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 single matrix multiplication (C = A * B) with a small K dimension
    """
    def __init__(self):
        super(Model, self).__init__()
    
    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        """
        Performs matrix multiplication.

        Args:
            A: Input tensor of shape (M, K).
            B: Input tensor of shape (K, N).

        Returns:
            Output tensor of shape (M, N).
        """
        return torch.matmul(A, B)

M = 16384 * 2
N = 16384 * 2
K = 32 * 2

def get_inputs():
    A = torch.rand(M, K)
    B = torch.rand(K, N)
    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 6 of 6 for this task. Your previous solution (turn 5) was:
```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

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

_cuda = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cuda.h>
#include <cuda_runtime.h>
#include <cuda_fp16.h>
#include <algorithm>

namespace {

constexpr int BM = 128;
constexpr int BN = 128;
constexpr int KK = 64;
constexpr int AS = 72;
constexpr int BS = 136;
constexpr int CS = 136;
constexpr int NT = 256;
constexpr int SMEM = 2 * (BM * AS + KK * BS) * sizeof(__half);

static_assert(SMEM >= BM * CS * sizeof(float),
              "Insufficient shared memory for output staging");

__device__ __forceinline__ unsigned half2_bits(__half2 value)
{
    return static_cast<unsigned>(__half_as_ushort(__low2half(value))) |
           (static_cast<unsigned>(__half_as_ushort(__high2half(value))) << 16);
}

__global__ void pack_planar(
    const float* __restrict__ A,
    const float* __restrict__ B,
    __half* __restrict__ high,
    __half* __restrict__ low,
    int64_t na,
    int64_t total)
{
    for (int64_t i = 4LL * (int64_t(blockIdx.x) * blockDim.x + threadIdx.x);
         i < total;
         i += 4LL * blockDim.x * gridDim.x) {
        const float4 x = i < na
            ? *reinterpret_cast<const float4*>(A + i)
            : *reinterpret_cast<const float4*>(B + i - na);

        const __half2 h0 = __floats2half2_rn(x.x, x.y);
        const __half2 h1 = __floats2half2_rn(x.z, x.w);
        const float2 f0 = __half22float2(h0);
        const float2 f1 = __half22float2(h1);

        const __half2 l0 = __floats2half2_rn(x.x - f0.x, x.y - f0.y);
        const __half2 l1 = __floats2half2_rn(x.z - f1.x, x.w - f1.y);

        *reinterpret_cast<uint2*>(high + i) =
            make_uint2(half2_bits(h0), half2_bits(h1));
        *reinterpret_cast<uint2*>(low + i) =
            make_uint2(half2_bits(l0), half2_bits(l1));
    }
}

__device__ __forceinline__ void copy16(
    __half* destination, const __half* source)
{
#if __CUDA_ARCH__ >= 800
    const unsigned address =
        static_cast<unsigned>(__cvta_generic_to_shared(destination));
    asm volatile(
        "cp.async.cg.shared.global [%0], [%1], 16;\n"
        :: "r"(address), "l"(source) : "memory");
#endif
}

__device__ __forceinline__ void load_a(
    unsigned (&r)[4], const __half* p)
{
#if __CUDA_ARCH__ >= 800
    const unsigned address =
        static_cast<unsigned>(__cvta_generic_to_shared(p));
    asm volatile(
        "ldmatrix.sync.aligned.m8n8.x4.shared.b16 "
        "{%0, %1, %2, %3}, [%4];"
        : "=r"(r[0]), "=r"(r[1]), "=r"(r[2]), "=r"(r[3])
        : "r"(address));
#endif
}

__device__ __forceinline__ void load_b(
    unsigned (&r)[2], const __half* p)
{
#if __CUDA_ARCH__ >= 800
    const unsigned address =
        static_cast<unsigned>(__cvta_generic_to_shared(p));
    asm volatile(
        "ldmatrix.sync.aligned.m8n8.x2.trans.shared.b16 "
        "{%0, %1}, [%2];"
        : "=r"(r[0]), "=r"(r[1])
        : "r"(address));
#endif
}

__device__ __forceinline__ void mma(
    float (&d)[4], const unsigned (&a)[4], const unsigned (&b)[2])
{
#if __CUDA_ARCH__ >= 800
    asm volatile(
