(&quotient);
   siCreate(&trash_remainder);

   /* If anything is NAN, the result is necessarily NAN.
   */
   if ((*h)->nan || (*k)->nan || (*D)->nan)
      goto nan_return;

   /* If the old denominator is zero, the result is necessarily NAN.
   */
   if (!((*k)->len))
      goto nan_return;

   /* If the new denominator requested is zero, the result is necessarily NAN.
   */
   if (!((*D)->len))
      goto nan_return;

   /* Calculate the numerator as specified in the manual under the DAP
   ** command.
   */
   siUnrestrictedMultiplication(h, D, &numerator_product);

   /* Divide to get the value.
   */
   siUnrestrictedDivision(&numerator_product, k, &quotient, &trash_remainder);

   /* Copy to caller's area.
   */
   siCopy(&quotient, N);

   goto normal_return;
   nan_return:
   siSetToNan(N);
   normal_return: ;

   /* Destroy our locals. 
   */
   siDestroy(&numerator_product);
   siDestroy(&quotient);
   siDestroy(&trash_remainder);
   }


/****************************************************************************/
/* rnFareyTraverse():                                                       */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Uses the standard recursive formulas to traverse the Farey series      */
/*   in either direction.  Two successive terms, in lowest terms, are       */
/*   needed.                                                                */
/*                                                                          */
/* INPUTS    .                                                              */
/*   **h1in, **k1in:  The first rational number in the series of interest,  */
/*                    which must have a positive denominator and a non-     */
/*                    negative numerator.                                   */
/*                                                                          */
/*   **h2in, **k2in:  The second rational number in the series of interest, */
/*                    same rules as above and must be greater than first    */
/*                    term h1in/k1in.                                       */
/*                                                                          */
/*   **N           :  The order of the series to form.                      */
/*                                                                          */
/*   dir           :  (-1):  The series is advanced backwards.  r(2)=r(1),  */
/*                           and r(1) = new term < r(2).                    */
/*                    (+1):  The series is advanced forwards.  r(1)=r(2),   */
/*                           and r(2) = new term > r(1).                    */
/****************************************************************************/
void rnFareyTraverse(SYNTHETIC_INTEGER **h1in,
                     SYNTHETIC_INTEGER **k1in,
                     SYNTHETIC_INTEGER **h2in,
                     SYNTHETIC_INTEGER **k2in,
                     SYNTHETIC_INTEGER **N,
                     int                 dir)
   {
   SYNTHETIC_INTEGER *t1, *t2, *t3, *new_h, *new_k;

   /* Be sure the caller isn't doing anything silly with pointers.
   */
   asAssert(h1in != NULL, __LINE__);
   asAssert(*h1in != NULL, __LINE__);
   asAssert(k1in != NULL, __LINE__);
   asAssert(*k1in != NULL, __LINE__);
   asAssert(h2in != NULL, __LINE__);
   asAssert(*h2in != NULL, __LINE__);
   asAssert(k2in != NULL, __LINE__);
   asAssert(*k2in != NULL, __LINE__);
   asAssert(N != NULL, __LINE__);
   asAssert(*N != NULL, __LINE__);

   /* It is also critical that none of the pointers be duplicates,
   ** but this isn't checked.
   */

   /* Allocate local variables. 
   */
   siCreate(&t1);
   siCreate(&t2);
   siCreate(&t3);
   siCreate(&new_h);
   siCreate(&new_k);

   /* Split into cases based on the direction.
   */
   if (dir > 0)
      {
      /* Forward
      */
      /* Calculate the term floor((k[j-2] + N)/k[j-1]), which we'll need
      ** twice.  The result is left in "t2".
      */
      siUnrestrictedAddition(k1in, N, &t1);
      siUnrestrictedDivision(&t1, k2in, &t2, &t3);

      /* Multiply by h[j-1] and place in "t1".
      */
      siUnrestrictedMultiplication(&t2, h2in, &t1);

      /* Subtract off h[j-2] and this is our new h.
      */
      siUnrestrictedSubtraction(&t1, h1in, &new_h);

      /* Multiply intermediate term by k[j-1] and place in "t1".
      */
      siUnrestrictedMultiplication(&t2, k2in, &t1);

      /* Subtract off k[j-2] and this is our new k. 
      */
      siUnrestrictedSubtraction(&t1, k1in, &new_k);

