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, adds a value, applies Swish, Tanh, GELU, and Hardtanh activation functions.
    """
    def __init__(self, in_features, out_features, add_value_shape):
        super(Model, self).__init__()
        self.matmul = nn.Linear(in_features, out_features)
        self.add_value = nn.Parameter(torch.randn(add_value_shape)) 

    def forward(self, x):
        x = self.matmul(x)
        x = x + self.add_value
        x = torch.sigmoid(x) * x # Swish
        x = torch.tanh(x)
        x = torch.nn.functional.gelu(x) # GELU
        x = torch.nn.functional.hardtanh(x, min_val=-1, max_val=1) # Hardtanh
        return x

batch_size = 1024
in_features = 8192
out_features = 8192
add_value_shape = (out_features,)

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

def get_init_inputs():
    return [in_features, out_features, add_value_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 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 os
import sys
import glob
import site

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


# ---------------------------------------------------------------------------
# Fused epilogue: + combined bias (linear bias + add_value), Swish, Tanh, GELU.
# Note: Tanh maps into (-1, 1) and GELU on (-1, 1) stays within [-0.17, 0.85],
# so the final Hardtanh([-1, 1]) is mathematically an identity and is omitted.
# Fast-but-accurate math: __expf-based sigmoid/tanh (inf-safe), and a Taylor
# polynomial for erf valid on |z| <= 0.7072 (max abs err ~1.2e-7).
# ---------------------------------------------------------------------------
epilogue_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

__device__ __forceinline__ float act_chain(float x) {
    // Swish: x * sigmoid(x)
    float sig = 1.0f / (1.0f + __expf(-x));
    float s = x * sig;
    // Tanh: 1 - 2 / (exp(2s) + 1)  (safe for exp overflow -> inf)
    float u = __expf(2.0f * s);
    float t = 1.0f - __fdividef(2.0f, u + 1.0f);
    // GELU: 0.5 * t * (1 + erf(t / sqrt(2))), |t| <= 1
    float z = t * 0.70710678118654752f;
    float z2 = z * z;
    float p = 0.000120553394f;
    p = fmaf(p, z2, -0.000854832702f);
    p = fmaf(p, z2, 0.00522397763f);
    p = fmaf(p, z2, -0.02686617065f);
    p = fmaf(p, z2, 0.11283791671f);
    p = fmaf(p, z2, -0.37612638903f);
    p = fmaf(p, z2, 1.12837916710f);
    float e = z * p;
    return 0.5f * t * (1.0f + e);
}

template<int UNROLL>
__global__ void fused_bias_act_vec4(const float4* __restrict__ x,
                                    const float4* __restrict__ b,
                                    float4* __restrict__ out,
                                    int n4, int row4) {
    int base = blockIdx.x * (blockDim.x * UNROLL) + threadIdx.x;
    bool pow2 = (row4 & (row4 - 1)) == 0;
    #pragma unroll
    for (int u = 0; u < UNROLL; ++u) {
        int i = base + u * blockDim.x;
        if (i < n4) {
            int j = pow2 ? (i & (row4 - 1)) : (i % row4);
            float4 v = __ldcs(&x[i]);
            float4 q = __ldg(&b[j]);
            v.x = act_chain(v.x + q.x);
            v.y = act_chain(v.y + q.y);
            v.z = act_chain(v.z + q.z);
            v.w = act_chain(v.w + q.w);
            __stcs(&out[i], v);
        }
    }
}

__global__ void fused_bias_act_scalar(const float* __restrict__ x,
                                      const float* __restrict__ b,
                                      float* __restrict__ out,
                                      int n, int row) {
    int i = blockIdx.x * blockDim.x + threadIdx.x;
    if (i < n) {
        out[i] = act_chain(__ldcs(&x[i]) + __ldg(&b[i % row]));
    }
}

torch::Tensor fused_bias_act(torch::Tensor x, torch::Tensor b) {
    TORCH_CHECK(x.is_cuda() && b.is_cuda(), "Inputs must be CUDA tensors");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "Input must be float32");
    auto xc = x.contiguous();
    auto bc = b.contiguous();
    auto out = torch::empty_like(xc);

    int64_t n = xc.numel();
    int64_t row = bc.numel();
    if (n == 0) return out;

    const int threads = 256;
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    if ((row % 4 == 0) && (n % 4 == 0)) {
        int64_t n4 = n / 4;
        int64_t row4 = row / 4;
        constexpr int UNROLL = 4;
        int64_t per_block = (int64_t)threads * UNROLL;
        int blocks = (int)((n4 + per_block - 1) / per_block);
        fused_bias_act_vec4<UNROLL><<<blocks, threads, 0, stream>>>(
            reinterpret_cast<const float4*>(xc.data_ptr<float>()),
            reinterpret_cast<const float4*>(bc.data_ptr<float>()),
            reinterpret_cast<float4*>(out.data_ptr<float>()),
            (int)n4, (int)row4);
    } else {
        int blocks = (int)((n + threads - 1) / threads);
        fused_bias_act_scalar<<<blocks, threads, 0, stream>>>(
            xc.data_ptr<float>(),
            bc.data_ptr<float>(),
            out.data_ptr<float>(),
            (int)n, (int)row);
    }
    return out;
}
"""

epilogue_cpp_source = "torch::Tensor fused_bias_act(torch::Tensor x, torch::Tensor b);"

epilogue_ext = None
try:
    epilogue_ext = load_inline(
        name="fused_bias_act_v6",
        cpp_sources=epilogue_cpp_source,
        cuda_sources=epilogue_source,
        functions=["fused_bias_act"],
        verbose=False,
        extra_cflags=["-O3"],
        extra_cuda_cflags=["-O3"],
    )
except Exception:
    epilogue_ext = None


