🦥 Unsloth: Will patch your computer to enable 2x faster free finetuning.
🦥 Unsloth Zoo will now patch everything to make training faster!
==((====))==  Unsloth 2025.10.4: Fast Gpt_Oss patching. Transformers: 4.56.2.
   \\   /|    Num GPUs = 1. Max memory: 79.318 GB. Platform: Linux.
O^O/ \_/ \    Torch: 2.8.0. Triton: 3.4.0
\        /    Bfloat16 = TRUE. FA [Xformers = None. FA2 = False]
 "-____-"     Free license: http://github.com/unslothai/unsloth
Unsloth: Fast downloading is enabled - ignore downloading bars which are red colored!
Unsloth: QLoRA and full finetuning all not selected. Switching to 16bit LoRA.

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Unsloth: Making `model.base_model.model.model` require gradients

[[-2.8313286   4.54613909 -7.95265309  6.53459836  2.87235103]
 [ 7.0739631   3.76278879  9.31565599 -8.52884711  9.96832952]
 [ 8.41214082  6.51136046 -3.79347975 -2.46773693 -2.32292989]
 [ 3.91302932  4.98335304 -5.33855089  5.71057634 -2.79871647]]
[[ 0.39218774 -9.6181377  -3.49736707]
 [-0.33354865 -1.05626139  3.87231208]
 [ 0.49494174  5.91863954 -6.83183693]
 [ 5.1465162  -7.51648113  1.00445384]
 [ 9.63213377 -4.92327556  3.323014  ]]
[[  54.73441488  -87.89725072   97.94605887]
 [  58.25238906   -1.8467447   -49.25453031]
 [ -35.82528794  -80.25394462   11.51225408]
 [  -0.33785799 -103.64132345   38.51974367]]

(7.105427357601002e-15, 4.6783406255758477e-29)

Only stdlib imports? False
{'stdlib': [], 'non_stdlib': ['numpy', 'torch'], 'relative_imports': 0}

<function matmul(A, B)>

Success
name 'np' is not defined

{'median_ns': 70895404,
, 'mean_ns': 70895404,
, 'stdev_ns': 0,
, 'exceptions': [],
, 'timeouts': 0}

Create a new fast matrix multiplication function using only native Python code.
You are given a list of list of numbers.
Output your new function in backticks using the format below:
```python
def matmul(A, B):
    return ...
```

<|start|>system<|message|>You are ChatGPT, a large language model trained by OpenAI.
Knowledge cutoff: 2024-06
Current date: 2025-10-17

Reasoning: low

# Valid channels: analysis, commentary, final. Channel must be included for every message.
Calls to these tools must go to the commentary channel: 'functions'.<|end|><|start|>user<|message|>Create a new fast matrix multiplication function using only native Python code.
You are given a list of list of numbers.
Output your new function in backticks using the format below:
```python
def matmul(A, B):
    return ...
```<|end|><|start|>assistant<|channel|>analysis<|message|>Need to implement efficient naive multiplication maybe using list comprehensions? Provide function.<|end|><|start|>assistant<|channel|>final<|message|>```python
def matmul(A, B):
    """
    Multiply two matrices A and B using only native Python code.

    Parameters
    ----------
    A : list[list[float]]
        Left matrix (m x n).
    B : list[list[float]]
        Right matrix (n x p).

    Returns
    -------
    list[list[float]]
        Product matrix (m x p).

    Notes
    -----
    The function assumes that the inner dimensions match (len(A[0]) == len(B)).
    It uses a straightforward triple loop but is written to maximize performance
    with Python's list comprehension and built‑in sum.
    """
    if not A or not B or not A[0] or not B[0]:
        raise ValueError("Matrices cannot be empty")

    m, n = len(A), len(A[0])
    n2, p = len(B), len(B[0])
    if n != n2:
        raise ValueError(f"Inner dimensions must agree: {n} vs {n2}")

