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):
    """
    A model that performs a 3D transposed convolution, followed by batch normalization, 
    two average pooling layers.
    """
    def __init__(self, in_channels, out_channels, kernel_size, stride, padding, bias_shape):
        super(Model, self).__init__()
        self.conv_transpose = nn.ConvTranspose3d(in_channels, out_channels, kernel_size, stride=stride, padding=padding)
        self.batch_norm = nn.BatchNorm3d(out_channels)
        self.avg_pool1 = nn.AvgPool3d(kernel_size=2)
        self.avg_pool2 = nn.AvgPool3d(kernel_size=2)

    def forward(self, x):
        x = self.conv_transpose(x)
        x = self.batch_norm(x)
        x = self.avg_pool1(x)
        x = self.avg_pool2(x)
        return x


batch_size = 64
in_channels = 3
out_channels = 16
depth, height, width = 32, 32, 32
kernel_size = 3
stride = 2
padding = 1
bias_shape = (out_channels, 1, 1, 1)

def get_inputs():
    return [torch.rand(batch_size, in_channels, depth, height, width)]

def get_init_inputs():
    return [in_channels, out_channels, kernel_size, stride, padding, bias_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 torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

# ---------------------------------------------------------------------------
# Math behind the fusion (kernel_size=3, stride=2, padding=1 transposed conv,
# followed by two stride-2 2x2x2 average pools):
#
#   convT output p receives x[i] with weight W[k] iff p = 2*i + k - 1.
#   Two avg pools with odd sizes (63 -> 31 -> 15) only use convT outputs
#   p in [0,59] per dim, i.e. final[q] = (1/64) * sum_{d in [0,3]^3} y[4q+d].
#   Gathering the 64 sub-terms gives a *valid* stride-2 kernel-3 convolution
#   on x[0..30] with an "effective" kernel E:
#       final[oc][q] = sum_ic sum_{j in [0,2]^3} E[oc][ic][j] * x[ic][2q+j]
#   where E folds W over the 64 pooled positions (and the 1/64 factor).
#   BatchNorm is per-channel affine, so it commutes with the average pool and
#   collapses into a per-channel scale/shift applied at the end.
#
# Train mode: batch statistics of the *full* convT output y (63^3 per slice)
# are computed on the fly (y is never materialized).  Each thread owns one py
# and an (even, odd) px pair, walks pz in (even, odd) pairs, register-blocks
# the <= 2x2x2x3 x values shared by the 4 y positions, and evaluates all 16
# output channels from shared-memory weights -- so each x tap is loaded once
# and reused for every tap/oc that needs it (~19x less load traffic than a
# direct evaluation).
# ---------------------------------------------------------------------------

cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define ROWS 8
#define TOTAL_ROWS 225
#define X_TILE_FLOATS (ROWS * 3 * 3 * 3 * 31)
#define SLICE_ELEMS 250047   // 63*63*63

// Build effective kernel E[oc][ic][27] from conv-transpose W[ic][oc][27].
// Valid (d,j)->k mapping per dim: j=0 -> k={1,2}, j=1 -> k={0,1,2}, j=2 -> k={0}.
__global__ void build_eff_kernel(const float* __restrict__ W,
                                 float* __restrict__ E,
                                 int IC, int OC) {
    int oc = blockIdx.x / IC;
    int ic = blockIdx.x % IC;
    int j = threadIdx.x;              // 0..26
    if (j >= 27) return;
    int jz = j / 9, jy = (j / 3) % 3, jx = j % 3;

    int kzl[3], kyl[3], kxl[3], nz, ny, nx;
    if (jz == 0)      { kzl[0]=1; kzl[1]=2; nz=2; }
    else if (jz == 1) { kzl[0]=0; kzl[1]=1; kzl[2]=2; nz=3; }
    else              { kzl[0]=0; nz=1; }
    if (jy == 0)      { kyl[0]=1; kyl[1]=2; ny=2; }
    else if (jy == 1) { kyl[0]=0; kyl[1]=1; kyl[2]=2; ny=3; }
    else              { kyl[0]=0; ny=1; }
    if (jx == 0)      { kxl[0]=1; kxl[1]=2; nx=2; }
    else if (jx == 1) { kxl[0]=0; kxl[1]=1; kxl[2]=2; nx=3; }
    else              { kxl[0]=0; nx=1; }

    const float* wb = W + ((long)ic * OC + oc) * 27;
    float acc = 0.f;
    for (int a = 0; a < nz; ++a)
        for (int b = 0; b < ny; ++b)
            for (int c = 0; c < nx; ++c)
                acc += wb[kzl[a] * 9 + kyl[b] * 3 + kxl[c]];
    E[((long)oc * IC + ic) * 27 + j] = acc * (1.0f / 64.0f);
}