# ---------------------------------------------------------------------------
# cuBLASLt GEMM wrapper with per-shape autotuning over heuristic algorithms,
# workspace sizes (so split-K variants become available) and weight layouts
# (TN via W, NN via a pre-transposed W). FP32 compute with TF32 tensor cores
# (CUBLAS_COMPUTE_32F_FAST_TF32), as permitted.
# ---------------------------------------------------------------------------
lt_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <cublasLt.h>
#include <vector>
#include <map>
#include <mutex>
#include <memory>
#include <string>

#define CULT_CHECK(x) do { cublasStatus_t _s = (x); TORCH_CHECK(_s == CUBLAS_STATUS_SUCCESS, "cublasLt error ", (int)_s); } while (0)

namespace {

struct LtPlan {
    cublasLtMatmulDesc_t opDesc = nullptr;
    cublasLtMatrixLayout_t Adesc = nullptr, Bdesc = nullptr, Cdesc = nullptr;
    std::vector<cublasLtMatmulHeuristicResult_t> heur;
    void* workspace = nullptr;
    size_t ws_bytes = 0;
    ~LtPlan() {
        if (opDesc) cublasLtMatmulDescDestroy(opDesc);
        if (Adesc) cublasLtMatrixLayoutDestroy(Adesc);
        if (Bdesc) cublasLtMatrixLayoutDestroy(Bdesc);
        if (Cdesc) cublasLtMatrixLayoutDestroy(Cdesc);
        if (workspace) cudaFree(workspace);
    }
};

std::string make_key(int64_t M, int64_t N, int64_t K, bool trans_w, size_t ws) {
    return std::to_string(M) + "_" + std::to_string(N) + "_" + std::to_string(K)
           + (trans_w ? "_t_" : "_n_") + std::to_string(ws);
}

cublasLtHandle_t lt_handle() {
    static cublasLtHandle_t h = nullptr;
    if (!h) CULT_CHECK(cublasLtCreate(&h));
    return h;
}

std::map<std::string, std::shared_ptr<LtPlan>>& plan_cache() {
    static std::map<std::string, std::shared_ptr<LtPlan>> c;
    return c;
}
std::mutex& cache_mutex() { static std::mutex m; return m; }

// Row-major C[M,N] = X[M,K] * W^T, computed as col-major:
// C_cm[N,M] = A[N,K] * B[K,M], where B is x viewed col-major and A is the
// weight viewed col-major (trans_w=false: W is [N,K] row-major, use OP_T;
// trans_w=true: Wt is [K,N] row-major, use OP_N).
std::shared_ptr<LtPlan> get_plan(int64_t M, int64_t N, int64_t K, bool trans_w, size_t ws_bytes) {
    std::string key = make_key(M, N, K, trans_w, ws_bytes);
    std::lock_guard<std::mutex> lock(cache_mutex());
    auto it = plan_cache().find(key);
    if (it != plan_cache().end()) return it->second;

    auto plan = std::make_shared<LtPlan>();
    CULT_CHECK(cublasLtMatmulDescCreate(&plan->opDesc, CUBLAS_COMPUTE_32F_FAST_TF32, CUDA_R_32F));
    cublasOperation_t opA = trans_w ? CUBLAS_OP_T : CUBLAS_OP_N;
    cublasOperation_t opB = CUBLAS_OP_N;
    CULT_CHECK(cublasLtMatmulDescSetAttribute(plan->opDesc, CUBLASLT_MATMUL_DESC_TRANSA, &opA, sizeof(opA)));
    CULT_CHECK(cublasLtMatmulDescSetAttribute(plan->opDesc, CUBLASLT_MATMUL_DESC_TRANSB, &opB, sizeof(opB)));

    if (trans_w) {
        CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Adesc, CUDA_R_32F, K, N, K));
    } else {
        CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Adesc, CUDA_R_32F, N, K, N));
    }
    CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Bdesc, CUDA_R_32F, K, M, K));
    CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Cdesc, CUDA_R_32F, N, M, N));

    plan->ws_bytes = ws_bytes;
    if (ws_bytes > 0) {
        TORCH_CHECK(cudaMalloc(&plan->workspace, ws_bytes) == cudaSuccess, "cublasLt workspace alloc failed");
    }

    cublasLtMatmulPreference_t pref = nullptr;
    CULT_CHECK(cublasLtMatmulPreferenceCreate(&pref));
    CULT_CHECK(cublasLtMatmulPreferenceSetAttribute(pref, CUBLASLT_MATMUL_PREF_MAX_WORKSPACE_BYTES,
                                                    &ws_bytes, sizeof(ws_bytes)));

    const int max_results = 64;
    plan->heur.resize(max_results);
    int returned = 0;
    CULT_CHECK(cublasLtMatmulAlgoGetHeuristic(lt_handle(), plan->opDesc,
                                              plan->Adesc, plan->Bdesc, plan->Cdesc, plan->Cdesc,
                                              pref, max_results, plan->heur.data(), &returned));
    plan->heur.resize(returned);
    cublasLtMatmulPreferenceDestroy(pref);

    std::vector<cublasLtMatmulHeuristicResult_t> good;
    good.reserve(plan->heur.size());
    for (auto& h : plan->heur) {
        if (h.state == CUBLAS_STATUS_SUCCESS && h.workspaceSize <= ws_bytes) good.push_back(h);
    }
    plan->heur.swap(good);
    TORCH_CHECK(!plan->heur.empty(), "cublasLt returned no algorithms");