    # Transpose B to improve cache locality
    Bt = list(zip(*B))   # (p x n) tuples

    # Compute the product
    return [
        [sum(a * b for a, b in zip(row, col)) for col in Bt]
        for row in A
    ]
```<|return|>

def matmul(A, B):
    return ...

(False,
, {'error': "SyntaxError: expected '(' (<unknown>, line 1)",
,  'stdlib': [],
,  'non_stdlib': [],
,  'relative_imports': 0})

np.float64(2.220446049250313e-16)

{'median_ns': 205566,
, 'mean_ns': 231173,
, 'stdev_ns': 39247,
, 'exceptions': [],
, 'timeouts': 0}

{'median_ns': 84237,
, 'mean_ns': 87442,
, 'stdev_ns': 4538,
, 'exceptions': [],
, 'timeouts': 0}

0.02440329071548132

3.333333333333333

49

{'prompt': [{'content': 'Create a new fast matrix multiplication function using only native Python code.\nYou are given a list of list of numbers.\nOutput your new function in backticks using the format below:\n```python\ndef matmul(A, B):\n    return ...\n```',
,   'role': 'user'}],
, 'answer': 0,
, 'reasoning_effort': 'low'}

Unsloth: We now expect `per_device_train_batch_size` to be a multiple of `num_generations`.
We will change the batch size of 1 to the `num_generations` of 2

The tokenizer has new PAD/BOS/EOS tokens that differ from the model config and generation config. The model config and generation config were aligned accordingly, being updated with the tokenizer's values. Updated tokens: {'bos_token_id': 199998}.
==((====))==  Unsloth - 2x faster free finetuning | Num GPUs used = 1
   \\   /|    Num examples = 1,000 | Num Epochs = 1 | Total steps = 100
O^O/ \_/ \    Batch size per device = 2 | Gradient accumulation steps = 1
\        /    Data Parallel GPUs = 1 | Total batch size (2 x 1 x 1) = 2
 "-____-"     Trainable parameters = 1,990,656 of 20,916,747,840 (0.01% trained)
`generation_config` default values have been modified to match model-specific defaults: {'max_length': 131072}. If this is not desired, please set these values explicitly.

def matmul(A, B):
    """
    Fast matrix multiplication using only native Python code.
    
    Parameters
    ----------
    A : list of list of numbers
        Left matrix of dimensions (m x p).
    B : list of list of numbers
        Right matrix of dimensions (p x n).
    
    Returns
    -------
    C : list of list of numbers
        Resulting matrix of dimensions (m x n) such that C = A × B.
    """
    # Transpose B to allow column access as rows.
    Bt = list(zip(*B))
    # Compute the dot product of each row from A with each column from B
    return [[sum(a * b for a, b in zip(row, col))
             for col in Bt]
            for row in A]
def matmul(A, B):
    return ...

Unsloth: Will smartly offload gradients to save VRAM!
def matmul(A, B):
    # ensure dimensions
    m = len(A)
    n = len(A[0])
    p = len(B[0]) if B else 0
    return [[sum(A[i][k] * B[k][j] for k in range(n)) for j in range(p)] for i in range(m)]
None
def matmul(A, B):
    # A: m x k
    # B: k x n
    # returns m x n
    ...
def matmul(A, B):
    # A: r x p, B: p x c
    r, p = len(A), len(A[0]) if A else 0
    p2, c = len(B), len(B[0]) if B else 0
    if r == 0 or c == 0 or p != p2:
        raise ValueError("Incompatible dimensions for multiplication")
    # transpose B to improve locality
    B_T = list(zip(*B))  # c x p
    result = [[0] * c for _ in range(r)]
    for i in range(r):
        Ai = A[i]
        Ri = result[i]
        for j, Bj in enumerate(B_T):
            s = 0
            for k in range(p):
                s += Ai[k] * Bj[k]
            Ri[j] = s
    return result
def matmul(A, B):
    """
    Multiply two matrices A and B.

    Parameters
    ----------
    A : list[list[float]]
        Left‑hand matrix with shape m x k.
    B : list[list[float]]
        Right‑hand matrix with shape k x n.

    Returns
    -------
    list[list[float]]
        Resulting matrix with shape m x n.

    Raises
    ------
    ValueError
        If dimensions are incompatible.
    """
    if not A or not B:
        return []

    # Dimensions
    rows_a, cols_a = len(A), len(A[0])
    rows_b, cols_b = len(B), len(B[0])

    if cols_a != rows_b:
        raise ValueError("Incompatible dimensions for multiplication")

    # Convert B into columns for faster row‑by‑row multiplication
    B_t = [list(col) for col in zip(*B)]

    # Compute each row of the result
    result = [
        [sum(a * b for a, b in zip(row_a, col_b)) for col_b in B_t]
        for row_a in A
    ]

    return result
None
def matmul(A, B):
    """Compute the matrix product of A and B using pure Python."""
    # Quick sanity checks
    m, n = len(A), len(A[0])
    p, q = len(B), len(B[0])
    assert n == p, "Number of columns of A must equal number of rows of B"
    # Prepare the result matrix with zeros
    result = [[0] * q for _ in range(m)]
    # This variant loops over the outermost index that is likely to be cache-friendly
    # and prefetches the row of B so that we access its elements in order.
    for i in range(m):
        row_a = A[i]
        for k in range(n):
            aik = row_a[k]
            if aik:                          # skip the zero case to save work
                row_b = B[k]
                for j in range(q):
                    result[i][j] += aik * row_b[j]
    return result
def matmul(A, B):
    m = len(A)
    if m==0: return []
    n = len(A[0]) or 0
    p = len(B[0]) if B else 0
    # ensure B has n rows
    # Use list comprehension summing product over k
    # Compute B transposed for column-wise access: BT = list(zip(*B))
    BT = list(zip(*B))
    return [[sum(a*b for a,b in zip(row,col)) for col in BT] for row in A]
def matmul(A, B):
    """
    Multiply two matrices where each matrix is represented as a list of lists
    and the elements are integers or floats.

    Parameters
    ----------
    A : list[list[int|float]]
        Left‑hand matrix of size (m × p).
    B : list[list[int|float]]
        Right‑hand matrix of size (p × n).

    Returns
    -------
    list[list[int|float]]
        The product matrix C = A @ B of size (m × n).
    """

    # Dimensions: A: m×p, B: p×n
    m, p = len(A), len(A[0])
    p2, n = len(B), len(B[0])
    if p != p2:
        raise ValueError("Inner dimensions must agree for matrix multiplication")

    # Pre‑transpose B so that column access is contiguous
    # This reduces random memory access during the dot product
    B_T = [list(col) for col in zip(*B)]

    result = []
    for i, row in enumerate(A):
        # Compute each entry of the i‑th row of the result
        result_row = []
        for j, col in enumerate(B_T):
            # use a local variable for speed
            dot = 0
            for a, b in zip(row, col):
                dot += a * b
            result_row.append(dot)
        result.append(result_row)

    return result
None
def matmul(A, B):
    """
    Multiply two matrices A and B using plain Python.
    A: list of lists (m × n), B: list of lists (n × p).
    Returns the product (m × p) as a new list of lists.
    """
    if not A or not B:
        return []

    m = len(A)
    n = len(A[0])
    p = len(B[0])

    # Quick compatibility check
    assert len(B) == n, "Incompatible matrix dimensions"

    # Allocate result matrix
    result = [[0] * p for _ in range(m)]

    # Standard triple‑loop multiplication, with a small speed‑up:
    # pull outer indices, cache row and column values locally,
    # and skip inner loop when the coefficient is zero.
    for i in range(m):
        Ai = A[i]
        Ri = result[i]
        for k in range(n):
            aik = Ai[k]
            if aik:                       # skip zero entries
                Bk = B[k]
                for j in range(p):
                    Ri[j] += aik * Bk[j]
    return result
def matmul(A, B):
    # number of rows in A
    m = len(A)
    # number of columns in B
    p = len(B[0]) if B else 0
    # number of columns in A (used as number of rows in B)
    k = len(A[0]) if A else 0

    # If shapes are incompatible, raise an error
    if len(B) != k:
        raise ValueError("Incompatible matrices: A.shape[1] != B.shape[0]")

    # Result matrix initialised with zeros
    result = [[0] * p for _ in range(m)]

    # Triple‑loop multiplication
    for i in range(m):
        for j in range(p):
            # Compute the dot product of row i of A and column j of B
            s = 0
            for t in range(k):
                s += A[i][t] * B[t][j]
            result[i][j] = s
    return result
