## C++ || Multi Digit, Decimal & Negative Number Infix To Postfix Conversion & Evaluation

The following is sample code which demonstrates the implementation of a multi digit, decimal, and negative number infix to postfix converter and evaluator using a Finite State Machine

REQUIRED KNOWLEDGE FOR THIS PROGRAM

```How To Convert Infix To Postfix How To Evaluate A Postfix Expression What Is A Finite State Machine? ```

Using a Finite State Machine, the program demonstrated on this page has the ability to convert and evaluate a single digit, multi digit, decimal number, and/or negative number infix equation. So for example, if the the infix equation of (19.87 * ~2) was entered into the program, the converted postfix expression of 19.87 ~2* would display to the screen, as well as the final evaluated answer of -39.74.

NOTE: In this program, negative numbers are represented by the “~” symbol.

This program has the following flow of control:

```• Get an infix expression from the user • Convert the infix expression to postfix • Use a Finite State Machine to isolate all of the math operators, multi digit, decimal, negative and single digit numbers that are found in the postfix expression • Evaluate the postfix expression using the tokens found from the above step • Display the evaluated answer to the screen ```

The above steps are implemented below.

``` Multi Digit Infix To Postfix Conversion & Evaluation C++ // ============================================================================ // Author: K Perkins // Taken From: http://programmingnotes.org/ // Date: Jan 31, 2014 // File: InToPostEval.cpp // Description: The following demonstrates the implementation of an infix to // postfix converter and evaluator. Using a Finite State Machine, this // program has the ability to convert and evaluate multi digit, decimal, // negative and positive values. // ============================================================================ #include <iostream> #include <cstdlib> #include <cmath> #include <cctype> #include <string> #include <vector> #include <stack> using namespace std; /* This holds the transition states for our Finite State Machine -- They are placed in numerical order for easy understanding within the FSM array, which is located below */ enum FSM_TRANSITIONS { REJECT = 0, INTEGER, REAL, NEGATIVE, OPERATOR, UNKNOWN, SPACE }; /* This is the Finite State Machine -- The zero represents a place holder, so the row in the array starts on row 1 instead of 0 integer, real, negative, operator, unknown, space */ int stateTable[][7] = {{0, INTEGER, REAL, NEGATIVE, OPERATOR, UNKNOWN, SPACE}, /* STATE 1 */ {INTEGER, INTEGER, REAL, REJECT, REJECT, REJECT, REJECT}, /* STATE 2 */ {REAL, REAL, REJECT, REJECT, REJECT, REJECT, REJECT}, /* STATE 3 */ {NEGATIVE, INTEGER, REAL, REJECT, REJECT, REJECT, REJECT}, /* STATE 4 */ {OPERATOR, REJECT, REJECT, REJECT, REJECT, REJECT, REJECT}, /* STATE 5 */ {UNKNOWN, REJECT, REJECT, REJECT, REJECT, UNKNOWN, REJECT}, /* STATE 6 */ {SPACE, REJECT, REJECT, REJECT, REJECT, REJECT, REJECT}}; // function prototypes void DisplayDirections(); string ConvertInfixToPostfix(string infix); bool IsMathOperator(char token); int OrderOfOperations(char token); vector<string> Lexer(string postfix); int Get_FSM_Col(char& currentChar); double EvaluatePostfix(const vector<string>& postfix); double Calculate(char token, double op1, double op2); int main() { // declare variables string infix = ""; string postfix = ""; double answer = 0; vector<string> tokens; // display directions to user DisplayDirections(); // get data from user cout<<"\nPlease enter an Infix expression: "; getline(cin, infix); postfix = ConvertInfixToPostfix(infix); // use the "Lexer" function to isolate multi digit, negative and decimal // numbers, aswell as single digit numbers and math operators tokens = Lexer(postfix); // display the found tokens to the screen // for(unsigned x = 0; x < tokens.size(); ++x) // { // cout<<tokens.at(x)<<endl; // } cout <<"\nThe Infix expression = "<<infix; cout <<"\nThe Postfix expression = "<<postfix<<endl; answer = EvaluatePostfix(tokens); cout<<"\nFinal answer = "<<answer<<endl; return 0; }// end of main void DisplayDirections() {// this function displays instructions to the screen cout << "\n==== Infix To Postfix Conversion & Evaluation ====\n" <<"\nMath Operators:\n" <<"+ || Addition\n" <<"- || Subtraction\n" <<"* || Multiplication\n" <<"/ || Division\n" <<"% || Modulus\n" <<"^ || Power\n" <<"\$ || Square Root\n" <<"s || Sine\n" <<"c || Cosine\n" <<"t || Tangent\n" <<"~ || Negative Number\n" <<"Sample Infix Equation: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23))\n"; // ((sin(-4^5)*1.4)/(sqrt(23+2)--2.8))*(cos(1%2)/(7.28*.1987)^(tan(23))) }// end of DisplayDirections string ConvertInfixToPostfix(string infix) {// this function converts an infix expression to postfix // declare function variables string postfix; stack<char> charStack; // loop thru array until there is no more data for(unsigned x = 0; x < infix.length(); ++x) { // place numbers (standard, decimal, & negative) // numbers onto the 'postfix' string if((isdigit(infix[x])) || (infix[x] == '.') || (infix[x] == '~')) { postfix += infix[x]; } else if(isspace(infix[x])) { continue; } else if(IsMathOperator(infix[x])) { postfix += " "; // use the 'OrderOfOperations' function to check equality // of the math operator at the top of the stack compared to // the current math operator in the infix string while((!charStack.empty()) && (OrderOfOperations(charStack.top()) >= OrderOfOperations(infix[x]))) { // place the math operator from the top of the // stack onto the postfix string and continue the // process until complete postfix += charStack.top(); charStack.pop(); } // push the remaining math operator onto the stack charStack.push(infix[x]); } // push outer parentheses onto stack else if(infix[x] == '(') { charStack.push(infix[x]); } else if(infix[x] == ')') { // pop the current math operator from the stack while((!charStack.empty()) && (charStack.top() != '(')) { // place the math operator onto the postfix string postfix += charStack.top(); // pop the next operator from the stack and // continue the process until complete charStack.pop(); } if(!charStack.empty()) // pop '(' symbol off the stack { charStack.pop(); } else // no matching '(' { cout<<"\nPARENTHESES MISMATCH #1\n"; exit(1); } } else { cout<<"\nINVALID INPUT #1\n"; exit(1); } } // place any