    plan_cache()[key] = plan;
    return plan;
}

} // namespace

int64_t lt_num_algos(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes) {
    auto plan = get_plan(M, N, K, trans_w, (size_t)ws_bytes);
    return (int64_t)plan->heur.size();
}

void lt_gemm(torch::Tensor x, torch::Tensor w, torch::Tensor out, bool trans_w, int64_t algo_idx, int64_t ws_bytes) {
    int64_t M = x.size(0);
    int64_t K = x.size(1);
    int64_t N = out.size(1);
    auto plan = get_plan(M, N, K, trans_w, (size_t)ws_bytes);
    TORCH_CHECK(algo_idx >= 0 && algo_idx < (int64_t)plan->heur.size(), "invalid algo index");

    float alpha = 1.0f;
    float beta = 0.0f;
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    CULT_CHECK(cublasLtMatmul(lt_handle(), plan->opDesc, &alpha,
                              w.data_ptr<float>(), plan->Adesc,
                              x.data_ptr<float>(), plan->Bdesc,
                              &beta,
                              out.data_ptr<float>(), plan->Cdesc,
                              out.data_ptr<float>(), plan->Cdesc,
                              &plan->heur[algo_idx].algo,
                              plan->workspace, plan->ws_bytes,
                              stream));
}

void lt_keep_only(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes) {
    std::string prefix = std::to_string(M) + "_" + std::to_string(N) + "_" + std::to_string(K) + "_";
    std::string keep = make_key(M, N, K, trans_w, (size_t)ws_bytes);
    std::lock_guard<std::mutex> lock(cache_mutex());
    auto& c = plan_cache();
    for (auto it = c.begin(); it != c.end();) {
        if (it->first.compare(0, prefix.size(), prefix) == 0 && it->first != keep) {
            it = c.erase(it);
        } else {
            ++it;
        }
    }
}
"""

lt_cpp_source = (
    "int64_t lt_num_algos(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes);\n"
    "void lt_gemm(torch::Tensor x, torch::Tensor w, torch::Tensor out, bool trans_w, int64_t algo_idx, int64_t ws_bytes);\n"
    "void lt_keep_only(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes);"
)


def _cublaslt_paths():
    inc_dirs, lib_dirs = [], []
    roots = []
    try:
        roots += site.getsitepackages()
    except Exception:
        pass
    try:
        roots.append(os.path.dirname(os.path.dirname(os.path.abspath(torch.__file__))))
    except Exception:
        pass
    roots.append(sys.prefix)
    roots.append(os.environ.get("CUDA_HOME", "/usr/local/cuda"))
    roots.append("/usr/local/cuda")
    for r in dict.fromkeys(roots):
        for sub in ("lib64", "lib",
                    os.path.join("nvidia", "cublas", "lib"),
                    os.path.join("torch", "lib")):
            p = os.path.join(r, sub)
            if os.path.isdir(p):
                lib_dirs.append(p)
        for sub in ("include", os.path.join("nvidia", "cublas", "include")):
            p = os.path.join(r, sub)
            if os.path.isdir(p):
                inc_dirs.append(p)
    lib_file = None
    for d in lib_dirs:
        for pat in ("libcublasLt.so", "libcublasLt.so.*"):
            hits = sorted(glob.glob(os.path.join(d, pat)))
            hits = [h for h in hits if os.path.isfile(h)]
            if hits:
                lib_file = hits[0]
                break
        if lib_file:
            break
    if lib_file:
        ldflags = [lib_file, "-Wl,-rpath," + os.path.dirname(lib_file)]
    else:
        ldflags = ["-lcublasLt"]
    return inc_dirs, ldflags


lt_ext = None
try:
    _inc, _ld = _cublaslt_paths()
    lt_ext = load_inline(
        name="cublaslt_gemm_autotune_v3",
        cpp_sources=lt_cpp_source,
        cuda_sources=lt_source,
        functions=["lt_num_algos", "lt_gemm", "lt_keep_only"],
        verbose=False,
        extra_cflags=["-O3"],
        extra_cuda_cflags=["-O3"],
        extra_include_paths=_inc,
        extra_ldflags=_ld,
    )
except Exception:
    lt_ext = None


class ModelNew(nn.Module):
    """
    Optimized model:
      * GEMM (FP32 with TF32 tensor cores) via cuBLASLt, autotuned per shape over
        heuristic algorithms, workspace sizes (unlocking split-K) and weight
        layouts (TN / pre-transposed NN). Falls back to the PyTorch cuBLAS path
        when cublasLt is unavailable.
      * Single fused CUDA epilogue kernel: combined bias (linear bias + add_value)
        + Swish + Tanh + GELU. The trailing Hardtanh([-1, 1]) is provably an
        identity after Tanh -> GELU and is dropped.
      * Whole forward is captured into a CUDA graph (keyed by input/parameter
        buffer signatures) and replayed on subsequent calls to remove host and
        launch overhead; any mismatch or failure falls back to the eager path.
    """
    def __init__(self, in_features, out_features, add_value_shape):
        super(ModelNew, self).__init__()
        # TF32 tensor-core math is explicitly permitted for FP32 workloads.
        torch.backends.cuda.matmul.allow_tf32 = True
        torch.backends.cudnn.allow_tf32 = True

        self.matmul = nn.Linear(in_features, out_features)
        self.add_value = nn.Parameter(torch.randn(add_value_shape))

        self._ws_list = [0, 32 << 20, 64 << 20, 128 << 20, 256 << 20]
        self._topk = 6
        self._gemm_choice = {}
        self._wt = None
        self._wt_ver = None
        self._comb = None
        self._comb_ver = None

        self._graphs = {}
        self._counts = {}
        self._bad = set()
        self._max_graphs = 16
        self._graphs_ok = hasattr(torch.cuda, "CUDAGraph") and hasattr(torch.cuda, "graph")

