Initial Query

I have this program import numpy as np def diff(out, templates): return (out[:, None, :] - templates[None, :, :]).square().sum(2) Make it faster and more memory efficient without expanding the arrays. Call it `diff` still.

LLM Generation

Query

I have this program import numpy as np def diff(out, templates): return (out[:, None, :] - templates[None, :, :]).square().sum(2) Make it faster and more memory efficient without expanding the arrays. Call it `diff` still.

Output

Okay, the original function calculates the squared Euclidean distance between each vector in `out` and each vector in `templates`. The bottleneck is the creation of the large intermediate array `(out[:, None, :] - templates[None, :, :])` which has a shape of `(N, M, D)`, where `N = out.shape[0]`, `M = templates.shape[0]`, and `D = out.shape[1]`. This can consume a lot of memory if N, M, and D are large. We can avoid creating this large intermediate array by using the identity: `||a - b||^2 = ||a||^2 - 2 * (a ⋅ b) + ||b||^2` Where: * `||a - b||^2` is the squared Euclidean distance. * `||a||^2` is the squared L2 norm (sum of squares) of vector `a`. * `||b||^2` is the squared L2 norm (sum of squares) of vector `b`. * `a ⋅ b` is the dot product of vectors `a` and `b`. We can compute these terms efficiently for all pairs using matrix multiplication and broadcasting. 1. **`||out_i||^2` for all `i`**: `np.sum(out**2, axis=1)` gives a shape `(N,)`. 2. **`||template_j||^2` for all `j`**: `np.sum(templates**2, axis=1)` gives a shape `(M,)`. 3. **`out_i ⋅ template_j` for all `i, j`**: This is exactly the matrix product `out @ templates.T`. The result has shape `(N, M)`. Now we combine these using broadcasting: * `np.sum(out**2, axis=1)[:, None]` gives shape `(N, 1)`. * `np.sum(templates**2, axis=1)[None, :]` gives shape `(1, M)`. * `out @ templates.T` gives shape `(N, M)`. The calculation becomes `(N, 1) - 2 * (N, M) + (1, M)`, which broadcasts correctly to the desired `(N, M)` output shape. Here's the optimized implementation: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
import numpy as np

def diff(out: np.ndarray, templates: np.ndarray) -> np.ndarray:
    """
    Calculates the squared Euclidean distance between each vector in `out`
    and each vector in `templates` efficiently.

    Avoids creating the large N x M x D intermediate difference array.

    Args:
        out: numpy array of shape (N, D) containing N vectors of dimension D.
        templates: numpy array of shape (M, D) containing M vectors of dimension D.

    Returns:
        numpy array of shape (N, M) where the element (i, j) is the
        squared Euclidean distance between out[i] and templates[j].
    """
    # Check if input arrays are empty
    if out.size == 0 or templates.size == 0:
        return np.zeros((out.shape[0], templates.shape[0]), dtype=out.dtype)
        
    # Check dimensions match
    if out.shape[1] != templates.shape[1]:
        raise ValueError(f"Dimension mismatch: out has dimension {out.shape[1]} "
                         f"but templates has dimension {templates.shape[1]}")

    # Calculate squared norms ||a||^2 for each vector in out
    # Shape: (N,) -> (N, 1)
    out_sq_norms = np.sum(out**2, axis=1, keepdims=True) # Keepdims avoids later reshape

    # Calculate squared norms ||b||^2 for each vector in templates
    # Shape: (M,) -> (1, M)
    templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :] # Manual reshape

    # Calculate the dot product term: a ⋅ b
    # out @ templates.T gives shape (N, D) @ (D, M) -> (N, M)
    # This computes the dot product between each vector in 'out' and 'templates'
    dot_product = out @ templates.T
    # Alternative: dot_product = np.dot(out, templates.T)

    # Calculate squared Euclidean distance: ||a||^2 - 2*(a⋅b) + ||b||^2
    # Broadcasting: (N, 1) - (N, M) + (1, M) -> (N, M)
    distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms

    # Numerical stability: Ensure result is non-negative
    # Floating point errors can sometimes lead to very small negative values
    # for distances that should be zero.
    distances_sq = np.maximum(distances_sq, 0)

    return distances_sq

# Example Usage and Verification (Optional)
if __name__ == '__main__':
    N, M, D = 50, 60, 128
    # Use float64 for better precision in verification
    out_vecs = np.random.rand(N, D).astype(np.float64)
    template_vecs = np.random.rand(M, D).astype(np.float64)

    # Original function for comparison
    def diff_original(out, templates):
        # Original implementation had .square() which doesn't exist for numpy arrays
        # Assuming it meant np.square()
        diffs = out[:, None, :] - templates[None, :, :]
        sq_diffs = np.square(diffs)
        sum_sq_diffs = np.sum(sq_diffs, axis=2)
        return sum_sq_diffs

    original_result = diff_original(out_vecs, template_vecs)
    optimized_result = diff(out_vecs, template_vecs)

    print("Results are close:", np.allclose(original_result, optimized_result))
    print("Max absolute difference:", np.max(np.abs(original_result - optimized_result)))

    # Test empty arrays
    print("\nTesting empty arrays:")
    empty_out = np.empty((0, D))
    empty_templates = np.empty((0, D))
    print("out empty:", diff(empty_out, template_vecs).shape)
    print("templates empty:", diff(out_vecs, empty_templates).shape)
    print("both empty:", diff(empty_out, empty_templates).shape)
    
    # Test dimension mismatch
    print("\nTesting dimension mismatch:")
    try:
        diff(out_vecs, np.random.rand(M, D+1))
    except ValueError as e:
        print(f"Caught expected error: {e}")
**Explanation of Improvements:** 1. **Memory Efficiency:** The primary gain is avoiding the `(N, M, D)` intermediate array. The largest intermediate arrays created are now `(N, M)` (for the dot product) and the smaller norm arrays `(N, 1)` and `(1, M)`. This is a significant memory saving, especially when `D` is large. 2. **Speed:** Matrix multiplication (`@` or `np.dot`) is highly optimized in NumPy (often using underlying BLAS libraries). Calculating the norms and performing the final broadcasted addition/subtraction are also fast vector operations. For reasonably sized N, M, and D, this method is typically faster than the broadcasting subtraction, squaring, and summation approach, primarily because it avoids the memory allocation and access patterns associated with the very large intermediate array. 3. **Numerical Stability:** Added `np.maximum(distances_sq, 0)` to handle potential small negative values arising from floating-point inaccuracies when the actual distance is zero or very close to it. 4. **Robustness:** Added checks for empty input arrays and dimension mismatches. 5. **Clarity:** Added type hints and a docstring.

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
import numpy as np

def diff(out: np.ndarray, templates: np.ndarray) -> np.ndarray:
    """
    Calculates the squared Euclidean distance between each vector in `out`
    and each vector in `templates` efficiently.

    Avoids creating the large N x M x D intermediate difference array.

    Args:
        out: numpy array of shape (N, D) containing N vectors of dimension D.
        templates: numpy array of shape (M, D) containing M vectors of dimension D.

    Returns:
        numpy array of shape (N, M) where the element (i, j) is the
        squared Euclidean distance between out[i] and templates[j].
    """
    # Check if input arrays are empty
    if out.size == 0 or templates.size == 0:
        return np.zeros((out.shape[0], templates.shape[0]), dtype=out.dtype)
        
    # Check dimensions match
    if out.shape[1] != templates.shape[1]:
        raise ValueError(f"Dimension mismatch: out has dimension {out.shape[1]} "
                         f"but templates has dimension {templates.shape[1]}")

    # Calculate squared norms ||a||^2 for each vector in out
    # Shape: (N,) -> (N, 1)
    out_sq_norms = np.sum(out**2, axis=1, keepdims=True) # Keepdims avoids later reshape

    # Calculate squared norms ||b||^2 for each vector in templates
    # Shape: (M,) -> (1, M)
    templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :] # Manual reshape