remaining math operators from the stack onto // the postfix array while(!charStack.empty()) { postfix += charStack.top(); charStack.pop(); } return postfix; }// end of ConvertInfixToPostfix bool IsMathOperator(char token) {// this function checks if operand is a math operator switch(tolower(token)) { case '+': case '-': case '*': case '/': case '%': case '^': case '\$': case 'c': case 's': case 't': return true; break; default: return false; break; } }// end of IsMathOperator int OrderOfOperations(char token) {// this function returns the priority of each math operator int priority = 0; switch(tolower(token)) { case 'c': case 's': case 't': priority = 5; break; case '^': case '\$': priority = 4; break; case '*': case '/': case '%': priority = 3; break; case '-': priority = 2; break; case '+': priority = 1; break; } return priority; }// end of OrderOfOperations vector<string> Lexer(string postfix) {// this function parses a postfix string using an FSM to generate // each individual token in the expression vector<string> tokens; char currentChar = ' '; int col = REJECT; int currentState = REJECT; string currentToken = ""; // use an FSM to parse multidigit and decimal numbers // also does error check for invalid input of decimals for(unsigned x = 0; x < postfix.length();) { currentChar = postfix[x]; // get the column number for the current character col = Get_FSM_Col(currentChar); // exit if the real number has multiple periods "." // in the expression (i.e: 19.3427.23) if((currentState == REAL) && (col == REAL)) { cerr<<"\nINVALID INPUT #2\n"; exit(1); } /* ======================================================== THIS IS WHERE WE CHECK THE FINITE STATE MACHINE TABLE USING THE "col" VARIABLE FROM ABOVE ^ ========================================================= */ // get the current state of our machine currentState = stateTable[currentState][col]; /* =================================================== THIS IS WHERE WE CHECK FOR A SUCESSFUL PARSE - If the current state in our machine == REJECT (the starting state), then we have sucessfully parsed a token, which is returned to its caller - ELSE we continue trying to find a sucessful token =================================================== */ if(currentState == REJECT) { if(currentToken != " ") // we dont care about whitespace { tokens.push_back(currentToken); } currentToken = ""; } else { currentToken += currentChar; ++x; } } // this ensures the last token gets saved when // we reach the end of the postfix string buffer if(currentToken != " ") // we dont care about whitespace { tokens.push_back(currentToken); } return tokens; }// end of Lexer int Get_FSM_Col(char& currentChar) {// this function determines the state of the type of character being examined // check for whitespace if(isspace(currentChar)) { return SPACE; } // check for integer numbers else if(isdigit(currentChar)) { return INTEGER; } // check for real numbers else if(currentChar == '.') { return REAL; } // check for negative numbers else if(currentChar == '~') { currentChar = '-'; return NEGATIVE; } // check for math operators else if(IsMathOperator(currentChar)) { return OPERATOR; } return UNKNOWN; }// end of Get_FSM_Col double EvaluatePostfix(const vector<string>& postfix) {// this function evaluates a postfix expression // declare function variables double op1 = 0; double op2 = 0; double answer = 0; stack<double> doubleStack; cout<<"\nCalculations:\n"; // loop thru array until there is no more data for(unsigned x = 0; x < postfix.size(); ++x) { // push numbers onto the stack if((isdigit(postfix[x][0])) || (postfix[x][0] == '.')) { doubleStack.push(atof(postfix[x].c_str())); } // push negative numbers onto the stack else if((postfix[x].length() > 1) && ((postfix[x][0] == '-') && (isdigit(postfix[x][1]) || (postfix[x][1] == '.')))) { doubleStack.push(atof(postfix[x].c_str())); } // if expression is a math operator, pop numbers from stack // & send the popped numbers to the 'Calculate' function else if(IsMathOperator(postfix[x][0]) && (!doubleStack.empty())) { char token = tolower(postfix[x][0]); // if expression is square root, sin, cos, // or tan operation only pop stack once if(token == '\$' || token == 's' || token == 'c' || token == 't') { op2 = 0; op1 = doubleStack.top(); doubleStack.pop(); answer = Calculate(token, op1, op2); doubleStack.push(answer); } else if(doubleStack.size() > 1) { op2 = doubleStack.top(); doubleStack.pop(); op1 = doubleStack.top(); doubleStack.pop(); answer = Calculate(token, op1, op2); doubleStack.push(answer); } } else // this should never execute, & if it does, something went really wrong { cout<<"\nINVALID INPUT #3\n"; exit(1); } } // pop the final answer from the stack, and return to main if(!doubleStack.empty()) { answer = doubleStack.top(); } return answer; }// end of EvaluatePostfix double Calculate(char token, double op1, double op2) {// this function carries out the actual math process double ans = 0; switch(tolower(token)) { case '+': cout<<op1<<token<<op2<<" = "; ans = op1 + op2; break; case '-': cout<<op1<<token<<op2<<" = "; ans = op1 - op2; break; case '*': cout<<op1<<token<<op2<<" = "; ans = op1 * op2; break; case '/': cout<<op1<<token<<op2<<" = "; ans = op1 / op2; break; case '%': cout<<op1<<token<<op2<<" = "; ans = ((int)op1%(int)op2)+modf(op1 , &op2); break; case '^': cout<<op1<<token<<op2<<" = "; ans = pow(op1, op2); break; case '\$': cout<<char(251)<<op1<<" = "; ans = sqrt(op1); break; case 'c': cout<<"cos("<<op1<<") = "; ans = cos(op1); break; case 's': cout<<"sin("<<op1<<") = "; ans = sin(op1); break; case 't': cout<<"tan("<<op1<<") = "; ans = tan(op1); break; default: ans = 0; break; } cout<<ans<<endl; return ans; }// http://programmingnotes.org/ 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454 // ============================================================================//   Author:  K Perkins//   Taken From: http://programmingnotes.org///   Date:  Jan 31, 2014//   File: InToPostEval.cpp//   Description: The following demonstrates the implementation of an infix to//     postfix converter and evaluator. Using a Finite State Machine, this//     program has the ability to convert and evaluate multi digit, decimal,//     negative and positive values.