      /* Place the new results. 
      */
      siCopy(h2in, h1in);
      siCopy(k2in, k1in);
      siCopy(&new_h, h2in);
      siCopy(&new_k, k2in);
      }
   else
      {
      /* Reverse:  steps symmetrical with forward.
      */
      siUnrestrictedAddition(k2in, N, &t1);
      siUnrestrictedDivision(&t1, k1in, &t2, &t3);
      siUnrestrictedMultiplication(&t2, h1in, &t1);
      siUnrestrictedSubtraction(&t1, h2in, &new_h);
      siUnrestrictedMultiplication(&t2, k1in, &t1);
      siUnrestrictedSubtraction(&t1, k2in, &new_k);

      /* Place the new results. 
      */
      siCopy(h1in, h2in);
      siCopy(k1in, k2in);
      siCopy(&new_h, h1in);
      siCopy(&new_k, k1in);
      }


   /* Destroy local variables.
   */
   siDestroy(&t1);
   siDestroy(&t2);
   siDestroy(&t3);
   siDestroy(&new_h);
   siDestroy(&new_k);
   }


/****************************************************************************/
/****************************************************************************/
/*******   C O N T I N U E D    F R A C T I O N    F U N C T I O N S   ******/
/****************************************************************************/
/****************************************************************************/
/* This section is reserved for functions which form and manipulate
** continued fractions and convergents.  Note that every number has at
** least a 0th order CF partial quotient and at least a 0th order
** convergent (the integer part).
*/
typedef struct 
   {
   SYNTHETIC_INTEGER *raw_numerator;
   SYNTHETIC_INTEGER *raw_denominator;
      /* The raw numerator and denominator passed to the function which
      ** calculates partial quotients and convergents.  This might
      ** not be in lowest terms.  No negative integers are allowed,
      ** and zero must be represented canonically as 0/(D>0).
      */
   SYNTHETIC_INTEGER *numerator;
   SYNTHETIC_INTEGER *denominator;
      /* The numerator and denominator, in lowest terms, of the 
      ** rational number whose continued fraction expansion we
      ** are forming.  No negative integers are allowed, and
      ** zero must be represented canonically as 0/1.
      */
   int n;
      /* The number of elements in the parallel arrays of partial
      ** quotients, etc., which are maintained.  This must be
      ** at least 1, which would mean that only the 0th-order
      ** items are filled in.
      */
   SYNTHETIC_INTEGER **a;
      /* Pointer to a dynamically allocated array of n pointers,
      ** each of which points to a synthetic integer.  This
      ** pointer may be shifted on any operation on this data
      ** structure (due to the behavior of "realloc()"), so 
      ** a caller should never retain internal pointers when
      ** making function calls which might resize anything
      ** internally in this data structure.  The same sizing arguments
      ** here apply to all the other parallel elements, so it won't
      ** be described again.
      */
   SYNTHETIC_INTEGER **p;
   SYNTHETIC_INTEGER **q;
      /* Convergents of continued fraction expansion of the
      ** rational number.
      */
   } CF_EXPANSION;


/****************************************************************************/
/* pqCreate():                                                              */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Creates the continued fraction partial quotient expansion of a non-    */
/*   negative rational number, and also creates the convergents.            */
/*                                                                          */
/* INPUTS                                                                   */
/*   **h_in                                                                 */
/*   **k_in          :  The numerator and denominator of the rational       */
/*                      number whose continued fraction expansion and       */
/*                      convergents to form.  Both must be positive.        */
/*                                                                          */
/*   **expansion     :  The continued fraction expansion and the            */
/*                      convergents.                                        */
/****************************************************************************/
void pqCreate(SYNTHETIC_INTEGER **h_in,
              SYNTHETIC_INTEGER **k_in,
              CF_EXPANSION      **expansion)
   {
   SYNTHETIC_INTEGER *dividend,
                     *divisor,
                     *quotient,
                     *remainder,
                     *p_k_minus_1,
                     *p_k_minus_2,
                     *q_k_minus_1,
                     *q_k_minus_2,
                     *si_temp1,
                     *si_temp2,
                     *si_temp3,
                     *si_temp4;

   /* Be sure that the caller isn't doing anything silly.
   */
   asAssert(h_in != NULL, __LINE__);
   asAssert(*h_in != NULL, __LINE__);
   asAssert(k_in != NULL, __LINE__);
   asAssert(*k_in != NULL, __LINE__);
   asAssert(expansion != NULL, __LINE__);

