Has anyone implemented a Float8 / Quarter type?

You're going a bit off the rails of what IEEE 754 defined, historically.

However, the new (2018) standard comes to the rescue with some clarity:

NOTE 3—For binary formats, the precision p should be at least 3, as some numerical properties do not hold for lower precisions.
Similarly, emax should be at least 2 to support the operations listed in 9.2.

So the significand should have at least three bits (one of which may be implicit), separate from the need to encode qNaN and sNaN.

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