// ============================================================================#include <iostream>#include <cstdlib>#include <cmath>#include <cctype>#include <string>#include <vector>#include <stack>using namespace std; /* This holds the transition states for our Finite State Machine    -- They are placed in numerical order for easy understanding within     the FSM array, which is located below */ enum FSM_TRANSITIONS{    REJECT = 0,    INTEGER,    REAL,    NEGATIVE,    OPERATOR,    UNKNOWN,    SPACE}; /* This is the Finite State Machine    -- The zero represents a place holder, so the row in the array         starts on row 1 instead of 0                            integer,  real,  negative, operator, unknown, space */int stateTable[][7] = {{0, INTEGER,  REAL, NEGATIVE, OPERATOR,  UNKNOWN,  SPACE},/* STATE 1 */   {INTEGER,  INTEGER,  REAL,   REJECT,  REJECT,   REJECT,  REJECT},/* STATE 2 */   {REAL,       REAL,  REJECT,  REJECT,  REJECT,   REJECT,  REJECT},/* STATE 3 */   {NEGATIVE, INTEGER,  REAL,   REJECT,  REJECT,   REJECT,  REJECT},/* STATE 4 */   {OPERATOR,  REJECT, REJECT,  REJECT,  REJECT,   REJECT,  REJECT},/* STATE 5 */   {UNKNOWN,   REJECT, REJECT,  REJECT,  REJECT,   UNKNOWN, REJECT},/* STATE 6 */   {SPACE,     REJECT, REJECT,  REJECT,  REJECT,   REJECT,  REJECT}}; // function prototypesvoid DisplayDirections();string ConvertInfixToPostfix(string infix);bool IsMathOperator(char token);int OrderOfOperations(char token);vector<string> Lexer(string postfix);int Get_FSM_Col(char& currentChar);double EvaluatePostfix(const vector<string>& postfix);double Calculate(char token, double op1, double op2); int main(){    // declare variables    string infix = "";    string postfix = "";    double answer = 0;    vector<string> tokens;     // display directions to user    DisplayDirections();     // get data from user    cout<<"\nPlease enter an Infix expression: ";    getline(cin, infix);     postfix = ConvertInfixToPostfix(infix);     // use the "Lexer" function to isolate multi digit, negative and decimal    // numbers, aswell as single digit numbers and math operators    tokens = Lexer(postfix);     // display the found tokens to the screen// for(unsigned x = 0; x < tokens.size(); ++x)// {//     cout<<tokens.at(x)<<endl;// }     cout <<"\nThe Infix expression = "<<infix;    cout <<"\nThe Postfix expression = "<<postfix<<endl;     answer = EvaluatePostfix(tokens);     cout<<"\nFinal answer = "<<answer<<endl;     return 0;}// end of main void DisplayDirections(){// this function displays instructions to the screen    cout << "\n==== Infix To Postfix Conversion & Evaluation ====\n"        <<"\nMath Operators:\n"        <<"+ || Addition\n"        <<"- || Subtraction\n"        <<"* || Multiplication\n"        <<"/ || Division\n"        <<"% || Modulus\n"        <<"^ || Power\n"        <<"\$ || Square Root\n"        <<"s || Sine\n"        <<"c || Cosine\n"        <<"t || Tangent\n"        <<"~ || Negative Number\n"        <<"Sample Infix Equation: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23))\n";        // ((sin(-4^5)*1.4)/(sqrt(23+2)--2.8))*(cos(1%2)/(7.28*.1987)^(tan(23)))}// end of DisplayDirections string ConvertInfixToPostfix(string infix){// this function converts an infix expression to postfix    // declare function variables    string postfix;    stack<char> charStack;     // loop thru array until there is no more data    for(unsigned x = 0; x < infix.length(); ++x)    {        // place numbers (standard, decimal, & negative)        // numbers onto the 'postfix' string        if((isdigit(infix[x])) || (infix[x] == '.') || (infix[x] == '~'))        {            postfix += infix[x];        }        else if(isspace(infix[x]))        {            continue;        }        else if(IsMathOperator(infix[x]))        {            postfix += " ";            // use the 'OrderOfOperations' function to check equality            // of the math operator at the top of the stack compared to            // the current math operator in the infix string            while((!charStack.empty()) &&                (OrderOfOperations(charStack.top()) >= OrderOfOperations(infix[x])))            {                // place the math operator from the top of the                // stack onto the postfix string and continue the                // process until complete                postfix += charStack.top();                charStack.pop();            }            // push the remaining math operator onto the stack            charStack.push(infix[x]);        }        // push outer parentheses onto stack        else if(infix[x] == '(')        {            charStack.push(infix[x]);        }        else if(infix[x] == ')')        {            // pop the current math operator from the stack            while((!charStack.empty()) && (charStack.top() != '('))            {                // place the math operator onto the postfix string                postfix += charStack.top();                // pop the next operator from the stack and                // continue the process until complete                charStack.pop();            }             if(!charStack.empty()) // pop '(' symbol off the stack            {                charStack.pop();            }            else // no matching '('            {                cout<<"\nPARENTHESES MISMATCH #1\n";                exit(1);            }        }        else        {            cout<<"\nINVALID INPUT #1\n";            exit(1);        }    }     // place any remaining math operators from the stack onto    // the postfix array    while(!charStack.empty())    {        postfix += charStack.top();        charStack.pop();    }     return postfix;}// end of ConvertInfixToPostfix bool IsMathOperator(char token){// this function checks if operand is a math operator    switch(tolower(token))    {        case '+': case '-': case '*': case '/':        case '%': case '^': case '\$': case 'c':        case 's': case 't':            return true;            break;        default:           return false;           break;    }}// end of IsMathOperator int OrderOfOperations(char token){// this function returns the priority of each math operator    int priority = 0;    switch(tolower(token))    {        case 'c': case 's': case 't':           priority = 5;           break;        case '^': case '\$':           priority = 4;           break;        case '*': case '/': case '%':           priority = 3;           break;        case '-':           priority = 2;           break;        case '+':           priority = 1;           break;    }    return priority;}// end of OrderOfOperations vector<string> Lexer(string postfix){// this function parses a postfix string using an FSM to generate //  each individual token in the expression    vector<string> tokens;    char currentChar = ' ';    int col = REJECT;    int currentState = REJECT;    string currentToken = "";     // use an FSM to parse multidigit and decimal numbers    // also does error check for invalid input of decimals    for(unsigned x = 0; x < postfix.length();)    {        currentChar = postfix[x];         // get the column number for the current character        col = Get_FSM_Col(currentChar);         // exit if the real number has multiple periods "."        // in the expression (i.e: 19.3427.23)        if((currentState == REAL) && (col == REAL))        {            cerr<<"\nINVALID INPUT #2\n";            exit(1);        }        /* ========================================================             THIS IS WHERE WE CHECK THE FINITE STATE MACHINE TABLE               USING THE "col" VARIABLE FROM ABOVE ^           ========================================================= */         // get the current state of our machine        currentState = stateTable[currentState][col];         /* ===================================================           THIS IS WHERE WE CHECK FOR A SUCESSFUL PARSE           - If the current state in our machine == REJECT             (the starting state), then we have sucessfully parsed             a token, which is returned to its caller             - ELSE we continue trying to find a sucessful token             =================================================== */        if(currentState == REJECT)        {            if(currentToken != " ") // we dont care about whitespace            {                tokens.push_back(currentToken);            }            currentToken = "";        }        else        {            currentToken += currentChar;            ++x;        }     }    // this ensures the last token gets saved when    // we reach the end of the postfix string buffer    if(currentToken != " ") // we dont care about whitespace    {        tokens.push_back(currentToken);    }    return tokens;}// end of Lexer int Get_FSM_Col(char& currentChar){// this function determines the state of the type of character being examined    // check for whitespace    if(isspace(currentChar))    {        return SPACE;    }     // check for integer numbers    else if(isdigit(currentChar))    {        return INTEGER;    }     // check for real numbers    else if(currentChar == '.')    {        return REAL;    }     // check for negative numbers    else if(currentChar == '~')    {        currentChar = '-';        return NEGATIVE;    }     // check for math operators    else if(IsMathOperator(currentChar))    {        return OPERATOR;    }    return UNKNOWN;}// end of Get_FSM_Col double EvaluatePostfix(const vector<string>& postfix){// this function evaluates a postfix expression    // declare function variables    double op1 = 0;    double op2 = 0;    double answer = 0;    stack<double> doubleStack;     cout<<"\nCalculations:\n";     // loop thru array until there is no more data    for(unsigned x = 0; x < postfix.size(); ++x)    {        // push numbers onto the stack        if((isdigit(postfix[x][0])) || (postfix[x][0] == '.'))        {           doubleStack.push(atof(postfix[x].c_str()));        }        // push negative numbers onto the stack        else if((postfix[x].length() > 1) && ((postfix[x][0] == '-') &&            (isdigit(postfix[x][1]) || (postfix[x][1] == '.'))))        {           doubleStack.push(atof(postfix[x].c_str()));        }        // if expression is a math operator, pop numbers from stack        // & send the popped numbers to the 'Calculate' function        else if(IsMathOperator(postfix[x][0]) && (!doubleStack.empty()))        {            char token = tolower(postfix[x][0]);             // if expression is square root, sin, cos,            // or tan operation only pop stack once            if(token == '\$' || token == 's' || token == 'c' || token == 't')            {                op2 = 0;                op1 = doubleStack.top();                doubleStack.pop();                answer = Calculate(token, op1, op2);                doubleStack.push(answer);            }            else if(doubleStack.size() > 1)            {                op2 = doubleStack.top();                doubleStack.pop();                op1 = doubleStack.top();                doubleStack.pop();                answer = Calculate(token, op1, op2);                doubleStack.push(answer);            }        }        else // this should never execute, & if it does, something went really wrong        {           cout<<"\nINVALID INPUT #3\n";           exit(1);        }    }    // pop the final answer from the stack, and return to main    if(!doubleStack.empty())    {        answer = doubleStack.top();    }    return answer;}// end of EvaluatePostfix double Calculate(char token, double op1, double op2){// this function carries out the actual math process    double ans = 0;    switch(tolower(token))    {        case '+':           cout<<op1<<token<<op2<<" = ";           ans = op1 + op2;           break;        case '-':           cout<<op1<<token<<op2<<" = ";           ans = op1 - op2;           break;        case '*':           cout<<op1<<token<<op2<<" = ";           ans = op1 * op2;           break;        case '/':           cout<<op1<<token<<op2<<" = ";           ans = op1 / op2;           break;        case '%':           cout<<op1<<token<<op2<<" = ";           ans = ((int)op1%(int)op2)+modf(op1 , &op2);           break;        case '^':           cout<<op1<<token<<op2<<" = ";           ans = pow(op1, op2);           break;        case '\$':           cout<<char(251)<<op1<<" = ";           ans = sqrt(op1);           break;        case 'c':            cout<<"cos("<<op1<<") = ";            ans = cos(op1);            break;        case 's':            cout<<"sin("<<op1<<") = ";            ans = sin(op1);            break;        case 't':            cout<<"tan("<<op1<<") = ";            ans = tan(op1);            break;        default:           ans = 0;           break;    }    cout<<ans<<endl;    return ans;}// http://programmingnotes.org/ ```

QUICK NOTES:
The highlighted lines are sections of interest to look out for.

The code is heavily commented, so no further insight is necessary. If you have any questions, feel free to leave a comment below.

The following is sample output.