None
def matmul(A, B):
    # Determine dimensions
    m, n = len(A), len(A[0])
    nB, p = len(B), len(B[0])
    assert n == nB, "Incompatible matrices"
    result = [[0]*p for _ in range(m)]
    # Multiply
    for i in range(m):
        Ai = A[i]
        for k in range(n):
            aik = Ai[k]
            if aik:
                Bk = B[k]
                for j in range(p):
                    result[i][j] += aik * Bk[j]
    return result
def matmul(A, B):
    """Fast matrix product using only native Python.

    Args:
        A (List[List[Number]]): left matrix (n × m)
        B (List[List[Number]]): right matrix (m × p)

    Returns:
        List[List[Number]]: the product A * B (n × p)

    Raises:
        ValueError: if the matrices cannot be multiplied
    """
    # Basic shape checks
    if not A or not A[0] or not B or not B[0]:
        raise ValueError("Input matrices must be non‑empty")
    n, m = len(A), len(A[0])
    if m != len(B):
        raise ValueError("Number of columns of A must equal number of rows of B")

    # Transpose B once for better locality
    B_T = list(zip(*B))  # now each element of B_T is a tuple representing a column

    # Compute the product
    result = []
    for row in A:
        # dot(row, col) for each column
        result.append([sum(a * b for a, b in zip(row, col)) for col in B_T])

    return result
None
def matmul(A, B):
    return ...
def matmul(A, B):
    ...
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    k = len(B)
    result = [[0]*m for _ in range(n)]
    for i in range(n):
        for j in range(m):
            s = 0
            for l in range(k):
                s += A[i][l] * B[l][j]
            result[i][j] = s
    return result
None
None
def matmul(A, B):
    m = len(A)
    n = len(B[0])
    p = len(B)
    return [[sum(A[i][k] * B[k][j] for k in range(p)) for j in range(n)] for i in range(m)]
None
def matmul(A, B):
    return ...
def matmul(A, B):
    import math
    # Assume square matrices of same size and power of 2
    n = len(A)
    if n == 0:
        return []
    def add(X, Y):
        return [[X[i][j] + Y[i][j] for j in range(n)] for i in range(n)]
    def sub(X, Y):
        return [[X[i][j] - Y[i][j] for j in range(n)] for i in range(n)]
    def split(M):
        k = n // 2
        return [ [row[:k] for row in M[:k]],
                 [row[:k] for row in M[k:]],
                 [row[k:] for row in M[:k]],
                 [row[k:] for row in M[k:]] ]
    def combine(A11, A12, A21, A22):
        k = len(A11)
        result = [ [0]* (k*2) for _ in range(k*2) ]
        for i in range(k):
            result[i][:k]   = A11[i]
            result[i][k:]   = A12[i]
            result[i+k][:k] = A21[i]
            result[i+k][k:] = A22[i]
        return result
    def strassen(X, Y):
        if n == 1:
            return [[X[0][0]*Y[0][0]]]
        a, b, c, d = split(X)
        e, f, g, h = split(Y)
        p1 = strassen(a, sub(f, h))
        p2 = strassen(add(a, b), h)
        p3 = strassen(add(c, d), e)
        p4 = strassen(d, sub(g, e))
        p5 = strassen(add(a, d), add(e, h))
        p6 = strassen(sub(b, d), add(g, h))
        p7 = strassen(sub(a, c), add(e, f))
        c11 = add(sub(add(p5, p4), p2), p6)
        c12 = add(p1, p2)
        c21 = add(p3, p4)
        c22 = sub(sub(add(p1, p5), p3), p7)
        return combine(c11, c12, c21, c22)
    return strassen(A, B)
def matmul(A, B):
    n = len(A)
    m = len(A[0])
    p = len(B[0])
    assert len(B) == m
    BT = list(zip(*B)) # transposed as tuples
    C = [[sum(Ai[k] * BTj[k] for k in range(m)) for BTj in BT] for Ai in A]
    return C
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    p = len(B)
    # transpose B
    B_T = list(map(list, zip(*B)))  # list of columns
    return [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
None
def matmul(A, B):
    return ...
def matmul(A, B):
    ...
def matmul(A, B):
    """Matrix multiplication using only native Python (no external libraries).

    Works for arbitrary sized matrices with compatible dimensions.
    The algorithm transposes matrix B to enhance cache locality,
    then uses a list‑comprehension to calculate the dot–product of
    corresponding rows and columns.

    Parameters
    ----------
    A : list[list[Number]]
        Left matrix of shape (m, n).
    B : list[list[Number]]
        Right matrix of shape (n, p).

    Returns
    -------
    list[list[Number]]
        Resulting product matrix of shape (m, p).
    """
    # Transpose B for efficient column access
    B_transposed = list(zip(*B))  # tuples, one per column of B
    return [
        [
            # dot product of row from A and column from B
            sum(a * b for a, b in zip(row_a, col_b))
            for col_b in B_transposed
        ]
        for row_a in A
    ]
None
None
def matmul(A, B):
    return ...
def matmul(A, B):
    return ...
def matmul(A, B):
    # check dimension matches
    nrows_a = len(A)
    ncols_a = len(A[0]) if A else 0
    nrows_b = len(B)
    ncols_b = len(B[0]) if B else 0
    if ncols_a != nrows_b:
        raise ValueError("Incompatible dimensions")
    # transpose B for cache locality
    BT = list(zip(*B))  # tuple of tuples used as rows
    result = [[0]*ncols_b for _ in range(nrows_a)]
    for i in range(nrows_a):
        ai = A[i]
        ri = result[i]
        for j in range(ncols_b):
            s = 0
            bj = BT[j]
            for k in range(ncols_a):
                s += ai[k] * bj[k]
            ri[j] = s
    return result
def matmul(A, B):
    B_T = list(zip(*B))
    res = [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
    return res
def matmul(A, B):
    """
    Multiply two square matrices A and B.

    Parameters
    ----------
    A : list[list[float]]
        First n × n matrix.
    B : list[list[float]]
        Second n × n matrix.

    Returns
    -------
    list[list[float]]
        The product matrix C = A @ B.
    """
    # Transpose B once for O(1) column access
    B_T = list(zip(*B))

    # Compute C[i][j] = dot(A[i], B_T[j])
    return [[sum(a * b for a, b in zip(row, col))
             for col in B_T]
            for row in A]
def matmul(A, B): return ...
def matmul(A, B):
    """
    Multiply two matrices A and B using native Python lists.
    `A` and `B` must be rectangular (i.e. all rows the same length).

    Returns a new matrix containing the product.
    Raises ValueError if the inner dimensions do not match.
    """
    if not A or not B:
        return []

    n_rows_A, n_cols_A = len(A), len(A[0])
    n_rows_B, n_cols_B = len(B), len(B[0])

    if n_cols_A != n_rows_B:
        raise ValueError("cannot multiply: inner dimensions do not match")

    # preallocate result matrix
    result = [[0] * n_cols_B for _ in range(n_rows_A)]

    for i in range(n_rows_A):
        # Local references for speed
        row_a = A[i]
        for k in range(n_cols_A):
            aik = row_a[k]
            if aik == 0:
                continue     # skip zero multiplications
            row_b = B[k]
            for j in range(n_cols_B):
                result[i][j] += aik * row_b[j]

    return result
None
def matmul(A, B):
    return ...
None
None
None
def matmul(A, B):
    return ...
None
None
None
None
def matmul(A, B):
    """Return the matrix product of A and B.

    A must be an `n×m` matrix, B an `m×p` matrix.
    Matrices are represented as nested lists of numbers.
    """
    if not A or not B:
        return []

    # Transpose B once so we can iterate over rows efficiently
    B_T = list(zip(*B))

    # Compute each entry using a dot‑product of corresponding rows
    return [
        [sum(a * b for a, b in zip(row_a, col_b)) for col_b in B_T]
        for row_a in A
    ]
None
def matmul(A, B):
    # A is m x p, B is p x n
    m = len(A)
    p = len(A[0])  # find p
    n = len(B[0])
    # create result m x n
    result = [... for i in ...]
def matmul(A, B):
    m, p = len(A), len(A[0]) if A else 0
    p2, n = len(B), len(B[0]) if B else 0
    if p != p2: raise ValueError("Incompatible dimensions")
    # Precompute transpose of B for cache-friendly access
    Bt = list(zip(*B))
    return [[sum(a * b for a, b in zip(row, col)) for col in Bt] for row in A]
def matmul(A, B): return ...
def matmul(A, B):
    return ...
None
def matmul(A, B):
    return ...
None
def matmul(A, B): return ...
None
def matmul(A, B):
    """
    Multiply two matrices A (m×n) and B (n×p) given as lists of lists.
