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):
    """
    Model that performs a matrix multiplication (Gemm), Batch Normalization, scaling, and Softmax.
    """
    def __init__(self, in_features, out_features, bn_eps=1e-5, bn_momentum=0.1, scale_shape=(1,)):
        super(Model, self).__init__()
        self.gemm = nn.Linear(in_features, out_features)
        self.bn = nn.BatchNorm1d(out_features, eps=bn_eps, momentum=bn_momentum)
        self.scale = nn.Parameter(torch.ones(scale_shape))
        self.softmax = nn.Softmax(dim=1)

    def forward(self, x):
        """
        Args:
            x (torch.Tensor): Input tensor of shape (batch_size, in_features).
        Returns:
            torch.Tensor: Output tensor of shape (batch_size, out_features).
        """
        x = self.gemm(x)
        x = self.bn(x)
        x = self.scale * x
        x = self.softmax(x)
        return x

batch_size = 1024
in_features = 8192
out_features = 8192
bn_eps = 1e-5
bn_momentum = 0.1
scale_shape = (1,)

def get_inputs():
    return [torch.rand(batch_size, in_features)]

def get_init_inputs():
    return [in_features, out_features, bn_eps, bn_momentum, scale_shape]
    ```
    
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
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline


_cpp_source = r"""
#include <torch/extension.h>

torch::Tensor fused_forward(
    torch::Tensor input,
    torch::Tensor linear_weight,
    torch::Tensor linear_bias,
    torch::Tensor bn_weight,
    torch::Tensor bn_bias,
    torch::Tensor running_mean,
    torch::Tensor running_var,
    torch::Tensor scale,
    bool training,
    double eps,
    double momentum);
"""

_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cublas_v2.h>
#include <cuda_runtime.h>
#include <cuda_fp16.h>
#include <stdint.h>
#include <math.h>

constexpr int TILE_ROWS = 128;
constexpr int THREADS = 256;

struct __align__(8) Half4 {
    __half2 lo;
    __half2 hi;
};

__global__ void convert_operands_vectorized(
    const float* __restrict__ input,
    const float* __restrict__ weight,
    __half* __restrict__ half_input,
    __half* __restrict__ half_weight,
    int64_t input_vectors,
    int64_t total_vectors)
{
    int64_t i = (int64_t)blockIdx.x * blockDim.x + threadIdx.x;
    int64_t stride = (int64_t)blockDim.x * gridDim.x;

    for (; i < total_vectors; i += stride) {
        float4 v;
        Half4* destination;

        if (i < input_vectors) {
            v = reinterpret_cast<const float4*>(input)[i];
            destination = reinterpret_cast<Half4*>(half_input) + i;
        } else {
            int64_t j = i - input_vectors;
            v = reinterpret_cast<const float4*>(weight)[j];
            destination = reinterpret_cast<Half4*>(half_weight) + j;
        }

        Half4 result;
        result.lo = __floats2half2_rn(v.x, v.y);
        result.hi = __floats2half2_rn(v.z, v.w);
        *destination = result;
    }
}

__global__ void convert_operands_scalar(
    const float* __restrict__ input,
    const float* __restrict__ weight,
    __half* __restrict__ half_input,
    __half* __restrict__ half_weight,
    int64_t input_elements,
    int64_t total_elements)
{
    int64_t i = (int64_t)blockIdx.x * blockDim.x + threadIdx.x;
    int64_t stride = (int64_t)blockDim.x * gridDim.x;

    for (; i < total_elements; i += stride) {
        if (i < input_elements) {
            half_input[i] = __float2half_rn(input[i]);
        } else {
            int64_t j = i - input_elements;
            half_weight[j] = __float2half_rn(weight[j]);
        }
    }
}

__global__ void partial_stats(
    const float* __restrict__ x,
    const float* __restrict__ linear_bias,
    float* __restrict__ partial,
    int rows, int cols, int parts)
{
    __shared__ float means[8][32];
    __shared__ float m2s[8][32];

    int tx = threadIdx.x;
    int ty = threadIdx.y;
    int col = blockIdx.x * 32 + tx;
    int start = blockIdx.y * TILE_ROWS;
    int end = min(start + TILE_ROWS, rows);

    float mean = 0.0f;
    float m2 = 0.0f;
    int count = 0;

    if (col < cols) {
        float b = linear_bias[col];
        for (int row = start + ty; row < end; row += 8) {
            float v = __fadd_rn(x[(int64_t)row * cols + col], b);
            ++count;
            float delta = v - mean;
            mean += delta / count;
            m2 = fmaf(delta, v - mean, m2);
        }
    }

    means[ty][tx] = mean;
    m2s[ty][tx] = m2;
    __syncthreads();

    if (ty == 0 && col < cols) {
        float total_mean = 0.0f;
        float total_m2 = 0.0f;
        int total_count = 0;

