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ExpressionProgram.java

/*************************************************************************
*                                                                        *
*   1) This source code file, in unmodified form, and compiled classes   *
*      derived from it can be used and distributed without restriction,  *
*      including for commercial use.  (Attribution is not required       *
*      but is appreciated.)                                              *
*                                                                        *
*    2) Modified versions of this file can be made and distributed       *
*       provided:  the modified versions are put into a Java package     *
*       different from the original package, edu.hws;  modified          *
*       versions are distributed under the same terms as the original;   *
*       and the modifications are documented in comments.  (Modification *
*       here does not include simply making subclasses that belong to    *
*       a package other than edu.hws, which can be done without any      *
*       restriction.)                                                    *
*                                                                        *
*   David J. Eck                                                         *
*   Department of Mathematics and Computer Science                       *
*   Hobart and William Smith Colleges                                    *
*   Geneva, New York 14456,   USA                                        *
*   Email: eck@hws.edu          WWW: http://math.hws.edu/eck/            *
*                                                                        *
*************************************************************************/

package edu.hws.jcm.data;

/**
 * An ExprssionProgram represents a mathematical expression such as "3" or "sin(x^2)", stored
 * in the form of a program for a stack machine.  The program consists of a sequence of 
 * commands that, when executed, will compute the value of the expression. 
 *
 * <p>Each command is encoded as an integer.  There are three types of commands that can
 * occur:  (1) A negative integer must be one of the 36 constant values PLUS, MINUS,..., CUBERT.
 * These constants represent unary and binary operators and standard functions.  (2) An integer in the range 
 * 0 <= n < 0x3FFFFFFF encodes an operation of the form "push a constant onto the stack".
 * The constant that is being pushed is encoded as an index in the array "constant",
 * which is a private member of this class that holds all the constants that
 * occur in this ExpressionProgram.  (3) An integer >= 0x3FFFFFFF represents
 * an ExpressionCommand object.  When 0x3FFFFFFF is subtracted from the integer, the
 * result is an index into the array "command", which is a private member of this
 * class that holds all the ExpressionCommands that occur in this
 * ExpressionProgram.
 */
00044 public class ExpressionProgram implements Expression {

   /**
    * Code for a unary or binary operator or a standard function.
    */
   public static final int
   
00051        PLUS = -1,            MINUS = -2,          TIMES = -3,
       DIVIDE = -4,          POWER = -5,          EQ = -6,
       NE = -7,              LT = -8,             GT = -9,
       LE = -10,             GE = -11,            AND = -12,
       OR = -13,             NOT = -14,           UNARY_MINUS = -15,
       FACTORIAL = -16,      SIN = -17,           COS = -18,
       TAN = -19,            COT = -20,           SEC = -21,
       CSC = -22,            ARCSIN = -23,        ARCCOS = -24,
       ARCTAN = -25,         ABS = -26,           SQRT = -27,
       EXP = -28,            LN = -29,            LOG2 = -30,
       LOG10 = -31,          TRUNC = -32,         ROUND = -33,
       FLOOR = -34,          CEILING = -35,       
       CUBERT = -36;
      
   /** 
    * If this is non-null, it is used as the print string
    * for this expression in the toString() method.  (When an
    * expression is created by a Parser by parsing a string,
    * the parse stores that string in this variable.)
    */
00071    public String sourceString;
   
   private int[] prog = new int[1]; // Contains the commands that make up this program.
                                    // (Constants and ExpressionCommands are encoded as
                                    // positive numbers, while negative numbers are
                                    // opCodes from the above list of constants.)
                                    // This array expands as necessary.

   private int progCt;   // The number of commands in prog.
   

   //--------------------- Methods for creating the program ------------------------------
   
   /**
    * Default constructor creates an initially empty program.
    */
00087    public ExpressionProgram() {
   }

   /**
    * Adds com as the next command in the program.  Among other things, for example, com can
    * be a Variable or Constant.  In that case, the meaning of the command is the stack
    * operation "push (value of com)".
    *
    * @param com added as next command in the program.
    */
00097    public void addCommandObject(ExpressionCommand com) {
      int loc = findCommand(com);
      addCommandCode(loc + 0x3FFFFFFF);
   }
   