    # ----- cached derived buffers -------------------------------------------
    def _get_wt(self):
        w = self.matmul.weight
        ver = (w.data_ptr(), w._version)
        if self._wt is None or self._wt_ver != ver:
            with torch.no_grad():
                self._wt = w.t().contiguous()
            self._wt_ver = ver
        return self._wt

    def _get_comb(self):
        b = self.matmul.bias
        av = self.add_value
        ver = (av.data_ptr(), av._version,
               0 if b is None else b.data_ptr(), -1 if b is None else b._version)
        if self._comb is None or self._comb_ver != ver:
            with torch.no_grad():
                self._comb = av.detach().clone() if b is None else (av + b).contiguous()
            self._comb_ver = ver
        return self._comb

    # ----- reference / eager paths ------------------------------------------
    def _reference(self, x):
        y = F.linear(x, self.matmul.weight, self.matmul.bias)
        y = y + self.add_value
        y = torch.sigmoid(y) * y
        y = torch.tanh(y)
        y = F.gelu(y)
        y = F.hardtanh(y, min_val=-1, max_val=1)
        return y

    @torch.no_grad()
    def _autotune(self, x):
        M, K = x.shape
        N = self.matmul.weight.shape[0]
        w = self.matmul.weight
        wt = self._get_wt()
        out = torch.empty(M, N, device=x.device, dtype=torch.float32)

        cands = []
        cands.append((("torch_tn",), lambda: torch.matmul(x, w.t(), out=out)))
        cands.append((("torch_nn",), lambda: torch.mm(x, wt, out=out)))

        if lt_ext is not None:
            xc = x if x.is_contiguous() else x.contiguous()
            for trans_w, wbuf in ((False, w), (True, wt)):
                for ws in self._ws_list:
                    try:
                        n_alg = int(lt_ext.lt_num_algos(M, N, K, trans_w, ws))
                    except Exception:
                        continue
                    for i in range(min(n_alg, self._topk)):
                        cands.append((("lt", trans_w, i, ws),
                                      (lambda wb=wbuf, tw=trans_w, ii=i, ww=ws:
                                       lt_ext.lt_gemm(xc, wb, out, tw, ii, ww))))

        best, best_t = None, float("inf")
        for tag, fn in cands:
            try:
                for _ in range(3):
                    fn()
                torch.cuda.synchronize(x.device)
                s = torch.cuda.Event(enable_timing=True)
                e = torch.cuda.Event(enable_timing=True)
                s.record()
                for _ in range(6):
                    fn()
                e.record()
                torch.cuda.synchronize(x.device)
                t = s.elapsed_time(e)
                if t < best_t:
                    best_t, best = t, tag
            except Exception:
                continue
        if best is None:
            best = ("torch_tn",)

        # Free losing cublasLt plans (and their workspaces) for this shape.
        if lt_ext is not None:
            try:
                if best[0] == "lt":
                    lt_ext.lt_keep_only(M, N, K, best[1], best[3])
                else:
                    lt_ext.lt_keep_only(M, N, K, False, -1)
            except Exception:
                pass
        return best

    def _run_gemm(self, x, choice):
        w = self.matmul.weight
        N = w.shape[0]
        M = x.shape[0]
        tag = choice[0]
        if tag == "lt":
            _, trans_w, idx, ws = choice
            if not x.is_contiguous():
                x = x.contiguous()
            wbuf = self._get_wt() if trans_w else w
            out = torch.empty(M, N, device=x.device, dtype=torch.float32)
            lt_ext.lt_gemm(x, wbuf, out, trans_w, idx, ws)
            return out
        if tag == "torch_nn":
            return torch.mm(x, self._get_wt())
        return torch.matmul(x, w.t())

    def _eager(self, x):
        key = (tuple(x.shape), x.device.index)
        choice = self._gemm_choice.get(key)
        if choice is None:
            choice = self._autotune(x)
            self._gemm_choice[key] = choice
        y = self._run_gemm(x, choice)
        if epilogue_ext is not None:
            return epilogue_ext.fused_bias_act(y, self._get_comb())
        y = y + self._get_comb()
        y = torch.sigmoid(y) * y
        y = torch.tanh(y)
        y = F.gelu(y)
        return y

    def _capture(self, x, key):
        # Warm up on a side stream (materializes workspaces, caches, etc.).
        s = torch.cuda.Stream(x.device)
        s.wait_stream(torch.cuda.current_stream(x.device))
        with torch.cuda.stream(s):
            for _ in range(3):
                self._eager(x)
        torch.cuda.current_stream(x.device).wait_stream(s)
        torch.cuda.synchronize(x.device)

        g = torch.cuda.CUDAGraph()
        with torch.cuda.graph(g):
            static_out = self._eager(x)
        self._graphs[key] = (g, static_out)