// Fully fused: effective stride-2 conv + BN affine + both avg pools.
// x: [N][IC][32][32][32], E: [OC][IC][27], out: [N][OC][15][15][15]
__global__ void fused_conv_bn_pool_kernel(
        const float* __restrict__ x,
        const float* __restrict__ E,
        const float* __restrict__ bn_w,
        const float* __restrict__ bn_b,
        const float* __restrict__ bn_mean,   // running_mean (eval) or batch mean (train)
        const float* __restrict__ bn_var,    // running_var  (eval) or batch biased var (train)
        const float* __restrict__ conv_bias,
        float* __restrict__ out,
        int N, int IC, int OC, float eps) {
    __shared__ float xs[X_TILE_FLOATS];
    __shared__ float Es[16 * 3 * 27];

    const int n   = blockIdx.z;
    const int row = blockIdx.x * ROWS + (threadIdx.x >> 4);  // 0..224
    const int qx  = threadIdx.x & 15;
    const int tid = threadIdx.x;

    // load effective kernel
    for (int i = tid; i < OC * IC * 27; i += blockDim.x)
        Es[i] = E[i];

    // load input tile: rows of this block -> x[n][ic][2qz+dz][2qy+dy][0..30]
    const int row0 = blockIdx.x * ROWS;
    const int nrows = min(ROWS, TOTAL_ROWS - row0);
    const long xbase = (long)n * IC * 32768;
    for (int idx = tid; idx < nrows * 3 * 3 * 3 * 31; idx += blockDim.x) {
        int xx  = idx % 31;
        int t   = idx / 31;
        int dy  = t % 3;
        int t2  = t / 3;
        int dz  = t2 % 3;
        int ic  = t2 / 3;
        int r   = ic / 3;  ic = ic % 3;
        int qz  = (row0 + r) / 15;
        int qy  = (row0 + r) % 15;
        int gz  = 2 * qz + dz;
        int gy  = 2 * qy + dy;
        xs[idx] = x[xbase + ((long)ic * 32 + gz) * 1024 + gy * 32 + xx];
    }
    __syncthreads();

    if (qx >= 15 || row >= TOTAL_ROWS) return;
    const int qz = row / 15;
    const int qy = row % 15;
    const int r  = threadIdx.x >> 4;

    for (int oc = 0; oc < OC; ++oc) {
        const float* ep = Es + oc * IC * 27;
        float acc = 0.f;
        for (int ic = 0; ic < IC; ++ic) {
            const float* xp = xs + ((r * 3 + ic) * 9) * 31 + 2 * qx;
            #pragma unroll
            for (int j = 0; j < 27; ++j) {
                int jz = j / 9, jy = (j / 3) % 3, jx = j % 3;
                acc += ep[j] * xp[(jz * 3 + jy) * 31 + jx];
            }
            ep += 27;
        }
        float scale = bn_w[oc] * rsqrtf(bn_var[oc] + eps);
        float shift = bn_b[oc] + scale * (conv_bias[oc] - bn_mean[oc]);
        out[(((long)n * OC + oc) * 225 + row) * 15 + qx] = acc * scale + shift;
    }
}

// Train-mode batch statistics of the full convT output y (63^3 per slice),
// computed on the fly: y is never written to global memory.
//   y[n][oc][pz][py][px] = bias[oc]
//       + sum_ic sum_{valid k taps} x[n][ic][iz][iy][ix] * W[ic][oc][k]
// Per dim: p even -> single tap (k=1, i=p/2);
//          p odd  -> two taps (k=0, i=(p+1)/2) and (k=2, i=(p-1)/2).
// block = (32, 8) threads: px = {2*tx, 2*tx+1}, py = 8*blockIdx.x + ty;
// pz walked in (even, odd) pairs so the <= 2x2x2 x values (x 3 input
// channels) sit in registers and feed all 4 y positions x 16 output channels.
// Per-block partials are reduced in shared memory and atomically accumulated.
__global__ void convt_stats_kernel(const float* __restrict__ x,
                                   const float* __restrict__ W,
                                   const float* __restrict__ bias,
                                   double* __restrict__ sum,
                                   double* __restrict__ sumsq,
                                   int IC, int OC) {
    __shared__ float Ws[16 * 3 * 27];
    __shared__ float bias_s[16];
    __shared__ double bs[256];
    __shared__ double bs2[256];

    const int n   = blockIdx.y;
    const int tid = threadIdx.x + (threadIdx.y << 5);

    for (int i = tid; i < OC * IC * 27; i += 256) {
        int oc = i / 81;
        int r  = i - oc * 81;
        int ic = r / 27;
        int k  = r - ic * 27;
        Ws[i] = W[((long)ic * OC + oc) * 27 + k];
    }
    if (tid < OC) bias_s[tid] = bias[tid];
    __syncthreads();

