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Aug 97 Challenge

Volume Number: 13 (1997)
Issue Number: 8
Column Tag: Programmer's Challenge

Programmers Challenge

by Bob Boonstra, Westford, MA


In recognition of Deep Blue's chess victory over Garry Kasparov (and I would have said "In celebration of...", except that I have mixed feelings about the outcome), we return to our series of board game tournament Challenges. This month, you will be writing a program to play the game Stratego. Stratego is a board game played on a rectangular 10x10 grid. Each player begins the game with 40 pieces, one of which is a Flag. The object of the game is to find and capture the opponent's flag.

The pieces provided to each player at the beginning of the game, in order of descending rank, are as follows (quantity in parentheses): Marshall (1), General (1), Colonels (2), Majors (3), Captains (4), Lieutenants (4), Seargent (4), Miners (5), Scouts (8), and Spy (1). In addition, each player is given Bombs (6) and a Flag (1). At the start of play, each player places his pieces in the four rows nearest his side of the board such that their rank, which is visible from one side of a piece but not from the other, is hidden from the opponent. Players alternate making moves, with Red moving first, then Blue. Each turn consists of either moving a piece into an open square, or striking an opponent's piece in an adjacent square. Except for the Scout, Flag, and Bomb pieces, each piece can move one square forward, backward, or sideways (but not diagonally) on a given turn. Flags and Bombs cannot move at all. Scouts can move any number of open squares forward, backward, or sideways (but again, not diagonally).

If a piece is moved into an adjacent square occupied by an opponent, the move is referred to as a "strike", which results in the lower ranking piece being removed from the board, and the higher ranking piece moving into the square formerly occupied by the losing piece. When both pieces involved in a strike are of equal rank, both pieces are removed. A Marshall defeats a General, a General defeats a Colonel, etc. The Spy is the lowest ranking piece, and is defeated by any other piece, except that a Spy defeats a Marshall when the Spy initiates the strike. Any piece except a Miner striking a Bomb is removed (and the Bomb remains in its original position). When a Miner strikes a Bomb, the Bomb is removed and the Miner moves into the square formerly occupied by the Bomb. The Flag and the Bomb cannot move and cannot initiate a strike. The game ends when a player strikes the opponent's Flag.

Your code will be competing in a round-robin tournament against other Challenge entries. The prototype for the code you should write is:

#define kBoardSize 10

typedef enum { kUnknown=0,
} PieceRank;

typedef enum {kRed=1,kBlue=2} PlayerColor;

typedef struct PieceType {
  PieceRank  thePieceRank;    /* rank of a piece */
  PlayerColor thePieceColor;  /* color of a piece */
} PieceType;

typedef PieceType Board[kBoardSize][kBoardSize];
/* Used to provide test code with board configuration.  Red starts 
  in rows 0..3, Blue starts in rows 6..9 */
/* Squares [4][2], [4][3], [4][6], [4][7] and [5][2], [5][3], [5][6], [5][7] are water 
  and cannot be occupied */

typedef struct PiecePosition {
  long row;  /* 0..9 */
  long col;  /* 0..9 */
} PiecePosition;

typedef struct MoveResult {
  PieceType attacker;        /* after a strike, returns rank of attacker */
  PieceType defender;        /* after a strike, returns rank of defender */
  Boolean attackerRemoved;   /* true after a strike against a piece of equal 
                                or greater rank, or against a bomb when the
                                attacker is not a Miner */
  Boolean defenderRemoved;   /* true after a strike by a piece of equal or
                                greater rank, or against a bomb when the
                                attacker is a Miner, or against a Marshall by a
                                Spy */
  Boolean victory;           /* true after a strike against the Flag */
  Boolean legalMove;         /* true unless you
         - move into an occupied square, or 
         - move or strike in a direction other than forward, backward, or
           sideways, or
         - move more than one square (except Scouts), or
         - move a Bomb or a Flag,
         - move into Water, or
         - strike a square not occupied by an opponent, or
         - make any other illegal move  */
} MoveResult;

void PositionPieces(
  void *privStorage,        /* 1MB of preinitialized storage for your use */
  PlayerColor playerColor,  /* you play red or blue, with red playing first */
  Board *theBoard           /* provide the initial position of your pieces */

typedef void (*ReportYourMove)( 
                    /* Callback to inform test code of move and get results */
  PiecePosition *moveFrom, /* piece you are moving or using to strike */
  PiecePosition *moveTo,   /* destination square or square being struck */
  Boolean strike,         /* false indicates a move, true indicates a strike */
  MoveResult *results     /* returns identity of struck piece and other info */

typedef void (*GetOpponentMove)(
                       /* Callback to get results of opponents last move */
  PiecePosition *moveFrom,/* piece opponent moved or used to strike */
  PiecePosition *moveTo,  /* destination square or square struck */
  Boolean *strike,        /* false indicates a move, true indicates a strike */
  MoveResult *results     /* returns identity of struck piece and other info */

Boolean /* TRUE claims victory or illegal opponent move */ MakeAMove(
  void *privStorage,        /* 1MB of storage from PositionPieces */
  PlayerColor playerColor,  /* you play red or blue, with red playing first */
  GetOpponentMove *GetMove, /* callback used to make a move */
  ReportYourMove *ReportMove /* callback used to find about opponents last move */

Your PositionPieces routine will be called at the beginning of each game so that you provide the initial position of your pieces. If your playerColor is kRed, you should occupy rows 0..3 of the board, otherwise you should occupy rows 6..9. You should set theBoard[row][col].thePieceColor to your playerColor and theBoard[row][col].thePieceRank to the PieceRank you are placing on each square. As indicated in the typedef for theBoard, eight squares in the middle rows of the board are water and cannot be occupied by pieces of either side.

Note that theBoard is used only to pass your initial position to the test code - you may choose your own data structure to keep track of your pieces and the knowledge gained about your opponent's pieces as the game progresses. PositionPieces will be provided with 1MB of preallocated storage pointed to by privStorage. This storage will be initialized to zero before PositionPieces is called and will persist between moves. You should not allocate any additional dynamic storage beyond that provided by privStorage. Small amounts of static storage are permitted.

The normal move sequence consists of finding out via the GetMove callback where your opponent moved on the previous turn, determining your next move, and finding out the results of that move via the ReportMove callback. Your MakeAMove code should include something like this:

  if (!firstMove || (playerColor==kBlue)) {
  /* respond to opponent move, and calculate your move */
  /* process results of move */

You provide ReportMove with the location from which and to which you are moving (moveFrom and moveTo, respectively), and set strike to TRUE if your move strikes one of your opponent's pieces. The callbacks will populate a MoveResult data structure to reflect the results of your opponent's last move or your current move. Flags are set to indicate whether the attacker or defender (or both) are removed as the result of an attack, and PieceType fields are populated when an attack reveals information about a piece. The key to winning a game is properly interpreting the movement and tactics of your opponent, deciding when and where to strike, and using the information gained in those strikes to best advantage. GetMove should return TRUE when you claim victory or when the callback indicates that an illegal move has been made.