        "mma.sync.aligned.m16n8k16.row.col.f32.f16.f16.f32 "
        "{%0, %1, %2, %3}, "
        "{%4, %5, %6, %7}, {%8, %9}, {%0, %1, %2, %3};"
        : "+f"(d[0]), "+f"(d[1]), "+f"(d[2]), "+f"(d[3])
        : "r"(a[0]), "r"(a[1]), "r"(a[2]), "r"(a[3]),
          "r"(b[0]), "r"(b[1]));
#endif
}

__global__ __launch_bounds__(NT, 2)
void multiply(
    const __half* __restrict__ AH,
    const __half* __restrict__ AL,
    const __half* __restrict__ BH,
    const __half* __restrict__ BL,
    float* __restrict__ C,
    int N)
{
#if __CUDA_ARCH__ >= 800
    extern __shared__ __align__(32) unsigned char storage[];
    __half* ah = reinterpret_cast<__half*>(storage);
    __half* al = ah + BM * AS;
    __half* bh = al + BM * AS;
    __half* bl = bh + KK * BS;

    const int tid = threadIdx.x;
    const int lane = tid & 31;
    const int warp = tid >> 5;
    const int row0 = blockIdx.y * BM;
    const int col0 = blockIdx.x * BN;
    const int wm = (warp >> 1) * 32;
    const int wn = (warp & 1) * 64;

    #pragma unroll
    for (int v = tid; v < BM * KK / 8; v += NT) {
        const int row = v / (KK / 8);
        const int k = (v % (KK / 8)) * 8;
        const int64_t src = int64_t(row0 + row) * KK + k;
        copy16(ah + row * AS + k, AH + src);
        copy16(al + row * AS + k, AL + src);
    }

    #pragma unroll
    for (int v = tid; v < KK * BN / 8; v += NT) {
        const int k = v / (BN / 8);
        const int col = (v % (BN / 8)) * 8;
        const int64_t src = int64_t(k) * N + col0 + col;
        copy16(bh + k * BS + col, BH + src);
        copy16(bl + k * BS + col, BL + src);
    }

    asm volatile("cp.async.commit_group;\n" ::: "memory");

    float accum[2][8][4];

    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            #pragma unroll
            for (int q = 0; q < 4; ++q)
                accum[i][j][q] = 0.0f;
        }
    }

    asm volatile("cp.async.wait_group 0;\n" ::: "memory");
    __syncthreads();

    #pragma unroll
    for (int kb = 0; kb < KK; kb += 16) {
        unsigned a_high[2][4];
        unsigned a_low[2][4];

        #pragma unroll
        for (int i = 0; i < 2; ++i) {
            const int offset =
                (wm + i * 16 + (lane & 15)) * AS +
                kb + (lane >> 4) * 8;
            load_a(a_high[i], ah + offset);
            load_a(a_low[i], al + offset);
        }

        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            unsigned b_high[2];
            unsigned b_low[2];
            const int offset =
                (kb + (lane & 15)) * BS + wn + j * 8;
            load_b(b_high, bh + offset);
            load_b(b_low, bl + offset);

            #pragma unroll
            for (int i = 0; i < 2; ++i) {
                mma(accum[i][j], a_low[i], b_high);
                mma(accum[i][j], a_high[i], b_low);
                mma(accum[i][j], a_high[i], b_high);
            }
        }
    }

    __syncthreads();

    float* tile = reinterpret_cast<float*>(storage);
    const int lane_row = lane >> 2;
    const int lane_col = (lane & 3) * 2;

    #pragma unroll
    for (int i = 0; i < 2; ++i) {
        const int row = wm + i * 16 + lane_row;

        #pragma unroll
        for (int j = 0; j < 8; ++j) {
            const int col = wn + j * 8 + lane_col;
            *reinterpret_cast<float2*>(tile + row * CS + col) =
                make_float2(accum[i][j][0], accum[i][j][1]);
            *reinterpret_cast<float2*>(tile + (row + 8) * CS + col) =
                make_float2(accum[i][j][2], accum[i][j][3]);
        }
    }

    __syncthreads();

    #pragma unroll
    for (int index = tid * 4; index < BM * BN; index += NT * 4) {
        const int row = index / BN;
        const int col = index % BN;
        const float4 value =
            *reinterpret_cast<const float4*>(tile + row * CS + col);
        *reinterpret_cast<float4*>(
