I'll provide the reasoning steps for generating this patch following the two-step process:

Step 1. Root Cause Analysis:

1. Input Analysis:
   - The function takes a `size` parameter of type `uint64_t`.
   - `size` is rounded up to the nearest multiple of PAGE_SIZE.

2. Vulnerability Identification:
   - The vulnerability occurs when calling `i915_gem_alloc_object(dev, size)`.
   - If `size` is 0 after rounding up, it could lead to an out-of-bounds write (CWE-787).

3. Potential Issues:
   - If `size` is 0 or very close to 0, rounding up might still result in 0.
   - Allocating an object with size 0 could cause undefined behavior or memory corruption.

4. Missing Checks:
   - There's no validation to ensure `size` is non-zero after rounding up.

Conclusion:
The root cause is the lack of a check for a zero-sized allocation after rounding up. This could lead to an out-of-bounds write if the allocation function doesn't handle zero-sized requests properly.

Step 2. Fixing Strategy:

1. Add a check for zero size:
   - After rounding up, verify that `size` is not zero.
   - If `size` is zero, return an error code to prevent the allocation.

2. Error Handling:
   - Use an appropriate error code (EINVAL) to indicate invalid input.

3. Preserve existing behavior:
   - Keep the rounding up operation.
   - Only add the new check and error return.

4. Implement the fix:
   - Add an if statement to check if `size` is zero after rounding up.
   - Return -EINVAL if `size` is zero.
   - Proceed with allocation only if `size` is non-zero.

Resulting Patch:
```
<     size = roundup(size, PAGE_SIZE);
<     obj = i915_gem_alloc_object(dev, size);
---
>     size = roundup(size, PAGE_SIZE);
>     if (size == 0)
>     {
>         return -EINVAL;
>     }
>     obj = i915_gem_alloc_object(dev, size);
```

This patch addresses the vulnerability by ensuring that no zero-sized allocations are attempted, preventing potential out-of-bounds writes.