The Challenge will be scored by a tournament where each entry plays against each other entry an even number of times, half playing first and half playing second. Another fair tournament schedule might be used if a large number of entries are received. For each game, the winner will earn game points for the victory, minus an amount based on the execution time used to compute the moves, as follows:

      game points = 10 - execution time in seconds

The loser will earn zero points and will not be penalized for the execution time expended in defeat. No points will be awarded for a tie. The player with the most total points for all games in the tournament will be the winner. This will be a native PowerPC Challenge, using the latest CodeWarrior environment. Solutions may be coded in C, C++, or Pascal.

Stratego is ® 1960 in the U.S. Patent Office and is © 1961 by the Milton Bradley Company.

Three Months Ago Winner

The Equation Evaluator Challenge posed in May required contestants to reproduce part of the functionality of Apple's Graphing Calculator. The problem was to parse an input equation and then evaluate it for combinations of up to three variables. Selecting a winner proved to be difficult, as none of the six entries I received proved to be completely correct. One of the fastest of the nearly correct entries needed only a trivial code change to be completely correct, met all of the accuracy requirements, was among the fastest of the nearly correct entries, and was the smallest of the top entries. Congratulations to Mark Day (Saratoga, CA) for submitting the winning entry, based on correctness, speed, and size.

Mark parses the input equation into a binary tree of TokenNode data structures, generates a sequence of PowerPC instructions in the CodeGen routine, and then executes those instructions to evaluate the input equation over the required set of values. As permitted by the problem statement, the CallGeneratedCode routine makes use of a small amount of assembly language to execute the generated code. The key to code generation is the subroutine CodeGenTree, which recursively produces the code corresponding to a given TokenNode, beginning with the root node. One thing to take note of as you look at the winning solution is the technique used to call library functions: a set of enums (e.g., kFuncSin) corresponds to an entry in the global function table gFunctions, which is accessed via a dedicated register (regFunctionBase) by the CallFunction code. Also note the use of MakeDataExecutable prior to execution of the generated code.

Turlough O'Connor also generates PowerPC instructions to solve the problem, but he demonstrates how to call the generated code without direct use of assembly language:

  codeBlock = NewPtr(kCodeBlockSize);
  // pageStart round up to 4K boundary
  pageStart = (Ptr) (((UInt32)codeBlock + 0xFFF) & ~0xFFF);  
  gPC = (UInt32 *)pageStart;
  prologStart = gPC;

  // [SNIP] - generate prolog starting at pageStart, and code starting at funcStart

                          (UInt32)gPC - (UInt32)pageStart);
  typedef struct {
    void *proc;
    void *rtoc;
  } TVector;

  typedef void (*GenFunc)(const Values *xP, const Values *yP, 
          const NValues *nP, Results *wP, 
          const ConstFuncTable *constTable);

  TVector ourVec = {funcStart, 0};
  GenFunc ourFunc = (GenFunc)&ourVec;

  (ourFunc)(xP, yP, nP, w, &gConstFuncTable);

Three of the remaining solutions (Willeke Rieken, Ernst Munter, and John Nevard) took an approach that did not involve generating PowerPC code. After parsing the input equation, these solutions evaluated the equation directly using a virtual machine of one sort or another. While this approach was generally slower, one of the solutions was competitive and could have won had it not suffered from an accuracy problem in a test case involving a truncated Taylor series expansion of a trig function.

I evaluated the entries using 17 test cases, designed to execute in comparable amounts of time, and summed the execution times to produce the final score. The test cases included combinations of arithmetic operations, trigonometric functions, hyperbolic functions, square roots, logarithms, and exponentiation. One test case included the truncated Taylor series mentioned above. Several test cases used polar coordinates as the equation input. Two of the entries, including the winner, had problems with polar coordinates: the winner calculated the angle as atan2(x,y) instead of atan2(y,x), while another entry used the atan function instead of the atan2 function. The former was easily corrected, while the latter was not. Three other entries did not meet the relative accuracy requirement specified in the problem, and one incorrectly processed some of the test cases. Only the last entry was disqualified.

The table below lists for each entry the total execution tine for all test cases, the severity of the errors in the solution, code and data sizes, and the programming language used. The number in parentheses after the entrant's name is the total number of Challenge points earned in all Challenges to date prior to this one.

Mark Day289atan2(x,y)73281364C/Asm
Willeke Rieken284accuracy167961779C++
Turlough O'Connor (7)284atan1317219283C++
Ernst Munter (242)341accuracy8904794C++
John Nevard (17)418accuracy45202126C++
M. N.239correctness163561799C++

Here is Mark's winning solution (with some formatting changes and deletions to reduce length - see the ftp site for the full code listing):

© 1997 Mark Day

//  Contains routines to parse an equation and drive the
//  evaluation of that equation over a set of input values.

#include <ctype.h>
#include <string.h>
#include <math.h>
#undef HUGE_VAL            // fp.h redefines this
#include <fp.h>
#include <Memory.h>
#include <OSUtils.h>
#include "Evaluate.h"

ulong gVariableFlags;

struct TokenLookup {
  int    token;
  int    which;
  char    *string;
typedef struct TokenLookup TokenLookup;

TokenLookup gTokenLookup[] = {
  tokLeftParen,    kLeftParen,      "(",
  tokRightParen,   kRightParen,     ")",
  tokAddOp,        kAdd,            "+",
  tokAddOp,        kSubtract,       "-",
  tokMulOp,        kMultiply,       "*",
  tokMulOp,        kDivide,         "/",
  tokPowerOp,      kExponent,       "^",
  tokRootOp,       kFuncSquareRoot, "\\",
  tokFactorial,    kFuncFactorial,  "!",
  tokFunction,     kFuncLogN,       "ln",
  tokFunction,     kFuncLogTen,     "log",
  //  Hyperbolic functions need to come first or else they
  //  would parse as the normal functions plus an extra
  //  "h" character (which makes for a syntax error)
  tokFunction,     kFuncSinh,    "sinh",
  tokFunction,     kFuncCosh,    "cosh",
  tokFunction,     kFuncTanh,    "tanh",
  tokFunction,     kFuncSech,    "sech",
  tokFunction,     kFuncCsch,    "csch",
  tokFunction,     kFuncCoth,    "coth",

  tokFunction,     kFuncSin,     "sin",
  tokFunction,     kFuncCos,     "cos",
  tokFunction,     kFuncTan,     "tan",
  tokFunction,     kFuncSec,     "sec",
  tokFunction,     kFuncCsc,     "csc",
  tokFunction,     kFuncCot,     "cot",

  tokFunction,     kFuncArcSin,  "asin",
  tokFunction,     kFuncArcCos,  "acos",
  tokFunction,     kFuncArcTan,  "atan",
  tokFunction,     kFuncArcSec,  "asec",
  tokFunction,     kFuncArcCsc,  "acsc",
  tokFunction,     kFuncArcCot,  "acot",

