[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data Dependence
   All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.

## 4. Example Code Snippet

### Example 1

```python
1    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.*;
2   import java.util.*;
3   public class Solution {
4       private static Scanner in;
5       public static double roundToDecimals(double value, int places) {
6           double scale = Math.pow(10, places);
7           return Math.round(value * scale) / scale;
8       }
9       public static void xuly(){
10          double a = in.nextDouble();
11          double[] centerX = {0.5,0,0};
12          double[] centerY = {0,0.5,0};
13          double[] centerZ = {0,0,0.5};
14          double[] x0 = {0.5, 0.5, 0.5, 0.5,-0.5,-0.5,-0.5,-0.5};
15          double[] y0 = {0.5, 0.5,-0.5,-0.5, 0.5, 0.5,-0.5,-0.5};
16          double[] z0 = {0.5,-0.5,-0.5, 0.5,-0.5, 0.5,-0.5, 0.5};
17          double i = 0;
18          boolean isFound = false;
19          while (i<=45){
20              isFound = false;
21              double rad = Math.toRadians(i);
22              double[] xr = new double[8];
23              double[] yr = new double[8];
24              double[] zr = new double[8];
25              for (int j = 0; j < 8; j++) {
26                  xr[j] = x0[j] * Math.cos(rad) - y0[j]* Math.sin(rad);
27                  yr[j] = x0[j] * Math.sin(rad) + y0[j]* Math.cos(rad);
28                  zr[j] = z0[j];
29              }
30              double x1 = 1,y1 = 1,x2 = -1,y2 = -1;
31              for (int j = 0; j < 8; j++) {
32                  x1 = Math.min(x1,xr[j]);
33                  x2 = Math.max(x2,xr[j]);
34                  y1 = Math.min(y1,zr[j]);
35                  y2 = Math.max(y2,zr[j]);
36              }
37              double dt = (x2-x1) * (y2-y1) ;
38              double diff = Math.abs(a - dt);
39              double[] cX = new double[3];
40              double[] cY = new double[3];
41              double[] cZ = new double[3];
42              for (int j = 0; j < 3; j++) {
43                  cX[j] = centerX[j] * Math.cos(rad) - centerY[j] * Math.sin(rad);
44                  cY[j] = centerX[j] * Math.sin(rad) + centerY[j] * Math.cos(rad);
45                  cZ[j] = centerZ[j];
46              }
47              if (diff <= 0.000001) {
48                  for (int j = 0; j < 3; j++) {
49                      System.out.println(cX[j] + " " + cY[j] + " " + cZ[j]);
50                  }
51                  break;
52              }
53              else{
54                  double[] xr0 = new double[8];
55                  double[] yr0 = new double[8];
56                  double[] zr0 = new double[8];
57                  for (int j = 0; j < 8; j++) {
58                      xr0[j] = xr[j];
59                      yr0[j] = yr[j];
60                      zr0[j] = zr[j];
61                  }
62                  double ii = 0;
63                  while (ii<=45) {
64                      isFound = false;
65                      double rad0 = Math.toRadians(ii);
66                      for (int j = 0; j < 8; j++) {
67                          xr[j] = xr0[j];
68                          zr[j] = yr0[j] * Math.sin(rad0) + zr0[j] * Math.cos(rad0);
69                      }
70                      x1 = 1;
71                      y1 = 1;
72                      x2 = -1;
73                      y2 = -1;
74                      for (int j = 0; j < 8; j++) {
75                          x1 = Math.min(x1, xr[j]);
76                          x2 = Math.max(x2, xr[j]);
77                          y1 = Math.min(y1, zr[j]);
78                          y2 = Math.max(y2, zr[j]);
79                      }
80                      dt = (x2 - x1) * (y2 - y1);
81                      diff = Math.abs(a - dt);
82                      if (diff <= 0.000001) {
83                          double[] cXn = new double[3];
84                          double[] cYn = new double[3];
85                          double[] cZn = new double[3];
86                          for (int j = 0; j < 3; j++) {
87                              cXn[j] = cX[j];
88                              cYn[j] = cY[j] * Math.cos(rad0) - cZ[j] * Math.sin(rad0);
89                              cZn[j] = cY[j] * Math.sin(rad0) + cZ[j] * Math.cos(rad0);
90                              System.out.println(cXn[j] + " " + cYn[j] + " " + cZn[j]);
91                          }
92                          isFound = true;
93                          break;
94                      }
95                      ii=roundToDecimals(ii+0.1,6);
96                  }
97              }
98              if (isFound) break;
99              i=roundToDecimals(i+0.1,6);
100          }
101          System.out.flush();
102      }
103      public static void main(String[] args) throws IOException {
104          in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
105          int numCases = in.nextInt();
106          for (int test = 1; test <= numCases; test++) {
107              System.out.print("Case #" + test + ": ");
108              xuly();
109          }
110      }
111  }
```


**Question**: Does `(rad,21)` have data dependence over `(x2,33)` in function `xuly`? If so, provide a trace.

**Output**:


Ground truth answer: yes