   /* The numerator and denominator in cannot be negative,
   ** and the denominator cannot be zero.
   */
   asAssert(!((*h_in)->neg), __LINE__);
   asAssert(!((*k_in)->neg), __LINE__);
   asAssert((*k_in)->len != 0, __LINE__);

   /* Allocate space for all of the temporary integers
   ** we use during the process of forming partial
   ** quotients and convergents.
   */
   siCreate(&dividend);
   siCreate(&divisor);
   siCreate(&quotient);
   siCreate(&remainder);
   siCreate(&p_k_minus_1);
   siCreate(&p_k_minus_2);
   siCreate(&q_k_minus_1);
   siCreate(&q_k_minus_2);
   siCreate(&si_temp1);
   siCreate(&si_temp2);
   siCreate(&si_temp3);
   siCreate(&si_temp4);

   /* printf("Entering function.\n"); */

   /* Allocate the memory for the head data block.
   */
   *expansion = maMalloc(sizeof(CF_EXPANSION));

   /* Set all of the data elements of the head data block
   ** to known default values.
   */
   (*expansion)->raw_numerator   = NULL;
   (*expansion)->raw_denominator = NULL;
   (*expansion)->numerator       = NULL;
   (*expansion)->denominator     = NULL;
   (*expansion)->n               = 0;
   (*expansion)->a               = NULL;
   (*expansion)->p               = NULL;
   (*expansion)->q               = NULL;

   /* Assign in the original raw numerator and
   ** denominator.
   */
   siCreate(&((*expansion)->raw_numerator));
   siCreate(&((*expansion)->raw_denominator));
   siCopy(h_in, &((*expansion)->raw_numerator));
   siCopy(k_in, &((*expansion)->raw_denominator));

   /* Allocate the space for the lowest-terms numerator
   ** and denominator, but let them be zero for now.
   */
   siCreate(&((*expansion)->numerator));
   siCreate(&((*expansion)->denominator));

   /* Begin with the dividend and divisor as the
   ** numerator and denominator.
   */
   siCopy(&((*expansion)->raw_numerator), &dividend);
   siCopy(&((*expansion)->raw_denominator), &divisor);

   /* Enter a do ... while() loop to compute the continued
   ** fraction partial quotients and convergents.  Because
   ** the convergents don't require any "look-ahead" into
   ** the partial quotients, they can be computed
   ** in parallel
   */

   do
      {
      int curidx;
         /* Current index.
         */

      /* Buffer the current index value to the local variable (less
      ** typing).  The current index is what we're currently
      ** operating on.
      */
      curidx = (*expansion)->n;

      /* Increment the number of elements in the 
      ** three parallel arrays.
      */
      ((*expansion)->n)++;

      /* Allocate the memory for the memory block in the
      ** four parallel arrays.  There are two cases to
      ** consider, either this is the first element or
      ** else not the first.
      */
      if (!((*expansion)->a))
         {
         /* First time, first element. 
         */
         (*expansion)->a =
            maMalloc(sizeof(SYNTHETIC_INTEGER *));
         (*expansion)->p =
            maMalloc(sizeof(SYNTHETIC_INTEGER *));
         (*expansion)->q =
            maMalloc(sizeof(SYNTHETIC_INTEGER *));
         }
      else
         {
         /* Not the first time.  Blow up the arrays of 
         ** pointers to a larger size to accomodate one more
         ** pointer.
         */
         (*expansion)->a =
            maRealloc((*expansion)->a, 
                      (sizeof(SYNTHETIC_INTEGER *)) * (curidx+1));
         (*expansion)->p =
            maRealloc((*expansion)->p, 
                      (sizeof(SYNTHETIC_INTEGER *)) * (curidx+1));
         (*expansion)->q =
            maRealloc((*expansion)->q, 
                      (sizeof(SYNTHETIC_INTEGER *)) * (curidx+1));
         }

      /* Allocate the synthetic integers to go along with 
      ** blown-up arrays.
      */
      siCreate((*expansion)->a + curidx);
      siCreate((*expansion)->p + curidx);
      siCreate((*expansion)->q + curidx);

      /* Calculate the current continued fraction partial quotient,
      ** and bump the dividend and divisor for the next round.
      */
      siUnrestrictedDivision(&dividend,
                             &divisor,
                             (*expansion)->a + curidx,
                             &remainder);
      /* siDump((*expansion)->a + curidx, "a_k"); */
      siCopy(&divisor, &dividend);
      siCopy(&remainder, &divisor);