    Uses only native Python code and is tuned for speed by transposing B.
    """
    if not A:
        return []

    n_rows_A, n_cols_A = len(A), len(A[0])
    # Basic consistency check – assume all rows have equal length
    # and B has compatible dimensions.
    n_rows_B, n_cols_B = len(B), len(B[0]) if B else 0

    # Transpose B to access its columns as tuples (faster indexing)
    B_T = list(zip(*B))          # shape: (p × n)

    result = []
    for row_A in A:              # iterate over rows of A
        # Compute the dot product of row_A with each column of B
        res_row = [sum(a * b for a, b in zip(row_A, col_B)) for col_B in B_T]
        result.append(res_row)

    return result
None
None
def matmul(A, B):
    """
    Multiply two matrices A and B (lists of lists) using native Python.
    The matrices are assumed to be square and of compatible dimensions.
    """
    n = len(A)
    # Transpose B to improve cache locality for the inner sum
    B_T = list(zip(*B))                       # each element is a tuple
    return [[sum(a * b for a, b in zip(row, col))
             for col in B_T]
            for row in A]
def matmul(A, B):
    """
    Multiply two matrices A and B using only native Python code.

    Parameters
    ----------
    A : list of lists (m x k)
    B : list of lists (k x n)

    Returns
    -------
    C : list of lists (m x n)
        Product matrix such that C[i][j] = sum(A[i][p] * B[p][j] for p in range(k))

    Raises
    ------
    ValueError
        If the number of columns in A does not equal the number of rows in B.
    """
    if not A or not B:
        raise ValueError("Both matrices must be non‑empty.")
    m, k1 = len(A), len(A[0])
    k2, n = len(B), len(B[0])
    if k1 != k2:
        raise ValueError(f"Incompatible shapes: {m}x{k1} multiplied by {k2}x{n}")
    # Pre‑compute columns of B for faster access
    B_cols = list(zip(*B))  # n tuples each of length k
    # Compute the product
    return [[sum(a * b for a, b in zip(row, col))
             for col in B_cols] for row in A]
def matmul(A, B):
    """
    Fast matrix multiplication for plain Python objects.
    A : list of m rows, each a list of n numbers
    B : list of n rows, each a list of p numbers
    Returns a new matrix of shape m × p.
    """
    m = len(A)
    if m == 0:
        return []
    n = len(A[0])
    # Verify dimension compatibility
    if len(B) != n or any(len(row) != n for row in A):
        raise ValueError("Inner dimensions must agree.")
    p = len(B[0])

    # Allocate result matrix
    result = [[0] * p for _ in range(m)]

    # Standard triple-loop, optimized for speed in pure Python
    for i in range(m):
        row_a = A[i]
        row_res = result[i]
        for k in range(n):
            a_val = row_a[k]
            row_b = B[k]
            # Use local variables for performance
            for j in range(p):
                row_res[j] += a_val * row_b[j]

    return result
def matmul(A, B):
    """Multiply two matrices A and B.

    A: list of m rows, each containing n elements.
    B: list of n rows, each containing p elements.
    Returns a new list of list representing the product matrix of shape (m, p).
    """
    # Basic sanity checks
    if not A or not B:
        raise ValueError("Input matrices must be non‑empty.")
    m, n = len(A), len(A[0])
    nB, p = len(B), len(B[0])
    if n != nB:
        raise ValueError("Number of columns of A must equal number of rows of B.")
    for row in A:
        if len(row) != n:
            raise ValueError("All rows of A must have the same length.")
    for row in B:
        if len(row) != p:
            raise ValueError("All rows of B must have the same length.")

    # Allocate the result matrix (m x p) initialized to 0
    result = [[0] * p for _ in range(m)]

    # Perform multiplication
    for i in range(m):
        rowA = A[i]
        rowR = result[i]
        for k in range(n):
            aik = rowA[k]
            if aik:  # Skip work for zero multiplication
                rowB = B[k]
                for j in range(p):
                    rowR[j] += aik * rowB[j]
    return result
None
None
def matmul(A, B):
    # Validate dimensions
    if not A or not B:
        return []
    if len(A[0]) != len(B):
        raise ValueError("Matrix dimensions do not match for multiplication.")
    B_cols = list(zip(*B))  # transpose B
    result = [[sum(a*b for a,b in zip(row, col)) for col in B_cols] for row in A]
    return result
None
def matmul(A, B):
    """Multiply matrices A × B (list‑of‑list format) using an optimized pure‑Python routine."""
    n, p = len(A), len(A[0])           # rows of A, columns of B (must match)
    m = len(B[0])                     # columns of B
    # Result matrix initialized with zeros
    result = [[0] * m for _ in range(n)]
    for i in range(n):
        rowA = A[i]
        rowR = result[i]
        for k in range(p):
            aik = rowA[k]
            if aik:                    # skip zero entries for a small extra speedup
                rowBk = B[k]
                for j in range(m):
                    rowR[j] += aik * rowBk[j]
    return result
def matmul(A, B):
    """Fast matrix multiplication using only native Python code.

    Parameters
    ----------
    A : list[list[float]]
        Left hand matrix of shape (n, m).
    B : list[list[float]]
        Right hand matrix of shape (m, q).

    Returns
    -------
    list[list[float]]
        The product matrix of shape (n, q).

    Notes
    -----
    The routine pre-allocates the result matrix and uses local variable
    bindings to reduce attribute look‑ups inside the innermost loop,
    which gives a noticeable speed boost for large matrices.
    """
    # Basic dimensions checking
    n, m = len(A), len(A[0])
    m2, q = len(B), len(B[0])
    if m != m2:
        raise ValueError("Number of columns in A must equal number of rows in B.")

    # Pre‑allocate result matrix with zeros
    result = [[0.0] * q for _ in range(n)]

    # Perform multiplication
    for i in range(n):
        rowA = A[i]
        rowR = result[i]
        for k in range(m):
            aik = rowA[k]
            rowB = B[k]
            for j in range(q):
                rowR[j] += aik * rowB[j]
    return result
def matmul(A, B):
    B_T = list(zip(*B))  # transpose B for inner product
    return [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
def matmul(A, B):
    # A: m x n, B: n x p
    m, n = len(A), len(A[0])
    n2, p = len(B), len(B[0])
    assert n==n2
    # initialize result matrix
    C = [[0]*p for _ in range(m)]
    # transpose B for better locality
    B_T = list(zip(*B))
    for i in range(m):
        Ai = A[i]
        Ci = C[i]
        for k in range(n):
            aik = Ai[k]
            Bk = B_T[k]
            for j in range(p):
                Ci[j] += aik * Bk[j]
    return C
def matmul(A, B):
    """Fast matrix multiplication using pure Python.

    Arguments:
        A: List of lists, the left matrix of size m x n.
        B: List of lists, the right matrix of size n x p.

    Returns a new matrix C of size m x p where C[i][j] = sum(A[i][k] * B[k][j] for k in range(n)).

    This implementation transposes B once to enable efficient column access
    and uses nested list comprehensions together with the built‑in `sum`
    function, which is implemented in C.
    """
    # Verify dimensions
    if not A or not B:
        raise ValueError("Matrices cannot be empty")
    n = len(A[0])
    if any(len(row) != n for row in A):
        raise ValueError("All rows in A must have the same length")
    if len(B) != n:
        raise ValueError("Number of columns in A must equal number of rows in B")
    p = len(B[0])
    if any(len(row) != p for row in B):
        raise ValueError("All rows in B must have the same length")

    # Transpose B to get columns as rows
    B_cols = list(zip(*B))

    # Compute product
    return [[sum(a * b for a, b in zip(row, col)) for col in B_cols] for row in A]
def matmul(A, B):
    BT = list(zip(*B))
    return [[sum(a*b for a,b in zip(row, col)) for col in BT] for row in A]
def matmul(A, B):
    """Multiplies matrix A by matrix B using pure native Python.

    Parameters
    ----------
    A : list[list[float]]
        The first matrix of shape (l, m).
    B : list[list[float]]
        The second matrix of shape (m, n).

    Returns
    -------
    list[list[float]]
        The product matrix of shape (l, n).
    """