```====== RUN 1 ====== ==== Infix To Postfix Conversion & Evaluation ====```

``` Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root s || Sine c || Cosine t || Tangent ~ || Negative Number Sample Infix Equation: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23)) Please enter an Infix expression: 12/3*9 The Infix expression = 12/3*9 The Postfix expression = 12 3 /9* Calculations: 12/3 = 4 4*9 = 36 Final answer = 36 ====== RUN 2 ====== ==== Infix To Postfix Conversion & Evaluation ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root s || Sine c || Cosine t || Tangent ~ || Negative Number Sample Infix Equation: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23)) Please enter an Infix expression: ~150.89996 - 87.56643 The Infix expression = ~150.89996 - 87.56643 The Postfix expression = ~150.89996 87.56643- Calculations: -150.9-87.5664 = -238.466 Final answer = -238.466 ====== RUN 3 ====== ==== Infix To Postfix Conversion & Evaluation ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root s || Sine c || Cosine t || Tangent ~ || Negative Number Sample Infix Equation: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23)) Please enter an Infix expression: ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23)) The Infix expression = ((s(~4^5)*1.4)/(\$(23+2)-~2.8))*(c(1%2)/(7.28*.1987)^(t23)) The Postfix expression = ~4 5^ s1.4* 23 2+ \$~2.8-/ 1 2% c7.28 .1987* 23t^/* Calculations: -4^5 = -1024 sin(-1024) = 0.158533 0.158533*1.4 = 0.221947 23+2 = 25 √25 = 5 5--2.8 = 7.8 0.221947/7.8 = 0.0284547 1%2 = 1 cos(1) = 0.540302 7.28*0.1987 = 1.44654 tan(23) = 1.58815 1.44654^1.58815 = 1.79733 0.540302/1.79733 = 0.300614 0.0284547*0.300614 = 0.00855389 Final answer = 0.00855389 ```

## C++ || Stack Based Postfix Evaluation (Single Digit)

This page consists of another homework assignment which was presented in a C++ Data Structures course. While the previously discussed program dealt with converting Infix expressions to Postfix, this program will demonstrate exactly how to evaluate them.

NOTE: Want to convert & evaluate multi digit, decimal, and negative numbers? Click here!

REQUIRED KNOWLEDGE FOR THIS PROGRAM

