mixed precision

Use proportional bit layouts and adjacent values around one to distinguish range from precision. Then choose formats from numerical tests and actual kernel support.

Watch a low-precision sum stop changing and a small weight update round away. Separate storage precision, accumulation, optimizer state and dynamic range before choosing a mixed-precision training policy.

Separate gradient range from update precision using a small numerical example, and make the training-state recipe explicit.

Separate FP8 input quantization from accumulation error. A small representable-value example shows when a shared scale loses small values, while the DeepSeek-V3 report supplies one carefully scoped training recipe.