  tokFunction,  kFuncAbs,    "abs",

  tokVariable, kPi,     "p",    tokVariable,  kE,           "e",
  tokVariable, kR,      "r",    tokVariable,  kTheta,       "t",
  tokVariable, kX,      "x",    tokVariable,  kY,           "y",
  tokVariable, kZ,      "z",    tokVariable,  kN,           "n",
  tokAssign,   kEquals, "=",    tokInvalid,   kInvalidToken, "",
//  These are single precision versions of functions, not
//  found in either math.h or fp.h
static float Factorial(float x)     { return gamma(x+1.0); }
static float Secant(float x)        { return 1.0 / cosf(x); }
static float CoSecant(float x)      { return 1.0 / sinf(x); }
static float CoTangent(float x)     { return 1.0 / tanf(x); }
static float ArcSecant(float x)     { return acosf(1.0/x); }
static float ArcCoSecant(float x)   { return asinf(1.0/x); }
static float ArcCoTangent(float x)  { return atanf(1.0/x); }
static float HypSecant(float x)     { return 1.0 / coshf(x); }
static float HypCoSecant(float x)   { return 1.0 / sinhf(x); }
static float HypCoTangent(float x)  { return 1.0 / tanhf(x); }

typedef float (*FloatFunc)(float); FloatFunc gFunctions[] = {
(FloatFunc) powf, fabsf, qrtf, Factorial, expf, logf, log10f, sinf, cosf, 
tanf, Secant,CoSecant, CoTangent, asinf, acosf, atanf, ArcSecant, 
ArcCoSecant, ArcCoTangent, sinhf, coshf, tanhf, HypSecant, HypCoSecant,
HypCoTangent, (FloatFunc) atan2f

//  Return the next token in the equation.  Adjust the
//  pointer to point just after the token.
int GetToken(char **equation, Constant *value)
  int      token = tokInvalid;
  char    *s = *equation;
  int      i, width;
  TokenLookup  *p;
  //  skip leading white space
  while (*s == ' ' || *s == '\t' || *s == '\n')  ++s;
  if (*s == '\0')        return tokInvalid;
  //  See if *s matches one of the token strings
  p = gTokenLookup;
  while (p->token != tokInvalid) {
    char *tokenString = p->string;
    size_t len = strlen(tokenString);
    if (!strncmp(s, tokenString, len)) {
      token = p->token;  value->which = p->which;
      s += len;            break;
    } else {
  //  Update the current position within the string.
  *equation = s;
  //  Try to parse a numeric constant.
  if (token == tokInvalid) {
    token = GetNumberToken(equation, value);
  return token;

//  GetNumberToken recognizes numeric constants of the forms
//  [0-9]+.[0-9]*  or  [0-9]*.[0-9]+
int GetNumberToken(char **equation, Constant *value)
  char    *s;
  char    c;           //  keep the next character in a register
  long    digits = 0;  //  used to determine whether we've seen any digits yet
  float  val;          //  the value of the constant
  float  divisor;      //  keeps track of place value of float digits
  int    token;        //  the kind of number we found
  token = tokInvalid;  //  assume we didn't find a number
  s = *equation;       //  point to start of string
  c = *s;              //  get first char
  //  Gather the integer part of the constant
  val = 0.0;
  while (isdigit(c)) {
    val = val*10.0+(c-'0');  // shift in the next digit
    ++digits;                // remember we found a digit
    c = *(++s);              // get next character
  //  If we found an integer part, prepare to return it.
  if (digits) {
    value->f = val;
    token = tokFloat;
  //  Now let's see if there is a trailing decimal point
  //  and optional digits.  If so, then this is a floating
  //  point constant.
  if (c == '.') {
    c = *(++s);              //  skip over decimal point
    divisor = 1.0;           //  haven't seen fraction part yet
    while (isdigit(c)) {
      val = val*10.0+(c-'0');  // shift in next digit
      divisor *= 10;           // remember to fix place value
      ++digits;                // we saw another digit
      c = *(++s);              // get next character
    //  If there were digits before or after the decimal
    //  point, then we have a valid float.  Otherwise,
    //  we only saw a decimal point.
    if (digits) {
      value->f = val / divisor; // this fixes the place value
      token = tokFloat;
    } else {
      //  We saw only a decimal point.  Point us back at the decimal point.
  *equation = s;            //  point beyond what we found
  return token;             //  return what we found

TokenNode *NewTokenNode(TokenNode *left, TokenNode *right, int token)
  TokenNode  *node;
  node = (TokenNode *) NewPtr(sizeof(TokenNode));
  node->left = left;  node->right = right;
  node->token = token;
  return node;

void FreeTree(TokenNode *root)
  if (root) {
    FreeTree(root->left);  FreeTree(root->right);
    DisposePtr((Ptr) root);

  A simple parser for our expression grammar.  The grammar
  looks like:
  expression  ->  term  ( ADD_OP  term )*
  term      ->  factor  ( [ MULTIPLY_OP ]  factor )*
  factor    ->  SQUARE_ROOT  factor
          |  signed  EXPONENT  factor
          |  signed
  signed    ->  MINUS  signed
          |  PLUS  signed
          |  gamma
  gamma    ->  simple_value  ( FACTORIAL )*
  simple_value-> NUMBER
          |  VARIABLE
          |  LEFT_PAREN  expression  RIGHT_PAREN
          |  FUNCTION   LEFT_PAREN  expression  RIGHT_PAREN
  statement->  VARIABLE  EQUALS  expression
  where (...)* means repeated zero or more times, and
  [...] means optional.

TokenNode *ParseExpression(char **s)
  char    *temp;
  TokenNode  *left, *right, *node;
  int      token;
  Constant  value;
  node = NULL;
  //  Get the first term
  left = ParseTerm(s);
  if (left == NULL)
    return left;
  node = left;

  //  Get successive terms, separated by tokAddOp's
  do {
    temp = *s;
    token = GetToken(&temp, &value);
    if (token == tokAddOp) {
      *s = temp;
      right = ParseTerm(s);
      if (right) {
        node = NewTokenNode(left, right, token);
        node->value.which = value.which;
        left = node;    //  this makes it left-associative
  } while (token == tokAddOp);
  return node;

TokenNode *ParseTerm(char **s)
  char    *temp;
  TokenNode  *left, *right, *node;
  int      token;
  Constant  value;
  node = NULL;
  left = ParseFactor(s);
  if (left == NULL)  return left;
  node = left;
  //  Get successive factors.  They may be separated by
  //  tokMulOp's, or they may be adjacent (implying
  //  multiplication).  If we find a tokAddOp, then we
  //  have to unwind; otherwise, an expression like
  //  "3 - 7" would be parsed as "3 * (-7)".
  do {
    temp = *s;
    token = GetToken(&temp, &value);
    switch (token) {
      //  Explicit multiply/divide
      case tokMulOp:
        *s = temp;
        right = ParseFactor(s);
        if (right) {
          node = NewTokenNode(left, right, token);
          node->value.which = value.which;
          left = node;  // this makes it left-associative
      //  Explicit add/subtract; don't do implicit multiply.
      //  Use the single factor only.
      case tokAddOp:
        node = left;
      //  Implicit multiply or single factor
        //  Try to find a second factor
        temp = *s;
        right = ParseFactor(s);
        if (right) {
          //  Got one, so do implicit multiply
          token = tokMulOp;  //  pretend there was an explicit multiply
          node = NewTokenNode(left, right, token);
          node->value.which = kMultiply;
          left = node;  // make it left-associative
        } else {
          //  Nope, just a single factor
          *s = temp;    // put back the tokens ParseFactor gobbled
          node = left;
  } while (token == tokMulOp);  
  return node;
// ParseFactor, PaarseNegative, ParseFactorial deleted for brevity
// See ftp site for complete code listing