    const int tx = threadIdx.x;                     // 0..31
    const int py = (blockIdx.x << 3) + threadIdx.y; // 0..63
    const bool py_valid = (py <= 62);
    const bool px1_ok   = (tx <= 30);               // px = 2*tx+1 <= 62

    const int ix0 = tx;
    const int ix1 = (tx + 1 <= 31) ? (tx + 1) : 31;  // clamped; unused at tx=31
    const bool py_odd = (py & 1) != 0;
    const int iyA = py_odd ? ((py - 1) >> 1) : (py >> 1);  // ky=2 (odd) / ky=1 (even)
    const int iyB = py_odd ? ((py + 1) >> 1) : (py >> 1);  // ky=0 (odd); == iyA if even

    float acc_s[16], acc_q[16];
    #pragma unroll
    for (int c = 0; c < 16; ++c) { acc_s[c] = 0.f; acc_q[c] = 0.f; }

    const long xn = (long)n * IC * 32768;

    if (py_valid) {
        for (int t = 0; t < 32; ++t) {
            const bool tail = (t == 31);   // only even pz = 62
            // register-block the x values shared by the (up to) 4 y positions
            float xr[2][2][2][3];          // [dz][dy][dx][ic]
            {
                const int zlim = tail ? 0 : 1;
                #pragma unroll
                for (int dz = 0; dz < 2; ++dz) {
                    if (dz > zlim) break;
                    const long zoff = xn + (long)(t + dz) * 1024;
                    #pragma unroll
                    for (int dy = 0; dy < 2; ++dy) {
                        const long yoff = zoff + (long)(dy ? iyB : iyA) * 32;
                        #pragma unroll
                        for (int dx = 0; dx < 2; ++dx) {
                            const float* xp = x + yoff + (dx ? ix1 : ix0);
                            #pragma unroll
                            for (int ic = 0; ic < 3; ++ic)
                                xr[dz][dy][dx][ic] = xp[(long)ic * 32768];
                        }
                    }
                }
            }

            float y0[16], y1[16], y2[16], y3[16];
            #pragma unroll
            for (int c = 0; c < 16; ++c) {
                float b = bias_s[c];
                y0[c] = b; y1[c] = b; y2[c] = b; y3[c] = b;
            }

            #pragma unroll
            for (int oc = 0; oc < 16; ++oc) {
                const float* wp = Ws + oc * 81;
                #pragma unroll
                for (int ic = 0; ic < 3; ++ic) {
                    const float* w = wp + ic * 27;
                    if (py_odd) {
                        float xa0 = xr[0][0][0][ic], xa1 = xr[0][0][1][ic];
                        float xb0 = xr[0][1][0][ic], xb1 = xr[0][1][1][ic];
                        // even pz (kz=1)
                        y0[oc] += w[16] * xa0 + w[7] * xb0;              // ky=2,ky=0 (kx=1)
                        y1[oc] += w[17] * xa0 + w[15] * xa1              // ky=2
                                + w[8]  * xb0 + w[6]  * xb1;             // ky=0
                        if (!tail) {
                            float xc0 = xr[1][0][0][ic], xc1 = xr[1][0][1][ic];
                            float xd0 = xr[1][1][0][ic], xd1 = xr[1][1][1][ic];
                            // odd pz: kz=2 (dz=0), kz=0 (dz=1)
                            y2[oc] += w[25] * xa0 + w[16] * xb0
                                    + w[7]  * xc0 + w[7 - 7 + 1] * 0.f;  // placeholder cleaned below
                            y2[oc] += w[1] * xc0 + w[1 + 9] * 0.f;       // cleaned below
                            y2[oc] += w[16 + 9] * 0.f;
                            y3[oc] += 0.f;
                        }
                    } else {
                        float xa0 = xr[0][0][0][ic], xa1 = xr[0][0][1][ic];
                        y0[oc] += w[13] * xa0;                           // ky=1, kx=1
                        y1[oc] += w[14] * xa0 + w[12] * xa1;             // kx=2, kx=0
                    }
                }
            }
        }
    }

    // placeholder (never compiled -- replaced below)
    bs[tid] = 0.0; bs2[tid] = 0.0;
    __syncthreads();
}

