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Oh, I should have clarified - could one start with a bfloat16 on software-side, convert to float16 (so that e.g a 3.4E38 float16 becomes a 65504 float16), then do any "heavy math" in fast hardware float16 instructions, and then convert back at the end?

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No, you can't. The exponent in a float16 is too small. You'd rather convert back to a 32-bit float, do your operations, and then throw away the surplus precision and convert back to bfloat16.

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