```What Is Postfix? How To Convert Infix To Postfix Equations Stack Data Structure Cin.getline How To Evaluate Postfix Expressions The Order Of Operations #include "ClassStackType.h"```

The title of this page is called – “Stack Based Postfix Evaluation (Single Digit).” Why “single digit?” The program demonstrated on this page has the ability to evaluate a postfix equation, but it only has the ability to evaluate single digit values. What do I mean by that? Consider the infix equation: 5+2. When that expression is converted to postfix, it will come out to be: 52+, and the answer will be 7 (5+2=7). But what if we have an equation like 12+2? When that expression is converted to postfix, it will come out to be: 122+. The postfix conversion is correct, but when you try to evaluate the expression, we do not know if the math operation should be 12+2 or 1+22, it can be read either way.

Question: So why is this program being displayed if it only works for single digits?
Answer: Because it demonstrates the process of evaluating postfix equations very well.

Want to convert & evaluate multi digit, decimal, and negative numbers? Click here!

Before we get into things, here is a helpful algorithm for evaluating a postfix expression in pseudo code:

``` // An algorithm for postfix evaluation. // For example, (1 + 2) / (5 + 6) translates to 1 2 + 5 6 + / // which equals the result of 0.272727 // Valid operands are single digits: 0-9 // Valid operators are: +, -, *, /, ^, \$ // Highest precedence: ^, \$ // Lowest precedence: +,- // the operators ')' and '('never goes on stack. double EvaluatePostfix(char* postfix) { while there is input { if input is a number push current number on stack else if input is a math operator and stack is not empty set operand2 to the top of the operand stack pop the stack set operand1 to the top of the operand stack pop the stack apply the math operation that represents to operand1 and operand2 push the result onto the stack else error } // When the loop is finished, the operand stack will contain one item, // the result of evaluating the expression pop the stack return the answer to the caller }// http://programmingnotes.org/ 1234567891011121314151617181920212223242526272829 // An algorithm for postfix evaluation.// For example,  (1 + 2) / (5 + 6) translates to  1 2 + 5 6 + /// which equals the result of 0.272727// Valid operands are single digits: 0-9// Valid operators are: +, -, *, /, ^, \$// Highest precedence: ^, \$// Lowest precedence: +,-// the operators ')' and '('never goes on stack. double EvaluatePostfix(char* postfix){  while there is input  { if input is a number push current number on stack else if input is a math operator and stack is not empty set operand2 to the top of the operand stack pop the stack set operand1 to the top of the operand stack pop the stack apply the math operation that represents to operand1 and operand2 push the result onto the stack else error  }     // When the loop is finished, the operand stack will contain one item, // the result of evaluating the expression  pop the stack  return the answer to the caller}// http://programmingnotes.org/ ```

Once you understand the process of converting from infix to postfix, adding the ability to evaluate multiple digits within this program should be doable.

======= POSTFIX EVALUATION =======

This program uses a custom template.h class. To obtain the code for that class, click here.

QUICK NOTES:
The highlighted lines are sections of interest to look out for.

Want to convert & evaluate multi digit, decimal, and negative numbers? Click here!

The code is heavily commented, so no further insight is necessary. If you have any questions, feel free to leave a comment below.

Once compiled, you should get this as your output
(Note: the code was compile three separate times to display different output)

`====== RUN 1 ======`

``` ==== Postfix Evaluation ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Postfix Equation: 45^14*232+\$2-/12%24*/* Please enter a postfix expression: 1 2 + 5 6 + / The postfix expression = 1 2 + 5 6 + / Calculations: 1+2 = 3 5+6 = 11 3/11 = 0.272727 Final answer = 0.272727 ====== RUN 2 ====== ==== Postfix Evaluation ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Postfix Equation: 45^14*232+\$2-/12%24*/* Please enter a postfix expression: 35*76^+ The postfix expression = 35*76^+ Calculations: 3*5 = 15 7^6 = 117649 15+117649 = 117664 Final answer = 117664 ====== RUN 3 ====== ==== Postfix Evaluation ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Postfix Equation: 45^14*232+\$2-/12%24*/* Please enter a postfix expression: 45^4*32+\$2-/12%24*/* The postfix expression = 45^4*32+\$2-/12%24*/* ```

```Calculations: 4^5 = 1024 1024*4 = 4096 3+2 = 5 √5 = 2.23607 2.23607-2 = 0.236068 4096/0.236068 = 17350.9 1%2 = 1 2*4 = 8 1/8 = 0.125 17350.9*0.125 = 2168.87 Final answer = 2168.87```

## C++ || Stack Based Infix To Postfix Conversion (Single Digit)

This page consists of another homework assignment which was presented in a C++ Data Structures course. No matter which institution you attend, it seems every instructor assigns a program similar to this at one time or another.

Want to convert & evaluate multi digit, decimal, and negative numbers? Click here!

REQUIRED KNOWLEDGE FOR THIS PROGRAM

```What Is Infix? What Is Postfix? Stack Data Structure Cin.getline How To Convert To Postfix The Order Of Operations #include "ClassStackType.h"```

The program demonstrated on this page has the ability to convert a normal infix equation to postfix equation, so for example, if the user enters the infix equation of (1*2)+3, the program will display the postfix result of 12*3+.