        #pragma unroll
        for (int lane = 0; lane < 8; ++lane) {
            int n = max(0, (end - start - lane + 7) / 8);
            if (n > 0) {
                int combined = total_count + n;
                float delta = means[lane][tx] - total_mean;
                total_m2 += m2s[lane][tx]
                    + delta * delta * ((float)total_count * n / combined);
                total_mean += delta * ((float)n / combined);
                total_count = combined;
            }
        }

        int64_t index = (int64_t)blockIdx.y * cols + col;
        partial[index] = total_mean;
        partial[(int64_t)parts * cols + index] = total_m2;
    }
}

__global__ void training_coefficients(
    const float* __restrict__ partial,
    const float* __restrict__ weight,
    const float* __restrict__ bias,
    float* __restrict__ running_mean,
    float* __restrict__ running_var,
    float* __restrict__ coefficients,
    int rows, int cols, int parts,
    float eps, float momentum)
{
    int col = blockIdx.x * blockDim.x + threadIdx.x;
    if (col >= cols) return;

    float mean = 0.0f;
    float m2 = 0.0f;
    int count = 0;

    for (int part = 0; part < parts; ++part) {
        int n = min(TILE_ROWS, rows - part * TILE_ROWS);
        int combined = count + n;
        int64_t index = (int64_t)part * cols + col;
        float delta = partial[index] - mean;

        m2 += partial[(int64_t)parts * cols + index]
            + delta * delta * ((float)count * n / combined);
        mean += delta * ((float)n / combined);
        count = combined;
    }

    coefficients[col] = mean;
    coefficients[cols + col] =
        weight[col] * rsqrtf(fmaxf(m2 / rows, 0.0f) + eps);
    coefficients[2 * cols + col] = bias[col];

    running_mean[col] =
        (1.0f - momentum) * running_mean[col] + momentum * mean;
    running_var[col] =
        (1.0f - momentum) * running_var[col]
        + momentum * (m2 / (rows - 1));
}

__global__ void inference_coefficients(
    const float* __restrict__ weight,
    const float* __restrict__ bias,
    const float* __restrict__ running_mean,
    const float* __restrict__ running_var,
    float* __restrict__ coefficients,
    int cols, float eps)
{
    int col = blockIdx.x * blockDim.x + threadIdx.x;
    if (col >= cols) return;

    coefficients[col] = running_mean[col];
    coefficients[cols + col] = weight[col] * rsqrtf(running_var[col] + eps);
    coefficients[2 * cols + col] = bias[col];
}

template<bool MAXIMUM>
__device__ __forceinline__ float block_reduce(float value)
{
    __shared__ float scratch[8];
    int lane = threadIdx.x & 31;
    int warp = threadIdx.x >> 5;

    #pragma unroll
    for (int offset = 16; offset > 0; offset >>= 1) {
        float other = __shfl_down_sync(0xffffffff, value, offset);
        if (MAXIMUM) value = fmaxf(value, other);
        else value += other;
    }

    if (lane == 0) scratch[warp] = value;
    __syncthreads();

    if (warp == 0) {
        value = lane < 8 ? scratch[lane] : (MAXIMUM ? -INFINITY : 0.0f);

        #pragma unroll
        for (int offset = 16; offset > 0; offset >>= 1) {
            float other = __shfl_down_sync(0xffffffff, value, offset);
            if (MAXIMUM) value = fmaxf(value, other);
            else value += other;
        }

        if (lane == 0) scratch[0] = value;
    }

    __syncthreads();
    float result = scratch[0];
    __syncthreads();
    return result;
}

template<int ITEMS>
__global__ void normalized_softmax(
    const float* __restrict__ x,
    const float* __restrict__ linear_bias,
    const float* __restrict__ coefficients,
    const float* __restrict__ scale,
    float* __restrict__ out,
    int cols, bool scalar_scale)
{
    int tid = threadIdx.x;
    int64_t base = (int64_t)blockIdx.x * cols;
    float scalar = scalar_scale ? scale[0] : 1.0f;

    float values[ITEMS];
    float maximum = -INFINITY;

