You write custom CUDA kernels to replace the pytorch operators in the given architecture to get speedups. 

    You have complete freedom to choose the set of operators you want to replace. You may make the decision to replace some operators with custom CUDA kernels and leave others unchanged. You may replace multiple operators with custom implementations, consider operator fusion opportunities (combining multiple operators into a single kernel, for example, combining matmul+relu), or algorithmic changes (such as online softmax). You are only limited by your imagination.


        Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is: 

        ``` 

        # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
import torch.nn.functional as F


class Model(nn.Module):
    def __init__(self) -> None:
        super().__init__()

    def forward(self, a, b):
        return a + b


def get_inputs():
    # randomly generate input tensors based on the model architecture
    a = torch.randn(1, 128).cuda()
    b = torch.randn(1, 128).cuda()
    return [a, b]


def get_init_inputs():
    # randomly generate tensors required for initialization based on the model architecture
    return []

        ``` 

        The example new arch with custom CUDA kernels looks like this:
        ```
        # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

# Define the custom CUDA kernel for element-wise addition
elementwise_add_source = """
#include <torch/extension.h>
#include <cuda_runtime.h>

__global__ void elementwise_add_kernel(const float* a, const float* b, float* out, int size) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx < size) {
        out[idx] = a[idx] + b[idx];
    }
}

torch::Tensor elementwise_add_cuda(torch::Tensor a, torch::Tensor b) {
    auto size = a.numel();
    auto out = torch::zeros_like(a);

    const int block_size = 256;
    const int num_blocks = (size + block_size - 1) / block_size;

    elementwise_add_kernel<<<num_blocks, block_size>>>(a.data_ptr<float>(), b.data_ptr<float>(), out.data_ptr<float>(), size);

    return out;
}
"""

elementwise_add_cpp_source = (
    "torch::Tensor elementwise_add_cuda(torch::Tensor a, torch::Tensor b);"
)

# Compile the inline CUDA code for element-wise addition
elementwise_add = load_inline(
    name="elementwise_add",
    cpp_sources=elementwise_add_cpp_source,
    cuda_sources=elementwise_add_source,
    functions=["elementwise_add_cuda"],
    verbose=True,
    extra_cflags=[""],
    extra_ldflags=[""],
)


class ModelNew(nn.Module):
    def __init__(self) -> None:
        super().__init__()
        self.elementwise_add = elementwise_add

    def forward(self, a, b):
        return self.elementwise_add.elementwise_add_cuda(a, b)

        ``` 

        
    You are given the following architecture: 

    ```
    # Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the license found in the
# LICENSE file in the root directory of this source tree.

import torch
import torch.nn as nn

class Model(nn.Module):
    """
    Simple model that performs a matrix multiplication (C = A * B) where A and B are lower triangular matrices. 
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, A, B):
        """
        Performs matrix multiplication of lower triangular matrices A and B.

        Args:
            A (torch.Tensor): Lower triangular matrix of shape (N, N).
            B (torch.Tensor): Lower triangular matrix of shape (N, N).

        Returns:
            torch.Tensor: The result of matrix multiplication C of shape (N, N).
        """
        return torch.tril(torch.matmul(A, B))

M = 4096

def get_inputs():
    A = torch.rand(M, M)
    B = torch.rand(M, M)
    A = torch.tril(A)
    B = torch.tril(B)
    return [A, B]

def get_init_inputs():
    return []  # No special initialization inputs needed
    ```
    
Optimize the architecture named Model with custom CUDA operators! Name your optimized output architecture ModelNew. Output the new code in codeblocks. Please generate real code, NOT pseudocode, make sure the code compiles and is fully functional. Just output the new model code, no other text, and NO testing code! 



This is turn 2 of 6 for this task. Your previous solution (turn 1) was:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

_cpp_source = r"""
#include <torch/extension.h>
torch::Tensor triangular_matmul_cuda(torch::Tensor A, torch::Tensor B);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAStream.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>

constexpr int TILE = 64;
constexpr int BK = 32;

__global__ void triangular_matmul_kernel(
    const float* __restrict__ A,
    const float* __restrict__ B,
    float* __restrict__ C,
    const int64_t N
) {
    const int tx = threadIdx.x;
    const int ty = threadIdx.y;
    const int tid = ty * 16 + tx;
    const int64_t row_base = int64_t(blockIdx.y) * TILE;
    const int64_t col_base = int64_t(blockIdx.x) * TILE;

    if (blockIdx.x > blockIdx.y) {
        #pragma unroll
        for (int i = 0; i < 4; ++i) {
            const int64_t row = row_base + ty + i * 16;
            #pragma unroll
            for (int j = 0; j < 4; ++j) {
                const int64_t col = col_base + tx + j * 16;
                if (row < N && col < N)
                    C[row * N + col] = 0.0f;
            }
        }
        return;
    }