TokenNode *ParseSimpleValue(char **s)
  char    *temp;
  TokenNode  *node, *left;
  int      token;
  Constant  value;

  left = NULL;    node = NULL;
  temp = *s;
  token = GetToken(s, &value);
  switch (token) {
    case tokFloat:
      left = NewTokenNode(NULL, NULL, token);
      left->value = value;

    case tokVariable:
      left = NewTokenNode(NULL, NULL, token);
      left->value = value;
      switch (value.which) {
        case kPi:  gVariableFlags |= kPiMask; break;
        case kE:  gVariableFlags |= kEMask;  break;
        case kR:
          // using r implies using x and y since r is computed from x and y
          gVariableFlags |= kRMask + kXMask + kYMask;
        case kTheta:
          // using theta implies using x and y since theta is computed from x & y
          gVariableFlags |= kThetaMask + kXMask + kYMask;
        case kX:  gVariableFlags |= kXMask;  break;
        case kY:  gVariableFlags |= kYMask;  break;
        case kN:  gVariableFlags |= kNMask;  break;
    case tokFunction:
      node = NewTokenNode(NULL, NULL, token);
      node->value.which = value.which;
      //  If the function was abs(), then convert to a 
      //  special token so it can be evaluated more
      //  efficiently.
      if (value.which == kFuncAbs)  node->token = tokAbs;

      token = GetToken(s, &value);
      if (token != tokLeftParen) {  // Missing left parenthesis
        node = NULL;
      //  Fall into processing of parethesized expression
    case tokLeftParen:
      left = ParseExpression(s);
      token = GetToken(s, &value);  // swallow right parenthesis
    case tokRightParen:
      //  A right parenthesis means we're doing doing
      //  a recursive parse of an expression.  In that
      //  case, pretend like we hit the end of string.
      *s = temp;    //  back up so caller finds right parenthesis

    case tokInvalid:  break;
    default:      //  An unexpected token was seen
  //  If it was a function call, then point to the argument.
  if (node == NULL)    node = left;
  else                    node->left = left;

  return node;
TokenNode *ParseStatement(char **s)
  TokenNode  *node, *left, *right;
  int      token;
  Constant  value;
  //  Get the variable
  token = GetToken(s, &value);
  if (token == tokVariable) {
    left = NewTokenNode(NULL, NULL, token);
    left->value = value;
    if (value.which == kZ)
      gVariableFlags |= kFunctionOfY;  // "z=" implies yP is valid input
  //  Get the assignment token
  token = GetToken(s, &value);
  if (token != tokAssign) {    //  Missing "="
    return NULL;
  //  Get the expression
  right = ParseExpression(s);
  //  Build the node
  if (right) {
    node = NewTokenNode(left, right, token);
    node->value.which = value.which;
  } else {
    node = NULL;
  return node;
//  Here's the main entry point for the challenge.
void Evaluate(
  char      *equation,    // null-terminated equation to evaluate
  const Values  *xP,      // input values for x
  const Values  *yP,      // input values for y
  const IntValues  *nP,    // input values for n
  Results      w[])      // preallocated storage for equation values
  char    *s = equation;
  TokenNode  *root;
  Values    nFloatValues;
  //  Set up the floating point values of n
  nFloatValues.first = nP->first; = nP->delta;
  nFloatValues.howMany = nP->howMany;
  //  Parse the input equation
  gVariableFlags = 0;
  root = ParseStatement(&s);
  //  Generate code to evaluate the equation
  MakeDataExecutable(gCodeBase, 32768);

  CallGeneratedCode(gCodeBase+4, xP, yP, nP,
            &nFloatValues, w,
            gConstants, gFunctions,
            2.718281828459, 3.1415926535898);
  //  Free up the memory we allocated.
  DisposePtr((Ptr) gCodeBase);
  gCodeBase = NULL;
  gNextInstruction = NULL;
  if (gConstants) {
    DisposePtr((Ptr) gConstants);
    gConstants = NULL;

//  Contains various routines used to produce PowerPC object code
//  (in memory) to evaluate an equation at a range of input values.

#include <Memory.h>
#include "Evaluate.h"

//  Some simple macros for producing the PowerPC instructions in the generated
//  code.  The order of the arguments matches the order you'd write those arguments
//  in assembly language source.
#define op_add(rD, rA, rB)      (0x7C000214 + \
                          (rD<<21) + (rA<<16) + (rB<<11))
#define op_addi(rD, rA, simm)  (0x38000000 + \
                    (rD<<21) + (rA<<16) + (simm & 0xFFFF))
#define op_b(offset)    (0x48000000 + (offset & 0x03FFFFFC))
#define op_bctr()        (0x4e800420)
#define op_bl(offset)    (0x48000001 + (offset & 0x03FFFFFC))
#define op_blr()        (0x4e800020)
#define op_bne(offset)    (0x40820000 + (offset & 0xFFFF))
#define op_cmplwi(rA, uimm)    (0x28000000 + \
                              (rA<<16) + (uimm & 0xFFFF))
#define op_fabs(frD, frB)  (0xFC000210 + (frD<<21)+(frB<<11))
#define op_fadds(frD, frA, frB) (0xEC00002A + (frD<<21) + \
                                (frA<<16) + (frB<<11))
#define op_fdivs(frD, frA, frB) (0xEC000024 + (frD<<21) + \
                                (frA<<16) + (frB<<11))
#define op_fmadds(frD, frA, frC, frB) (0xEC00003A + (frD<<21) + \
                        (frA<<16) + (frB<<11) + (frC<<6))
#define op_fmr(frD, frS)  (0xFC000090 + (frD<<21) + (frS<<11))
#define op_fmuls(frD, frA, frC) (0xEC000032 + (frD<<21) + \
                                (frA<<16) + (frC<<6))
#define op_fneg(frD, frB) (0xFC000050 + (frD<<21) + (frB<<11))
#define op_fsubs(frD, frA, frB) (0xEC000028 + (frD<<21) + \
                                (frA<<16) + (frB<<11))
#define op_lfs(frD, d, rA)    (0xC0000000 + (frD<<21) + \
                            (rA<<16) + d)
#define op_lwz(rD, d, rA)    (0x80000000 + (rD<<21) + \
                            (rA<<16) + d)
#define op_mflr(rD)          (0x7c0802a6 + (rD<<21))
#define op_mtctr(rS)        (0x7c0903a6 + (rS<<21))
#define op_mtlr(rS)          (0x7c0803a6 + (rS<<21))
#define op_nop()            (0x60000000)
#define op_stfs(frS, d, rA)  (0xD0000000 + (frS<<21) + \
                            (rA<<16) + d)
#define op_stfsu(frS, d, rA)  (0xD4000000 + (frS<<21) + \
                            (rA<<16) + d)
#define op_stwu(rS, d, rA)    (0x94000000 + (rS<<21) + \
                            (rA<<16) + d)
#define op_subi(rD, rA, simm)  op_addi(rD, rA, -simm)