    #pragma unroll
    for (int i = 0; i < ITEMS; ++i) {
        int col = tid + i * THREADS;
        float value = -INFINITY;

        if (col < cols) {
            float z = __fadd_rn(x[base + col], linear_bias[col]);
            value = fmaf(
                z - coefficients[col],
                coefficients[cols + col],
                coefficients[2 * cols + col]
            ) * (scalar_scale ? scalar : scale[col]);
        }

        values[i] = value;
        maximum = fmaxf(maximum, value);
    }

    maximum = block_reduce<true>(maximum);
    float sum = 0.0f;

    #pragma unroll
    for (int i = 0; i < ITEMS; ++i) {
        values[i] = __expf(values[i] - maximum);
        sum += values[i];
    }

    float inverse_sum = 1.0f / block_reduce<false>(sum);

    #pragma unroll
    for (int i = 0; i < ITEMS; ++i) {
        int col = tid + i * THREADS;
        if (col < cols)
            out[base + col] = values[i] * inverse_sum;
    }
}

torch::Tensor fused_forward(
    torch::Tensor input,
    torch::Tensor linear_weight,
    torch::Tensor linear_bias,
    torch::Tensor bn_weight,
    torch::Tensor bn_bias,
    torch::Tensor running_mean,
    torch::Tensor running_var,
    torch::Tensor scale,
    bool training,
    double eps,
    double momentum)
{
    const c10::cuda::CUDAGuard guard(input.device());
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();

    int rows = input.size(0);
    int inner = input.size(1);
    int cols = linear_weight.size(0);

    auto x = torch::empty({rows, cols}, input.options());
    auto out = torch::empty_like(x);
    if (rows == 0) return out;

    auto half_input = torch::empty(
        {rows, inner}, input.options().dtype(torch::kFloat16));
    auto half_weight = torch::empty(
        {cols, inner}, input.options().dtype(torch::kFloat16));

    int64_t nx = input.numel();
    int64_t nw = linear_weight.numel();

    const float* input_ptr = input.data_ptr<float>();
    const float* weight_ptr = linear_weight.data_ptr<float>();
    __half* half_input_ptr =
        reinterpret_cast<__half*>(half_input.data_ptr<at::Half>());
    __half* half_weight_ptr =
        reinterpret_cast<__half*>(half_weight.data_ptr<at::Half>());

    bool vectorized =
        nx % 4 == 0 && nw % 4 == 0 &&
        reinterpret_cast<uintptr_t>(input_ptr) % 16 == 0 &&
        reinterpret_cast<uintptr_t>(weight_ptr) % 16 == 0;

    if (vectorized) {
        int64_t count = (nx + nw) / 4;
        int blocks = static_cast<int>(
            std::min<int64_t>((count + 255) / 256, 65535));
        convert_operands_vectorized<<<blocks, 256, 0, stream>>>(
            input_ptr, weight_ptr, half_input_ptr, half_weight_ptr,
            nx / 4, count);
    } else {
        int64_t count = nx + nw;
        int blocks = static_cast<int>(
            std::min<int64_t>((count + 255) / 256, 65535));
        convert_operands_scalar<<<blocks, 256, 0, stream>>>(
            input_ptr, weight_ptr, half_input_ptr, half_weight_ptr,
            nx, count);
    }

    cublasHandle_t handle = at::cuda::getCurrentCUDABlasHandle();
    TORCH_CHECK(
        cublasSetStream(handle, stream) == CUBLAS_STATUS_SUCCESS,
        "cublasSetStream failed");

    const float alpha = 1.0f;
    const float beta = 0.0f;

    // FP16 tensor-core operands with FP32 accumulation and FP32 output.
    cublasStatus_t status = cublasGemmEx(
        handle,
        CUBLAS_OP_T, CUBLAS_OP_N,
        cols, rows, inner,
        &alpha,
        half_weight_ptr, CUDA_R_16F, inner,
        half_input_ptr, CUDA_R_16F, inner,
        &beta,
        x.data_ptr<float>(), CUDA_R_32F, cols,
        CUBLAS_COMPUTE_32F,
        CUBLAS_GEMM_DEFAULT_TENSOR_OP);

    TORCH_CHECK(status == CUBLAS_STATUS_SUCCESS,
                "cublasGemmEx failed: ", static_cast<int>(status));

    auto coefficients = torch::empty({3, cols}, input.options());

    if (training) {
        int parts = (rows + TILE_ROWS - 1) / TILE_ROWS;
        auto partial = torch::empty({2, parts, cols}, input.options());

        partial_stats<<<
            dim3((cols + 31) / 32, parts), dim3(32, 8), 0, stream
        >>>(
            x.data_ptr<float>(), linear_bias.data_ptr<float>(),
            partial.data_ptr<float>(), rows, cols, parts);

