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 5 of 6 for this task. Your previous solution (turn 4) was:
```python
# 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
from torch.utils.cpp_extension import load_inline

# Blocked lower-triangular matmul:
#
#   For lower-triangular A, B the product C = tril(A @ B) satisfies
#       C[i, j] = sum_{s=j}^{i} A[i, s] * B[s, j],
#   so, with column blocks of width bs, only the following block-GEMMs are
#   required (every strictly above-diagonal block is exactly zero):
#
#       C[k:, :k2] += A[k:, k:k2] @ B[k:k2, :k2]        k = 0, bs, 2*bs, ...
#
#   This performs ~4x fewer FLOPs than the dense matmul + tril reference
#   while still running entirely on cuBLAS TF32 tensor-core GEMMs (TF32 is
#   allowed, as in the PyTorch reference).  Details that make it fast:
#     * the loop runs in a single C++ call with raw cublasGemmEx calls
#       (no per-iteration Python slicing / dispatcher overhead);
#     * block k = 0 uses beta = 0 (first touch of its panel), so only
#       columns [bs:, N) need pre-zeroing;
#     * that cudaMemset2DAsync runs on a side stream, overlapped with the
#       first GEMM (they touch disjoint columns of C);
#     * no tril / zero-upper pass is needed afterwards: off-diagonal
#       *blocks* are never written (they are zeroed by the memset), and
#       the strict upper triangle inside each diagonal block is written
#       by its GEMM with exact 0.0f values (every product term has a zero
#       factor).
_CUDA_SRC = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <cublas_v2.h>
#include <cuda_runtime.h>
#include <algorithm>

#define CUBLAS_CHECK(expr)                                          \
  do {                                                              \
    cublasStatus_t _st = (expr);                                    \
    TORCH_CHECK(_st == CUBLAS_STATUS_SUCCESS,                       \
                "cuBLAS error ", (int)_st, " at ", __FILE__, ":",   \
                __LINE__);                                          \
  } while (0)

#define CUDA_CHECK(expr)                                            \
  do {                                                              \
    cudaError_t _err = (expr);                                      \
    TORCH_CHECK(_err == cudaSuccess, "CUDA error: ",                \
                cudaGetErrorString(_err));                          \
  } while (0)

torch::Tensor trimul_lower(torch::Tensor A, torch::Tensor B, int64_t block_size) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "inputs must be CUDA tensors");
    TORCH_CHECK(A.scalar_type() == at::kFloat && B.scalar_type() == at::kFloat,
                "only float32 is supported");
    TORCH_CHECK(A.dim() == 2 && A.size(0) == A.size(1), "A must be square");
    TORCH_CHECK(B.dim() == 2 && B.size(0) == A.size(0) && B.size(1) == A.size(1),
                "B must have the same shape as A");

    const c10::cuda::CUDAGuard device_guard(A.device());
    auto Ac = A.contiguous();
    auto Bc = B.contiguous();
    const int64_t n = Ac.size(0);
    auto C = at::empty({n, n}, Ac.options());
    if (n == 0) {
        return C;
    }
    const int64_t bs = std::max<int64_t>(1, std::min(block_size, n));

    float* __restrict__ Cp = C.data_ptr<float>();
    const float* __restrict__ Ap = Ac.data_ptr<float>();
    const float* __restrict__ Bp = Bc.data_ptr<float>();

    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    cublasHandle_t handle = at::cuda::getCurrentCUDABlasHandle();
    CUBLAS_CHECK(cublasSetStream(handle, stream));

    // TF32 tensor-core math on sm80+, plain FP32 otherwise.
    auto* prop = at::cuda::getCurrentDeviceProperties();
    const cublasComputeType_t ctype =
        (prop->major >= 8) ? CUBLAS_COMPUTE_32F_FAST_TF32 : CUBLAS_COMPUTE_32F;

    const size_t elt = sizeof(float);

    // Pre-zero C[:, bs:] (columns written with beta=1 later).  This runs on a
    // side stream so it overlaps with the first (beta=0) GEMM, which only
    // touches C[:, :bs].  Block 0's panel is fully overwritten, so it never
    // needs zero-initialization.
    cudaStream_t zero_stream = nullptr;
    cudaEvent_t fork_ev, join_ev;
    bool have_side = (bs < n);
    if (have_side) {
        CUDA_CHECK(cudaEventCreateWithFlags(&fork_ev, cudaEventDisableTiming));
        CUDA_CHECK(cudaEventCreateWithFlags(&join_ev, cudaEventDisableTiming));
        zero_stream = at::cuda::getStreamFromPool(/*isHighPriority=*/false,
                                                  A.device().index());
        CUDA_CHECK(cudaEventRecord(fork_ev, stream));
        CUDA_CHECK(cudaStreamWaitEvent(zero_stream, fork_ev, 0));
        CUDA_CHECK(cudaMemset2DAsync(Cp + bs, n * elt, 0, (n - bs) * elt, n,
                                     zero_stream));
        CUDA_CHECK(cudaEventRecord(join_ev, zero_stream));
    }

    const float alpha = 1.0f;
    for (int64_t k = 0; k < n; k += bs) {
        const int64_t k2 = std::min(k + bs, n);
        if (k == bs && have_side) {
            // Ensure the pre-zeroing finished before the first beta=1 GEMM.