    # Pre‑get dimensions for speed
    l = len(A)           # Number of rows in A
    m = len(A[0])        # Number of columns in A / rows in B
    n = len(B[0])        # Number of columns in B

    # The result will have shape (l, n)
    result = [[0.0] * n for _ in range(l)]

    for i in range(l):
        Ai = A[i]
        for k in range(m):
            aik = Ai[k]
            Bk = B[k]
            # Unroll column loop to reduce attribute lookups
            for j in range(n):
                result[i][j] += aik * Bk[j]

    return result
def matmul(A, B):
    """
    Multiply two matrices A (n x m) and B (m x p) using only native Python.
    Returns the resulting matrix as a list of lists.
    """
    n = len(A)
    m = len(A[0])
    if len(B) != m:
        raise ValueError("Number of columns in A must equal number of rows in B")
    p = len(B[0])

    # Pre‑allocate result matrix with zeros
    result = [[0] * p for _ in range(n)]

    for i in range(n):
        row_a = A[i]
        for k in range(m):
            aik = row_a[k]          # A[i][k]
            row_b = B[k]            # The k-th row of B
            for j in range(p):
                result[i][j] += aik * row_b[j]  # C[i][j] += A[i][k] * B[k][j]
    return result
def matmul(A, B):
    return ...
def matmul(A, B):
    return ...
None
def matmul(A, B):
    # assume A dims m x n, B n x p
def matmul(A, B):
    return ...
def matmul(A, B):
    # Should handle maybe rectangular matrices
    ...
def matmul(A, B): return ...
def matmul(A, B): return ...
def matmul(A, B):
    result = []
    for i in range(len(A)):
        res_row = []
        for j in range(len(B[0])):
            sum_val = 0
            for k in range(len(B)):
                sum_val += A[i][k] * B[k][j]
            res_row.append(sum_val)
        result.append(res_row)
    return result
def matmul(A, B): return ...
def matmul(A, B):
    """
    Multiply two matrices represented as lists of lists using only
    standard Python code. Raises a ValueError if the matrices cannot
    be multiplied.
    
    Parameters
    ----------
    A : List[List[Number]]
        The left-hand-side matrix of shape (m, n).
    B : List[List[Number]]
        The right-hand-side matrix of shape (n, p).

    Returns
    -------
    C : List[List[Number]]
        The product matrix of shape (m, p).
    """
    if not A or not B:
        raise ValueError("Empty matrices cannot be multiplied")

    # Verify inner dimensions match
    n = len(A[0])
    for row in A:
        if len(row) != n:
            raise ValueError("All rows of A must have the same length")
    if len(B) != n:
        raise ValueError("Number of columns in A must equal number of rows in B")

    p = len(B[0])
    for row in B:
        if len(row) != p:
            raise ValueError("All rows of B must have the same length")

    # Transpose B so that columns can be accessed as tuples
    B_T = list(zip(*B))        # Each element is a tuple of length n

    # Compute the product
    C = []
    for a_row in A:
        c_row = []
        for b_col in B_T:
            dot_product = sum(x * y for x, y in zip(a_row, b_col))
            c_row.append(dot_product)
        C.append(c_row)

    return C
def matmul(A, B):
    """
    Multiply two matrices A and B represented as nested lists.
    
    Parameters:
        A (list[list[float]]): Matrix of size (m x n).
        B (list[list[float]]): Matrix of size (n x p).
    
    Returns:
        list[list[float]]: Resultant matrix of size (m x p).
    """
    # Pre‑compute the columns of B
    B_cols = list(zip(*B))
    
    # Compute the product row by row
    return [
        [
            sum(a * b for a, b in zip(row, col))
            for col in B_cols
        ]
        for row in A
    ]
def matmul(A, B):
    # Basic dimension check
    if not A or not B:
        return []
    n_rows_a = len(A)
    n_cols_a = len(A[0])
    n_rows_b = len(B)
    n_cols_b = len(B[0])
    if n_cols_a != n_rows_b:
        raise ValueError("Incompatible dimensions for matrix multiplication")
    result = [[0] * n_cols_b for _ in range(n_rows_a)]
    for i in range(n_rows_a):
        for k in range(n_cols_a):
            aik = A[i][k]
            if aik == 0:
                continue
            for j in range(n_cols_b):
                result[i][j] += aik * B[k][j]
    return result
def matmul(A, B): return ...
def matmul(A, B):
    return ...
def matmul(A, B):
    # A: m x n, B: n x p
    # returns m x p
    # check sizes
    m, n = len(A), len(A[0])
    assert len(B) == n
    p = len(B[0])
    # Precompute transpose of B
    Bt = list(zip(*B))
    res = [[sum(a*b for a,b in zip(row, col)) for col in Bt] for row in A]
    return res
def matmul(A, B):
    return ...
def matmul(A, B):
    """Multiply two matrices A and B (list of lists) purely in plain Python."""
    m, n = len(A), len(A[0])
    nB, p = len(B), len(B[0])
    if n != nB:
        raise ValueError("Inner matrix dimensions must agree.")
    # Pre‑allocate result matrix
    C = [[0] * p for _ in range(m)]
    # Perform multiplication in the ctr order: i, k, j
    for i in range(m):
        rowA = A[i]
        rowC = C[i]
        for k in range(n):
            aik = rowA[k]
            if aik:  # skip zero multiplications
                rowB = B[k]
                for j in range(p):
                    rowC[j] += aik * rowB[j]
    return C
def matmul(A, B): return ...
def matmul(A, B):
    return ...
def matmul(A, B):
    B_T = list(zip(*B))
    return [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
def matmul(A, B):
    return ...
None
def matmul(A, B):
    ...
def matmul(A, B):
    ...
def matmul(A, B):
    ...
None
def matmul(A, B):
    # assume A rows, B columns
    m, k1 = len(A), len(A[0]) if A else 0
    k2, n = len(B), len(B[0]) if B else 0
    assert k1 == k2, "Inner dimensions must match"
    B_T = list(zip(*B))  # transposed B
    result = [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
    return result
None
def matmul(A, B):
    """
    Multiply two matrices A and B (both given as lists of lists) using
    only native Python constructs.

    The function first validates that the inner dimensions match (number of
    columns in A must equal the number of rows in B).  Then it computes the
    product using a straightforward triple‑loop but expressed in a compact
    list‑comprehension.  The columns of B are accessed by transposing B
    once via ``zip(*B)``, which avoids explicit indexing and gives good
    cache locality in Python.

    Parameters
    ----------
    A : List[List[Number]]
        Left‑hand matrix of size m × n.

    B : List[List[Number]]
        Right‑hand matrix of size n × p.

    Returns
    -------
    List[List[Number]]
        The matrix product of A and B, which has shape m × p.

    Raises
    ------
    ValueError
        If the matrices cannot be multiplied due to incompatible dimensions.
    """
    # Validate dimensions
    if not A or not B:
        raise ValueError("Matrices cannot be empty")
    n_cols_A = len(A[0])
    n_rows_B = len(B)
    if n_cols_A != n_rows_B:
        raise ValueError("Inner dimensions must match: "
                         f"{n_cols_A} != {n_rows_B}")

    # Transpose B once to make column access efficient
    B_cols = list(zip(*B))

    # Compute product using list comprehensions
    return [
        [sum(a * b for a, b in zip(row, col)) for col in B_cols]
        for row in A
    ]
def matmul(A, B):
    if not A or not B:
        return []
    m, p = len(A), len(A[0])
    n = len(B[0])
    # ensure inner dimension matches
    result = [[0]*n for _ in range(m)]
    for i in range(m):
        for k in range(p):
            aik = A[i][k]
            for j in range(n):
                result[i][j] += aik * B[k][j]
    return result
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    p = len(B)
    return [[sum(A[i][k] * B[k][j] for k in range(p)) for j in range(m)] for i in range(n)]
def matmul(A, B):
    # Basic check dims
    n = len(A)
    assert n > 0
    m = len(A[0])
    # B has size m x p
    assert len(B) == m
    p = len(B[0])
    # Pre-allocate result
    C = [[0]*p for _ in range(n)]
    # Compute transpose of B for better locality
    B_T = list(map(list, zip(*B)))
    for i in range(n):
        Ai = A[i]
        Ci = C[i]
        for j in range(p):
            Bj = B_T[j]
            s = 0
            for k in range(m):
                s += Ai[k] * Bj[k]
            Ci[j] = s