   /**
    * Add the number d as the next command in the program.  The meaning of this command is
    * actually the stack operation "push d".
    *
    * @param d added as next command in program.
    */
00108    public void addConstant(double d) {
      int loc = findConstant(d);
      addCommandCode(loc);
   }
   
   /**
    * Add a command code to the program, where code is one of the opCode constants
    * that are public final members of this class, from CUBERT to PLUS.  Each code
    * represents either a binary or unary operator or a standard function that operates on the stack by
    * poping its argument(s) from the stack, perfroming the operation, and pushing
    * the result back onto the stack.
    */
00120    public void addCommand(int code) {
      if (code >= 0 || code < CUBERT)
         throw new IllegalArgumentException("Internal Error.  Illegal command code.");
      addCommandCode(code);
   }
   
   /**
    * To save space, cut the arrays that holds the program data down to the actual
    * amount of data that they contain.  This should be called after the complete 
    * program has been generated.
    */
00131    public void trim() {
      if (progCt != prog.length) {
         int[] temp = new int[progCt];
         System.arraycopy(prog,0,temp,0,progCt);
         prog = temp;
      }
      if (commandCt != command.length) {
         ExpressionCommand[] temp = new ExpressionCommand[commandCt];
         System.arraycopy(command,0,temp,0,commandCt);
         command = temp;
      }
      if (constantCt != constant.length) {  
         double[] temp = new double[constantCt];
         System.arraycopy(constant,0,temp,0,constantCt);
         constant = temp;
      }
   }
   
   private void addCommandCode(int code) {  // System.out.println("Add code " + code + " " + progCt);
      if (progCt == prog.length) {
         int[] temp = new int[ prog.length * 2 ];
         System.arraycopy(prog,0,temp,0,progCt);
         prog = temp;
      }
      prog[progCt++] = code;
   }


   //------------------- Methods for evaluating the program -----------------------
   
   private Cases cases;  // If this is non-null while the expression is being evaluated, then
                         // information about "cases" is recorded in it as the evaluation is
                         // done.  See the Cases class for more information.

   private StackOfDouble stack = new StackOfDouble();  
              // An ExpressionProgram is a program for a stack machien.  This is the
              // stack that is used when the program is executed.
   
   /**
    * Run the ExprssionProgram and return the value that it computes.
    */
00172    synchronized public double getVal() {
      cases = null;
      return basicGetValue();
   }
   
   /**
    * Run the ExprssionProgram and return the value that it computes.
    * If the Cases object, c, is non-null, then information about "cases" is recorded in c.
    * This information can be used to help detect possible "discontinuities"
    * between two evaluations.  See the Cases class for more information.
    */
00183    synchronized public double getValueWithCases(Cases c) {
      cases = c;
      double d = basicGetValue();
      cases = null;
      return d;
   }
   
   private double basicGetValue() {
          // Run the program and return its value.
      stack.makeEmpty();
      for (int pc = 0; pc < progCt; pc++) {
         int code = prog[pc];
         if (code < 0) {
            double ans = applyCommandCode(code);
            if (Double.isNaN(ans) || Double.isInfinite(ans)) {
               if (cases != null)
                  cases.addCase(0);
               return Double.NaN;
            }
            stack.push(ans);
         }
         else if (code < 0x3FFFFFFF) {
            stack.push(constant[code]);
         }
         else {
            command[code-0x3FFFFFFF].apply(stack,cases);
         }
      }
      if (stack.size() != 1)
         throw new IllegalArgumentException("Internal Error:  Improper stack state after expression evaluation.");
      double val = stack.pop();
      if (cases != null)
         cases.addCase( Double.isNaN(val) ? 0 : 1 );
      return val;
   }
   
   private void addCase(int c) {
         // If the member variable cases is not null, record c as the next item of case information in cases.
      if (cases != null)
         cases.addCase(c);
   }
   