// compute batch mean/var and update running stats like aten batch_norm
__global__ void bn_finalize_kernel(const double* __restrict__ sum,
                                   const double* __restrict__ sumsq,
                                   float* __restrict__ running_mean,
                                   float* __restrict__ running_var,
                                   float* __restrict__ out_mean,
                                   float* __restrict__ out_var,
                                   long count, double momentum, float eps, int C) {
    int c = blockIdx.x * blockDim.x + threadIdx.x;
    if (c >= C) return;
    double m  = sum[c] / (double)count;
    double sq = sumsq[c] / (double)count;
    double v  = sq - m * m;
    if (v < 0.0) v = 0.0;
    out_mean[c] = (float)m;
    out_var[c]  = (float)v;
    double unb = (count > 1) ? v * ((double)count / (double)(count - 1)) : 0.0;
    running_mean[c] = (float)((1.0 - momentum) * (double)running_mean[c] + momentum * m);
    running_var[c]  = (float)((1.0 - momentum) * (double)running_var[c]  + momentum * unb);
}

torch::Tensor build_eff(torch::Tensor W, int64_t IC, int64_t OC) {
    auto E = torch::empty({OC, IC, 27}, W.options());
    auto stream = at::cuda::getCurrentCUDAStream();
    dim3 grid(OC * IC);
    build_eff_kernel<<<grid, 32, 0, stream>>>(W.data_ptr<float>(), E.data_ptr<float>(), (int)IC, (int)OC);
    return E;
}

torch::Tensor fused_eval(torch::Tensor x, torch::Tensor E,
                         torch::Tensor bn_w, torch::Tensor bn_b,
                         torch::Tensor bn_mean, torch::Tensor bn_var,
                         torch::Tensor conv_bias,
                         int64_t N, int64_t IC, int64_t OC, double eps) {
    auto out = torch::empty({N, OC, 15, 15, 15}, x.options());
    auto stream = at::cuda::getCurrentCUDAStream();
    dim3 grid((TOTAL_ROWS + ROWS - 1) / ROWS, 1, N);
    fused_conv_bn_pool_kernel<<<grid, 128, 0, stream>>>(
        x.data_ptr<float>(), E.data_ptr<float>(),
        bn_w.data_ptr<float>(), bn_b.data_ptr<float>(),
        bn_mean.data_ptr<float>(), bn_var.data_ptr<float>(),
        conv_bias.data_ptr<float>(), out.data_ptr<float>(),
        (int)N, (int)IC, (int)OC, (float)eps);
    return out;
}

// train-mode: on-the-fly convT batch stats (y never materialized),
// then fused effective-conv + BN + pools with the batch stats.
torch::Tensor fused_train(torch::Tensor x, torch::Tensor E, torch::Tensor W,
                          torch::Tensor bn_w, torch::Tensor bn_b,
                          torch::Tensor running_mean, torch::Tensor running_var,
                          torch::Tensor conv_bias,
                          int64_t N, int64_t IC, int64_t OC,
                          double eps, double momentum) {
    auto stream = at::cuda::getCurrentCUDAStream();
    auto opts_d = torch::TensorOptions().dtype(torch::kFloat64).device(x.device());
    auto sum    = torch::zeros({OC}, opts_d);
    auto sumsq  = torch::zeros({OC}, opts_d);

    dim3 sgrid(8, (unsigned)N, 1);
    dim3 sblock(32, 8);
    convt_stats_kernel<<<sgrid, sblock, 0, stream>>>(
        x.data_ptr<float>(), W.data_ptr<float>(), conv_bias.data_ptr<float>(),
        sum.data_ptr<double>(), sumsq.data_ptr<double>(), (int)IC, (int)OC);

    auto bmean = torch::empty({OC}, x.options());
    auto bvar  = torch::empty({OC}, x.options());
    long count = (long)N * SLICE_ELEMS;
    bn_finalize_kernel<<<(unsigned)((OC + 63) / 64), 64, 0, stream>>>(
        sum.data_ptr<double>(), sumsq.data_ptr<double>(),
        running_mean.data_ptr<float>(), running_var.data_ptr<float>(),
        bmean.data_ptr<float>(), bvar.data_ptr<float>(),
        count, momentum, (float)eps, (int)OC);

    auto out = torch::empty({N, OC, 15, 15, 15}, x.options());
    dim3 grid((TOTAL_ROWS + ROWS - 1) / ROWS, 1, N);
    fused_conv_bn_pool_kernel<<<grid, 128, 0, stream>>>(
        x.data_ptr<float>(), E.data_ptr<float>(),
        bn_w.data_ptr<float>(), bn_b.data_ptr<float>(),
        bmean.data_ptr<float>(), bvar.data_ptr<float>(),
        conv_bias.data_ptr<float>(), out.data_ptr<float>(),
        (int)N, (int)IC, (int)OC, (float)eps);
    return out;
}
"""
```
Result of evaluating it:
The solution failed to compile. Error output:
```