  Some missing optimizations:
    * Loop unrolling (** probably the most imporant opportunity **)
    * Common sub-expression elimination
      * Example: It's usually faster to compute both cos(x) and sin(x)
             in a single operation than computing them separately.
             The trick is recognizing the opportunity; CSE is a start.
    * Take advantage of trig identities
    * Redundant load/store
    * Sometimes moves values to and from non-volatile storage when the
        intervening operations don't change volatile registers.
    * Doesn't combine multiply and add into a single instruction.
    * Negation should be combined with previous fmul or fabs instruction.

ulong regUsage[32];      // non-zero means register[x] is currently in use
ulong numMemoryVars;    // number of temporaries not in registers
ulong gNumConstants;
float *gConstants = NULL;
ulong *gCodeBase,*gNextInstruction;

void InitCodeGen(ulong *codeStart)
  int i;
  gCodeBase = codeStart;
  gNextInstruction = codeStart;

  for (i=0; i<32; i++)
    regUsage[i] = 0;    // mark registers as not in use (i.e. available)
  // put constants in overflow register pool
  numMemoryVars = gNumConstants; 
  //  For variables that are used, mark them
  if (gVariableFlags & kNMask)  regUsage[regN] = 1;
  if (gVariableFlags & kXMask)  regUsage[regX] = 1;
  if (gVariableFlags & kYMask)  regUsage[regY] = 1;
  if (gVariableFlags & kRMask)  regUsage[regR] = 1;
  if (gVariableFlags & kThetaMask)  regUsage[regTheta] = 1;
  if (gVariableFlags & kEMask)  regUsage[regE] = 1;
  if (gVariableFlags & kPiMask)  regUsage[regPi] = 1;
  //  Allocate memory for constants
  gConstants = (float *) NewPtr(gNumConstants * sizeof(float));
  gNumConstants = 0;

//  AllocateRegister: reserve a non-volatile register or memory location
//  to hold some value.
static ulong AllocateRegister(void)
  int i;
  //  Try allocating a real register.
  for (i=regFreeBase; i<regMax; i++) {
    if (regUsage[i] == 0) {
      return i;
  //  Didn't work.  Need to use memory.
  return regMemory + numMemoryVars++;

//  UseRegister: Mark a register as in use.  If the given register
//  is a non-volatile register (not a memory location), then increment
//  it's usage count.  This is similar to AllocateRegister, but when
//  you already have the register number.
static void UseRegister(ulong which)
  if (which >= regFreeBase && which < regMax) {

// FreeRegister, StoreRegister, NonVolatileRegister have been
// deleted for brevity.  See ftp site for full code listing.

//  LoadRegister: Make sure the source register/memory is in a real
//  register.  If not, move it to the destination (which is a real
//  register) via a lfs instruction.
static ulong LoadRegister(ulong source, ulong destination)
  ulong  displacement;
  if (source >= regMemory) {
    displacement = (source-regMemory)*4;
    *(gNextInstruction++) = 
      op_lfs(destination, displacement, regVariableBase);
    return destination;
  } else {
    return source;

//  TemporaryResult: Generate a (real) register number to be a destination register
//  for some instruction.  The destination has already been determined and is in 
//   "which".
//  If "which" is memory, then use a volatile register, which will be stored later.
static ulong TemporaryResult(ulong which)
  if (which >= regMemory)  return fpTempResult;
  else                          return which;

//  StoreResult: the second part of TemporaryResult.  If TemporaryResult was passed
//  a memory location, then this function takes care of actually storing the result
//  (after it has been put in the temporary volatile register).
static void StoreResult(ulong which)
  if (which >= regMemory) {
    StoreRegister(which, fpTempResult);

//MoveRegister: Move register to memory, register to register, or memory to register.
static void MoveRegister(ulong destination, ulong source)
  if (source >= regMax) {        //  Source is memory.
    if (destination < regMax)
      LoadRegister(source, destination);
  if (destination >= regMax) {
    //  Destination is in memory, so generate a stfs instruction
    StoreRegister(destination, source);
  } else {
    //  Source and destination are registers, so generate fmr instruction
    *(gNextInstruction++) = op_fmr(destination, source);

//  CallFunction: Call a function.  Put the descriptor pointer into R12 and
//  use the function call glue at 0 to jump to the desired routine.
static void CallFunction(ulong which)
  long  displacement;
  //  Load the pointer to the desired function into r12
  displacement = which*4;
  *(gNextInstruction++) = 
      op_lwz(regFunctionGlue, displacement, regFunctionBase);
  //  Generate a bl instruction to the glue at the start of the generated code.
  displacement = (ulong) gCodeBase - (ulong) gNextInstruction;
  *(gNextInstruction++) = op_bl(displacement);

    root        (sub)tree to generate code for
    resultRegister  desired register for result (0 = any)
    Register containing result.  If resultRegister was non-zero, then
    this will be the same as resultRegister.

int CodeGenTree(TokenNode *root, int resultRegister)
  char  *opname;
  ulong  temp, temp2, arg1, arg2;
  int  result;
  if (root == NULL) {
    return 0;
  switch (root->token) {
    case tokFloat:
      //  Stuff the constant in memory in the first part of the overflow register area.
      gConstants[gNumConstants] = root->value.f;
      //  If the result needs to go into a specific register, then load that register from 
      //  memory.  Otherwise, return an overflow register number (which will get 
      //  loaded into a temporary registerjust before being used).
      if (resultRegister) {
        //  Explicitly move value to a given register
        result = resultRegister;
        LoadRegister(  regMemory + gNumConstants, 
      } else {  //  Tell caller where the value is so they can fetch it
        result = regMemory + gNumConstants;
      return result;

    case tokAddOp:
      temp = CodeGenTree(root->left, 0);
      temp = NonVolatileRegister(temp);  
      //  make sure it's in a non-volatile reg.
      temp2 = CodeGenTree(root->right, 0);
      //  so we can evaluate second arg
      if (resultRegister)    result = resultRegister;
      else                  result = AllocateRegister();
      arg1 = LoadRegister(temp, fpTemp1);
      arg2 = LoadRegister(temp2, fpTemp2);
      if (root->value.which == kAdd) {
        *(gNextInstruction++) = 
          op_fadds(TemporaryResult(result), arg1, arg2);
      } else {
        *(gNextInstruction++) = 
          op_fsubs(TemporaryResult(result), arg1, arg2);
      return result;

    case tokMulOp:
      temp = CodeGenTree(root->left, 0);
      temp = NonVolatileRegister(temp);
      //  make sure it's in a non-volatile reg.
      temp2 = CodeGenTree(root->right, 0);
      //  so we can evaluate second arg
      if (resultRegister)    result = resultRegister;
      else                  result = AllocateRegister();
      arg1 = LoadRegister(temp, fpTemp1);
      arg2 = LoadRegister(temp2, fpTemp2);
      if (root->value.which == kMultiply) {
        *(gNextInstruction++) = 
            op_fmuls(TemporaryResult(result), arg1, arg2);
      } else {
        *(gNextInstruction++) = 
            op_fdivs(TemporaryResult(result), arg1, arg2);
      return result;