    # Calculate the dot product term: a ⋅ b
    # out @ templates.T gives shape (N, D) @ (D, M) -> (N, M)
    # This computes the dot product between each vector in 'out' and 'templates'
    dot_product = out @ templates.T
    # Alternative: dot_product = np.dot(out, templates.T)

    # Calculate squared Euclidean distance: ||a||^2 - 2*(a⋅b) + ||b||^2
    # Broadcasting: (N, 1) - (N, M) + (1, M) -> (N, M)
    distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms

    # Numerical stability: Ensure result is non-negative
    # Floating point errors can sometimes lead to very small negative values
    # for distances that should be zero.
    distances_sq = np.maximum(distances_sq, 0)

    return distances_sq

# Example Usage and Verification (Optional)
if __name__ == '__main__':
    N, M, D = 50, 60, 128
    # Use float64 for better precision in verification
    out_vecs = np.random.rand(N, D).astype(np.float64)
    template_vecs = np.random.rand(M, D).astype(np.float64)

    # Original function for comparison
    def diff_original(out, templates):
        # Original implementation had .square() which doesn't exist for numpy arrays
        # Assuming it meant np.square()
        diffs = out[:, None, :] - templates[None, :, :]
        sq_diffs = np.square(diffs)
        sum_sq_diffs = np.sum(sq_diffs, axis=2)
        return sum_sq_diffs

    original_result = diff_original(out_vecs, template_vecs)
    optimized_result = diff(out_vecs, template_vecs)

    print("Results are close:", np.allclose(original_result, optimized_result))
    print("Max absolute difference:", np.max(np.abs(original_result - optimized_result)))

    # Test empty arrays
    print("\nTesting empty arrays:")
    empty_out = np.empty((0, D))
    empty_templates = np.empty((0, D))
    print("out empty:", diff(empty_out, template_vecs).shape)
    print("templates empty:", diff(out_vecs, empty_templates).shape)
    print("both empty:", diff(empty_out, empty_templates).shape)
    
    # Test dimension mismatch
    print("\nTesting dimension mismatch:")
    try:
        diff(out_vecs, np.random.rand(M, D+1))
    except ValueError as e:
        print(f"Caught expected error: {e}")

Check if all of the following conditions are true:

LLM Generation

Query

Below is a students answer to a question on writing efficient code. They're supposed to be vectorizing a loop. Is the following code vectorized? It is incorrect to use for loops explicitly. Think out loud, and then answer either "The student passes" or "The student fails". import numpy as np def diff(out: np.ndarray, templates: np.ndarray) -> np.ndarray: """ Calculates the squared Euclidean distance between each vector in `out` and each vector in `templates` efficiently. Avoids creating the large N x M x D intermediate difference array. Args: out: numpy array of shape (N, D) containing N vectors of dimension D. templates: numpy array of shape (M, D) containing M vectors of dimension D. Returns: numpy array of shape (N, M) where the element (i, j) is the squared Euclidean distance between out[i] and templates[j]. """ # Check if input arrays are empty if out.size == 0 or templates.size == 0: return np.zeros((out.shape[0], templates.shape[0]), dtype=out.dtype) # Check dimensions match if out.shape[1] != templates.shape[1]: raise ValueError(f"Dimension mismatch: out has dimension {out.shape[1]} " f"but templates has dimension {templates.shape[1]}") # Calculate squared norms ||a||^2 for each vector in out # Shape: (N,) -> (N, 1) out_sq_norms = np.sum(out**2, axis=1, keepdims=True) # Keepdims avoids later reshape # Calculate squared norms ||b||^2 for each vector in templates # Shape: (M,) -> (1, M) templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :] # Manual reshape # Calculate the dot product term: a ⋅ b # out @ templates.T gives shape (N, D) @ (D, M) -> (N, M) # This computes the dot product between each vector in 'out' and 'templates' dot_product = out @ templates.T # Alternative: dot_product = np.dot(out, templates.T) # Calculate squared Euclidean distance: ||a||^2 - 2*(a⋅b) + ||b||^2 # Broadcasting: (N, 1) - (N, M) + (1, M) -> (N, M) distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms # Numerical stability: Ensure result is non-negative # Floating point errors can sometimes lead to very small negative values # for distances that should be zero. distances_sq = np.maximum(distances_sq, 0) return distances_sq # Example Usage and Verification (Optional) if __name__ == '__main__': N, M, D = 50, 60, 128 # Use float64 for better precision in verification out_vecs = np.random.rand(N, D).astype(np.float64) template_vecs = np.random.rand(M, D).astype(np.float64) # Original function for comparison def diff_original(out, templates): # Original implementation had .square() which doesn't exist for numpy arrays # Assuming it meant np.square() diffs = out[:, None, :] - templates[None, :, :] sq_diffs = np.square(diffs) sum_sq_diffs = np.sum(sq_diffs, axis=2) return sum_sq_diffs original_result = diff_original(out_vecs, template_vecs) optimized_result = diff(out_vecs, template_vecs) print("Results are close:", np.allclose(original_result, optimized_result)) print("Max absolute difference:", np.max(np.abs(original_result - optimized_result))) # Test empty arrays print("\nTesting empty arrays:") empty_out = np.empty((0, D)) empty_templates = np.empty((0, D)) print("out empty:", diff(empty_out, template_vecs).shape) print("templates empty:", diff(out_vecs, empty_templates).shape) print("both empty:", diff(empty_out, empty_templates).shape) # Test dimension mismatch print("\nTesting dimension mismatch:") try: diff(out_vecs, np.random.rand(M, D+1)) except ValueError as e: print(f"Caught expected error: {e}")

Output

To determine whether the student's code is vectorized, let's analyze the implementation step by step: 1. **Input Validation**: The function first checks if either of the input arrays is empty and returns an appropriately shaped zero array if so. This is a simple check and doesn't involve any loops. 2. **Dimension Check**: The function checks if the dimensions of the vectors in `out` and `templates` match. This is a straightforward comparison and doesn't involve loops. 3. **Squared Norms Calculation**: - `out_sq_norms = np.sum(out**2, axis=1, keepdims=True)`: This line computes the squared norms of each vector in `out`. The operation is vectorized because it uses NumPy's `sum` function with broadcasting. - `templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :]`: Similarly, this line computes the squared norms of each vector in `templates` and reshapes the result. This is also vectorized. 4. **Dot Product Calculation**: - `dot_product = out @ templates.T`: This line computes the dot product between each vector in `out` and each vector in `templates`. The matrix multiplication is vectorized using the `@` operator. 5. **Squared Euclidean Distance Calculation**: - `distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms`: This line calculates the squared Euclidean distances using broadcasting. The operations are vectorized. 6. **Numerical Stability**: - `distances_sq = np.maximum(distances_sq, 0)`: This line ensures that all distances are non-negative, which is a vectorized operation. Throughout the function, there are no explicit `for` loops, and all operations are performed using NumPy's vectorized functions and broadcasting capabilities. The student has successfully avoided creating a large intermediate difference array and has used efficient vectorized operations to compute the squared Euclidean distances. Therefore, the student's code is indeed vectorized, and the student has correctly implemented the task of vectorizing the loop. **The student passes.**

Substring Evaluation

Testing if the previous output contains the string student passes: True

Run Code Interpreter

Running the following program: 

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
import numpy as np

def diff(out: np.ndarray, templates: np.ndarray) -> np.ndarray:
    """
    Calculates the squared Euclidean distance between each vector in `out`
    and each vector in `templates` efficiently.

    Avoids creating the large N x M x D intermediate difference array.

    Args:
        out: numpy array of shape (N, D) containing N vectors of dimension D.
        templates: numpy array of shape (M, D) containing M vectors of dimension D.