   /**
    * Apply the stack operation represented by code (a number < 0) to the stack.
    */
00228    protected double applyCommandCode(int code) {
      double ans;
      if (code < OR)
         ans = eval(code,stack.pop());
      else
         ans = eval(code,stack.pop(),stack.pop());
      return ans;
   }
   
   private double eval(int commandCode, double x) {
          // Compute the value of the unary operator represented by commandCode, when
          // applied to the value x.
      switch (commandCode) {
         case NOT: return (x == 0)? 1 : 0;
         case UNARY_MINUS:  return -x;
         case FACTORIAL: return factorial(x);
         case SIN: return Math.sin(x);
         case COS: return Math.cos(x);
         case TAN: addCase((int)Math.floor((x-Math.PI/2.0)/Math.PI)); return Math.tan(x);
         case COT: addCase((int)Math.floor(x/Math.PI)); return Math.cos(x)/Math.sin(x);
         case SEC: addCase((int)Math.floor((x-Math.PI/2.0)/Math.PI)); return 1/Math.cos(x);
         case CSC: addCase((int)Math.floor(x/Math.PI)); return 1/Math.sin(x);
         case ARCSIN: return Math.asin(x);
         case ARCCOS: return Math.acos(x);
         case ARCTAN: return Math.atan(x);
         case ABS: addCase((x > 0)? 1 : ( (x < 0)? -1 : 0 )); return Math.abs(x);
         case SQRT: return (x < 0)? Double.NaN : Math.sqrt(x);
         case EXP: return Math.exp(x);
         case LN: return (x <= 0)? Double.NaN : Math.log(x);
         case LOG2: return (x <= 0)? Double.NaN : Math.log(x)/Math.log(2);
         case LOG10: return (x <= 0)? Double.NaN : Math.log(x)/Math.log(10);
         case TRUNC: addCase((int)x); return (long)x;
         case ROUND: addCase((int)Math.floor(x+0.5)); return Math.floor(x+0.5);
         case FLOOR: addCase((int)Math.floor(x)); return Math.floor(x);
         case CEILING: addCase((int)Math.floor(x)); return Math.ceil(x);
         case CUBERT: addCase((x > 0)? 1 : -1); return (x >= 0)? Math.pow(x,1.0/3.0) : -Math.pow(-x,1.0/3.0);
         default: return Double.NaN;
      }
   }
   
   private double factorial(double x) {
          // Compute x!.  x is rounded to the nearest integer.  If x > 170, then the
          // answer is too big to represent in a value of type double, so the value
          // is given as Double.NaN.
      if (x <= -0.5 || x > 170.5) {
         addCase(-1);
         return Double.NaN;
      }
      int n = (int)x;
      addCase(n);
      double ans = 1;
      for (int i = 1; i <= n; i++)
         ans *= i;
      return ans;
  }

   private double eval(int commandCode, double y, double x) {
          // Compute the value of the unary operator represented by commandCode, when
          // applied to the values x and y.  (Note that the second operand comes
          // first in the parameter list, since that is the way they were popped
          // off the stack.)
      switch (commandCode) {
         case PLUS: return x+y;
         case MINUS: return x-y;
         case TIMES: return x*y;
         case DIVIDE: addCase( (y > 0)? 1 : ( (y < 0)? -1 : 0 ) ); return x/y;
         case POWER: return Math.pow(x,y);
         case EQ: return (x == y) ? 1 : 0;
         case NE: return (x != y) ? 1 : 0;
         case GT: return (x > y) ? 1 : 0;
         case LT: return (x < y) ? 1 : 0;
         case GE: return (x >= y) ? 1 : 0;
         case LE: return (x <= y) ? 1 : 0;
         case AND: return ((x != 0) && (y != 0)) ? 1 : 0;
         case OR: return ((x != 0) || (y != 0)) ? 1 : 0;
         default: return Double.NaN;
      }
   }
   

   //------------- Getting the print string of this expression ------------------------
   
   /**
    * If a source string has been saved, use it as the print string.  (When a Parser creates
    * an expression by parsing a string, it saves the source string in the ExpressionProgram.)
    * Otherwise, construct the print string based on the commands in the program.
    */
00315    public String toString() {
      if (sourceString != null)
         return sourceString;
      else {
         StringBuffer buffer = new StringBuffer();
         appendOutputString(progCt-1,buffer);
         return buffer.toString();
      }
   }