        training_coefficients<<<(cols + 255) / 256, 256, 0, stream>>>(
            partial.data_ptr<float>(),
            bn_weight.data_ptr<float>(), bn_bias.data_ptr<float>(),
            running_mean.data_ptr<float>(), running_var.data_ptr<float>(),
            coefficients.data_ptr<float>(),
            rows, cols, parts, (float)eps, (float)momentum);
    } else {
        inference_coefficients<<<(cols + 255) / 256, 256, 0, stream>>>(
            bn_weight.data_ptr<float>(), bn_bias.data_ptr<float>(),
            running_mean.data_ptr<float>(), running_var.data_ptr<float>(),
            coefficients.data_ptr<float>(), cols, (float)eps);
    }

    bool scalar_scale = scale.numel() == 1;

    #define LAUNCH(N) \
        normalized_softmax<N><<<rows, THREADS, 0, stream>>>( \
            x.data_ptr<float>(), linear_bias.data_ptr<float>(), \
            coefficients.data_ptr<float>(), scale.data_ptr<float>(), \
            out.data_ptr<float>(), cols, scalar_scale)

    if      (cols <= 256)   { LAUNCH(1); }
    else if (cols <= 512)   { LAUNCH(2); }
    else if (cols <= 1024)  { LAUNCH(4); }
    else if (cols <= 2048)  { LAUNCH(8); }
    else if (cols <= 4096)  { LAUNCH(16); }
    else if (cols <= 8192)  { LAUNCH(32); }
    else                   { LAUNCH(64); }

    #undef LAUNCH

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return out;
}
"""

_ops = load_inline(
    name="half_gemm_fp32_bn_scale_softmax_v3",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["fused_forward"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    extra_ldflags=["-lcublas"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(
        self,
        in_features,
        out_features,
        bn_eps=1e-5,
        bn_momentum=0.1,
        scale_shape=(1,),
    ):
        super().__init__()
        self.gemm = nn.Linear(in_features, out_features)
        self.bn = nn.BatchNorm1d(
            out_features, eps=bn_eps, momentum=bn_momentum
        )
        self.scale = nn.Parameter(torch.ones(scale_shape))
        self.softmax = nn.Softmax(dim=1)

    def forward(self, x):
        cols = self.gemm.out_features
        tensors = (
            self.gemm.weight,
            self.gemm.bias,
            self.bn.weight,
            self.bn.bias,
            self.bn.running_mean,
            self.bn.running_var,
            self.scale,
        )

        supported = (
            x.is_cuda
            and x.dtype == torch.float32
            and x.ndim == 2
            and x.is_contiguous()
            and x.size(1) == self.gemm.in_features
            and x.size(1) > 0
            and 0 < cols <= 16384
            and not torch.is_autocast_enabled()
            and self.bn.affine
            and self.bn.track_running_stats
            and (not self.bn.training or x.size(0) > 1)
            and tuple(self.scale.shape)
            in {(), (1,), (cols,), (1, 1), (1, cols)}
            and all(
                t is not None
                and t.device == x.device
                and t.dtype == torch.float32
                and t.is_contiguous()
                for t in tensors
            )
        )

        if not supported:
            return self.softmax(self.scale * self.bn(self.gemm(x)))

        momentum = self.bn.momentum
        if self.bn.training:
            self.bn.num_batches_tracked.add_(1)
            if momentum is None:
                momentum = 1.0 / self.bn.num_batches_tracked.item()
        elif momentum is None:
            momentum = 0.0

        return _ops.fused_forward(
            x,
            self.gemm.weight,
            self.gemm.bias,
            self.bn.weight,
            self.bn.bias,
            self.bn.running_mean,
            self.bn.running_var,
            self.scale,
            self.bn.training,
            self.bn.eps,
            momentum,
        )
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 1.1 ms; the PyTorch reference (with TF32 enabled) takes 1.34 ms; speedup 1.218x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| void cutlass::Kernel2<cutlass_80_tensorop_s16816gemm_f16_128x128_32x4_tn_align8>(cutlass_8 | 559.1 | 1 |
| convert_operands_vectorized(float const*, float const*, __half*, __half*, long, long) | 338.5 | 1 |
| void normalized_softmax<32>(float const*, float const*, float const*, float const*, float* | 61.1 | 1 |
| partial_stats(float const*, float const*, float*, int, int, int) | 44.9 | 1 |
| training_coefficients(float const*, float const*, float const*, float*, float*, float*, in | 8.6 | 1 |
| void at::native::vectorized_elementwise_kernel<2, at::native::CUDAFunctorOnSelf_add<long>, | 2.3 | 1 |
| Memset (Device) | 1.2 | 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.