    case tokPowerOp:
      //  Evaluate second argument, put in any register.
      temp = CodeGenTree(root->right, 0);
      //  make sure it's in a non-volatile reg.
      temp = NonVolatileRegister(temp);
      //  Evaluate first argument, put into fpArg1.
      CodeGenTree(root->left, fpArg1);
      //  Move second argument to fpArg2 If first argument didn't require a 
      //  function call, we could have put second argument directly into fpArg2
      MoveRegister(fpArg2, temp);
      //  Call power function
      //  Return result in desired register
      temp = fpResult;
      if (resultRegister && resultRegister != temp) {
        MoveRegister(resultRegister, temp);
        temp = resultRegister;
      return temp;

    case tokNegate:
      temp = CodeGenTree(root->left, 0);
      if (resultRegister)    result = resultRegister;
      else                  result = AllocateRegister();
      arg1 = LoadRegister(temp, fpTemp1);
      *(gNextInstruction++) = 
          op_fneg(TemporaryResult(result), arg1);
      return result;

    case tokAbs:
      temp = CodeGenTree(root->left, 0);
      if (resultRegister)    result = resultRegister;
      else                  result = AllocateRegister();
      arg1 = LoadRegister(temp, fpTemp1);
      *(gNextInstruction++) = 
          op_fabs(TemporaryResult(result), arg1);
      return result;

    case tokFunction:
      //  Evaluate the argument (root->left), putting it in fp1.
      temp = CodeGenTree(root->left, fpArg1);
      //  Call the function
      //  Put the result in the desired location
      temp = fpResult;
      if (resultRegister && resultRegister != temp) {
        MoveRegister(resultRegister, temp);
        temp = resultRegister;
      return temp;

    case tokVariable:
      switch (root->value.which) {
        case kPi:  temp = regPi;    break;
        case kE:    temp = regE;    break;
        case kR:    temp = regR;    break;
        case kTheta:  temp = regTheta; break;
        case kX:    temp = regX;    break;
        case kY:    temp = regY;    break;
        case kN:    temp = regN;    break;
      if (resultRegister && resultRegister != temp) {
        MoveRegister(resultRegister, temp);
        temp = resultRegister;
      UseRegister(temp);  // balance FreeRegister when this value is used
      return temp;
      //  Should never get here.
      return 0;
  //  Should never get here.
  return 0;

//  If r or theta (t) is used to evaluate the equation, generate some code
//  to set up their values based on the values of x and y.
static void CodeGenRTheta(void)
  if (gVariableFlags & kRMask) {
    //  r = sqrtf(x*x + y*y)
    *(gNextInstruction++)  = op_fmuls(fpArg1, regX, regX);
    *(gNextInstruction++)    = op_fmadds(fpArg1, regY, regY, 
    MoveRegister(regR, fpResult);
  if (gVariableFlags & kThetaMask) {
    // JRB correction - theta = atan2(y,x), not atan2(x,y)
    MoveRegister(fpArg1, regY);
    MoveRegister(fpArg2, regX);
    // end correction
    MoveRegister(regTheta, fpResult);

//  Generate code to store one equation result plus the variables
//  used to produce that result.  Assumes that the wP register points
//  to 4 bytes BEFORE the location to start storing (so we can use the
//  store-with-update instructions).
static void CodeGenStoreResult(ulong result)
  //  Store equation result
  *(gNextInstruction++) = op_stfsu(result, 4, regResults);
  //  Store x
  *(gNextInstruction++) = op_stfsu(regX, 4, regResults);
  if (gVariableFlags & kFunctionOfY) {
    //  Store both y and n
    *(gNextInstruction++) = op_stfsu(regY, 4, regResults);
    *(gNextInstruction++) = op_stwu(regNInteger, 4, 
  } else {    //  Skip over y and store n
    *(gNextInstruction++) = op_stwu(regNInteger, 8, 

//  Generate code to initialize the loop variables. Load the first value, load the count 
//  (howMany), then branch to the test part of the loop (after the body of the loop). 
//  Since we don't know where the test will be, we just skip an instruction word and 
//  will insert the branch when we generate the test.
static void CodeGenLoopInit(ulong variableFlags, ulong 
                **xBranch, ulong **yBranch, ulong **nBranch)
  //  Generate the loop for x first
  if (variableFlags & kXMask) {
    *(gNextInstruction++) = op_lfs(regX, 0, regXValues);
    // x = xP->first
    *(gNextInstruction++) = op_lwz(regXCount, 8, regXValues);
    // xCount = xP->howMany
    *xBranch = gNextInstruction;
    // remember branch address
    *(gNextInstruction++) = op_nop();
    // will be patched later
  //  Now generate the loop for y, if function was of the form "z=..."
  if ((variableFlags & kFunctionOfY) && (variableFlags &         kYMask)) {
    *(gNextInstruction++) = op_lfs(regY, 0, regYValues);
    // y = yP->first
    *(gNextInstruction++) = op_lwz(regYCount, 8, regYValues);
    // yCount = yP->howMany
    *yBranch = gNextInstruction;
    *(gNextInstruction++) = op_nop();
    // will be patched later
  //  And lastly, n.  N is a bit different since we maintain
  //  both the integer and floating point values simultaneously.
  if (variableFlags & kNMask) {
  *(gNextInstruction++) = op_lfs(regN, 0, regNValues);
    // n = nFloat->first
  *(gNextInstruction++) = op_lwz(regNInteger, 0, regNIntValues);
    // nInteger = nP->first
  *(gNextInstruction++) = op_lwz(regNCount, 8, regNIntValues);
    // nCount = nP->howMany
  *nBranch = gNextInstruction;
  *(gNextInstruction++) = op_nop();
    // will be patched later

