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-scalar multiplication (C = A * s)
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, A: torch.Tensor, s: float) -> torch.Tensor:
        """
        Performs matrix-scalar multiplication.

        Args:
            A: Input matrix of shape (M, N)
            s: Scalar value

        Returns:
            C: Resulting matrix of shape (M, N)
        """
        return A * s

M = 16384 * 4
N = 4096 * 4

def get_inputs():
    A = torch.rand(M, N)
    s = 3.14
    return [A, s]

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 4 of 6 for this task. Your previous solution (turn 3) 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

# Custom CUDA kernel for matrix-scalar multiplication (C = A * s).
# This op is purely memory-bandwidth bound, so the kernel is tuned for
# maximum sustained DRAM throughput:
#   * 128-bit vectorized loads/stores (float4)
#   * 8 independent float4 loads in flight per thread (deep memory-level
#     parallelism to hide latency and saturate the memory controllers)
#   * streaming cache hints (__ldcs / __stcs) since there is zero data reuse
#   * exact grid coverage (no grid-stride loop overhead); only the final
#     block takes a bounds-checked tail path
#   * minimal host-side work (no dispatch, no TensorIterator setup)
scalar_mul_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>
#include <cstdint>
#include <climits>

namespace {

constexpr int  kThreads    = 256;
constexpr int  kUnroll     = 8;                          // float4 per thread
constexpr long kTileFloat4 = (long)kThreads * kUnroll;   // float4 per block

__global__ void __launch_bounds__(kThreads)
scalar_mul_vec4_kernel(const float4* __restrict__ A,
                       float4* __restrict__ C,
                       const float s,
                       const long n4) {
    const long base = (long)blockIdx.x * kTileFloat4;
    const int  t    = threadIdx.x;

    if (base + kTileFloat4 <= n4) {
        // Fast path: full tile, no bounds checks, 8 outstanding 16B loads.
        float4 v[kUnroll];
        #pragma unroll
        for (int i = 0; i < kUnroll; ++i) {
            v[i] = __ldcs(A + (base + t + i * kThreads));
        }
        #pragma unroll
        for (int i = 0; i < kUnroll; ++i) {
            v[i].x *= s;
            v[i].y *= s;
            v[i].z *= s;
            v[i].w *= s;
        }
        #pragma unroll
        for (int i = 0; i < kUnroll; ++i) {
            __stcs(C + (base + t + i * kThreads), v[i]);
        }
    } else {
        // Tail path (at most one block): bounds-checked.
        for (long idx = base + t; idx < n4; idx += kThreads) {
            float4 v = __ldcs(A + idx);
            v.x *= s;
            v.y *= s;
            v.z *= s;
            v.w *= s;
            __stcs(C + idx, v);
        }
    }
}

// Scalar grid-stride fallback for unaligned / non-multiple-of-4 sizes.
__global__ void scalar_mul_fallback_kernel(const float* __restrict__ A,
                                           float* __restrict__ C,
                                           const float s,
                                           const long n) {
    const long stride = (long)gridDim.x * blockDim.x;
    for (long idx = (long)blockIdx.x * blockDim.x + threadIdx.x; idx < n; idx += stride) {
        C[idx] = A[idx] * s;
    }
}

}  // namespace

torch::Tensor scalar_mul_cuda(torch::Tensor A, double s) {
    if (!A.is_cuda() || A.scalar_type() != at::kFloat) {
        return at::mul(A, at::Scalar(s));
    }

    auto Ac = A.is_contiguous() ? A : A.contiguous();
    auto C  = at::empty_like(Ac);

    const long n = (long)Ac.numel();
    if (n == 0) {
        return C;
    }

    const float* a_ptr = Ac.data_ptr<float>();
    float*       c_ptr = C.data_ptr<float>();
    const float  sf    = static_cast<float>(s);

    cudaStream_t stream = at::cuda::getCurrentCUDAStream();

    const bool aligned =
        (((reinterpret_cast<uintptr_t>(a_ptr)) | (reinterpret_cast<uintptr_t>(c_ptr))) & 0xF) == 0;

    if (aligned && ((n & 3) == 0)) {
        const long n4     = n >> 2;
        const long blocks = (n4 + kTileFloat4 - 1) / kTileFloat4;
        if (blocks <= (long)INT_MAX) {
            scalar_mul_vec4_kernel<<<(unsigned int)blocks, kThreads, 0, stream>>>(
                reinterpret_cast<const float4*>(a_ptr),
                reinterpret_cast<float4*>(c_ptr),
                sf, n4);
            C10_CUDA_KERNEL_LAUNCH_CHECK();
            return C;
        }
    }

    long blocks = (n + kThreads - 1) / kThreads;
    const long cap = 1L << 20;
    if (blocks > cap) blocks = cap;
    scalar_mul_fallback_kernel<<<(unsigned int)blocks, kThreads, 0, stream>>>(
        a_ptr, c_ptr, sf, n);
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return C;
}
"""

scalar_mul_cpp_source = "torch::Tensor scalar_mul_cuda(torch::Tensor A, double s);"

# Compile the inline CUDA extension
scalar_mul_ext = load_inline(
    name="scalar_mul_ext_v3",
    cpp_sources=scalar_mul_cpp_source,
    cuda_sources=scalar_mul_cuda_source,
    functions=["scalar_mul_cuda"],
    verbose=False,
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3", "--use_fast_math"],
)


class ModelNew(nn.Module):
    """
    Optimized model that performs a matrix-scalar multiplication (C = A * s)
    using a custom, vectorized, bandwidth-maximizing CUDA kernel.
    """
    def __init__(self):
        super(ModelNew, self).__init__()
        self._mul_scalar = scalar_mul_ext.scalar_mul_cuda

    def forward(self, A: torch.Tensor, s: float) -> torch.Tensor:
        return self._mul_scalar(A, s)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 6.44 ms; the PyTorch reference (with TF32 enabled) takes 6.24 ms; speedup 0.969x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| (anonymous namespace)::scalar_mul_vec4_kernel(float4 const*, float4*, float, long) | 6167.6 | 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.
