```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline


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

torch::Tensor triangular_mm_cuda(torch::Tensor A, torch::Tensor B);
"""

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

template<int R>
__global__ __launch_bounds__(256)
void triangular_mm_kernel(
    const float* __restrict__ A,
    const float* __restrict__ B,
    float* __restrict__ C,
    int N
) {
    constexpr int TILE = 16 * R;
    constexpr int BK = 16;

    const int tid = threadIdx.x;
    const int tr = tid / 16;
    const int tc = tid % 16;
    const int row_base = blockIdx.y * TILE;
    const int col_base = blockIdx.x * TILE;

    if (blockIdx.x > blockIdx.y) {
        for (int index = tid; index < TILE * TILE; index += 256) {
            const int row = row_base + index / TILE;
            const int col = col_base + index % TILE;
            if (row < N && col < N)
                C[(int64_t)row * N + col] = 0.0f;
        }
        return;
    }

    __shared__ float As[BK][TILE + 1];
    __shared__ float Bs[BK][TILE];

    float accum[R][R];

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

    const int k_end = min(row_base + TILE, N);

    for (int k_base = col_base; k_base < k_end; k_base += BK) {
        #pragma unroll
        for (int index = tid; index < TILE * BK; index += 256) {
            const int local_row = index / BK;
            const int local_k = index % BK;
            const int row = row_base + local_row;
            const int k = k_base + local_k;

            As[local_k][local_row] =
                (row < N && k < N)
                ? A[(int64_t)row * N + k]
                : 0.0f;
        }

        #pragma unroll
        for (int index = tid; index < BK * TILE; index += 256) {
            const int local_k = index / TILE;
            const int local_col = index % TILE;
            const int k = k_base + local_k;
            const int col = col_base + local_col;

            Bs[local_k][local_col] =
                (k < N && col < N)
                ? B[(int64_t)k * N + col]
                : 0.0f;
        }

        __syncthreads();

        #pragma unroll
        for (int k = 0; k < BK; ++k) {
            float av[R];
            float bv[R];

            #pragma unroll
            for (int i = 0; i < R; ++i)
                av[i] = As[k][tr + 16 * i];

            #pragma unroll
            for (int j = 0; j < R; ++j)
                bv[j] = Bs[k][tc + 16 * j];

            #pragma unroll
            for (int i = 0; i < R; ++i) {
                #pragma unroll
                for (int j = 0; j < R; ++j)
                    accum[i][j] = fmaf(av[i], bv[j], accum[i][j]);
            }
        }

        __syncthreads();
    }

    #pragma unroll
    for (int i = 0; i < R; ++i) {
        const int row = row_base + tr + 16 * i;

        #pragma unroll
        for (int j = 0; j < R; ++j) {
            const int col = col_base + tc + 16 * j;
            if (row < N && col < N) {
                C[(int64_t)row * N + col] =
                    row >= col ? accum[i][j] : 0.0f;
            }
        }
    }
}

torch::Tensor triangular_mm_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "Inputs must be CUDA tensors");
    TORCH_CHECK(A.device() == B.device(), "Inputs must share a device");
    TORCH_CHECK(
        A.scalar_type() == at::kFloat && B.scalar_type() == at::kFloat,
        "Inputs must be float32"
    );
    TORCH_CHECK(
        A.dim() == 2 && B.dim() == 2 &&
        A.size(0) == A.size(1) && B.sizes() == A.sizes(),
        "Inputs must be square matrices of the same shape"
    );

    const c10::cuda::CUDAGuard device_guard(A.device());

    A = A.contiguous();
    B = B.contiguous();

    auto C = torch::empty_like(A);
    const int N = static_cast<int>(A.size(0));
    if (N == 0)
        return C;

    const auto stream = at::cuda::getCurrentCUDAStream();

    if (N >= 2048) {
        const int tiles = (N + 127) / 128;
        triangular_mm_kernel<8><<<dim3(tiles, tiles), 256, 0, stream>>>(
            A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(), N
        );
    } else {
        const int tiles = (N + 63) / 64;
        triangular_mm_kernel<4><<<dim3(tiles, tiles), 256, 0, stream>>>(
            A.data_ptr<float>(), B.data_ptr<float>(), C.data_ptr<float>(), N
        );
    }

    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return C;
}
"""

_triangular_mm = load_inline(
    name="triangular_mm_tiled_cuda",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["triangular_mm_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, A, B):
        if (
            A.is_cuda
            and B.is_cuda
            and A.dtype == torch.float32
            and B.dtype == torch.float32
        ):
            return _triangular_mm.triangular_mm_cuda(A, B)
        return torch.tril(torch.matmul(A, B))
```