Arithmetica

An arithmetic interpreter that allows users to enter, change, combine, and evaluate integer arithmetic expressions interactively. The name arithmetica was suggested by former Duke Professor Michael Littman as a pun on the popular mathematics interpreter Mathematica.

An example session might look like the following (the user's input is in bold): 

Welcome to Arithmetica
1-> 3 + 7 * 2
17
2-> (3 + 7) * 2
20
3-> + 10
30
4-> * 10
300
5-> 1 - ans
-299
6-> a = 2 * 4 * 2
16
7-> b = 16
16
8-> c = a - b
0
9-> b = 2 * 16
32
10-> c
-16
11-> a = -b
-32
12-> c
-64
13-> z
0
14-> 1 - ans
300
15-> .

Expressions

Your program should recognize the following expressions:
expression
syntax
 semantics
examples
Constant
<any integer>
[0-9]+
 an integer constant
87
13
0
Variable
<any string>
[a-zA-z]+
 an expression represented by a word
a
bugs
total_squished
Assignment
var = expr
 assigns an expression to a variable
a = 87
bugs = 12 * a
a = b
Unary Operator
op expr
 prefixes an expression  
-   // negation
-13
-(a + b * c)
-bugs
Binary Operator
expr op expr
 combines two expressions into a single expression (in precedence order) 
%   // remainder
*   // times
/   // divide
+   // plus
-   // minus
a + b
a % 2
a + 10 * c
Unary Functions
fun(expr)
 a function that takes an expression as its single argument 
abs    // absolute value
sqrt   // square root
abs(a * b)
sqrt(x) - y ^ 2
Multi-Argument Functions
fun(expr,...)
 a function that takes one or more expressions as arguments (each separated by a comma) 
sum
average
mean
sum(a, 1, b + 1)
Parentheses
(expr)
 raises an expression's precedence
(a + b) * 3
sqrt(3*(bugs+t))

Operators have the following precedence from left to right (listed from highest to lowest):

() parentheses
- unary operators
*, /, % arithmetic operators
+, - arithmetic operators
= assignment

Extra Credit

Some suggested extensions follow (listed roughly in order from easiest to hardest):

BigInt allow expressions of integers with unbounded precision (implementation available here)
Right Associative Binary Operator
expr op expr
 combines two expressions into a single expression (in precedence order) 
^   // exponentiation
a ^ b
a ^ b ^ c
Postfix Unary Operator
expr op
  an expression  
!   // factorial
5!
(a + b)!
Comments?