            C + int64_t(row0 + row) * N + col0 + col) = value;
    }
#endif
}

} // namespace

torch::Tensor small_k_mm(torch::Tensor A, torch::Tensor B)
{
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "CUDA inputs required");
    TORCH_CHECK(A.device() == B.device(), "Device mismatch");
    TORCH_CHECK(A.scalar_type() == at::kFloat &&
                B.scalar_type() == at::kFloat, "Expected float32");
    TORCH_CHECK(A.dim() == 2 && B.dim() == 2, "Expected matrices");
    TORCH_CHECK(A.is_contiguous() && B.is_contiguous(),
                "Expected contiguous inputs");
    TORCH_CHECK(A.size(1) == KK && B.size(0) == KK, "Expected K=64");
    TORCH_CHECK(A.size(0) % BM == 0 && B.size(1) % BN == 0,
                "Dimensions must be multiples of 128");

    const c10::cuda::CUDAGuard guard(A.device());
    const int M = static_cast<int>(A.size(0));
    const int N = static_cast<int>(B.size(1));
    auto C = torch::empty({M, N}, A.options());

    if (M == 0 || N == 0)
        return C;

    C10_CUDA_CHECK(cudaFuncSetAttribute(
        multiply, cudaFuncAttributeMaxDynamicSharedMemorySize, SMEM));

    const int64_t na = A.numel();
    const int64_t total = na + B.numel();
    auto packed = torch::empty({total}, A.options());
    auto* high = reinterpret_cast<__half*>(packed.data_ptr<float>());
    auto* low = high + total;

    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    const int blocks = static_cast<int>(
        std::min<int64_t>((total + 1023) / 1024, 4096));

    pack_planar<<<blocks, 256, 0, stream>>>(
        A.data_ptr<float>(), B.data_ptr<float>(), high, low, na, total);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    multiply<<<dim3(N / BN, M / BM), NT, SMEM, stream>>>(
        high, low, high + na, low + na, C.data_ptr<float>(), N);
    C10_CUDA_KERNEL_LAUNCH_CHECK();

    return C;
}
"""

_extension = load_inline(
    name="small_k_vectorized_mma_v5",
    cpp_sources=_cpp,
    cuda_sources=_cuda,
    functions=["small_k_mm"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()
        self._device_support = {}

    def forward(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
        if (
            A.is_cuda
            and B.is_cuda
            and A.device == B.device
            and A.dtype == torch.float32
            and B.dtype == torch.float32
            and A.ndim == 2
            and B.ndim == 2
            and A.is_contiguous()
            and B.is_contiguous()
            and A.shape[1] == 64
            and B.shape[0] == 64
            and A.shape[0] % 128 == 0
            and B.shape[1] % 128 == 0
            and A.shape[0] <= 128 * 65535
            and B.shape[1] < 2**31
            and A.data_ptr() % 16 == 0
            and B.data_ptr() % 16 == 0
            and not (
                torch.is_grad_enabled()
                and (A.requires_grad or B.requires_grad)
            )
        ):
            device = A.device.index
            if device not in self._device_support:
                props = torch.cuda.get_device_properties(device)
                self._device_support[device] = (
                    props.major >= 8
                    and props.shared_memory_per_block_optin >= 71680
                )
            if self._device_support[device]:
                return _extension.small_k_mm(A, B)
        return torch.matmul(A, B)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 4.67 ms; the PyTorch reference (with TF32 enabled) takes 3.19 ms; speedup 0.683x.
Profiler, top device kernels of one forward pass of your solution:
| device kernel | total device time (us) | calls |
|---|---|---|
| (anonymous namespace)::multiply(__half const*, __half const*, __half const*, __half const* | 4790.2 | 1 |
| (anonymous namespace)::pack_planar(float const*, float const*, __half*, __half*, long, lon | 28.5 | 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.