Before we get into things, here is a helpful algorithm for converting from infix to postfix in pseudo code:

======= INFIX TO POSTFIX CONVERSION =======

This program uses a custom template.h class. To obtain the code for that class, click here.

``` Infix To Postfix Conversion (Single Digit) C++ // ============================================================================ // Author: K Perkins // Date: Mar 23, 2012 // Taken From: http://programmingnotes.org/ // File: PostfixConversion.cpp // Description: Demonstrate the use of a stack based infix to // postfix conversion. // ============================================================================ #include <iostream> #include <cctype> #include <cstdlib> #include <cstring> #include "ClassStackType.h" using namespace std; // function prototypes void DisplayDirections(); void ConvertInfixToPostfix(char* infix); int OrderOfOperations(char token); bool IsMathOperator(char token); int main() { // declare variables char expression[50]; // array holding the infix data // display directions to user DisplayDirections(); // get data from user cout<<"\nPlease enter an infix expression: "; cin.getline(expression, sizeof(expression)); cout <<"\nThe Infix expression = "<<expression<<endl; ConvertInfixToPostfix(expression); cout<<"The Postfix expression = "<<expression<<endl; return 0; }// end of main void DisplayDirections() { cout << "\n==== Infix to Postfix Conversion ====\n" <<"\nMath Operators:\n" <<"+ || Addition\n" <<"- || Subtraction\n" <<"* || Multiplication\n" <<"/ || Division\n" <<"% || Modulus\n" <<"^ || Power\n" <<"\$ || Square Root\n" <<"Sample Infix Equation: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4)\n"; }// end of DisplayDirections void ConvertInfixToPostfix(char* infix) { // declare function variables int infixCounter = 0; int postfixCounter = 0; char token = 'a'; char postfix[50]; StackType<char> charStack; // loop thru array until there is no more data while(infix[infixCounter] != '\0') { // push numbers/letters onto 'postfix' array if(isdigit(infix[infixCounter]) || isalpha(infix[infixCounter])) { postfix[postfixCounter] = infix[infixCounter]; ++postfixCounter; } else if(isspace(infix[infixCounter])) { // DO NOTHING } else if(IsMathOperator(infix[infixCounter])) { // if stack is empty, place first math operator onto stack token = infix[infixCounter]; if(charStack.IsEmpty()) { charStack.Push(token); } else { // get the current math operator from the top of the stack token = charStack.Top(); charStack.Pop(); // use the 'OrderOfOperations' function to check equality // of the math operators while(OrderOfOperations(token) >= OrderOfOperations(infix[infixCounter])) { // if stack is empty, do nothing if(charStack.IsEmpty()) { break; } // place the popped math operator from above ^ // onto the postfix array else { postfix[postfixCounter] = token; ++postfixCounter; // pop the next operator from the stack and // continue the process until complete token = charStack.Top(); charStack.Pop(); } } // push any remainding math operators onto the stack charStack.Push(token); charStack.Push(infix[infixCounter]); } } // push outer parentheses onto stack else if(infix[infixCounter] == '(') { charStack.Push(infix[infixCounter]); } else if(infix[infixCounter] == ')') { // pop the current math operator from the stack token = charStack.Top(); charStack.Pop(); while(token != '(' && !charStack.IsEmpty()) { // place the math operator onto the postfix array postfix[postfixCounter] = token; ++postfixCounter; // pop the next operator from the stack and // continue the process until complete token = charStack.Top(); charStack.Pop(); } } else { cout<<"\nINVALID INPUT\n"; exit(1); } ++infixCounter; } // place any remaining math operators from the stack onto // the postfix array while(!charStack.IsEmpty()) { postfix[postfixCounter] = charStack.Top(); ++postfixCounter; charStack.Pop(); } postfix[postfixCounter] = '\0'; // copy the data from the postfix array into the infix array // the data in the infix array gets sent back to main // since the array is passed by reference strcpy(infix,postfix); }// end of ConvertInfixToPostfix int OrderOfOperations(char token) {// this function checks priority of each math operator int priority = 0; if(token == '^'|| token == '\$') { priority = 4; } else if(token == '*' || token == '/' || token == '%') { priority = 3; } else if(token == '-') { priority = 2; } else if(token == '+') { priority = 1; } return priority; }// end of OrderOfOperations bool IsMathOperator(char token) {// this function checks if operand is a math operator switch(token) { case '+': return true; break; case '-': return true; break; case '*': return true; break; case '/': return true; break; case '%': return true; break; case '^': return true; break; case '\$': return true; break; default: return false; break; } }// http://programmingnotes.org/ 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216 // ============================================================================//     Author: K Perkins//     Date:   Mar 23, 2012//     Taken From: http://programmingnotes.org///     File:  PostfixConversion.cpp//     Description: Demonstrate the use of a stack based infix to //         postfix conversion.