//  Generate the iteration step and test/branch for the variable loops.
//  Just before we generate the test/branch instructions, patch up the
//  branch instructions inserted by CodeGenLoopInit.
//  The general form generates something like:
//  iteration:    x += xP->delta;
//          xCount-;
//  test:      if (xCount != 0) goto loop_body
//  where loop_body is the instruction following the forward branch that gets patched.
static void CodeGenLoopEnd(ulong variableFlags, ulong *xBranch, ulong *yBranch, ulong *nBranch)
  ulong  offset;
  //  Generate the part for n.  Remember that we must increment both the integer
  //  and floating point values of n.
  if (variableFlags & kNMask) {
  *(gNextInstruction++) = op_lfs(fpTemp1, 4, regNValues);
    //  get nFloatValues->delta
  *(gNextInstruction++) = op_fadds(regN, regN, fpTemp1);
    //  increment n (float)
  *(gNextInstruction++) = op_lwz(regIntTemp1, 4, regNIntValues);
    //  get nP->delta
  *(gNextInstruction++) = 
          op_add(regNInteger, regNInteger, regIntTemp1);
    //  increment n
  *(gNextInstruction++) = op_subi(regNCount, regNCount, 1);
    //  nCount-
    offset = (int) gNextInstruction - (int) nBranch;
  *nBranch = op_b(offset);
    //  patch branch at start of loop
  *(gNextInstruction++) = op_cmplwi(regNCount, 0);
    //  nCount == 0?
    offset = (int) (nBranch+1) - (int) gNextInstruction;
  *(gNextInstruction++) = op_bne(offset);
    //  if not, branch to loop body
  //  Next comes y, if a loop was generated for it (i.e. equation was "z=...")
  if ((variableFlags & kFunctionOfY) && (variableFlags & kYMask)) 
  *(gNextInstruction++) = op_lfs(fpTemp1, 4, regYValues);
    //  get yP->delta
  *(gNextInstruction++) = op_fadds(regY, regY, fpTemp1);
    //  increment y
  *(gNextInstruction++) = op_subi(regYCount, regYCount, 1);
    //  yCount-
    offset = (int) gNextInstruction - (int) yBranch;
  *yBranch = op_b(offset);
    //  patch branch at start of loop
  *(gNextInstruction++) = op_cmplwi(regYCount, 0);
    //  yCount == 0?
    offset = (int) (yBranch+1) - (int) gNextInstruction;
    *(gNextInstruction++) = op_bne(offset);
    //  if not, branch to loop body
  //  Lastly comes x.
  if (variableFlags & kXMask) {
  *(gNextInstruction++) = op_lfs(fpTemp1, 4, regXValues);
    //  get xP->delta
  *(gNextInstruction++) = op_fadds(regX, regX, fpTemp1);
    //  increment x
    *(gNextInstruction++) = op_subi(regXCount, regXCount, 1);
    //  xCount-
    offset = (int) gNextInstruction - (int) xBranch;
  *xBranch = op_b(offset);
    //  patch branch at start of loop
  *(gNextInstruction++) = op_cmplwi(regXCount, 0);
    //  xCount == 0?
    offset = (int) (xBranch+1) - (int) gNextInstruction;
  *(gNextInstruction++) = op_bne(offset);
    //  if not, branch to loop body

//  Build up a function that loops over all the variable input values, evaluates
//  the function, and stores the results.
//  If the loops were written in C, they'd look like this:
//    for (x=xP->first, xCount=xP->howMany;      // loop initialization
//       xCount != 0;                    // loop condition
//       x+=xP->delta, xCount-)            // iteration step
//    {
//      evaluate and store the result
//    }
void CodeGen(TokenNode *root)
  ulong  result;
  ulong  *generatedCode;
  ulong  *xBranch, *yBranch, *nBranch;
  generatedCode = (ulong *) NewPtr(32768);

  if (generatedCode != NULL) {
    //  Generate the pointer glue for making indirect function calls
    *(gNextInstruction++) = op_lwz(regR0, 0, regFunctionGlue);
      // lwz r0, 0(r12)
    *(gNextInstruction++) = op_lwz(regTOC, 4, regFunctionGlue);
      // lwz r2, 4(r12)
    *(gNextInstruction++) = op_mtctr(regR0);  // mtcr r0
    *(gNextInstruction++) = op_bctr();        // bctr
    //  Generate the function prolog - save the link register
    *(gNextInstruction++) = op_mflr(regSavedLR);
    // mflr r15
    //  Generate loop initialization for any variables that _are_
    //  used to evaluate the function
    CodeGenLoopInit(gVariableFlags, &xBranch, &yBranch, &nBranch);
    //  If r or theta are used to evaluate the equation, then compute them
    //  from x and y.
    //  Generate the code to evaluate the equation.  The equation can be put
    //  in any register.  But make sure it ends up in a real register (not memory).
    result = CodeGenTree(root, 0);
    result = LoadRegister(result, fpResult);
    //  Generate loop initialization for any variables that _are not_ used
    //  to evaluate the function.  This just causes the result to be stored
    //  several times in a row, without re-evaluating it.
    CodeGenLoopInit(gVariableFlags ^ (kXMask+kYMask+kNMask), 
        &xBranch, &yBranch, &nBranch);
    //  Generate the code to store one result
    //  Generate the iteration and test parts of the variable loops
    CodeGenLoopEnd(gVariableFlags ^ (kXMask+kYMask+kNMask), 
                              xBranch, yBranch, nBranch);
    CodeGenLoopEnd(gVariableFlags, xBranch, yBranch, nBranch);
    //  Generate the function epilog - restore link register and return
    *(gNextInstruction++) = op_mtlr(regSavedLR);
      // mtlr r15
    *(gNextInstruction++) = op_blr();
      // blr
//  Contains a glue routine used to call the PowerPC code produced
//  by CodeGen.c

#include "Evaluate.h"

typedef struct {
  unsigned long  savedSP;
  unsigned long  savedCR;
  unsigned long  savedLR;
  unsigned long  reserved1;
  unsigned long  reserved2;
  unsigned long  savedTOC;
} Linkage;

typedef struct {
  Linkage  linkage;
  long  arguments[8];      //  set aside maximum room for called functions
  long  savedGPR[20];      //  r13-r31    (last location is not used)
  double  savedFPR[18];    //  fp14-fp31
} StackFrame;

enum { StackFrameSize = sizeof(StackFrame) };