    def forward(self, x):
        if (self._graphs_ok and x.is_cuda and x.dim() == 2
                and x.dtype == torch.float32 and self.matmul.weight.is_cuda):
            w = self.matmul.weight
            b = self.matmul.bias
            key = (x.data_ptr(), tuple(x.shape), tuple(x.stride()), x.device.index,
                   w.data_ptr(), w._version,
                   0 if b is None else b.data_ptr(), -1 if b is None else b._version,
                   self.add_value.data_ptr(), self.add_value._version)
            ent = self._graphs.get(key)
            if ent is not None:
                try:
                    ent[0].replay()
                    return ent[1]
                except Exception:
                    self._graphs.pop(key, None)
                    self._bad.add(key)
            elif key not in self._bad:
                n = self._counts.get(key, 0) + 1
                self._counts[key] = n
                if n >= 3 and len(self._graphs) < self._max_graphs:
                    try:
                        self._capture(x, key)
                        ent = self._graphs.get(key)
                        if ent is not None:
                            ent[0].replay()
                            return ent[1]
                    except Exception:
                        self._graphs.pop(key, None)
                        self._bad.add(key)
            return self._eager(x)
        # Non-CUDA fallback: replicate the reference computation exactly.
        return self._reference(x)
```
Result of evaluating it:
The solution is correct on the test inputs. Mean runtime 1.12 ms; the PyTorch reference (with TF32 enabled) takes 1.45 ms; speedup 1.295x.
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_256x128_32x3_nn_align4>(cutlass_80_ten | 1080.3 | 1 |
| void fused_bias_act_vec4<4>(float4 const*, float4 const*, float4*, int, int) | 43.6 | 1 |
| Memset (Unknown) | 1.2 | 1 |

The fastest correct solution so far (1.295x speedup) is:
```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 os
import sys
import glob
import site

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


# ---------------------------------------------------------------------------
# Fused epilogue: + combined bias (linear bias + add_value), Swish, Tanh, GELU.
# Note: Tanh maps into (-1, 1) and GELU on (-1, 1) stays within [-0.17, 0.85],
# so the final Hardtanh([-1, 1]) is mathematically an identity and is omitted.
# Fast-but-accurate math: __expf-based sigmoid/tanh (inf-safe), and a Taylor
# polynomial for erf valid on |z| <= 0.7072 (max abs err ~1.2e-7).
# ---------------------------------------------------------------------------
epilogue_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

__device__ __forceinline__ float act_chain(float x) {
    // Swish: x * sigmoid(x)
    float sig = 1.0f / (1.0f + __expf(-x));
    float s = x * sig;
    // Tanh: 1 - 2 / (exp(2s) + 1)  (safe for exp overflow -> inf)
    float u = __expf(2.0f * s);
    float t = 1.0f - __fdividef(2.0f, u + 1.0f);
    // GELU: 0.5 * t * (1 + erf(t / sqrt(2))), |t| <= 1
    float z = t * 0.70710678118654752f;
    float z2 = z * z;
    float p = 0.000120553394f;
    p = fmaf(p, z2, -0.000854832702f);
    p = fmaf(p, z2, 0.00522397763f);
    p = fmaf(p, z2, -0.02686617065f);
    p = fmaf(p, z2, 0.11283791671f);
    p = fmaf(p, z2, -0.37612638903f);
    p = fmaf(p, z2, 1.12837916710f);
    float e = z * p;
    return 0.5f * t * (1.0f + e);
}

template<int UNROLL>
__global__ void fused_bias_act_vec4(const float4* __restrict__ x,
                                    const float4* __restrict__ b,
                                    float4* __restrict__ out,
                                    int n4, int row4) {
    int base = blockIdx.x * (blockDim.x * UNROLL) + threadIdx.x;
    bool pow2 = (row4 & (row4 - 1)) == 0;
    #pragma unroll
    for (int u = 0; u < UNROLL; ++u) {
        int i = base + u * blockDim.x;
        if (i < n4) {
            int j = pow2 ? (i & (row4 - 1)) : (i % row4);
            float4 v = __ldcs(&x[i]);
            float4 q = __ldg(&b[j]);
            v.x = act_chain(v.x + q.x);
            v.y = act_chain(v.y + q.y);
            v.z = act_chain(v.z + q.z);
            v.w = act_chain(v.w + q.w);
            __stcs(&out[i], v);
        }
    }
}

__global__ void fused_bias_act_scalar(const float* __restrict__ x,
                                      const float* __restrict__ b,
                                      float* __restrict__ out,
                                      int n, int row) {
    int i = blockIdx.x * blockDim.x + threadIdx.x;
    if (i < n) {
        out[i] = act_chain(__ldcs(&x[i]) + __ldg(&b[i % row]));
    }
}

torch::Tensor fused_bias_act(torch::Tensor x, torch::Tensor b) {
    TORCH_CHECK(x.is_cuda() && b.is_cuda(), "Inputs must be CUDA tensors");
    TORCH_CHECK(x.scalar_type() == at::kFloat, "Input must be float32");
    auto xc = x.contiguous();
    auto bc = b.contiguous();
    auto out = torch::empty_like(xc);

    int64_t n = xc.numel();
    int64_t row = bc.numel();
    if (n == 0) return out;

    const int threads = 256;
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    if ((row % 4 == 0) && (n % 4 == 0)) {
        int64_t n4 = n / 4;
        int64_t row4 = row / 4;
        constexpr int UNROLL = 4;
        int64_t per_block = (int64_t)threads * UNROLL;
        int blocks = (int)((n4 + per_block - 1) / per_block);
        fused_bias_act_vec4<UNROLL><<<blocks, threads, 0, stream>>>(
            reinterpret_cast<const float4*>(xc.data_ptr<float>()),
            reinterpret_cast<const float4*>(bc.data_ptr<float>()),
            reinterpret_cast<float4*>(out.data_ptr<float>()),
            (int)n4, (int)row4);
    } else {
        int blocks = (int)((n + threads - 1) / threads);
        fused_bias_act_scalar<<<blocks, threads, 0, stream>>>(
            xc.data_ptr<float>(),
            bc.data_ptr<float>(),
            out.data_ptr<float>(),
            (int)n, (int)row);
    }
    return out;
}
"""

epilogue_cpp_source = "torch::Tensor fused_bias_act(torch::Tensor x, torch::Tensor b);"

epilogue_ext = None
try:
    epilogue_ext = load_inline(
        name="fused_bias_act_v4",
        cpp_sources=epilogue_cpp_source,
        cuda_sources=epilogue_source,
        functions=["fused_bias_act"],
        verbose=False,
        extra_cflags=["-O3"],
        extra_cuda_cflags=["-O3"],
    )
except Exception:
    epilogue_ext = None