    return C
def matmul(A, B):
    """Return the product of two matrices in Theta(m^3/mn)*n^2 time,
    optimizing cache usage if --precache-multiple was selected.
    """
    _check_input(A, B)
    m = len(A); n = len(A[0]); k = len(B[0]);   # (m x n) * (n x k) => (m x k)
    _use_a = (m / n <= 100.0 * n / k)
    M_local_cache_avail_prefs = sympy.cache.get('M_local_cache_avail_prefs', 0)
    M_local_cache_avail_nonp = sympy.cache.get('M_local_cache_avail_nonp', 0)

    if sympy.cache and M_local_cache_avail_nonp and _use_a:
        local_cache_key_A = (m,n,1)
        local_cache_key_B = (n,k,1)
        if local_cache_key_A not in sympy.cache:
            sympy.cache[local_cache_key_A] = [(r,c) for r in range(m) for c in range(n)]
        if local_cache_key_B not in sympy.cache:
            sympy.cache[local_cache_key_B] = [(r,c) for r in range(n) for c in range(k)]
        A_local, B_local = sympy.cache[local_cache_key_A], sympy.cache[local_cache_key_B]
    else:
        A_local, B_local = None, None

    def fast_matmul(A_local, B_local, _):
        result = [[sum(A_local[i][j] * B_local[j][l] for j in range(n))
                   for l in range(k)]
                  for i in range(m)]
        return result

    if A_local and B_local:
        return fast_matmul(A_local, B_local, k)
    else:
        return fast_matmul(A, B, k)
def matmul(A, B):
    n, m = len(A), len(B)
    assert all(len(row)==m for row in A) and all(len(row)==len(B[0]) for row in B)
    # maybe compute B transposed
    BT = list(zip(*B))
    result = [[sum(a*b for a,b in zip(row, col)) for col in BT] for row in A]
    return result
def matmul(A, B):
    m = len(A)
    n = len(A[0])
    p = len(B[0])
    # compute result matrix C(m x p)
    result = [[sum(A[i][k]*B[k][j] for k in range(n)) for j in range(p)] for i in range(m)]
    return result
None
None
None
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    p = len(B)
    result = [[0]*m for _ in range(n)]
    for i in range(n):
        Ai=A[i]
        Ri=result[i]
        for k in range(p):
            aik=Ai[k]
            Bk=B[k]
            for j in range(m):
                Ri[j]+=aik*Bk[j]
    return result
def matmul(A, B):
    """
    Multiply two matrices A (n x m) and B (m x p) without using external libraries.
    """
    n=len(A)
    m=len(A[0])
    p=len(B[0])
    # Check compatibility
    if len(B)!=m:
        raise ValueError("Incompatible dimensions")
    # Transpose B for cache-friendly access
    B_T=[list(col) for col in zip(*B)]
    # Prepare result matrix
    result=[[0]*p for _ in range(n)]
    for i in range(n):
        row=A[i]
        # local assignments
        res_row=result[i]
        for j in range(p):
            col=B_T[j]
            s=0
            for a, b in zip(row, col):
                s += a*b
            res_row[j]=s
    return result
def matmul(A, B):
    """Multiply two matrices A (m x n) and B (n x p) using pure Python.

    Args:
        A: list of m lists, each of length n.
        B: list of n lists, each of length p.

    Returns:
        C: list of m lists, each of length p, the product.
    """
    m = len(A); n = len(A[0]); p = len(B[0])
    # Preallocate result
    C = [[0]*p for _ in range(m)]
    # transpose B for better cache (although Python's memory model)
    B_T = [list(col) for col in zip(*B)]
    # iterate over rows of A and rows of B_T
    for i in range(m):
        Ai = A[i]
        Ci = C[i]
        for k in range(p):
            s = 0
            Bk = B_T[k]
            for j in range(n):
                s += Ai[j] * Bk[j]
            Ci[k] = s
    return C
def matmul(A, B):
    # A: m x n, B: n x p
    m, n = len(A), len(A[0])
    n2, p = len(B), len(B[0])
    assert n == n2
    # Precompute columns of B
    Bcols = list(zip(*B))
    return [[sum(a*b for a,b in zip(row,c)) for c in Bcols] for row in A]
None
def matmul(A, B):
    """
    Multiply two matrices using pure Python.

    Parameters
    ----------
    A : List[List[Number]]
        The first matrix (m × n).
    B : List[List[Number]]
        The second matrix (n × p).

    Returns
    -------
    List[List[Number]]
        The product matrix (m × p).

    Raises
    ------
    ValueError
        If the inner dimensions of A and B do not match.
    """
    if not A or not B or not B[0]:
        return []

    m = len(A)
    n = len(A[0])
    if len(B) != n:
        raise ValueError("Inner dimensions must agree: A: {}, B: {}.".format(n, len(B)))

    p = len(B[0])
    # column count of B must be p
    # Use list comprehensions for a compact, Python‑native implementation
    return [[sum(A[i][k] * B[k][j] for k in range(n)) for j in range(p)] for i in range(m)]
def matmul(A, B):
    ...
None
def matmul(A, B):
    return ...
def matmul(A, B):
    return ...
def matmul(A, B): return ...
def matmul(A, B):
    # matrix multiplication using only native Python
    ...
    return ...
def matmul(A, B):
    n_rows = len(A)
    n_cols = len(B[0])
    k = len(B)  # check match
    if any(len(row)!=k for row in A):
        raise ValueError("A dimensions don't match B")
    # transpose B for better cache
    B_T = list(zip(*B))  # returns tuples but we can keep
    result = [[0]*n_cols for _ in range(n_rows)]
    for i in range(n_rows):
        Ai = A[i]
        Ri = result[i]
        for j, Bj in enumerate(B_T):
            s = 0
            for a,b in zip(Ai, Bj):
                s += a*b
            Ri[j] = s
    return result
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    p = len(B)
    # assert p == len(A[0])? we can compute
    result = [[0]*m for _ in range(n)]
    for i in range(n):
        ai = A[i]
        for k in range(p):
            aik = ai[k]
            if aik:
                bk = B[k]
                for j in range(m):
                    result[i][j] += aik * bk[j]
    return result
None
None
def matmul(A, B):
    # Ensure both are lists of lists, etc. 
    return [[sum(a*b for a,b in zip(row,col)) for col in zip(*B)] for row in A]
None
None
def matmul(A, B): return ...
None
def matmul(A, B): return ...
None
def matmul(A, B):
    """
    Multiplies two matrices A and B using pure Python lists.
    Expects A to be m x n and B to be n x p.
    Returns the resulting m x p matrix.
    """
    return [[sum(a * b for a, b in zip(rowA, colB)) for colB in zip(*B)] for rowA in A]
def matmul(A, B):
    """
    Multiply two matrices A and B using only native Python constructs.
    
    Parameters
    ----------
    A : list[list[Number]]
        Left‑hand matrix.
    B : list[list[Number]]
        Right‑hand matrix.
    
    Returns
    -------
    list[list[Number]]
        The product matrix A * B.
    """
    # Transpose B to ease column access
    Bt = list(zip(*B))
    # Compute each entry as the dot product of a row of A and a column of B
    return [[sum(a * b for a, b in zip(row, col))
             for col in Bt] for row in A]
None
def matmul(A, B):
    # assume A: m x n, B: n x p
    n = len(A)
    m = len(A[0])
    p = len(B[0])
    # precompute columns of B via zip
    columns_B = list(zip(*B))
    result = [[sum(a*b for a,b in zip(row, col)) for col in columns_B] for row in A]
    return result
None
def matmul(A, B):
    """
    Multiply two matrices A and B using plain Python.
    A      : list of m rows, each a list of p numbers
    B      : list of p rows, each a list of n numbers
    Returns a list of m rows, each a list of n numbers
    """
    m, p = len(A), len(A[0])           # dimensions of A
    assert p == len(B), "Incompatible matrix dimensions"
    n = len(B[0])                      # number of columns of B

    # initialise result matrix with zeros
    C = [[0] * n for _ in range(m)]

    # iterate over the shared dimension first
    for k in range(p):
        B_row = B[k]
        for i in range(m):
            aik = A[i][k]
            if aik:                   # skip multiplies by zero
                C_row = C[i]
                # the inner loop that does the real work
                for j in range(n):
                    C_row[j] += aik * B_row[j]
    return C
def matmul(A, B):
    m = len(A); n = len(A[0]); p = len(B[0])
    # validate B has proper shape
    result = [[0]*p for _ in range(m)]
    for i in range(m):
        rowA = A[i]
        row_res = result[i]
        for k in range(n):
            aik = rowA[k]
            if aik:
                rowBk = B[k]
                for j in range(p):
                    row_res[j] += aik * rowBk[j]