   /**
    * Add a string representing part of the expression to the output buffer.
    * You probably are only interested in this if you write a ParserExtension
    * or ExpressionCommand.
    * (The command at position index in the program represents a
    * subexpression.  It could be a constant or a variable, for example,
    * which is complete subexpression in itself.  Or it could be the
    * final operator in a larger subexpression.  In that case, the operands,
    * which are located in lower positions in the program, are considered to
    * be part of the expression.   This routine appends a print string
    * for the entire subexpression to the buffer.  When this is called
    * with index = progCt-1 (the last command in the program), it processes
    * the entire program. 
    * Note that the hard part here is deciding when to put in parentheses.
    * This is done based on the precedence of the operators.  The result is not always pretty.
    */
00341    public void appendOutputString(int index, StringBuffer buffer) {
      if (prog[index] >= 0x3FFFFFFF) {
         command[prog[index] - 0x3FFFFFFF].appendOutputString(this,index,buffer);
      }
      else if (prog[index] >= 0) {
         buffer.append(NumUtils.realToString(constant[prog[index]]));
      }
      else if (prog[index] >= OR) {
         int op2 = index-1;            // Position of command representing the second operand.
         int op1 = op2 - extent(op2);  // Position of command representing the first operand.
         if (precedence(prog[op1]) < precedence(prog[index]) ||
                   (prog[index] == POWER && precedence(prog[op1]) == precedence(prog[index]))) {
             buffer.append('(');
             appendOutputString(op1,buffer);
             buffer.append(')');
         }
         else
             appendOutputString(op1,buffer);
         switch (prog[index]) {
            case PLUS: buffer.append(" + "); break;
            case MINUS: buffer.append(" - "); break;
            case TIMES: buffer.append("*"); break;
            case DIVIDE: buffer.append("/"); break;
            case POWER: buffer.append("^"); break;
            case AND: buffer.append(" AND "); break;
            case OR: buffer.append(" OR "); break;
            case EQ: buffer.append(" = "); break;
            case NE: buffer.append(" <> "); break;
            case GE: buffer.append(" >= "); break;
            case LE: buffer.append(" <= "); break;
            case GT: buffer.append(" > "); break;
            case LT: buffer.append(" < "); break;
         }
         if (prog[op2] == UNARY_MINUS ||
                precedence(prog[op2]) < precedence(prog[index]) ||
                   ((prog[index] == MINUS || prog[index] == DIVIDE) && precedence(prog[op2]) == precedence(prog[index]))) {
             buffer.append('(');
             appendOutputString(op2,buffer);
             buffer.append(')');
         }
         else
             appendOutputString(op2,buffer);
      }
      else if (prog[index] <= SIN) {
         buffer.append(StandardFunction.standardFunctionName(prog[index]));
         buffer.append('(');
         appendOutputString(index-1,buffer);
         buffer.append(')');
      }
      else if (prog[index] == UNARY_MINUS) {
         buffer.append('-');
         if (precedence(prog[index-1]) <= precedence(UNARY_MINUS)) {
            buffer.append('(');
            appendOutputString(index-1,buffer);
            buffer.append(')');
         }
         else
            appendOutputString(index-1,buffer);
      }
      else if (prog[index] == NOT) {
         buffer.append("NOT (");
         appendOutputString(index-1,buffer);
         buffer.append(')');
      }
      else if (prog[index] == FACTORIAL) {
         if (prog[index-1] >= 0 &&
              (prog[index-1] < 0x3FFFFFFF || 
                command[prog[index-1] - 0x3FFFFFFF] instanceof Variable ||
                  command[prog[index-1] - 0x3FFFFFFF] instanceof Constant)) {
            appendOutputString(index-1,buffer);
         }
         else {
            buffer.append('(');
            appendOutputString(index-1,buffer);
            buffer.append(')');
         }
         buffer.append('!');
      }
   }
   
   private int precedence(int opcode) {
          // Return the precedence of the operator.  This is used
          // by the prceding method to decide when parentheses are needed.
      if (opcode >= 0) {
         if (opcode >= 0x3FFFFFFF && (command[opcode-0x3FFFFFFF] instanceof ConditionalExpression))
            return 0;
         else
            return 7;
      }
      else switch (opcode) {
         case FACTORIAL:
         case OR: 
         case AND: return 1; 
         case GE:
         case LE:
         case GT:
         case EQ:
         case NE:
         case LT: return 2;
         case PLUS:
         case MINUS:
         case UNARY_MINUS: return 3; 
         case TIMES:
         case DIVIDE: return 4;
         case POWER: return 6;
         default: return 7;
      }
   }
   

   //----------------------- Methods for computing the derivative -------------------------

   /**   
    * Compute the derivative of this expression with respect to the Variable wrt.
    * The value returned is actually an ExpressionProgram.
    */
00457    public Expression derivative(Variable wrt) {
       ExpressionProgram deriv = new ExpressionProgram();
       compileDerivative(progCt-1,deriv,wrt);
       deriv.trim();
       return deriv;
   }
   