    __shared__ float As[TILE][BK];
    __shared__ float Bs[BK][TILE];

    float acc[4][4] = {};

    const int64_t k_end =
        (row_base + TILE < N) ? row_base + TILE : N;

    // Lower-triangular inputs allow only col <= k <= row.
    for (int64_t k_base = col_base; k_base < k_end; k_base += BK) {
        #pragma unroll
        for (int offset = 0; offset < TILE * BK; offset += 256) {
            const int index = tid + offset;
            const int ar = index / BK;
            const int ak = index % BK;
            const int bk = index / TILE;
            const int bc = index % TILE;

            const int64_t a_row = row_base + ar;
            const int64_t a_col = k_base + ak;
            const int64_t b_row = k_base + bk;
            const int64_t b_col = col_base + bc;

            As[ar][ak] = (a_row < N && a_col < N)
                ? A[a_row * N + a_col] : 0.0f;
            Bs[bk][bc] = (b_row < N && b_col < N)
                ? B[b_row * N + b_col] : 0.0f;
        }
        __syncthreads();

        #pragma unroll
        for (int k = 0; k < BK; ++k) {
            float av[4];
            float bv[4];

            #pragma unroll
            for (int i = 0; i < 4; ++i)
                av[i] = As[ty + i * 16][k];

            #pragma unroll
            for (int j = 0; j < 4; ++j)
                bv[j] = Bs[k][tx + j * 16];

            #pragma unroll
            for (int i = 0; i < 4; ++i) {
                #pragma unroll
                for (int j = 0; j < 4; ++j)
                    acc[i][j] = fmaf(av[i], bv[j], acc[i][j]);
            }
        }
        __syncthreads();
    }

    #pragma unroll
    for (int i = 0; i < 4; ++i) {
        const int64_t row = row_base + ty + i * 16;
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const int64_t col = col_base + tx + j * 16;
            if (row < N && col < N)
                C[row * N + col] = col <= row ? acc[i][j] : 0.0f;
        }
    }
}

torch::Tensor triangular_matmul_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "Expected CUDA tensors");
    TORCH_CHECK(A.device() == B.device(), "Inputs must share a device");
    TORCH_CHECK(
        A.scalar_type() == at::kFloat && B.scalar_type() == at::kFloat,
        "Expected float32 tensors"
    );
    TORCH_CHECK(
        A.dim() == 2 && B.dim() == 2 &&
        A.size(0) == A.size(1) &&
        B.size(0) == A.size(0) && B.size(1) == A.size(1),
        "Expected equally sized square matrices"
    );

    const c10::cuda::CUDAGuard guard(A.device());
    A = A.contiguous();
    B = B.contiguous();

    const int64_t N = A.size(0);
    auto C = torch::empty({N, N}, A.options());
    if (N == 0)
        return C;

    const int64_t tiles = (N + TILE - 1) / TILE;
    TORCH_CHECK(tiles <= 65535, "Matrix exceeds supported grid dimensions");

    triangular_matmul_kernel<<<
        dim3(static_cast<unsigned>(tiles), static_cast<unsigned>(tiles)),
        dim3(16, 16),
        0,
        c10::cuda::getCurrentCUDAStream().stream()
    >>>(
        A.data_ptr<float>(),
        B.data_ptr<float>(),
        C.data_ptr<float>(),
        N
    );
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return C;
}
"""

_triangular_ops = load_inline(
    name="triangular_matmul_tiled_fp32",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["triangular_matmul_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, A, B):
        if (
            A.is_cuda
            and B.is_cuda
            and A.dtype == torch.float32
            and B.dtype == torch.float32
        ):
            return _triangular_ops.triangular_matmul_cuda(A, B)
        return torch.tril(torch.matmul(A, B))
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 2.09 ms; the PyTorch reference (with TF32 enabled) takes 1.12 ms; speedup 0.536x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| triangular_matmul_kernel(float const*, float const*, float*, long) | 2065.7 | 1 |

Write an improved complete ModelNew: fix any error, and if the solution is already correct, make it faster. Output the new code in one ```python codeblock. Just the model code, no other text, and NO testing code.