    return result
None
None
def matmul(A, B):
    if not A or not B:
        return []
    n, m = len(A), len(A[0])
    p, q = len(B), len(B[0])
    if m != p:
        raise ValueError("Inner matrix dimensions must agree.")
    # compute C with zeros
    C = [[0]*q for _ in range(n)]
    for i in range(n):
        Ai = A[i]
        Ci = C[i]
        for k in range(m):
            aik = Ai[k]
            Bk = B[k]
            for j in range(q):
                Ci[j] += aik * Bk[j]
    return C
def matmul(A, B):
    """
    Multiply two matrices A (n × p) and B (p × m) using a cache‑friendly
    block‑style algorithm that reduces Python overhead compared to a
    straightforward triple loop.
    """
    n = len(A)            # rows of A
    p = len(B)            # shared dimension
    m = len(B[0])         # columns of B

    # Result matrix initialised with zeros
    C = [[0] * m for _ in range(n)]

    # Optimised loop ordering: i → k → j
    for i in range(n):
        Ai = A[i]
        Ci = C[i]
        for k in range(p):
            aik = Ai[k]
            if aik:                     # skip zero multiplications
                Bk = B[k]
                for j in range(m):
                    Ci[j] += aik * Bk[j]
    return C
def matmul(A, B):
    # ensure convertible shapes
    m, n = len(A), len(A[0])
    p = len(B[0])
    # Precompute transpose of B to improve cache locality
    BT = list(zip(*B))
    result = [[0]*p for _ in range(m)]
    for i in range(m):
        Ai = A[i]
        Ri = result[i]
        for j, Bj in enumerate(BT):
            s = 0
            for k in range(n):
                s += Ai[k] * Bj[k]
            Ri[j] = s
    return result
None
def matmul(A, B): return ...
None
None
None
None
None
None
def matmul(A, B):
    # ensure shape
    if not A: return []
    m, k = len(A), len(A[0])
    k2, n = len(B), len(B[0])
    assert k == k2
    # transpose B
    B_T = list(zip(*B))
    # compute
    return [[sum(a*b for a,b in zip(row, col)) for col in B_T] for row in A]
None
def matmul(A, B):
    # A is an m x n matrix, B is an n x p matrix
    m = len(A)
    n = len(A[0]) if A else 0
    p = len(B[0]) if B else 0

    # In case the dimensions are incompatible, raise an error
    if n != len(B):
        raise ValueError("Incompatible matrix dimensions: A has %d cols but B has %d rows." % (n, len(B)))

    # Prepare the result matrix filled with zeros
    C = [[0] * p for _ in range(m)]

    # Perform the multiplication using a cache-friendly triple loop
    for i in range(m):
        Ai = A[i]
        Ci = C[i]
        for k in range(n):
            aik = Ai[k]
            Bk = B[k]
            for j in range(p):
                Ci[j] += aik * Bk[j]
    return C
None
def matmul(A, B):
    """
    Multiply two matrices A and B using plain Python lists.

    Parameters
    ----------
    A : list[list[float]]
        Left matrix of shape (m, n).
    B : list[list[float]]
        Right matrix of shape (n, p).

    Returns
    -------
    list[list[float]]
        The product matrix of shape (m, p).

    Raises
    ------
    ValueError
        If the inner dimensions do not agree.
    """
    # Validate input
    if not A or not B:
        return []

    m, n_A = len(A), len(A[0])
    n_B, p = len(B), len(B[0])

    if n_A != n_B:
        raise ValueError("Inner matrix dimensions must agree: "
                         f"{n_A} != {n_B}")

    # Pre‑allocate result matrix with zeros
    C = [[0.0 for _ in range(p)] for _ in range(m)]

    # Classic triple‑loop multiplication
    for i in range(m):
        Ai = A[i]
        Ci = C[i]
        for k in range(n_A):
            aik = Ai[k]
            Bk = B[k]
            # Unroll the inner loop over p
            for j in range(p):
                Ci[j] += aik * Bk[j]

    return C
None
None
def matmul(A, B):
    """
    Multiplies two matrices A and B represented as lists of lists using
    plain Python code. Handles generic rectangular matrices.
    """
    # Basic size validation
    if not A or not B or not A[0] or not B[0]:
        raise ValueError("Matrices cannot be empty")
    rows_a = len(A)
    cols_a = len(A[0])
    rows_b = len(B)
    cols_b = len(B[0])

    if cols_a != rows_b:
        raise ValueError("Inner dimensions must match for multiplication")

    # Compute the product using a straightforward triple loop
    result = [
        [sum(A[i][k] * B[k][j] for k in range(cols_a)) for j in range(cols_b)]
        for i in range(rows_a)
    ]

    return result
def matmul(A, B):
    return ...
def matmul(A, B):
    """
    Multiply two matrices using a naïve algorithm implemented in pure Python.
    """
    # Get dimensions
    na = len(A)
    ma = len(A[0]) if A else 0
    nb = len(B)
    mb = len(B[0]) if B else 0
    
    if ma != nb:
        raise ValueError("Inner dimensions must match for multiplication.")
    
    # Create result matrix
    result = [[0 for _ in range(mb)] for _ in range(na)]
    
    # Standard triple‑loop multiplication
    for i in range(na):
        Ai = A[i]
        for j in range(mb):
            sum = 0
            for k in range(ma):
                sum += Ai[k] * B[k][j]
            result[i][j] = sum
    return result
def matmul(A, B):
    """
    Multiply two matrices A and B and return the product.

    Parameters
    ----------
    A : list[list[Number]]
        Left matrix of shape (m, n).
    B : list[list[Number]]
        Right matrix of shape (n, p).

    Returns
    -------
    C : list[list[Number]]
        Matrix product of shape (m, p).

    Raises
    ------
    ValueError
        If the input matrices cannot be multiplied due to incompatible shapes.
    """
    # Validate inputs
    if not A or not B:
        return []

    # Dimensions
    m, n = len(A), len(A[0]) if A else 0
    n_b, p = len(B), len(B[0]) if B else 0

    if n != n_b:
        raise ValueError(f"Incompatible dimensions: A is {m}x{n}, B is {n_b}x{p}")

    # Pre-allocate result matrix with zeros
    C = [[0] * p for _ in range(m)]

    # Main multiplication loop
    for i in range(m):
        a_row = A[i]
        c_row = C[i]
        for k in range(n):
            aik = a_row[k]
            if aik == 0:
                continue  # Skip multiplication by zero
            b_row_k = B[k]
            # Unroll the inner loop for potential speed
            for j in range(p):
                c_row[j] += aik * b_row_k[j]
    return C
def matmul(A, B):
    return ...
def matmul(A, B):
    return ...
def matmul(A, B):
    """Multiply two matrices using only native Python lists."""
    # Basic dimension checks
    if not A or not B or not A[0] or not B[0]:
        raise ValueError("Input matrices must be non-empty")
    if len(A[0]) != len(B):
        raise ValueError("Number of columns in A must equal number of rows in B")

    # Transpose B to get its columns efficiently
    BT = list(zip(*B))  # each is a tuple: column of B

    # Compute the product
    result = [[sum(a * b for a, b in zip(row, col)) for col in BT] for row in A]
    return result
def matmul(A, B):
    m = len(A)
    p = len(A[0])
    if p==0:
        return [[] for _ in range(m)]
    n = len(B[0])
    # transpose B for cache
    Bt = [[B[i][j] for i in range(p)] for j in range(n)]
    res = [[0]*n for _ in range(m)]
    for i in range(m):
        Ai = A[i]
        Ri = res[i]
        for k in range(p):
            aik = Ai[k]
            Bk = Bt  # not right; we need to use Bt[j][k].
def matmul(A, B): return ...
def matmul(A, B):
    ...
None
def matmul(A, B):
    # A: m x n, B: n x p
    m, n = len(A), len(A[0])
    assert n == len(B), "Incompatible matrices"
    p = len(B[0])
    # pre-transpose B to speed up
    B_T = list(map(list, zip(*B)))  # p x n
    return [[sum(a*b for a, b in zip(row, col)) for col in B_T] for row in A]
None
def matmul(A, B): return ...
def matmul(A, B):
    """Return `A * B` for two 2‑D lists `A` and `B` using pure Python."""
    # Ensure dimensions are compatible
    if not A or not B or not B[0]:
        return []

    # Transpose B once to obtain column access in O(1)
    B_t = list(zip(*B))

    # Compute the matrix product using list comprehensions
    return [
        [sum(a * b for a, b in zip(row, col))
         for col in B_t]  # compute dot product of row with each column of B
        for row in A
    ]
def matmul(A, B):
    """
    Compute the matrix product C = A @ B with pure Python.

    Parameters
    ----------
    A : List[List[float]]
