```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_matmul_cuda(torch::Tensor A, torch::Tensor B);
"""

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

constexpr int TILE = 64;
constexpr int BK = 32;

__global__ void triangular_matmul_kernel(
    const float* __restrict__ A,
    const float* __restrict__ B,
    float* __restrict__ C,
    const int64_t N
) {
    const int tx = threadIdx.x;
    const int ty = threadIdx.y;
    const int tid = ty * 16 + tx;
    const int64_t row_base = int64_t(blockIdx.y) * TILE;
    const int64_t col_base = int64_t(blockIdx.x) * TILE;

    if (blockIdx.x > blockIdx.y) {
        #pragma unroll
        for (int i = 0; i < 4; ++i) {
            const int64_t row = row_base + ty + i * 16;
            #pragma unroll
            for (int j = 0; j < 4; ++j) {
                const int64_t col = col_base + tx + j * 16;
                if (row < N && col < N)
                    C[row * N + col] = 0.0f;
            }
        }
        return;
    }

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

    float acc[4][4] = {};

    const int64_t k_end =
        (row_base + TILE < N) ? row_base + TILE : N;

    // Lower-triangular inputs allow only col <= k <= row.
    for (int64_t k_base = col_base; k_base < k_end; k_base += BK) {
        #pragma unroll
        for (int offset = 0; offset < TILE * BK; offset += 256) {
            const int index = tid + offset;
            const int ar = index / BK;
            const int ak = index % BK;
            const int bk = index / TILE;
            const int bc = index % TILE;

            const int64_t a_row = row_base + ar;
            const int64_t a_col = k_base + ak;
            const int64_t b_row = k_base + bk;
            const int64_t b_col = col_base + bc;

            As[ar][ak] = (a_row < N && a_col < N)
                ? A[a_row * N + a_col] : 0.0f;
            Bs[bk][bc] = (b_row < N && b_col < N)
                ? B[b_row * N + b_col] : 0.0f;
        }
        __syncthreads();

        #pragma unroll
        for (int k = 0; k < BK; ++k) {
            float av[4];
            float bv[4];

            #pragma unroll
            for (int i = 0; i < 4; ++i)
                av[i] = As[ty + i * 16][k];

            #pragma unroll
            for (int j = 0; j < 4; ++j)
                bv[j] = Bs[k][tx + j * 16];

            #pragma unroll
            for (int i = 0; i < 4; ++i) {
                #pragma unroll
                for (int j = 0; j < 4; ++j)
                    acc[i][j] = fmaf(av[i], bv[j], acc[i][j]);
            }
        }
        __syncthreads();
    }

    #pragma unroll
    for (int i = 0; i < 4; ++i) {
        const int64_t row = row_base + ty + i * 16;
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const int64_t col = col_base + tx + j * 16;
            if (row < N && col < N)
                C[row * N + col] = col <= row ? acc[i][j] : 0.0f;
        }
    }
}

torch::Tensor triangular_matmul_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "Expected 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,
        "Expected float32 tensors"
    );
    TORCH_CHECK(
        A.dim() == 2 && B.dim() == 2 &&
        A.size(0) == A.size(1) &&
        B.size(0) == A.size(0) && B.size(1) == A.size(1),
        "Expected equally sized square matrices"
    );

    const c10::cuda::CUDAGuard guard(A.device());
    A = A.contiguous();
    B = B.contiguous();

    const int64_t N = A.size(0);
    auto C = torch::empty({N, N}, A.options());
    if (N == 0)
        return C;

    const int64_t tiles = (N + TILE - 1) / TILE;
    TORCH_CHECK(tiles <= 65535, "Matrix exceeds supported grid dimensions");

    triangular_matmul_kernel<<<
        dim3(static_cast<unsigned>(tiles), static_cast<unsigned>(tiles)),
        dim3(16, 16),
        0,
        c10::cuda::getCurrentCUDAStream().stream()
    >>>(
        A.data_ptr<float>(),
        B.data_ptr<float>(),
        C.data_ptr<float>(),
        N
    );
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return C;
}
"""

_triangular_ops = load_inline(
    name="triangular_matmul_tiled_fp32",
    cpp_sources=_cpp_source,
    cuda_sources=_cuda_source,
    functions=["triangular_matmul_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_ops.triangular_matmul_cuda(A, B)
        return torch.tril(torch.matmul(A, B))
```