ModelNew not defined (or syntax error in candidate source)
```

The fastest correct solution so far (9.654x 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 torch
import torch.nn as nn
import torch.nn.functional as F
from torch.utils.cpp_extension import load_inline

# ---------------------------------------------------------------------------
# Math behind the fusion (kernel_size=3, stride=2, padding=1 transposed conv,
# followed by two stride-2 2x2x2 average pools):
#
#   convT output p receives x[i] with weight W[k] iff p = 2*i + k - 1.
#   Two avg pools with odd sizes (63 -> 31 -> 15) only use convT outputs
#   p in [0,59] per dim, i.e. final[q] = (1/64) * sum_{d in [0,3]^3} y[4q+d].
#   Gathering the 64 sub-terms gives a *valid* stride-2 kernel-3 convolution
#   on x[0..30] with an "effective" kernel E:
#       final[oc][q] = sum_ic sum_{j in [0,2]^3} E[oc][ic][j] * x[ic][2q+j]
#   where E folds W over the 64 pooled positions (and the 1/64 factor).
#   BatchNorm is per-channel affine, so it commutes with the average pool and
#   collapses into a per-channel scale/shift applied at the end.
#
# Train mode: batch statistics of the *full* convT output are computed by a
# dedicated kernel that evaluates y = convT(x) on the fly (each output element
# costs <= 8 taps x IC MACs; parity of p determines the taps) and accumulates
# per-channel sum / sum-of-squares directly in registers + shared memory, so
# the 1 GB intermediate tensor is never materialized.
# ---------------------------------------------------------------------------

cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <cuda_runtime.h>

#define ROWS 8
#define TOTAL_ROWS 225
#define X_TILE_FLOATS (ROWS * 3 * 3 * 3 * 31)
#define SLICE_ELEMS 250047   // 63*63*63

// Build effective kernel E[oc][ic][27] from conv-transpose W[ic][oc][27].
// Valid (d,j)->k mapping per dim: j=0 -> k={1,2}, j=1 -> k={0,1,2}, j=2 -> k={0}.
__global__ void build_eff_kernel(const float* __restrict__ W,
                                 float* __restrict__ E,
                                 int IC, int OC) {
    int oc = blockIdx.x / IC;
    int ic = blockIdx.x % IC;
    int j = threadIdx.x;              // 0..26
    if (j >= 27) return;
    int jz = j / 9, jy = (j / 3) % 3, jx = j % 3;

    int kzl[3], kyl[3], kxl[3], nz, ny, nx;
    if (jz == 0)      { kzl[0]=1; kzl[1]=2; nz=2; }
    else if (jz == 1) { kzl[0]=0; kzl[1]=1; kzl[2]=2; nz=3; }
    else              { kzl[0]=0; nz=1; }
    if (jy == 0)      { kyl[0]=1; kyl[1]=2; ny=2; }
    else if (jy == 1) { kyl[0]=0; kyl[1]=1; kyl[2]=2; ny=3; }
    else              { kyl[0]=0; ny=1; }
    if (jx == 0)      { kxl[0]=1; kxl[1]=2; nx=2; }
    else if (jx == 1) { kxl[0]=0; kxl[1]=1; kxl[2]=2; nx=3; }
    else              { kxl[0]=0; nx=1; }

    const float* wb = W + ((long)ic * OC + oc) * 27;
    float acc = 0.f;
    for (int a = 0; a < nz; ++a)
        for (int b = 0; b < ny; ++b)
            for (int c = 0; c < nx; ++c)
                acc += wb[kzl[a] * 9 + kyl[b] * 3 + kxl[c]];
    E[((long)oc * IC + ic) * 27 + j] = acc * (1.0f / 64.0f);
}

// Fully fused: effective stride-2 conv + BN affine + both avg pools.
// x: [N][IC][32][32][32], E: [OC][IC][27], out: [N][OC][15][15][15]
__global__ void fused_conv_bn_pool_kernel(
        const float* __restrict__ x,
        const float* __restrict__ E,
        const float* __restrict__ bn_w,
        const float* __restrict__ bn_b,
        const float* __restrict__ bn_mean,   // running_mean (eval) or batch mean (train)
        const float* __restrict__ bn_var,    // running_var  (eval) or batch biased var (train)
        const float* __restrict__ conv_bias,
        float* __restrict__ out,
        int N, int IC, int OC, float eps) {
    __shared__ float xs[X_TILE_FLOATS];
    __shared__ float Es[16 * 3 * 27];

    const int n   = blockIdx.z;
    const int row = blockIdx.x * ROWS + (threadIdx.x >> 4);  // 0..224
    const int qx  = threadIdx.x & 15;
    const int tid = threadIdx.x;

    // load effective kernel
    for (int i = tid; i < OC * IC * 27; i += blockDim.x)
        Es[i] = E[i];

    // load input tile: rows of this block -> x[n][ic][2qz+dz][2qy+dy][0..30]
    const int row0 = blockIdx.x * ROWS;
    const int nrows = min(ROWS, TOTAL_ROWS - row0);
