Design

Metal source

src/tandem_mlx/tandem.metal is the shader source of tandem-metal, commit 3c0b1be, unchanged. A test checks its hash. It holds the building blocks, the derived draws and the fills as functions in namespace tandem, and kernels for a Metal host, which tandem-mlx leaves out by defining TANDEM_NO_KERNELS. tm.metal_source() returns that header, so other mx.fast.metal_kernel code can use it.

Fills

One thread runs one chunk for its K steps, and the eight threads of a group store one contiguous 128-byte row per step. Normals from an odd first draw pair the last word of a block with the first word of the next block through threadgroup memory.

Normals and exponentials

The normals use the polynomial logarithm, sine and cosine of tandem-c, and the exponentials tandem-c's two-part -log, within 0.58 ulp, all with explicit fma and metal::precise::divide and metal::precise::sqrt. They are bit for bit equal to tandem-c under every Metal math mode. Plain sqrt is not correctly rounded even in Metal's safe mode, and plain / is not in fast mode. MLX compiles custom kernels in safe mode by default, and tandem-mlx passes compile_options={"math_mode": "safe"}, which needs MLX 0.32.