# ---------------------------------------------------------------------------
# cuBLASLt GEMM wrapper with per-shape autotuning over heuristic algorithms
# and over weight layouts (TN via W, NN via a pre-transposed W). FP32 compute
# with TF32 tensor cores (CUBLAS_COMPUTE_32F_FAST_TF32), as permitted.
# ---------------------------------------------------------------------------
lt_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>
#include <cublasLt.h>
#include <vector>
#include <map>
#include <mutex>
#include <memory>
#include <string>

#define CULT_CHECK(x) do { cublasStatus_t _s = (x); TORCH_CHECK(_s == CUBLAS_STATUS_SUCCESS, "cublasLt error ", (int)_s); } while (0)

namespace {

struct LtPlan {
    cublasLtMatmulDesc_t opDesc = nullptr;
    cublasLtMatrixLayout_t Adesc = nullptr, Bdesc = nullptr, Cdesc = nullptr;
    std::vector<cublasLtMatmulHeuristicResult_t> heur;
    void* workspace = nullptr;
    size_t ws_bytes = 0;
    ~LtPlan() {
        if (opDesc) cublasLtMatmulDescDestroy(opDesc);
        if (Adesc) cublasLtMatrixLayoutDestroy(Adesc);
        if (Bdesc) cublasLtMatrixLayoutDestroy(Bdesc);
        if (Cdesc) cublasLtMatrixLayoutDestroy(Cdesc);
        if (workspace) cudaFree(workspace);
    }
};

cublasLtHandle_t lt_handle() {
    static cublasLtHandle_t h = nullptr;
    if (!h) CULT_CHECK(cublasLtCreate(&h));
    return h;
}

std::map<std::string, std::shared_ptr<LtPlan>>& plan_cache() {
    static std::map<std::string, std::shared_ptr<LtPlan>> c;
    return c;
}
std::mutex& cache_mutex() { static std::mutex m; return m; }

// Row-major C[M,N] = X[M,K] * W^T, computed as col-major:
// C_cm[N,M] = A[N,K] * B[K,M], where B is x viewed col-major and A is the
// weight viewed col-major (trans_w=true: W is [N,K] row-major, use OP_T;
// trans_w=false: Wt is [K,N] row-major, use OP_N).
std::shared_ptr<LtPlan> get_plan(int64_t M, int64_t N, int64_t K, bool trans_w, size_t ws_bytes) {
    std::string key = std::to_string(M) + "_" + std::to_string(N) + "_" + std::to_string(K)
                      + (trans_w ? "_t_" : "_n_") + std::to_string(ws_bytes);
    std::lock_guard<std::mutex> lock(cache_mutex());
    auto it = plan_cache().find(key);
    if (it != plan_cache().end()) return it->second;

    auto plan = std::make_shared<LtPlan>();
    CULT_CHECK(cublasLtMatmulDescCreate(&plan->opDesc, CUBLAS_COMPUTE_32F_FAST_TF32, CUDA_R_32F));
    cublasOperation_t opA = trans_w ? CUBLAS_OP_T : CUBLAS_OP_N;
    cublasOperation_t opB = CUBLAS_OP_N;
    CULT_CHECK(cublasLtMatmulDescSetAttribute(plan->opDesc, CUBLASLT_MATMUL_DESC_TRANSA, &opA, sizeof(opA)));
    CULT_CHECK(cublasLtMatmulDescSetAttribute(plan->opDesc, CUBLASLT_MATMUL_DESC_TRANSB, &opB, sizeof(opB)));

    if (trans_w) {
        CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Adesc, CUDA_R_32F, K, N, K));
    } else {
        CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Adesc, CUDA_R_32F, N, K, N));
    }
    CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Bdesc, CUDA_R_32F, K, M, K));
    CULT_CHECK(cublasLtMatrixLayoutCreate(&plan->Cdesc, CUDA_R_32F, N, M, N));

    plan->ws_bytes = ws_bytes;
    if (ws_bytes > 0) {
        TORCH_CHECK(cudaMalloc(&plan->workspace, ws_bytes) == cudaSuccess, "cublasLt workspace alloc failed");
    }

    cublasLtMatmulPreference_t pref = nullptr;
    CULT_CHECK(cublasLtMatmulPreferenceCreate(&pref));
    CULT_CHECK(cublasLtMatmulPreferenceSetAttribute(pref, CUBLASLT_MATMUL_PREF_MAX_WORKSPACE_BYTES,
                                                    &ws_bytes, sizeof(ws_bytes)));

    const int max_results = 64;
    plan->heur.resize(max_results);
    int returned = 0;
    CULT_CHECK(cublasLtMatmulAlgoGetHeuristic(lt_handle(), plan->opDesc,
                                              plan->Adesc, plan->Bdesc, plan->Cdesc, plan->Cdesc,
                                              pref, max_results, plan->heur.data(), &returned));
    plan->heur.resize(returned);
    cublasLtMatmulPreferenceDestroy(pref);
    TORCH_CHECK(returned > 0, "cublasLt returned no algorithms");