   /**
    * The command at position index in the program represents a subexpression of
    * the whole expression.  This routine adds commands to deriv for computing the
    * derivative of that subexpression with respect to the variable wrt.
    * You probably are not interested in this unless you write a ParserExtension
    * or an ExpressionCommand.
    */
00471    public void compileDerivative(int index, ExpressionProgram deriv, Variable wrt) {
      if (!dependsOn(index,wrt))
         deriv.addConstant(0);
      else if (prog[index] >= 0x3FFFFFFF)
         command[prog[index]-0x3FFFFFFF].compileDerivative(this,index,deriv,wrt);
      else if (prog[index] >= 0)
         deriv.addConstant(0);
      else if (prog[index] >= POWER) {
         int indexOp2 = index - 1;
         int indexOp1 = indexOp2 - extent(indexOp2);
         doBinDeriv(prog[index],indexOp1,indexOp2,deriv,wrt);
      }
      else if (prog[index] <= SIN) {
         doFuncDeriv(prog[index],index-1,deriv,wrt);
      }
      else if (prog[index] == UNARY_MINUS) {
         compileDerivative(index-1,deriv,wrt);
         deriv.addCommand(UNARY_MINUS);
      }
      else if (prog[index] == FACTORIAL) {
         deriv.addConstant(Double.NaN);
      }
      else if (prog[index] >= NOT)
         throw new IllegalArgumentException("Internal Error: Attempt to take the derivative of a logical-valued expression.");
      else
         throw new IllegalArgumentException("Internal Error: Unknown opcode.");
   }
   
   /**
    * The command at position index in the program represents a subexpression of
    * the whole expression.  This routine finds and returns the number of commands
    * in the program that are part of that subexpression.  That is, the subexpresssion
    * occupies the part of the program between index - extent + 1 and index.
    * You probably are not interested in this unless you write a ParserExtension
    * or an ExpressionCommand.
    */
00507    public int extent(int index) {
      if (prog[index] <= NOT)
         return 1 + extent(index-1);
      else if (prog[index] < 0) {
         int extentOp1 = extent(index - 1);              // Extent of second operand (which is on top in the program).
         int extentOp2 = extent(index - 1 - extentOp1);  // Extent of second operand.
         return extentOp1 + extentOp2 + 1;
      }
      else if (prog[index] < 0x3FFFFFFF)
         return 1;
      else
         return command[prog[index]-0x3FFFFFFF].extent(this,index);
   }
   
   /**
    * The command at position index in the program represents a subexpression of
    * the whole expression.  This routine copies the commands for the entire
    * subexpression to the destination program.
    * You probably are not interested in this unless you write a ParserExtension
    * or an ExpressionCommand.
    */
00528    public void copyExpression(int index, ExpressionProgram destination) {
      int size = extent(index);
      for (int i = index-size+1; i <= index; i++) {
          if (prog[i] < 0)
             destination.addCommand(prog[i]);
          else if (prog[i] >= 0x3FFFFFFF)
             destination.addCommandObject(command[prog[i]-0x3FFFFFFF]);
          else
             destination.addConstant(constant[prog[i]]);
      }
   }
   
   /**
    * The command at position index in the program represents a subexpression of
    * the whole expression.  If that subexpression includes some dependence on
    * the variable x, then true is returned.  If the subexpression is constant 
    * with respect to the variable x, then false is returned.
    * You probably are not interested in this unless you write a ParserExtension
    * or an ExpressionCommand.
    */
00548    public boolean dependsOn(int index, Variable x) {
      int size = extent(index);
      for (int i = index-size+1; i <= index; i++) {
          if (prog[i] >= 0x3FFFFFFF) {
              ExpressionCommand c = command[prog[i]-0x3FFFFFFF];
              if (c == x || c.dependsOn(x))
                 return true;
          }
      }
      return false;
   }
   
