The spec is only 12 pages, much shorter than for IEEE (while it references it), and the similarities and differences with IEEE floats are intriguing:
6.5 Conversions between posit format and IEEE Std 754™ float format
[…] NaR converts to quiet NaN […]
The standard describes posit8 as the smallest “example”, but I believe you can (and want?) to have smaller. At least I think that size may be very valuable (and e.g. posit4, maybe unsigned?, would only need 64-bit quire [operations]), and better than Float8 (available in a package), and [b]float16 in some cases.
5.7 Functions of three posit value arguments
fMM(posit1, posit2, posit3) returns posit1 × posit2 × posit3, rounded.
[unlike IEEE, not defined there unless I missed something]
and its footnote:
Because multiplication is commutative and associative, any permutation of the inputs will return the same rounded result.
[And note, there’s no FMA on three posit inputs, similar to FMA with 3 floats in IEEE. But there is “FMA”, when one input of 3 is from the quire, so I think it amounts to the same.]
Should Julia support fMM (and Posits) at some point? I’m trying to think why is it defined, anyone know, or wants to guess? I suppose it can be an important operation, and while rounded, it must have something to do with the quire which I think is a type of fixed-point accumulator with guard bits to prevent overflow:
quire value A real number representable using a quire format described in this standard, or NaR.
quire sum limit The minimum number of additions of posit values that can overflow the quire format.
Note, this footnote:
The product of two posit values in a format of precision 𝑛 is always exactly expressible in a posit format of precision 2𝑛, but the quire format obviates such temporary doubling of precision when computing sums and differences of products. Sums of posit values using the quire are guaranteed exact up to 2^23+4𝑛 terms, per the quire sum limit
Quire format can represent the exact dot product of two posit vectors having at most 2^31 (approximately two billion) terms without the possibility of rounding or overflow.
Note, the minimum “quire format precision” is 16𝑛 bit (changed from nbits^2/2 in the draft standard), e.g. 128 bits for posit8 (no longer 32 bits), up to 512 bits for posit32, and no longer 2048 bits for posit64 (would be 1024 but posit64 is dropped as an example since the draft, maybe not too useful or needed format?).
fused rounded only after an exact evaluation of an expression involving more than one operation.
5.11 Functions involving quire value arguments
With the exception of qToP which returns a posit value result, these functions return a quire value result.13
qSubQ(quire1, quire2) returns quire1 − quire2.
qMulAdd(quire, posit1, posit2) returns quire + (posit1 × posit2).
qMulSub(quire, posit1, posit2) returns quire − (posit1 × posit2).
qToP(quire) returns quire rounded to posit format per Section 4.1.
Other functions of quire values may be provided, but are not required for compliance. They may be required in a future revision of this standard.
Discussion in footnote 13, on e.g. complex-values, and elsewhere is intriguing.
NaR Not a real. Umbrella value for anything not mathematically definable as a unique real number.
julia> NaN == NaN false
A test of equality between NaR values returns True.
exception A special case in the interpretation of representations in posit format: 0 or NaR.
Posit representation exceptions do not imply a need for status flags or heavyweight operating system or language runtime handling.
regime The power-of-16 scaling determined by the regime bits. It is a signed integer.
next(posit) returns the posit value of the lexicographic successor of posit’s representation.8
prior(posit) returns the posit value of the lexicographic predecessor of posit’s representation.9
the footnote a bit scare (depending on what “if necessary” mean here):
wrapping around, if necessary
6.2 Conversions involving quire values
A posit compliant system only needs to support rounding from quire to posit values and conversion of posit to s in the matching precision, per Section 5.11.