        Left matrix of size m × p.
    B : List[List[float]]
        Right matrix of size p × n.

    Returns
    -------
    List[List[float]]
        Resulting matrix of size m × n.
    """
    m, pA = len(A), len(A[0])
    pB, n = len(B), len(B[0])

    if pA != pB:
        raise ValueError("Inner matrix dimensions must agree (got %d×%d and %d×%d) "
                         % (m, pA, pB, n))

    # Allocate result matrix
    C = [[0.0] * n for _ in range(m)]

    # Blocked matrix multiplication for better cache locality.
    # Choose a block size that works well on most systems
    block = 64

    for i0 in range(0, m, block):
        i_max = min(i0 + block, m)
        for k0 in range(0, pA, block):
            k_max = min(k0 + block, pA)
            for j0 in range(0, n, block):
                j_max = min(j0 + block, n)

                for i in range(i0, i_max):
                    ai = A[i]
                    ci = C[i]
                    for k in range(k0, k_max):
                        aik = ai[k]
                        bk = B[k]
                        for j in range(j0, j_max):
                            ci[j] += aik * bk[j]
    return C
None
def matmul(A, B):
    if not A or not B:
        return []
    m = len(A)
    n = len(A[0])
    p = len(B[0])
    # check compatibility: len(B)==n
    # compute result matrix with nested loops
    res = [[0]*p for _ in range(m)]
    for i in range(m):
         Ai = A[i]
         for j in range(p):
             s = 0
             for k in range(n):
                  s += Ai[k]*B[k][j]
             res[i][j] = s
    return res
def matmul(A, B):
    # check dimensions
    ...
    # compute product
def matmul(A, B):
    # Extract dimensions
    if not A or not B or not A[0] or not B[0]:
        return []  # or raise ValueError...
    # Determine dimensions
    p = len(A[0])  # columns of A, rows of B
    if p != len(B):
        raise ValueError("Incompatible matrix sizes.")
    # Transpose B for efficient column access
    BT = list(zip(*B))
    # Perform multiplication
    result = []
    for row in A:
        new_row = []
        for col in BT:
            # Dot product
            sum_val = 0
            for a, b in zip(row, col):
                sum_val += a * b
            new_row.append(sum_val)
        result.append(new_row)
    return result
def matmul(A, B):
    # optimize: transpose B
    BT = list(zip(*B))  # tuple of tuples (row of B^T)
    return [[sum(a*b for a, b in zip(rowA, colB)) for colB in BT] for rowA in A]
def matmul(A, B):
    n = len(A)
    m = len(B[0])
    p = len(B)
    return [[sum(A[i][k]*B[k][j] for k in range(p)) for j in range(m)] for i in range(n)]
def matmul(A, B):
    """
    Multiply two matrices A and B using pure Python.

    Parameters
    ----------
    A : list[list[float | int | complex]]
        The first matrix of dimensions (m, p).
    B : list[list[float | int | complex]]
        The second matrix of dimensions (p, n).

    Returns
    -------
    list[list[float | int | complex]]
        The product matrix of dimensions (m, n).

    Raises
    ------
    ValueError
        If the inner dimensions do not match.
    """
    # Basic sanity checks for empty/corrupt inputs
    if not A or not B:
        raise ValueError("Input matrices cannot be empty")

    p = len(A[0])
    for row in A:
        if len(row) != p:
            raise ValueError("All rows in A must have the same length")
    q = len(B)
    for row in B:
        if len(row) != len(B[0]):
            raise ValueError("All rows in B must have the same length")

    # Ensure the inner dimensions match
    if p != q:
        raise ValueError("Inner matrix dimensions must agree: len(A[0]) != len(B)")

    # Transpose B to avoid repeated look‑ups in the inner loop
    B_T = list(zip(*B))  # Each element is a tuple representing a column

    # Perform matrix multiplication
    return [[sum(a * b for a, b in zip(row, col)) for col in B_T] for row in A]
def matmul(A, B):
    """Return the matrix product of A and B using native Python."""
    # Transpose B once to avoid repeated column lookups.
    B_T = list(zip(*B))
    # Compute each element of the product using a generator expression.
    return [[sum(a * b for a, b in zip(row, col)) for col in B_T] for row in A]
None
None
None
None
def matmul(A, B):
    if not A or not B: 
        return []
    nrowA = len(A)
    ncolA = len(A[0])
    nrowB = len(B)
    ncolB = len(B[0])
    if ncolA != nrowB:
        raise ValueError("Incompatible dimensions for matrix multiplication")
    # Precompute columns of B
    colsB = list(zip(*B))
    return [
        [sum(a*b for a, b in zip(row, col)) for col in colsB]
        for row in A
    ]

TrainOutput(global_step=100, training_loss=3.209321222243488e-05, metrics={'train_runtime': 5865.3354, 'train_samples_per_second': 0.034, 'train_steps_per_second': 0.017, 'total_flos': 0.0, 'train_loss': 3.209321222243488e-05})

<|start|>system<|message|>You are ChatGPT, a large language model trained by OpenAI.
Knowledge cutoff: 2024-06
Current date: 2025-10-17

Reasoning: low

# Valid channels: analysis, commentary, final. Channel must be included for every message.
Calls to these tools must go to the commentary channel: 'functions'.<|end|><|start|>user<|message|>Create a new fast matrix multiplication function using only native Python code.
You are given a list of list of numbers.
Output your new function in backticks using the format below:
```python
def matmul(A, B):
    return ...
```<|end|><|start|>assistant<|channel|>analysis<|message|>We need to implement fast matmul in pure Python. Could use Strassen or naive? Since only native Python, maybe implement Strassen. Output function code.<|end|><|start|>assistant<|channel|>final<|message|>```python
def matmul(A, B):
    """Fast matrix multiplication using Strassen's algorithm (naive fallback for small sizes)."""
    # Helper to pad matrices to the next power of two
    def _next_pow2(n):
        return 1 << (n - 1).bit_length()

    def _pad(mat, new_size):
        padded = [[0] * new_size for _ in range(new_size)]
        for i, row in enumerate(mat):
            padded[i][:len(row)] = row
        return padded

    def _unpad(mat, rows, cols):
        return [row[:cols] for row in mat[:rows]]

    # Base case: small matrices use simple multiplication
    def _simple_mul(X, Y):
        n, m, p = len(X), len(Y[0]), len(Y)
        return [[sum(X[i][k] * Y[k][j] for k in range(p)) for j in range(m)] for i in range(n)]

    # Strassen recursion
    def _strassen(X, Y):
        n = len(X)
        if n <= 64:  # threshold for switching to naive multiplication
            return _simple_mul(X, Y)

        k = n // 2

        # Divide matrices into quadrants
        A11 = [row[:k] for row in X[:k]]
        A12 = [row[k:] for row in X[:k]]
        A21 = [row[:k] for row in X[k:]]
        A22 = [row[k:] for row in X[k:]]

        B11 = [row[:k] for row in Y[:k]]
        B12 = [row[k:] for row in Y[:k]]
        B21 = [row[:k] for row in Y[k:]]
        B22 = [row[k:] for row in Y[k:]]

        # Compute the 7 products, recursively
        M1 = _strassen(_add(A11, A22), _add(B11, B22))
        M2 = _strassen(_add(A21, A22), B11)
        M3 = _strassen(A11, _sub(B12, B22))
        M4 = _strassen(A22, _sub(B21, B11))
        M5 = _strassen(_add(A11, A12), B22)
        M6 = _strassen(_sub(A21, A11), _add(B11, B12))
        M7 = _strassen(_sub(A12, A22), _add(B21, B22))

        # Combine intermediate products into final quadrants
        C11 = _add(_sub(_add(M1, M4), M5), M7)
        C12 = _add(M3, M5)
        C21 = _add(M2, M4)
        C22 = _add(_sub(_add(M1, M3), M2), M6)

        # Merge quadrants
        C = []
        for i in range(k):
            C.append(C11[i] + C12[i])
        for i in range(k):
            C.append(C21[i] + C22[i])
        return C

    # Basic matrix operations
    def _add(X, Y):
        return [[X[i][j] + Y[i][j] for j in range(len(X[0]))] for i in range(len(X))]

    def _sub(X, Y):
        return [[X[i][j] - Y[i][j] for j in range(len(X[0]))] for i in range(len(X))]

    # Prepare matrices
    n, m = len(A), len(A[0])
    p = len(B[0])
    if len(B) != m:
        raise ValueError("Inner dimensions must agree")
    size = _next_pow2(max(n, m, p))
    A_pad = _pad(A, size)
    B_pad = _pad(B, size)
    # Compute
    C_pad = _strassen(A_pad, B_pad)
    # Trim to original size
    return _unpad(C_pad, n, p)
```<|return|>