    const long xbase = (long)n * IC * 32768;
    for (int idx = tid; idx < nrows * 3 * 3 * 3 * 31; idx += blockDim.x) {
        int xx  = idx % 31;
        int t   = idx / 31;
        int dy  = t % 3;
        int t2  = t / 3;
        int dz  = t2 % 3;
        int ic  = t2 / 3;
        int r   = ic / 3;  ic = ic % 3;
        int qz  = (row0 + r) / 15;
        int qy  = (row0 + r) % 15;
        int gz  = 2 * qz + dz;
        int gy  = 2 * qy + dy;
        xs[idx] = x[xbase + ((long)ic * 32 + gz) * 1024 + gy * 32 + xx];
    }
    __syncthreads();

    if (qx >= 15 || row >= TOTAL_ROWS) return;
    const int qz = row / 15;
    const int qy = row % 15;
    const int r  = threadIdx.x >> 4;

    for (int oc = 0; oc < OC; ++oc) {
        const float* ep = Es + oc * IC * 27;
        float acc = 0.f;
        for (int ic = 0; ic < IC; ++ic) {
            const float* xp = xs + ((r * 3 + ic) * 9) * 31 + 2 * qx;
            #pragma unroll
            for (int j = 0; j < 27; ++j) {
                int jz = j / 9, jy = (j / 3) % 3, jx = j % 3;
                acc += ep[j] * xp[(jz * 3 + jy) * 31 + jx];
            }
            ep += 27;
        }
        float scale = bn_w[oc] * rsqrtf(bn_var[oc] + eps);
        float shift = bn_b[oc] + scale * (conv_bias[oc] - bn_mean[oc]);
        out[(((long)n * OC + oc) * 225 + row) * 15 + qx] = acc * scale + shift;
    }
}

// Train-mode batch statistics of the full convT output y (63^3 per slice),
// computed on the fly: y is never written to global memory.
//   y[n][oc][pz][py][px] = bias[oc]
//       + sum_ic sum_{valid k taps} x[n][ic][iz][iy][ix] * W[ic][oc][k]
// For each dim: p even -> single tap (k=1, i=p/2);
//               p odd  -> two taps (k=0, i=(p+1)/2) and (k=2, i=(p-1)/2).
// block = (64, 4) threads -> px = threadIdx.x, py = blockIdx.x*4 + threadIdx.y,
// loop over pz.  Per-block partial sums are reduced in shared memory and
// atomically accumulated per output channel.
__global__ void convt_stats_kernel(const float* __restrict__ x,
                                   const float* __restrict__ W,
                                   const float* __restrict__ bias,
                                   double* __restrict__ sum,
                                   double* __restrict__ sumsq,
                                   int IC, int OC) {
    __shared__ float Ws[8 * 27];
    __shared__ double bs[256];
    __shared__ double bs2[256];

    const int oc  = blockIdx.z;
    const int n   = blockIdx.y;
    const int tid = threadIdx.x + threadIdx.y * blockDim.x;

    for (int i = tid; i < IC * 27; i += 256) {
        int ic = i / 27, k = i - ic * 27;
        Ws[i] = W[((long)ic * OC + oc) * 27 + k];
    }
    __syncthreads();

    const int px = threadIdx.x;
    const int py = blockIdx.x * 4 + threadIdx.y;

    double s = 0.0, s2 = 0.0;
    if (px < 63 && py < 63) {
        int ix[2], kx[2], iy[2], ky[2], nx, ny;
        if (px & 1) { ix[0]=(px+1)>>1; kx[0]=0; ix[1]=(px-1)>>1; kx[1]=2; nx=2; }
        else        { ix[0]=px>>1;     kx[0]=1;                       nx=1; }
        if (py & 1) { iy[0]=(py+1)>>1; ky[0]=0; iy[1]=(py-1)>>1; ky[1]=2; ny=2; }
        else        { iy[0]=py>>1;     ky[0]=1;                       ny=1; }

        const float b = bias[oc];
        const long xn = (long)n * IC * 32768;

        for (int pz = 0; pz < 63; ++pz) {
            int iz[2], kz[2], nz;
            if (pz & 1) { iz[0]=(pz+1)>>1; kz[0]=0; iz[1]=(pz-1)>>1; kz[1]=2; nz=2; }
            else        { iz[0]=pz>>1;     kz[0]=1;                       nz=1; }

            float acc = b;
            for (int ic = 0; ic < IC; ++ic) {
                const float* xp = x + xn + (long)ic * 32768;
                const float* wp = Ws + ic * 27;
                for (int a = 0; a < nz; ++a) {
                    const float* xz = xp + (long)iz[a] * 1024;
                    const int kza = kz[a] * 9;
                    for (int bb = 0; bb < ny; ++bb) {
                        const float* xy = xz + iy[bb] * 32;
                        const int kb = kza + ky[bb] * 3;
                        #pragma unroll
                        for (int c = 0; c < 2; ++c) {
                            if (c < nx) acc += xy[ix[c]] * wp[kb + kx[c]];
                        }
                    }
                }