//  Saves registers and sets up parameter area for generated code. The generated code 
//  may save the link register in the linkage area. Since the generated code never uses 
//  the TOC register itself, it is sufficient to save and restore it here in case the 
//  generated code calls to a different fragment.
asm float CallGeneratedCode(
  register void *addr,  //  first instruction to execute
  register const Values *xValues, 
  register const Values *yValues,
  register const IntValues *nIntValues, 
  register const Values *nValues,
  register Results *resultsPtr,
  register float *constants, register void *functions,
  register float eValue, register float piValue)
    //  Save our return address and TOC
    mflr  r0
    stw    r2,Linkage.savedTOC(SP)
    stw    r0,Linkage.savedLR(SP)
    //  Make new stack frame
    stwu  SP,-StackFrameSize(SP)
    //  Save non-volatile FPRs
    stfd  fp14,StackFrame.savedFPR[0](SP)
    stfd  fp15,StackFrame.savedFPR[1](SP)
    stfd  fp16,StackFrame.savedFPR[2](SP)
    stfd  fp17,StackFrame.savedFPR[3](SP)
    stfd  fp18,StackFrame.savedFPR[4](SP)
    stfd  fp19,StackFrame.savedFPR[5](SP)
    stfd  fp20,StackFrame.savedFPR[6](SP)
    stfd  fp21,StackFrame.savedFPR[7](SP)
    stfd  fp22,StackFrame.savedFPR[8](SP)
    stfd  fp23,StackFrame.savedFPR[9](SP)
    stfd  fp24,StackFrame.savedFPR[10](SP)
    stfd  fp25,StackFrame.savedFPR[11](SP)
    stfd  fp26,StackFrame.savedFPR[12](SP)
    stfd  fp27,StackFrame.savedFPR[13](SP)
    stfd  fp28,StackFrame.savedFPR[14](SP)
    stfd  fp29,StackFrame.savedFPR[15](SP)
    stfd  fp30,StackFrame.savedFPR[16](SP)
    stfd  fp31,StackFrame.savedFPR[17](SP)
    //  Save non-volatile GPRs
    stw    r13,StackFrame.savedGPR[0](SP)
    stw    r14,StackFrame.savedGPR[1](SP)
    stw    r15,StackFrame.savedGPR[2](SP)
    stw    r16,StackFrame.savedGPR[3](SP)
    stw    r17,StackFrame.savedGPR[4](SP)
    stw    r18,StackFrame.savedGPR[5](SP)
    stw    r19,StackFrame.savedGPR[6](SP)
    stw    r20,StackFrame.savedGPR[7](SP)
    stw    r21,StackFrame.savedGPR[8](SP)
    stw    r22,StackFrame.savedGPR[9](SP)
    stw    r23,StackFrame.savedGPR[10](SP)
    stw    r24,StackFrame.savedGPR[11](SP)
    //  Pass input parameters to generated code
    fmr    fp30,eValue
    fmr    fp31,piValue
    mr      r13,constants
    mr      r14,functions
    mr      r16,xValues
    mr      r17,yValues
    mr      r18,nValues
    mr      r19,nIntValues
    subi  r20,resultsPtr,4  // set up results-4
    //  Call function at addr
    mtctr  addr
    bl    GoToCTR    //  this sets up LR to return to us
    //  Restore GPRs
    lwz    r13,StackFrame.savedGPR[0](SP)
    lwz    r14,StackFrame.savedGPR[1](SP)
    lwz    r15,StackFrame.savedGPR[2](SP)
    lwz    r16,StackFrame.savedGPR[3](SP)
    lwz    r17,StackFrame.savedGPR[4](SP)
    lwz    r18,StackFrame.savedGPR[5](SP)
    lwz    r19,StackFrame.savedGPR[6](SP)
    lwz    r20,StackFrame.savedGPR[7](SP)
    lwz    r21,StackFrame.savedGPR[8](SP)
    lwz    r22,StackFrame.savedGPR[9](SP)
    lwz    r23,StackFrame.savedGPR[10](SP)
    lwz    r24,StackFrame.savedGPR[11](SP)
    //  Restore FPRs
    lfd    fp14,StackFrame.savedFPR[0](SP)
    lfd    fp15,StackFrame.savedFPR[1](SP)
    lfd    fp16,StackFrame.savedFPR[2](SP)
    lfd    fp17,StackFrame.savedFPR[3](SP)
    lfd    fp18,StackFrame.savedFPR[4](SP)
    lfd    fp19,StackFrame.savedFPR[5](SP)
    lfd    fp20,StackFrame.savedFPR[6](SP)
    lfd    fp21,StackFrame.savedFPR[7](SP)
    lfd    fp22,StackFrame.savedFPR[8](SP)
    lfd    fp23,StackFrame.savedFPR[9](SP)
    lfd    fp24,StackFrame.savedFPR[10](SP)
    lfd    fp25,StackFrame.savedFPR[11](SP)
    lfd    fp26,StackFrame.savedFPR[12](SP)
    lfd    fp27,StackFrame.savedFPR[13](SP)
    lfd    fp28,StackFrame.savedFPR[14](SP)
    lfd    fp29,StackFrame.savedFPR[15](SP)
    lfd    fp30,StackFrame.savedFPR[16](SP)
    lfd    fp31,StackFrame.savedFPR[17](SP)
    //  Unwind stack frame
    addi  SP,SP,StackFrameSize
    //  Restore TOC and return
    lwz    r0,Linkage.savedLR(SP)
    lwz    r2,Linkage.savedTOC(SP)
    mtlr  r0

// Prototypes deleted - see code on ftp site

  These are types and function prototypes used internally
extern ulong gVariableFlags;
extern ulong gNumConstants;    // Number of constants found/stored
extern float *gConstants;      // Array of constant values from equation
extern ulong *gCodeBase;        // Address of start of generated code
extern ulong *gNextInstruction;
                    // Where next generated instruction will be stored

union Constant {
  float  f;      // a float constant
  long  which;    // which one of a group (kVariable, kAddOp, etc.)
typedef union Constant Constant;

struct TokenNode {
  int          token;
  Constant      value;
  struct TokenNode  *left;
  struct TokenNode  *right;    // unused for unary operators, functions
typedef struct TokenNode TokenNode;

//  Return the next token in the equation.  Adjust the pointer to
//  point just after the token.
enum {
  tokFloat,      tokLeftParen,      tokRightParen,
  tokAddOp,      tokMulOp,        tokPowerOp,
  tokNegate,      tokAbs,          tokRootOp,
  tokFunction,  // ln, sin, tan, etc.
  tokVariable,  // x, y, r, t, etc.
  tokAssign,    // =

//  Indices for the various functions
enum {
  kFuncPower,        kFuncAbs,  kFuncSquareRoot,
  kFuncFactorial,    kFuncExp,  kFuncLogN,
  kFuncLogTen,      kFuncSin,  kFuncCos,
  kFuncTan,        kFuncSec,  kFuncCsc,
  kFuncCot,        kFuncArcSin,
  kFuncArcCos,      kFuncArcTan,
  kFuncArcSec,      kFuncArcCsc,
  kFuncArcCot,      kFuncSinh,  kFuncCosh,
  kFuncTanh,        kFuncSech,  kFuncCsch,
  kFuncCoth,        kFuncAtan2

//  Values of "which" for misc. tokens
enum {
  kFloat,      kInteger,    kLeftParen,
  kRightParen,  kAdd,        kSubtract,
  kMultiply,    kDivide,      kExponent,
  //  "variables"
  kPi, kE, kR, kTheta, kX, kY, kZ, kN,
  kEquals,    kInvalidToken

//  Constants for bits in gVariableFlags
enum {
  kRMask        = 0x0001,    //  r was used in the equation
  kThetaMask  = 0x0002,      //  t was used in the equation
  kXMask        = 0x0004,    //  x was used in the equation
  kYMask        = 0x0008,    //  y was used in the equation
  kNMask        = 0x0010,    //  n was used in the equation
  kEMask        = 0x0020,    //  e was used in the equation
  kPiMask      = 0x0040,     //  p was used in the equation
  kFunctionOfY= 0x0080       //  function was of the form "z="

//  The various integer and floating point registers used
enum {
  regR0              = 0,    //  used for function calls
  regTOC             = 2,    //  TOC is pointer to globals
  regFunctionGlue    = 12,
  regVariableBase  = 13,  //  points to (temporary) values stored in memory
  regFunctionBase  = 14,  //  points to array of function pointers
  regSavedLR         = 15,        //  we save caller's LR here
  regXValues         = 16,        regYValues   = 17,
  regNValues         = 18,        regNIntValues   = 19,
  regResults         = 20,        regNInteger  = 21,
  regXCount          = 22,        regYCount   = 23,
  regNCount          = 24,
  fpResult      = 1,      //  results always in fp1
  fpArg1        = 1,      //  first fp argument in fp1
  fpArg2        = 2,      //  second fp argument in fp2
  fpTemp1       = 3,
  fpTemp2       = 4,
  fpTempResult  = 5,
  regIntTemp1   = 4,      //  a volatile integer register
  regFreeBase   = 14,    //  first non-volatile register
  regMax        = 32,    //  number of fp registers
  regN          = 25,    regX      = 26,  regY  = 27,    
  regR          = 28,    regTheta  = 29,  regE  = 30,
  regPi         = 31,
  regMemory     = 100    // fake register number for memory locations
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