   /**
    * Checks whether the expression as a whole has any dependence on the variable x.
    */
00563    public boolean dependsOn(Variable x) {
      return dependsOn(progCt - 1, x);
   }
   
   private void doBinDeriv(int opCode, int op1, int op2, ExpressionProgram deriv, Variable wrt) {
          // Add commands to deriv to compute the derivative of a subexpression of this program
          // with respect to the variable wrt, where the main operation in the subexpression is 
          // the binary operator opCode, the position of the first operand of the operator is
          // at the index op1 in the program, and the position of second operand is op2.
      switch (opCode) {
         case PLUS:
              if (!dependsOn(op1,wrt))
                 compileDerivative(op2,deriv,wrt);
              else if (!dependsOn(op2,wrt))
                 compileDerivative(op1,deriv,wrt);
              else {
                 compileDerivative(op1,deriv,wrt);
                 compileDerivative(op2,deriv,wrt);
                 deriv.addCommand(PLUS);
              }
            break;
         case MINUS:
              if (!dependsOn(op1,wrt)) {
                 compileDerivative(op2,deriv,wrt);
                 deriv.addCommand(UNARY_MINUS);
              }
              else if (!dependsOn(op2,wrt))
                 compileDerivative(op1,deriv,wrt);
              else {
                 compileDerivative(op1,deriv,wrt);
                 compileDerivative(op2,deriv,wrt);
                 deriv.addCommand(MINUS);
              }
            break;
         case TIMES: {
              int cases = 0;
              if (dependsOn(op2,wrt)) {
                 copyExpression(op1,deriv);
                 if (prog[op2] < 0x3FFFFFFF || command[prog[op2]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op2,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 cases++;
              }
              if (dependsOn(op1,wrt)) {
                 copyExpression(op2,deriv);
                 if (prog[op1] < 0x3FFFFFFF || command[prog[op1]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op1,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 cases++;
              }
              if (cases == 2)
                 deriv.addCommand(PLUS);
            }
            break;
         case DIVIDE:
              if (!dependsOn(op2,wrt)) {
                 compileDerivative(op1,deriv,wrt);
                 copyExpression(op2,deriv);
                 deriv.addCommand(DIVIDE);
              }
              else if (!dependsOn(op1,wrt)) {
                 copyExpression(op1,deriv);
                 deriv.addCommand(UNARY_MINUS);
                 copyExpression(op2,deriv);
                 deriv.addConstant(2);
                 deriv.addCommand(POWER);
                 deriv.addCommand(DIVIDE);
                 if (prog[op2] < 0x3FFFFFFF || command[prog[op2]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op2,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
              }
              else {
                 copyExpression(op2,deriv);
                 if (prog[op1] < 0x3FFFFFFF || command[prog[op1]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op1,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 copyExpression(op1,deriv);
                 if (prog[op2] < 0x3FFFFFFF || command[prog[op2]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op2,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 deriv.addCommand(MINUS);
                 copyExpression(op2,deriv);
                 deriv.addConstant(2);
                 deriv.addCommand(POWER);
                 deriv.addCommand(DIVIDE);
              }
            break;
         case POWER:
              if (!dependsOn(op2,wrt)) {
                 copyExpression(op2,deriv);
                 copyExpression(op1,deriv);
                 if ( prog[op2] >= 0 && prog[op2] < 0x3FFFFFFF) {
                    if (constant[prog[op2]] != 2) {
                       deriv.addConstant(constant[prog[op2]] - 1);
                       deriv.addCommand(POWER);
                    }
                 }
                 else if (prog[op2] == UNARY_MINUS && prog[op2-1] >= 0 && prog[op2-1] < 0x3FFFFFFF) {
                    deriv.addConstant( constant[prog[op2-1]] + 1 );
                    deriv.addCommand(UNARY_MINUS);
                    deriv.addCommand(POWER);
                 }
                 else {
                     copyExpression(op2,deriv);
                     deriv.addConstant(1);
                     deriv.addCommand(MINUS);
                     deriv.addCommand(POWER);
                 }
                 deriv.addCommand(TIMES);
                 if (prog[op1] < 0x3FFFFFFF || command[prog[op1]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op1,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
              }
              else if (!dependsOn(op1,wrt)) {
                 copyExpression(op1,deriv);
                 copyExpression(op2,deriv);
                 deriv.addCommand(POWER);
                 if ( ! (prog[op1] >= 0x3FFFFFFF && command[prog[op1]-0x3FFFFFFF] instanceof Constant 
                           && ((Constant)command[prog[op1]-0x3FFFFFFF]).getVal() == Math.E) ) {
                    copyExpression(op1,deriv);
                    deriv.addCommand(LN);
                    deriv.addCommand(TIMES);         
                 }
                 if (prog[op2] < 0x3FFFFFFF || command[prog[op2]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op2,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
              }
              else {
                 copyExpression(op1,deriv);
                 copyExpression(op2,deriv);
                 deriv.addCommand(POWER);
                 boolean eq = true;
                 int ext1 = extent(op1);
                 int ext2 = extent(op2);
                 if (ext1 != ext2)
                    eq = false;
                 else
                    for (int i = 0; i < ext1; i++)
                       if (prog[op1-i] != prog[op2-i]) {
                          eq = false;
                          break;
                       }
                 if (eq)
                    deriv.addConstant(1);
                 else {
                    copyExpression(op2,deriv);
                    copyExpression(op1,deriv);
                    deriv.addCommand(DIVIDE);
                 }
                 if (prog[op1] < 0x3FFFFFFF || command[prog[op1]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op1,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 copyExpression(op1,deriv);
                 deriv.addCommand(LN);
                 if (prog[op2] < 0x3FFFFFFF || command[prog[op2]-0x3FFFFFFF] != wrt) {
                    compileDerivative(op2,deriv,wrt);
                    deriv.addCommand(TIMES);
                 }
                 deriv.addCommand(PLUS);
                 deriv.addCommand(TIMES);
              }
            break;
      }
   }