            }
            s  += (double)acc;
            s2 += (double)acc * (double)acc;
        }
    }

    bs[tid] = s; bs2[tid] = s2;
    __syncthreads();
    for (int off = 128; off > 0; off >>= 1) {
        if (tid < off) {
            bs[tid]  += bs[tid + off];
            bs2[tid] += bs2[tid + off];
        }
        __syncthreads();
    }
    if (tid == 0) {
        atomicAdd(&sum[oc], bs[0]);
        atomicAdd(&sumsq[oc], bs2[0]);
    }
}

// compute batch mean/var and update running stats like aten batch_norm
__global__ void bn_finalize_kernel(const double* __restrict__ sum,
                                   const double* __restrict__ sumsq,
                                   float* __restrict__ running_mean,
                                   float* __restrict__ running_var,
                                   float* __restrict__ out_mean,
                                   float* __restrict__ out_var,
                                   long count, double momentum, float eps, int C) {
    int c = blockIdx.x * blockDim.x + threadIdx.x;
    if (c >= C) return;
    double m  = sum[c] / (double)count;
    double sq = sumsq[c] / (double)count;
    double v  = sq - m * m;
    if (v < 0.0) v = 0.0;
    out_mean[c] = (float)m;
    out_var[c]  = (float)v;
    double unb = (count > 1) ? v * ((double)count / (double)(count - 1)) : 0.0;
    running_mean[c] = (float)((1.0 - momentum) * (double)running_mean[c] + momentum * m);
    running_var[c]  = (float)((1.0 - momentum) * (double)running_var[c]  + momentum * unb);
}

torch::Tensor build_eff(torch::Tensor W, int64_t IC, int64_t OC) {
    auto E = torch::empty({OC, IC, 27}, W.options());
    auto stream = at::cuda::getCurrentCUDAStream();
    dim3 grid(OC * IC);
    build_eff_kernel<<<grid, 32, 0, stream>>>(W.data_ptr<float>(), E.data_ptr<float>(), (int)IC, (int)OC);
    return E;
}

torch::Tensor fused_eval(torch::Tensor x, torch::Tensor E,
                         torch::Tensor bn_w, torch::Tensor bn_b,
                         torch::Tensor bn_mean, torch::Tensor bn_var,
                         torch::Tensor conv_bias,
                         int64_t N, int64_t IC, int64_t OC, double eps) {
    auto out = torch::empty({N, OC, 15, 15, 15}, x.options());
    auto stream = at::cuda::getCurrentCUDAStream();
    dim3 grid((TOTAL_ROWS + ROWS - 1) / ROWS, 1, N);
    fused_conv_bn_pool_kernel<<<grid, 128, 0, stream>>>(
        x.data_ptr<float>(), E.data_ptr<float>(),
        bn_w.data_ptr<float>(), bn_b.data_ptr<float>(),
        bn_mean.data_ptr<float>(), bn_var.data_ptr<float>(),
        conv_bias.data_ptr<float>(), out.data_ptr<float>(),
        (int)N, (int)IC, (int)OC, (float)eps);
    return out;
}

// train-mode: on-the-fly convT batch stats (y never materialized),
// then fused effective-conv + BN + pools with the batch stats.
torch::Tensor fused_train(torch::Tensor x, torch::Tensor E, torch::Tensor W,
                          torch::Tensor bn_w, torch::Tensor bn_b,
                          torch::Tensor running_mean, torch::Tensor running_var,
                          torch::Tensor conv_bias,
                          int64_t N, int64_t IC, int64_t OC,
                          double eps, double momentum) {
    auto stream = at::cuda::getCurrentCUDAStream();
    auto opts_d = torch::TensorOptions().dtype(torch::kFloat64).device(x.device());
    auto sum    = torch::zeros({OC}, opts_d);
    auto sumsq  = torch::zeros({OC}, opts_d);

    dim3 sgrid(16, (unsigned)N, (unsigned)OC);
    dim3 sblock(64, 4);
    convt_stats_kernel<<<sgrid, sblock, 0, stream>>>(
        x.data_ptr<float>(), W.data_ptr<float>(), conv_bias.data_ptr<float>(),
        sum.data_ptr<double>(), sumsq.data_ptr<double>(), (int)IC, (int)OC);

    auto bmean = torch::empty({OC}, x.options());
    auto bvar  = torch::empty({OC}, x.options());
    long count = (long)N * SLICE_ELEMS;
    bn_finalize_kernel<<<(unsigned)((OC + 63) / 64), 64, 0, stream>>>(
        sum.data_ptr<double>(), sumsq.data_ptr<double>(),
        running_mean.data_ptr<float>(), running_var.data_ptr<float>(),
        bmean.data_ptr<float>(), bvar.data_ptr<float>(),
        count, momentum, (float)eps, (int)OC);

    auto out = torch::empty({N, OC, 15, 15, 15}, x.options());
    dim3 grid((TOTAL_ROWS + ROWS - 1) / ROWS, 1, N);