// ============================================================================#include <iostream>#include <cctype>#include <cstdlib>#include <cstring>#include "ClassStackType.h"using namespace std; // function prototypesvoid DisplayDirections();void ConvertInfixToPostfix(char* infix);int OrderOfOperations(char token);bool IsMathOperator(char token); int main(){ // declare variables char expression[50]; // array holding the infix data // display directions to user DisplayDirections(); // get data from user cout<<"\nPlease enter an infix expression: "; cin.getline(expression, sizeof(expression)); cout <<"\nThe Infix expression = "<<expression<<endl; ConvertInfixToPostfix(expression); cout<<"The Postfix expression = "<<expression<<endl; return 0;}// end of main void DisplayDirections(){ cout << "\n==== Infix to Postfix Conversion ====\n" <<"\nMath Operators:\n" <<"+ || Addition\n" <<"- || Subtraction\n" <<"* || Multiplication\n" <<"/ || Division\n" <<"% || Modulus\n" <<"^ || Power\n" <<"\$ || Square Root\n" <<"Sample Infix Equation: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4)\n";}// end of DisplayDirections void ConvertInfixToPostfix(char* infix){ // declare function variables int infixCounter = 0; int postfixCounter = 0; char token = 'a'; char postfix[50]; StackType<char> charStack; // loop thru array until there is no more data while(infix[infixCounter] != '\0') { // push numbers/letters onto 'postfix' array if(isdigit(infix[infixCounter]) || isalpha(infix[infixCounter])) { postfix[postfixCounter] = infix[infixCounter]; ++postfixCounter; } else if(isspace(infix[infixCounter])) { // DO NOTHING } else if(IsMathOperator(infix[infixCounter])) { // if stack is empty, place first math operator onto stack token = infix[infixCounter]; if(charStack.IsEmpty()) { charStack.Push(token); } else { // get the current math operator from the top of the stack token = charStack.Top(); charStack.Pop(); // use the 'OrderOfOperations' function to check equality // of the math operators while(OrderOfOperations(token) >= OrderOfOperations(infix[infixCounter])) { // if stack is empty, do nothing if(charStack.IsEmpty()) { break; } // place the popped math operator from above ^ // onto the postfix array else { postfix[postfixCounter] = token; ++postfixCounter; // pop the next operator from the stack and // continue the process until complete token = charStack.Top(); charStack.Pop(); } } // push any remainding math operators onto the stack charStack.Push(token); charStack.Push(infix[infixCounter]); } } // push outer parentheses onto stack else if(infix[infixCounter] == '(') { charStack.Push(infix[infixCounter]); } else if(infix[infixCounter] == ')') { // pop the current math operator from the stack token = charStack.Top(); charStack.Pop(); while(token != '(' && !charStack.IsEmpty()) { // place the math operator onto the postfix array postfix[postfixCounter] = token; ++postfixCounter; // pop the next operator from the stack and // continue the process until complete token = charStack.Top(); charStack.Pop(); } } else { cout<<"\nINVALID INPUT\n"; exit(1); } ++infixCounter; } // place any remaining math operators from the stack onto // the postfix array while(!charStack.IsEmpty()) { postfix[postfixCounter] = charStack.Top(); ++postfixCounter; charStack.Pop(); } postfix[postfixCounter] = '\0'; // copy the data from the postfix array into the infix array // the data in the infix array gets sent back to main // since the array is passed by reference strcpy(infix,postfix);}// end of ConvertInfixToPostfix int OrderOfOperations(char token){// this function checks priority of each math operator int priority = 0; if(token == '^'|| token == '\$') { priority = 4; } else if(token == '*' || token == '/' || token == '%') { priority = 3; } else if(token == '-') { priority = 2; } else if(token == '+') { priority = 1; } return priority; }// end of OrderOfOperations bool IsMathOperator(char token){// this function checks if operand is a math operator     switch(token)     { case '+': return true; break; case '-': return true; break; case '*': return true; break; case '/': return true; break; case '%': return true; break; case '^': return true; break; case '\$': return true; break;   default: return false; break; }}// http://programmingnotes.org/ ```

QUICK NOTES:
The highlighted lines are sections of interest to look out for.

Want to convert & evaluate multi digit, decimal, and negative numbers? Click here!

The code is heavily commented, so no further insight is necessary. If you have any questions, feel free to leave a comment below.

`====== RUN 1 ======`
``` ==== Infix to Postfix Conversion ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Infix Equation: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4) Please enter an infix expression: ((a+b)+c)/(d^e) The Infix expression = ((a+b)+c)/(d^e) The Postfix expression = ab+c+de^/ ====== RUN 2 ====== ==== Infix to Postfix Conversion ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Infix Equation: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4) Please enter an infix expression: (3*5)+(7^6) The Infix expression = (3*5)+(7^6) The Postfix expression = 35*76^+ ====== RUN 3 ====== ==== Infix to Postfix Conversion ==== Math Operators: + || Addition - || Subtraction * || Multiplication / || Division % || Modulus ^ || Power \$ || Square Root Sample Infix Equation: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4) ```
```Please enter an infix expression: (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4) The Infix expression = (((4^5)*14)/(\$(23+2)-2))*(1%2)/(2*4) The Postfix expression = 45^14*232+\$2-/12%24*/*```