   private void doFuncDeriv(int opCode, int op, ExpressionProgram deriv, Variable wrt) {
          // Add commands to deriv to compute the derivative of a subexpression of this program
          // with respect to the variable wrt, where the main operation in the subexpression is 
          // the standard function represented by  opCode,  and the position of the operand of the 
          // standard function is  at the index op in the program.
/*      if (opCode == TRUNC || opCode == ROUND || opCode == FLOOR || opCode == CEILING) {
            // I'm pretending that the derivative is zero, even though it's undefined at
            // certain values.
         deriv.addConstant(0);
         return;
      }
*/
      switch (opCode) {
         case FLOOR:
         case CEILING:
         case TRUNC: 
         case ROUND:
               copyExpression(op,deriv);
               if (opCode == ROUND) {
                  deriv.addConstant(0.5);
                  deriv.addCommand(PLUS);
               }
               deriv.addCommand(ROUND);
               copyExpression(op,deriv);
               if (opCode == ROUND) {
                  deriv.addConstant(0.5);
                  deriv.addCommand(PLUS);
               }
               deriv.addCommand(NE);
               if (opCode == TRUNC) {
                  copyExpression(op,deriv);
                  deriv.addConstant(0);
                  deriv.addCommand(EQ);
                  deriv.addCommand(OR);
               }
               ExpressionProgram zero = new ExpressionProgram();
               zero.addConstant(0);
               deriv.addCommandObject( new ConditionalExpression(zero , null) );
            return;
         case SIN:
               copyExpression(op,deriv);
               deriv.addCommand(COS);
            break;
         case COS:
               copyExpression(op,deriv);
               deriv.addCommand(SIN);
               deriv.addCommand(UNARY_MINUS);
            break;
         case TAN:
               copyExpression(op,deriv);
               deriv.addCommand(SEC);
               deriv.addConstant(2);
               deriv.addCommand(POWER);
            break;
         case COT:
               copyExpression(op,deriv);
               deriv.addCommand(CSC);
               deriv.addConstant(2);
               deriv.addCommand(POWER);
               deriv.addCommand(UNARY_MINUS);
            break;
         case SEC:
               copyExpression(op,deriv);
               deriv.addCommand(SEC);
               copyExpression(op,deriv);
               deriv.addCommand(TAN);
               deriv.addCommand(TIMES);
            break;
         case CSC:
               copyExpression(op,deriv);
               deriv.addCommand(CSC);
               copyExpression(op,deriv);
               deriv.addCommand(COT);
               deriv.addCommand(TIMES);
               deriv.addCommand(UNARY_MINUS);
            break;
         case ARCSIN:
         case ARCCOS:
               deriv.addConstant(1);
               if (opCode == ARCCOS)
                  deriv.addCommand(UNARY_MINUS);
               deriv.addConstant(1);
               copyExpression(op,deriv);
               deriv.addConstant(2);
               deriv.addCommand(POWER);
               deriv.addCommand(MINUS);
               deriv.addCommand(SQRT);
               deriv.addCommand(DIVIDE);
            break;
         case ARCTAN:
               deriv.addConstant(1);
               deriv.addConstant(1);
               copyExpression(op,deriv);
               deriv.addConstant(2);
               deriv.addCommand(POWER);
               deriv.addCommand(PLUS);
               deriv.addCommand(DIVIDE);
            break;
         case ABS: {
               ExpressionProgram pos = new ExpressionProgram();
               ExpressionProgram neg = new ExpressionProgram();
               compileDerivative(op,pos,wrt);
               compileDerivative(op,neg,wrt);
               neg.addCommand(UNARY_MINUS);
               ExpressionProgram negTest = new ExpressionProgram();
               copyExpression(op,negTest);
               negTest.addConstant(0);
               negTest.addCommand(LT);
               negTest.addCommandObject( new ConditionalExpression(neg,null) );
               copyExpression(op,deriv);
               deriv.addConstant(0);
               deriv.addCommand(GT);
               deriv.addCommandObject( new ConditionalExpression( pos, negTest ) );
            }
            return;
         case SQRT:
               deriv.addConstant(1);
               deriv.addConstant(2);
               copyExpression(op,deriv);
               deriv.addCommand(SQRT);
               deriv.addCommand(TIMES);
               deriv.addCommand(DIVIDE);
            break;
         case EXP:
               copyExpression(op,deriv);
               deriv.addCommand(EXP);
            break;
         case LN:
         case LOG2:
         case LOG10:
               ExpressionProgram d = new ExpressionProgram();
               d.addConstant(1);
               copyExpression(op,d);
               d.addCommand(DIVIDE);
               if (opCode != LN) {
                  d.addConstant( (opCode == LOG2)? 2 : 10);
                  d.addCommand(LN);
                  d.addCommand(DIVIDE);               
               }
               copyExpression(op,deriv);
               deriv.addConstant(0);
               deriv.addCommand(GT);
               deriv.addCommandObject( new ConditionalExpression(d,null) );
               break;
         case CUBERT:
               deriv.addConstant(1);
               deriv.addConstant(3);
               copyExpression(op,deriv);
               deriv.addConstant(2);
               deriv.addCommand(POWER);
               deriv.addCommand(CUBERT);
               deriv.addCommand(TIMES);
               deriv.addCommand(DIVIDE);
            break;
      }
      if (prog[op] < 0x3FFFFFFF || command[prog[op]-0x3FFFFFFF] != wrt) {
         compileDerivative(op,deriv,wrt);
         deriv.addCommand(TIMES);
      }
   }
   