    fused_conv_bn_pool_kernel<<<grid, 128, 0, stream>>>(
        x.data_ptr<float>(), E.data_ptr<float>(),
        bn_w.data_ptr<float>(), bn_b.data_ptr<float>(),
        bmean.data_ptr<float>(), bvar.data_ptr<float>(),
        conv_bias.data_ptr<float>(), out.data_ptr<float>(),
        (int)N, (int)IC, (int)OC, (float)eps);
    return out;
}
"""

cpp_source = r"""
torch::Tensor build_eff(torch::Tensor W, int64_t IC, int64_t OC);
torch::Tensor fused_eval(torch::Tensor x, torch::Tensor E,
                         torch::Tensor bn_w, torch::Tensor bn_b,
                         torch::Tensor bn_mean, torch::Tensor bn_var,
                         torch::Tensor conv_bias,
                         int64_t N, int64_t IC, int64_t OC, double eps);
torch::Tensor fused_train(torch::Tensor x, torch::Tensor E, torch::Tensor W,
                          torch::Tensor bn_w, torch::Tensor bn_b,
                          torch::Tensor running_mean, torch::Tensor running_var,
                          torch::Tensor conv_bias,
                          int64_t N, int64_t IC, int64_t OC,
                          double eps, double momentum);
"""

_fused = load_inline(
    name="fused_convT_bn_avgpool_v2",
    cpp_sources=cpp_source,
    cuda_sources=cuda_source,
    functions=["build_eff", "fused_eval", "fused_train"],
    verbose=False,
    extra_cuda_cflags=["-O3", "--use_fast_math"],
)


class ModelNew(nn.Module):
    """
    Optimized version of Model.  The conv-transpose (k=3, s=2, p=1), batch-norm
    and both 2x2x2 average pools are fused into custom CUDA kernels by
    algebraically collapsing the whole chain into one valid stride-2 kernel-3
    convolution with an effective kernel, plus a per-channel affine transform.
    In training mode the batch statistics of the transposed-conv output are
    computed on the fly (the 63^3-per-channel intermediate is never
    materialized), which removes the dominant cuDNN dgrad + memset + stats
    passes entirely.
    """

    def __init__(self, in_channels, out_channels, kernel_size, stride, padding, bias_shape):
        super(ModelNew, self).__init__()
        self.conv_transpose = nn.ConvTranspose3d(
            in_channels, out_channels, kernel_size, stride=stride, padding=padding
        )
        self.batch_norm = nn.BatchNorm3d(out_channels)
        self._in_channels = in_channels
        self._out_channels = out_channels

    def _fallback(self, x):
        x = self.conv_transpose(x)
        x = self.batch_norm(x)
        x = F.avg_pool3d(x, kernel_size=2)
        x = F.avg_pool3d(x, kernel_size=2)
        return x

    def forward(self, x):
        ic, oc = self._in_channels, self._out_channels
        bn = self.batch_norm
        ct = self.conv_transpose

        fast_ok = (
            x.is_cuda and x.dtype == torch.float32 and x.dim() == 5
            and x.size(1) == ic and x.size(2) == 32 and x.size(3) == 32 and x.size(4) == 32
            and ic == 3 and oc == 16
            and ct.bias is not None and bn.affine and bn.track_running_stats
        )
        if not fast_ok:
            return self._fallback(x)

        x = x.contiguous()
        w = ct.weight.contiguous()
        conv_bias = ct.bias.contiguous()
        N = x.shape[0]
        E = _fused.build_eff(w, ic, oc)

        if not self.training:
            return _fused.fused_eval(
                x, E,
                bn.weight.contiguous(), bn.bias.contiguous(),
                bn.running_mean.contiguous(), bn.running_var.contiguous(),
                conv_bias, N, ic, oc, float(bn.eps),
            )
        else:
            return _fused.fused_train(
                x, E, w,
                bn.weight.contiguous(), bn.bias.contiguous(),
                bn.running_mean, bn.running_var,
                conv_bias, N, ic, oc, float(bn.eps), float(bn.momentum),
            )


batch_size = 64
in_channels = 3
out_channels = 16
depth, height, width = 32, 32, 32
kernel_size = 3
stride = 2
padding = 1
bias_shape = (out_channels, 1, 1, 1)


def get_inputs():
    return [torch.rand(batch_size, in_channels, depth, height, width)]


def get_init_inputs():
    return [in_channels, out_channels, kernel_size, stride, padding, bias_shape]
```

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().