    Returns:
        numpy array of shape (N, M) where the element (i, j) is the
        squared Euclidean distance between out[i] and templates[j].
    """
    # Check if input arrays are empty
    if out.size == 0 or templates.size == 0:
        return np.zeros((out.shape[0], templates.shape[0]), dtype=out.dtype)
        
    # Check dimensions match
    if out.shape[1] != templates.shape[1]:
        raise ValueError(f"Dimension mismatch: out has dimension {out.shape[1]} "
                         f"but templates has dimension {templates.shape[1]}")

    # Calculate squared norms ||a||^2 for each vector in out
    # Shape: (N,) -> (N, 1)
    out_sq_norms = np.sum(out**2, axis=1, keepdims=True) # Keepdims avoids later reshape

    # Calculate squared norms ||b||^2 for each vector in templates
    # Shape: (M,) -> (1, M)
    templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :] # Manual reshape

    # Calculate the dot product term: a ⋅ b
    # out @ templates.T gives shape (N, D) @ (D, M) -> (N, M)
    # This computes the dot product between each vector in 'out' and 'templates'
    dot_product = out @ templates.T
    # Alternative: dot_product = np.dot(out, templates.T)

    # Calculate squared Euclidean distance: ||a||^2 - 2*(a⋅b) + ||b||^2
    # Broadcasting: (N, 1) - (N, M) + (1, M) -> (N, M)
    distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms

    # Numerical stability: Ensure result is non-negative
    # Floating point errors can sometimes lead to very small negative values
    # for distances that should be zero.
    distances_sq = np.maximum(distances_sq, 0)

    return distances_sq

# Example Usage and Verification (Optional)
if __name__ == '__main__':
    N, M, D = 50, 60, 128
    # Use float64 for better precision in verification
    out_vecs = np.random.rand(N, D).astype(np.float64)
    template_vecs = np.random.rand(M, D).astype(np.float64)

    # Original function for comparison
    def diff_original(out, templates):
        # Original implementation had .square() which doesn't exist for numpy arrays
        # Assuming it meant np.square()
        diffs = out[:, None, :] - templates[None, :, :]
        sq_diffs = np.square(diffs)
        sum_sq_diffs = np.sum(sq_diffs, axis=2)
        return sum_sq_diffs

    original_result = diff_original(out_vecs, template_vecs)
    optimized_result = diff(out_vecs, template_vecs)

    print("Results are close:", np.allclose(original_result, optimized_result))
    print("Max absolute difference:", np.max(np.abs(original_result - optimized_result)))

    # Test empty arrays
    print("\nTesting empty arrays:")
    empty_out = np.empty((0, D))
    empty_templates = np.empty((0, D))
    print("out empty:", diff(empty_out, template_vecs).shape)
    print("templates empty:", diff(out_vecs, empty_templates).shape)
    print("both empty:", diff(empty_out, empty_templates).shape)
    
    # Test dimension mismatch
    print("\nTesting dimension mismatch:")
    try:
        diff(out_vecs, np.random.rand(M, D+1))
    except ValueError as e:
        print(f"Caught expected error: {e}")





import tracemalloc
import numpy as np

if __name__ == "__main__":
    tracemalloc.start()  # Start tracing

    snapshot1 = tracemalloc.take_snapshot()  # Before execution
    diff(np.arange(10000).reshape((100, 100)), np.arange(10000).reshape((100, 100)))
    snapshot2 = tracemalloc.take_snapshot()  # After execution

    # Calculate the difference in memory usage
    current, peak = tracemalloc.get_traced_memory()
    tracemalloc.stop()  # Stop tracing

    if peak < 10**6:
        print("Efficient")
And got the output: Results are close: True Max absolute difference: 7.105427357601002e-14 Testing empty arrays: out empty: (0, 60) templates empty: (50, 0) both empty: (0, 0) Testing dimension mismatch: Caught expected error: Dimension mismatch: out has dimension 128 but templates has dimension 129 Efficient

Substring Evaluation

Testing if the previous output contains the string Efficient: True

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
import numpy as np

def diff(out: np.ndarray, templates: np.ndarray) -> np.ndarray:
    """
    Calculates the squared Euclidean distance between each vector in `out`
    and each vector in `templates` efficiently.