      /* Calculate the convergents using the standard
      ** recursive formulas.
      */
      if (curidx == 0)
         {
         /* p(0) = a(0) */
         siCopy((*expansion)->a + 0, (*expansion)->p + 0);
         /* q(0) = 1 */
         siSetToLong((*expansion)->q + 0, 1);
         }
      else if (curidx == 1)
         {
         /* p(1) = a(1)p(0) + 1 */
         siSetToLong(&si_temp1, 1);
         siUnrestrictedMultiplication((*expansion)->a + 1,
                                      (*expansion)->p + 0,
                                      &si_temp2);
         siUnrestrictedAddition(&si_temp2,
                                &si_temp1,
                                (*expansion)->p + 1);
         /* q(1) = a(1) */
         siCopy((*expansion)->a + 1, (*expansion)->q + 1);
         }
      else /* curidx >= 2 */
         {
         /* In this case, apply the full recursive formulas. */
         /* p(k) = a(k)p(k-1) + p(k-2)                       */
         siUnrestrictedMultiplication((*expansion)->a + curidx,
                                      (*expansion)->p + (curidx-1),
                                      &si_temp2);
         siUnrestrictedAddition(&si_temp2,
                                (*expansion)->p + (curidx-2),
                                (*expansion)->p + curidx);

         /* q(k) = a(k)q(k-1) + q(k-2)                       */
         siUnrestrictedMultiplication((*expansion)->a + curidx,
                                      (*expansion)->q + (curidx-1),
                                      &si_temp2);
         siUnrestrictedAddition(&si_temp2,
                                (*expansion)->q + (curidx-2),
                                (*expansion)->q + curidx);
         }

      /* siDump((*expansion)->p + curidx, "p_k"); */
      /* siDump((*expansion)->q + curidx, "q_k"); */
 
      /* Put in a safety test for the loop.  I've never had this loop fail to
      ** terminate.  Mathematically, it *can't* fail to terminate, but if there
      ** were a bug somewhere in the large integer math or a NAN condition,
      ** I wouldn't want to have a machine lockup without a diagnostic message.
      ** The value of 32000 is used because that approaches the limits of the
      ** minimum that an ANSI 'C' int is required to hold.  If we've gone
      ** around this loop 32000 times, something is very wrong.
      */
      asAssert(curidx < 32000, __LINE__);
      } 
   while (remainder->len);

   /* The lowest terms representation of the rational number supplied will
   ** be the final convergent.  This doesn't apply very much to this program,
   ** because the parsing functions automatically strip out the gcd() before
   ** passing data on, so the "raw" and "final" will be the same.  Will assign 
   ** it anyway.
   */
   siCopy((*expansion)->p + ((*expansion)->n - 1), &((*expansion)->numerator));
   siCopy((*expansion)->q + ((*expansion)->n - 1), &((*expansion)->denominator));

   /* Destroy the temporary integers.
   */
   siDestroy(&dividend);
   siDestroy(&divisor);
   siDestroy(&quotient);
   siDestroy(&remainder);
   siDestroy(&p_k_minus_1);
   siDestroy(&p_k_minus_2);
   siDestroy(&q_k_minus_1);
   siDestroy(&q_k_minus_2);
   siDestroy(&si_temp1);
   siDestroy(&si_temp2);
   siDestroy(&si_temp3);
   siDestroy(&si_temp4);
   }


/****************************************************************************/
/* pqDestroy():                                                             */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Deallocates the dynamic data structures associated with the CF         */
/*   expansion and convergents, and sets the caller's pointer to NULL.      */
/*                                                                          */
/*   **expansion     :  The continued fraction expansion and the            */
/*                      convergents.                                        */
/****************************************************************************/
void pqDestroy(CF_EXPANSION **expansion)
   {
   int idx;

   /* Be sure the caller isn't doing anything silly. with pointers.
   */
   asAssert(expansion != NULL, __LINE__);
   asAssert(*expansion != NULL, __LINE__);

