Definitions
 

Infix: The operator is placed between the two operands: 3 + 5

Prefix: The operator is placed before the two operands: + 3 5

Postfix: The operator is placed after the two operands: 3 5 +

 

Calculator algorithm

1. Convert infix to postfix
2. Use stack to evaluate postfix
3. Output top of stack (should be the only element)

Evaluate postfix algorithm

1. Given a sequence of tokens s
2. While s is not empty:
   1. Let t = next token.
   2. If t is an operand, push it;
   3. If t is an operator:
      1. put the top into rhs, pop it;
      2. put the top into lhs, pop it;
      3. calculate lhs t rhs;
      4. push the result
3. The top of the stack is the result.

Convert infix to postfix algorithm

1. Given a sequence of tokens s and a result r
2. While s is not empty:
   1. Let t = next token.
   2. If t is an operand, r = r + t;
   3. If t is an open parenthesis, push it.
   4. If t is a close parenthesis:
      1. while top <> open parenthesis
         1. r = r + top
         2. pop
      2. pop // removes the open parenthesis
   5. If t is an operator
      1. while not (top is of lower precedence than t OR t is right associative and top is of equal precedence)
         1. r = r + top
         2. pop
      2. push t
3. while stack not empty
1. r = r + top
2. pop