            CUDA_CHECK(cudaStreamWaitEvent(stream, join_ev, 0));
        }
        const float beta = (k == 0) ? 0.0f : 1.0f;
        const float* Asub = Ap + k * n + k;   // (n-k) x (k2-k), ld = n
        const float* Bsub = Bp + k * n;       // (k2-k) x k2,    ld = n
        float* Csub = Cp + k * n;             // (n-k) x k2,     ld = n
        // Row-major C = A @ B  <=>  column-major C' = B' @ A'.
        CUBLAS_CHECK(cublasGemmEx(handle, CUBLAS_OP_N, CUBLAS_OP_N,
                                  (int)k2, (int)(n - k), (int)(k2 - k),
                                  &alpha,
                                  Bsub, CUDA_R_32F, (int)n,
                                  Asub, CUDA_R_32F, (int)n,
                                  &beta,
                                  Csub, CUDA_R_32F, (int)n,
                                  ctype, CUBLAS_GEMM_DEFAULT));
    }

    if (have_side) {
        CUDA_CHECK(cudaEventRecord(fork_ev, stream));
        CUDA_CHECK(cudaStreamWaitEvent(zero_stream, fork_ev, 0));
        CUDA_CHECK(cudaEventDestroy(join_ev));
        CUDA_CHECK(cudaEventDestroy(fork_ev));
    }
    return C;
}
"""

_CPP_SRC = "torch::Tensor trimul_lower(torch::Tensor A, torch::Tensor B, int64_t block_size);"

try:
    _ext = load_inline(
        name="trimul_lower_cublas_ext",
        cpp_sources=_CPP_SRC,
        cuda_sources=_CUDA_SRC,
        functions=["trimul_lower"],
        verbose=False,
        extra_cflags=["-O3"],
        extra_cuda_cflags=["-O3"],
        extra_ldflags=["-lcublas", "-lcudart"],
    )
except Exception:
    _ext = None


class ModelNew(nn.Module):
    """
    Optimized C = tril(A @ B) for lower-triangular A, B.

    Evaluates the product as ~N/(2*bs) cuBLAS (TF32 tensor-core) block-GEMMs
    covering only the lower block triangle (~4x fewer FLOPs than dense
    matmul + tril), with the output pre-zeroing overlapped on a side CUDA
    stream.  Falls back to the reference computation for unsupported inputs.
    """

    def __init__(self, block_size: int = 512) -> None:
        super().__init__()
        self.block_size = int(block_size)

    def forward(self, A, B):
        if (
            _ext is not None
            and A.is_cuda and B.is_cuda
            and A.dtype == torch.float32 and B.dtype == torch.float32
            and A.dim() == 2 and A.size(0) == A.size(1)
            and B.dim() == 2 and B.size(0) == A.size(0) and B.size(1) == A.size(1)
        ):
            return _ext.trimul_lower(A, B, self.block_size)
        return torch.tril(torch.matmul(A, B))
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 0.505 ms; the PyTorch reference (with TF32 enabled) takes 1.12 ms; speedup 2.218x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void cutlass::Kernel2<cutlass_80_tensorop_s1688gemm_128x128_16x5_nn_align4>(cutlass_80_ten | 238.2 | 4 |
| void cutlass::Kernel2<cutlass_80_tensorop_s1688gemm_128x128_16x4_nn_align4>(cutlass_80_ten | 123.7 | 2 |
| Memset (Device) | 41.1 | 1 |
| void cutlass::Kernel2<cutlass_80_tensorop_s1688gemm_256x128_16x3_nn_align4>(cutlass_80_ten | 37.5 | 1 |
| void cutlass::Kernel2<cutlass_80_tensorop_s1688gemm_256x128_32x3_nn_align4>(cutlass_80_ten | 35.4 | 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.


Precision rule: the timed forward must compute in FP32 (TF32 tensor-core math is allowed, as in the PyTorch reference). Do not cast inputs, weights or intermediate results to float16 or bfloat16 and do not call half-precision kernels in forward().
