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In particular, IEEE 754 already uses "canonical NaN" with the meaning of "canonical encoding of a NaN" (e.g. "isCanonical(x) is true if and only if x is a finite number, infinity, or NaN that is canonical." page 38, but also for totalOrder page 42), thus a different meaning from what is used here. Please help clarify the section.
However, float in Python, Ruby, PHP, and OCaml and single in versions of Octave before 3.2 refer to double-precision numbers. ... ±infinity: NaN (quiet, signalling)
negative infinity By default, 1/3 rounds down like for double precision , because of the odd number of bits in the significand. The bits beyond the rounding point are 0101 ... which is less than 1/2 of a unit in the last place .
NaN is sortable. NaN is treated as if it had a larger absolute value than Infinity (or any other floating-point numbers). (−NaN < −Infinity; +Infinity < +NaN.) qNaN and sNaN are treated as if qNaN had a larger absolute value than sNaN. (−qNaN < −sNaN; +sNaN < +qNaN.) NaN is then sorted according to the payload.
The graphic demonstrates the addition of even smaller (1.3.2.3)-minifloats with 6 bits. This floating-point system follows the rules of IEEE 754 exactly. NaN as operand produces always NaN results. Inf − Inf and (−Inf) + Inf results in NaN too (green area). Inf can be augmented and decremented by finite values without change.
An exceptional result is represented by a special code called a NaN, for "Not a Number". All NaNs in IEEE 754-1985 have this format: sign = either 0 or 1. biased exponent = all 1 bits. fraction = anything except all 0 bits (since all 0 bits represents infinity).
NaN (quiet, signalling) The minimum strictly positive (subnormal) value is 2 −262378 ≈ 10 −78984 and has a precision of only one bit. The minimum positive normal value is 2 −262142 ≈ 2.4824 × 10 −78913 .
The most significant two bits of the exponent are limited to the range of 0−2, and the most significant 4 bits of the significand are limited to the range of 0−9. The 30 possible combinations are encoded in a 5-bit field, along with special forms for infinity and NaN.