**Evaluating TF32 and FP32 differences** I'm trying to figure out if there's a notable difference between TF32 errors and exact comparisons. It seems FP32 could be a bit too high, like relative 1e-3. I need to check if PyTorch is using the default settings for TF32 and ensure the reference matches the expected tolerances — usually 1e-4 for FP32. I should confirm if cublasStrmm is available and ensure that things like with_cuda=True are correctly set up in the build parameters. Maybe I'll explore if custom kernels will fit the requirements. **Considering performance options** I'm trying to figure out how to use both triangular matrices effectively, aiming for N^3/6 performance. It seems like there might be a challenge, especially since the performance of custom simple tiled FP32 CUDA cores is lower than what I’d get with cuBLAS. I wonder how to optimize this better. Do I need to rethink my approach or maybe adjust my strategy to improve performance? There’s definitely a lot to consider here! **Calculating tile performance** I’m working through some calculations related to tile design. The total number of tiles sums up to about 2.83 million, which is equivalent to 11.6 billion fused multiply-add (FMA) operations. Each 16x16 block seems to have 256 threads, running around 85 iterations, so we're looking at about 1,360 FMA per tile. I’m considering shared loads and register occupancy; performance should be moderate. There’s also redundancy in using __syncthreads, which could lead to overhead. Lastly, I’m pondering block configurations for optimal performance. **Evaluating CUDA integration** I'm considering whether to include the cublas host in the cuda_sources. If I'm using cublas, it seems like I can't have a custom kernel. I'm thinking of including a custom zero for the upper output, but then I hesitate. Maybe I'll add a custom kernel and choose cublasStrmm in my wrapper. Ultimately, my goal is to see if cublasStrmm produces the exact upper zeros I need. Let's clarify that! **Evaluating matrix computations** I’m looking into how lower entries might compute the sum of B_col(r,k)*A_col(k,c) for k. It seems with upper triangular matrices, I need to consider conditions like r > c and if k satisfies r <= k <= c. I might skip structural terms if necessary, and I wonder if cublas will write the lower part properly. It's good that cublas TRMM should ideally compute and write the full C matrix. If there's a speed issue, I could consider using a custom kernel. Finally, ensuring my output is in JSON format with one tool call is crucial. **Considering handle optimization** I’m thinking about potential name collisions in caching across submissions, and it might be best to use unique names. I could optimize the handle creation overhead by instantiating it during the first forward, but that could make the cublasCreate overhead quite significant. Maybe I can create a handle at module load through a function. I should define an init_cublas() that can be called after load_inline in Python to set up the handle before timing. This could include a guard to manage device contexts effectively and minimize overhead during evaluations. **Optimizing handle management** Alright, if there are multiple static handles, I’ll need to associate them with device 0, especially since matrix A is on device 0 too. I should confirm if the cublas handle can be linked to the device. If I initialize it before any device settings, that should default to device 0, which is fine. I also want to ensure my initialization step is in place. I need to check if the PyTorch timing accurately reflects the module load, as that compilation might not be measured.