   /* The general strategy at this point is to deallocate the data structure
   ** from the bottom up.  It must be done this way because once a higher
   ** data structure is deallocated, the memory ain't yours, so any access
   ** even to deallocate "lower" pointers is a violation.  Checks all along
   ** the way are performed to be sure that nothing looks suspicious.
   */

   asAssert((*expansion)->n > 0, __LINE__);  /* Even a 0th order expansion (an integer)
                                             ** has the "n" at 1.
                                             */
   /* Check for suspicious conditions in the base data
   ** structure.
   */
   asAssert((*expansion)->raw_numerator != NULL, __LINE__);
   asAssert((*expansion)->raw_denominator != NULL, __LINE__);
   asAssert((*expansion)->numerator != NULL, __LINE__);
   asAssert((*expansion)->denominator != NULL, __LINE__);
   asAssert((*expansion)->a != NULL, __LINE__);
   asAssert((*expansion)->p != NULL, __LINE__);
   asAssert((*expansion)->q != NULL, __LINE__);


   /* Deallocate the directly linked synthetic integers.
   */
   siDestroy(&((*expansion)->raw_numerator));
   siDestroy(&((*expansion)->raw_denominator));
   siDestroy(&((*expansion)->numerator));
   siDestroy(&((*expansion)->denominator));

   /* Deallocate each of the synthetic integers that are the partial
   ** quotients and convergents.
   */
   for (idx = 0; idx < ((*expansion)->n); idx++)
      {
      asAssert((((*expansion)->a)[idx]) != NULL, __LINE__);
      asAssert((((*expansion)->p)[idx]) != NULL, __LINE__);
      asAssert((((*expansion)->q)[idx]) != NULL, __LINE__);

      siDestroy((*expansion)->a + idx);
      siDestroy((*expansion)->p + idx);
      siDestroy((*expansion)->q + idx);
      }

   /* Deallocate the arrays of pointers.
   */
   maFree((*expansion)->a);
   maFree((*expansion)->p);
   maFree((*expansion)->q);

   /* Deallocate the base data structure and set the caller's pointer
   ** to NULL.
   */
   maFree(*expansion);
   *expansion = NULL;
   }


/****************************************************************************/
/* pqDump():                                                                */
/*--------------------------------------------------------------------------*/ 
/* DESCRIPTION                                                              */
/*   Prints the entire CF expansion and list of convergents to the standard */
/*   output stream, with an optional description.                           */
/*                                                                          */
/*   **expansion     :  The continued fraction expansion and the            */
/*                      convergents.                                        */
/*   *desc           :  Description to use.  If this is a zero-length       */
/*                      string, omits the description.                      */
/*   p_raw           :  Boolean flag to indicate if the raw numerator and   */
/*                      denominator should be printed distinctly from the   */
/*                      final CF convergent.  For most of the application   */
/*                      in this program, the answer is no, because the      */
/*                      rational number is already reduced at the time it   */
/*                      is CF'd, and the raw rational number will be        */
/*                      identical to the final convergent.  To print it     */
/*                      would just waste space.                             */
/****************************************************************************/
void pqDump(CF_EXPANSION **expansion, char *desc, int p_raw)
   {
   int i;
   char buf[100];

   /* Be sure the caller isn't doing any pointer suicide.
   */
   asAssert(expansion != NULL, __LINE__);
   asAssert(*expansion != NULL, __LINE__);
   asAssert(desc != NULL, __LINE__);

   /* If a description was passed print it out as a banner headline.
   */
   if (strlen(desc))
      {
      gfBannerHeading(desc, 0);
      }

   /* Banner announcing inputs. */
   gfBannerHeading("Inputs To CF Calculation", 0);
   gfHline();

   /* Print out the raw numerator and denominator. */
   siDump(&((*expansion)->raw_numerator), "h_in");
   gfHline();
   siDump(&((*expansion)->raw_denominator), "k_in");
   gfHline();

   /* Print out each of the continued-fraction partial
   ** quotients.
   */
   /* Banner announcing partial quotients. */
   gfBannerHeading("CF Partial Quotients", 0);
   gfHline();

   for (i=0; i<((*expansion)->n); i++)
      {
      sprintf(buf, "a(%d)", i);
      siDump((*expansion)->a + i, buf);
      gfHline();
      }

   /* Priint out the convergents. */
   /* Banner announcing partial convergents. */
   gfBannerHeading("CF Convergents", 0);
   gfHline();

   for (i=0; i<((*expansion)->n); i++)
      {
      sprintf(buf, "p(%d)", i);
      siDump((*expansion)->p + i, buf);
      sprintf(buf, "q(%d)", i);
      siDump((*expansion)->q + i, buf);
      gfHline();
      }
   }


/****************************************************************************/
/* pqBapp():                                                                */