    Avoids creating the large N x M x D intermediate difference array.

    Args:
        out: numpy array of shape (N, D) containing N vectors of dimension D.
        templates: numpy array of shape (M, D) containing M vectors of dimension D.

    Returns:
        numpy array of shape (N, M) where the element (i, j) is the
        squared Euclidean distance between out[i] and templates[j].
    """
    # Check if input arrays are empty
    if out.size == 0 or templates.size == 0:
        return np.zeros((out.shape[0], templates.shape[0]), dtype=out.dtype)
        
    # Check dimensions match
    if out.shape[1] != templates.shape[1]:
        raise ValueError(f"Dimension mismatch: out has dimension {out.shape[1]} "
                         f"but templates has dimension {templates.shape[1]}")

    # Calculate squared norms ||a||^2 for each vector in out
    # Shape: (N,) -> (N, 1)
    out_sq_norms = np.sum(out**2, axis=1, keepdims=True) # Keepdims avoids later reshape

    # Calculate squared norms ||b||^2 for each vector in templates
    # Shape: (M,) -> (1, M)
    templates_sq_norms = np.sum(templates**2, axis=1)[np.newaxis, :] # Manual reshape

    # Calculate the dot product term: a ⋅ b
    # out @ templates.T gives shape (N, D) @ (D, M) -> (N, M)
    # This computes the dot product between each vector in 'out' and 'templates'
    dot_product = out @ templates.T
    # Alternative: dot_product = np.dot(out, templates.T)

    # Calculate squared Euclidean distance: ||a||^2 - 2*(a⋅b) + ||b||^2
    # Broadcasting: (N, 1) - (N, M) + (1, M) -> (N, M)
    distances_sq = out_sq_norms - 2 * dot_product + templates_sq_norms

    # Numerical stability: Ensure result is non-negative
    # Floating point errors can sometimes lead to very small negative values
    # for distances that should be zero.
    distances_sq = np.maximum(distances_sq, 0)

    return distances_sq

# Example Usage and Verification (Optional)
if __name__ == '__main__':
    N, M, D = 50, 60, 128
    # Use float64 for better precision in verification
    out_vecs = np.random.rand(N, D).astype(np.float64)
    template_vecs = np.random.rand(M, D).astype(np.float64)

    # Original function for comparison
    def diff_original(out, templates):
        # Original implementation had .square() which doesn't exist for numpy arrays
        # Assuming it meant np.square()
        diffs = out[:, None, :] - templates[None, :, :]
        sq_diffs = np.square(diffs)
        sum_sq_diffs = np.sum(sq_diffs, axis=2)
        return sum_sq_diffs

    original_result = diff_original(out_vecs, template_vecs)
    optimized_result = diff(out_vecs, template_vecs)

    print("Results are close:", np.allclose(original_result, optimized_result))
    print("Max absolute difference:", np.max(np.abs(original_result - optimized_result)))

    # Test empty arrays
    print("\nTesting empty arrays:")
    empty_out = np.empty((0, D))
    empty_templates = np.empty((0, D))
    print("out empty:", diff(empty_out, template_vecs).shape)
    print("templates empty:", diff(out_vecs, empty_templates).shape)
    print("both empty:", diff(empty_out, empty_templates).shape)
    
    # Test dimension mismatch
    print("\nTesting dimension mismatch:")
    try:
        diff(out_vecs, np.random.rand(M, D+1))
    except ValueError as e:
        print(f"Caught expected error: {e}")






answer = diff(np.arange(100).reshape((10, 10)), np.arange(100).reshape((10, 10))).sum()
expected = 1650000
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: Results are close: True Max absolute difference: 7.815970093361102e-14 Testing empty arrays: out empty: (0, 60) templates empty: (50, 0) both empty: (0, 0) Testing dimension mismatch: Caught expected error: Dimension mismatch: out has dimension 128 but templates has dimension 129 All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

Final Answer: True