19033번 - Exact Arithmetic 다국어

Let $X$ be the set of all rational numbers and all numbers of form $q \sqrt{r}$, where $q$ is a non-zero rational number and $r$ is an integer greater than $1$. Here, $r$ must not have a square number except for $1$ as its divisor. Also, let $X^{*}$ be the set of all numbers which can be expressed as a sum of one or more elements in $X$.

Consider the machine $Y$ which is a stack-based calculator operating on the values in $X^{*}$ and has the instructions shown below.

• push $n$: pushes an integer specified in the operand onto the stack.
• add: pops two values $x_1$ and $x_2$ from the top of the stack in this order, then pushes $(x_2 + x_1)$.
• sub: pops two values $x_1$ and $x_2$ from the top of the stack in this order, then pushes $(x_2 - x_1)$.
• mul: pops two values $x_1$ and $x_2$ from the top of the stack in this order, then pushes $(x_2 \cdot x_1)$.
• div: pops two values $x_1$ and $x_2$ from the top of the stack in this order, then pushes $(x2 / x1)$. Here, $x_1$ must be a non-zero value from $X$ (not from $X^{*}$).
• sqrt: pops one value $x$ from the stack and pushes the square root of $x$. Here, $x$ must be a non-negative rational number.
• disp: pops one value $x$ from the stack, and outputs the string representation of the value $x$ to the display. The representation rules are stated later.
• stop: terminates calculation. The stack must be empty when this instruction is called.

Initially, the stack is empty. A sufficient number of values must exist in the stack on execution of every instruction. In addition, due to the limitation of the machine $Y$, no more values can be pushed when the stack already stores as many as $256$ values. Also, there exist several restrictions on values to be pushed onto the stack:

• For rational numbers, neither numerator nor denominator in the  irreducible form may exceed $32\,768$ in its absolute value.
• For any element in $X$ of the form $q \sqrt{r} = (a / b) \sqrt{r}$, the following must hold: $|a \sqrt{r}| \le 32\,768$ and $|b| \le 32\,768$.

For any element in $X^{*}$, each term in the sum must satisfy the above conditions.

The rules for the string representations of the values (on the machine $Y$) are as follows:

• A rational number is represented as either an integer or an irreducible fraction with a denominator greater than $1$.
• A fraction is represented as $\mathit{num}$/$\mathit{den}$. Here, $\mathit{num}$ is the numerator, and $\mathit{den}$ is the denominator. The sign character - precedes negative numbers.
• A number of the form $q \sqrt{r}$ is represented as $\mathit{rep}$*sqrt($r$) except for the case with $|q| = 1$, in which the number is represented as sqrt($r$) for $q = 1$ or -sqrt($r$) for $q = -1$. Here, $\mathit{rep}$ is the string representation of $q$.
• For a sum of two or more elements of $X$, string representations of all the (non-zero) elements are connected using the binary operator +. In this case, all terms with the same rooted number are merged into a single term, and the terms must be shown in the ascending order of their root component. For the purpose of this rule, all rational numbers are regarded to accompany $\sqrt{1}$. There is exactly one space character before and after each of the binary operators +. No space characters appear at any other place.

The followings are a few examples of valid string representations:

0
1
-1/10
2*sqrt(2) + 1/2*sqrt(3) + -1/2*sqrt(5)
1/2 + sqrt(10) + -sqrt(30)

Your task is to write a program that simulates the machine $Y$.

The input is a sequence of no more than 3000 instructions. Each line contains a single instruction. You may assume that every instruction is called in a legal way. The instruction "stop" appears only once, at the end of the entire input.

Output the strings which the machine $Y$ will display. Write each string on a separate line.