    plan_cache()[key] = plan;
    return plan;
}

} // namespace

int64_t lt_num_algos(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes) {
    auto plan = get_plan(M, N, K, trans_w, (size_t)ws_bytes);
    return (int64_t)plan->heur.size();
}

void lt_gemm(torch::Tensor x, torch::Tensor w, torch::Tensor out, bool trans_w, int64_t algo_idx, int64_t ws_bytes) {
    int64_t M = x.size(0);
    int64_t K = x.size(1);
    int64_t N = out.size(1);
    auto plan = get_plan(M, N, K, trans_w, (size_t)ws_bytes);
    TORCH_CHECK(algo_idx >= 0 && algo_idx < (int64_t)plan->heur.size(), "invalid algo index");

    float alpha = 1.0f;
    float beta = 0.0f;
    cudaStream_t stream = at::cuda::getCurrentCUDAStream();
    CULT_CHECK(cublasLtMatmul(lt_handle(), plan->opDesc, &alpha,
                              w.data_ptr<float>(), plan->Adesc,
                              x.data_ptr<float>(), plan->Bdesc,
                              &beta,
                              out.data_ptr<float>(), plan->Cdesc,
                              out.data_ptr<float>(), plan->Cdesc,
                              &plan->heur[algo_idx].algo,
                              plan->workspace, plan->ws_bytes,
                              stream));
}
"""

lt_cpp_source = (
    "int64_t lt_num_algos(int64_t M, int64_t N, int64_t K, bool trans_w, int64_t ws_bytes);\n"
    "void lt_gemm(torch::Tensor x, torch::Tensor w, torch::Tensor out, bool trans_w, int64_t algo_idx, int64_t ws_bytes);"
)


def _cublaslt_paths():
    inc_dirs, lib_dirs = [], []
    roots = []
    try:
        roots += site.getsitepackages()
    except Exception:
        pass
    try:
        roots.append(os.path.dirname(os.path.dirname(os.path.abspath(torch.__file__))))
    except Exception:
        pass
    roots.append(sys.prefix)
    roots.append(os.environ.get("CUDA_HOME", "/usr/local/cuda"))
    roots.append("/usr/local/cuda")
    for r in dict.fromkeys(roots):
        for sub in ("lib64", "lib",
                    os.path.join("nvidia", "cublas", "lib"),
                    os.path.join("torch", "lib")):
            p = os.path.join(r, sub)
            if os.path.isdir(p):
                lib_dirs.append(p)
        for sub in ("include", os.path.join("nvidia", "cublas", "include")):
            p = os.path.join(r, sub)
            if os.path.isdir(p):
                inc_dirs.append(p)
    lib_file = None
    for d in lib_dirs:
        for pat in ("libcublasLt.so", "libcublasLt.so.*"):
            hits = sorted(glob.glob(os.path.join(d, pat)))
            hits = [h for h in hits if os.path.isfile(h)]
            if hits:
                lib_file = hits[0]
                break
        if lib_file:
            break
    if lib_file:
        ldflags = [lib_file, "-Wl,-rpath," + os.path.dirname(lib_file)]
    else:
        ldflags = ["-lcublasLt"]
    return inc_dirs, ldflags


lt_ext = None
try:
    _inc, _ld = _cublaslt_paths()
    lt_ext = load_inline(
        name="cublaslt_gemm_autotune_v1",
        cpp_sources=lt_cpp_source,
        cuda_sources=lt_source,
        functions=["lt_num_algos", "lt_gemm"],
        verbose=False,
        extra_cflags=["-O3"],
        extra_cuda_cflags=["-O3"],
        extra_include_paths=_inc,
        extra_ldflags=_ld,
    )
except Exception:
    lt_ext = None


class ModelNew(nn.Module):
    """
    Optimized model:
      * GEMM (FP32 with TF32 tensor cores) via cuBLASLt, autotuned per shape over
        heuristic algorithms and weight layouts (TN / pre-transposed NN), with a
        large workspace so split-K algorithms are available. Falls back to the
        PyTorch cuBLAS path when cublasLt is unavailable.
      * Single fused CUDA epilogue kernel: combined bias (linear bias + add_value)
        + Swish + Tanh + GELU. The trailing Hardtanh([-1, 1]) is provably an
        identity after Tanh -> GELU and is dropped.
      * Whole forward is captured into a CUDA graph (keyed by input/parameter
        buffer signatures) and replayed on subsequent calls to remove host and
        launch overhead; any mismatch or failure falls back to the eager path.
    """
    def __init__(self, in_features, out_features, add_value_shape):
        super(ModelNew, self).__init__()
        # TF32 tensor-core math is explicitly permitted for FP32 workloads.
        torch.backends.cuda.matmul.allow_tf32 = True
        torch.backends.cudnn.allow_tf32 = True

        self.matmul = nn.Linear(in_features, out_features)
        self.add_value = nn.Parameter(torch.randn(add_value_shape))

        self._ws_bytes = 32 * 1024 * 1024
        self._gemm_choice = {}
        self._wt = None
        self._wt_ver = None
        self._comb = None
        self._comb_ver = None

        self._graphs = {}
        self._counts = {}
        self._bad = set()
        self._max_graphs = 8
        self._graphs_ok = hasattr(torch.cuda, "CUDAGraph") and hasattr(torch.cuda, "graph")

