numerics

Every format is a trade between how big a number can be and how finely it is spaced. The bit layouts, the largest and smallest values derived from them, why fp16 needed loss scaling and bf16 did not, and why fp8 comes in two flavors with a scale factor attached.

Normalization reads a row and writes a row, so a copy sets the ceiling and everything else is overhead you can remove. The six changes in the order a reviewer wants them, what each is worth, and the one that is a correctness fix rather than a speed fix, with the measured error that proves it.

A float format's resolution is relative, so the damage depends on how large a running total grows against the terms being added. The measured error for a 4,096-term dot product in each direction, the weight update that vanishes entirely, and why bf16 removed loss scaling but not master weights.

A bug that scales with the number of tiles is invisible in a unit test that fits in one tile. The three candidates that produce exactly this signature, the measurement that separates them, and a test design that would have caught all three before the kernel shipped.

A loss that stops moving has five different causes and one of them is not a training problem at all. The checks in order: is the step counter moving, what are the learning rate and the grad norm doing, is the data repeating, has the loss scaler collapsed, and is one rank sending zeros.