   //--------- private  stuff for storing constants and ExpressionCommands ----------

   private double[] constant = new double[1];  // Holds all the constants that have been added
                                               // to this ExpressionProgram.  In a program, a 
                                               // constant is represented by an index into this
                                               // array.
                                                       
   private int constantCt;  // The number of values in the constant array.
   
   private ExpressionCommand[] command = new ExpressionCommand[1];  // Holds all the ExpressionCommands that have been added
                                                                     // to this ExpressionProgram.  In a program, an
                                                                     // ExpressionCommand is represented by an index into this
                                                                     // array, with 0x3FFFFFFF added to the index to give
                                                                     // a number >= 0x3FFFFFFF. 
                                                                            
   private int commandCt;  // The number of items in the command array.

   private int findConstant(double d) {
          // Find the index of d in the constant array, adding it if it is not already there.
          // The array will be expanded if necessary.
      for (int i = 0; i < constantCt; i++)
         if (constant[i] == d) {  
            return i;}
      if (constantCt == constant.length) {
         double[] temp = new double[ constant.length * 2 ];
         System.arraycopy(constant,0,temp,0,constantCt);
         constant = temp;
      }
      constant[constantCt++] = d;  
      return constantCt - 1;
   }

   private int findCommand(ExpressionCommand com) {
          // Find the index of com in the command array, adding it if it is not already there.
          // The array will be expanded if necessary.
      for (int i = 0; i < commandCt; i++)
         if (command[i] == com)
            return i;
      if (commandCt == command.length) {
         ExpressionCommand[] temp = new ExpressionCommand[ command.length * 2 ];
         System.arraycopy(command,0,temp,0,commandCt);
         command = temp;
      }
      command[commandCt++] = com;
      return commandCt - 1;
   }
} // end class ExpressionProgram


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