    # ----- cached derived buffers -------------------------------------------
    def _get_wt(self):
        w = self.matmul.weight
        ver = (w.data_ptr(), w._version)
        if self._wt is None or self._wt_ver != ver:
            with torch.no_grad():
                self._wt = w.t().contiguous()
            self._wt_ver = ver
        return self._wt

    def _get_comb(self):
        b = self.matmul.bias
        av = self.add_value
        ver = (av.data_ptr(), av._version,
               0 if b is None else b.data_ptr(), -1 if b is None else b._version)
        if self._comb is None or self._comb_ver != ver:
            with torch.no_grad():
                self._comb = av.detach().clone() if b is None else (av + b).contiguous()
            self._comb_ver = ver
        return self._comb

    # ----- reference / eager paths ------------------------------------------
    def _reference(self, x):
        y = F.linear(x, self.matmul.weight, self.matmul.bias)
        y = y + self.add_value
        y = torch.sigmoid(y) * y
        y = torch.tanh(y)
        y = F.gelu(y)
        y = F.hardtanh(y, min_val=-1, max_val=1)
        return y

    @torch.no_grad()
    def _autotune(self, x):
        M, K = x.shape
        N = self.matmul.weight.shape[0]
        w = self.matmul.weight
        wt = self._get_wt()
        out = torch.empty(M, N, device=x.device, dtype=torch.float32)

        cands = []
        cands.append((("torch_tn",), lambda: torch.matmul(x, w.t(), out=out)))
        cands.append((("torch_nn",), lambda: torch.mm(x, wt, out=out)))

        if lt_ext is not None:
            xc = x.contiguous()
            for trans_w, wbuf in ((False, w), (True, wt)):
                try:
                    n_alg = int(lt_ext.lt_num_algos(M, N, K, trans_w, self._ws_bytes))
                except Exception:
                    n_alg = 0
                for i in range(min(n_alg, 16)):
                    cands.append((("lt", trans_w, i),
                                  (lambda wb=wbuf, tw=trans_w, ii=i:
                                   lt_ext.lt_gemm(xc, wb, out, tw, ii, self._ws_bytes))))

        best, best_t = None, float("inf")
        for tag, fn in cands:
            try:
                for _ in range(3):
                    fn()
                torch.cuda.synchronize(x.device)
                s = torch.cuda.Event(enable_timing=True)
                e = torch.cuda.Event(enable_timing=True)
                s.record()
                for _ in range(5):
                    fn()
                e.record()
                torch.cuda.synchronize(x.device)
                t = s.elapsed_time(e)
                if t < best_t:
                    best_t, best = t, tag
            except Exception:
                continue
        if best is None:
            best = ("torch_tn",)
        return best

    def _run_gemm(self, x, choice):
        w = self.matmul.weight
        N = w.shape[0]
        M = x.shape[0]
        tag = choice[0]
        if tag == "lt":
            _, trans_w, idx = choice
            if not x.is_contiguous():
                x = x.contiguous()
            wbuf = self._get_wt() if trans_w else w
            out = torch.empty(M, N, device=x.device, dtype=torch.float32)
            lt_ext.lt_gemm(x, wbuf, out, trans_w, idx, self._ws_bytes)
            return out
        if tag == "torch_nn":
            return torch.mm(x, self._get_wt())
        return torch.matmul(x, w.t())

    def _eager(self, x):
        key = (tuple(x.shape), x.device.index)
        choice = self._gemm_choice.get(key)
        if choice is None:
            choice = self._autotune(x)
            self._gemm_choice[key] = choice
        y = self._run_gemm(x, choice)
        if epilogue_ext is not None:
            return epilogue_ext.fused_bias_act(y, self._get_comb())
        y = y + self._get_comb()
        y = torch.sigmoid(y) * y
        y = torch.tanh(y)
        y = F.gelu(y)
        return y

    def _capture(self, x, key):
        # Warm up on a side stream (materializes workspaces, caches, etc.).
        s = torch.cuda.Stream(x.device)
        s.wait_stream(torch.cuda.current_stream(x.device))
        with torch.cuda.stream(s):
            for _ in range(2):
                self._eager(x)
        torch.cuda.current_stream(x.device).wait_stream(s)
        torch.cuda.synchronize(x.device)

        g = torch.cuda.CUDAGraph()
        with torch.cuda.graph(g):
            static_out = self._eager(x)
        self._graphs[key] = (g, static_out)

    def forward(self, x):
        if (self._graphs_ok and x.is_cuda and x.dtype == torch.float32
                and self.matmul.weight.is_cuda):
            w = self.matmul.weight
            b = self.matmul.bias
            key = (x.data_ptr(), tuple(x.shape), tuple(x.stride()), x.device.index,
                   w.data_ptr(), w._version,
                   0 if b is None else b.data_ptr(), -1 if b is None else b._version,
                   self.add_value.data_ptr(), self.add_value._version)
            ent = self._graphs.get(key)
            if ent is not None:
                try:
                    ent[0].replay()
                    return ent[1]
                except Exception:
                    self._graphs.pop(key, None)
                    self._bad.add(key)
            elif key not in self._bad:
                n = self._counts.get(key, 0) + 1
                self._counts[key] = n
                if n >= 3 and len(self._graphs) < self._max_graphs:
                    try:
                        self._capture(x, key)
                        ent = self._graphs.get(key)
                        if ent is not None:
                            ent[0].replay()
                            return ent[1]
                    except Exception:
                        self._graphs.pop(key, None)
                        self._bad.add(key)
            return self._eager(x)
        # Non-CUDA fallback: replicate the reference computation exactly.
        return self._reference(x)
```

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().
