Exponent(inf) trouble

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The significant bits in a NaN (except the leading bit distinguishing between quiet and signalling NaN) are called “payload”.
This is what Perplexity.ai gave me for How do NaN payloads work in IEEE 754?:

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NaN payloads in IEEE 754 provide additional bits in the significand field of NaN values for diagnostic or application-specific data, beyond just signaling invalid operations. en.wikipedia

NaN Structure

NaNs share the all-1s exponent with infinities but have a non-zero significand (fraction/mantissa).

  • Sign bit: Ignored or application-defined (often 0).
  • Leading significand bit: Distinguishes types.
    • Quiet NaN (qNaN): MSB=1 (propagates silently).
    • Signaling NaN (sNaN): MSB=0 (may raise exceptions if enabled).
  • Payload: Remaining significand bits (e.g., 22 bits for single precision binary32, 51 for binary64 double). stackoverflow
Format Total Bits Exponent Sign + Type Bit Payload Bits
binary32 32 8 2 22
binary64 64 11 2 51
binary128 128 15 2 111

Example binary32 qNaN: 0 11111111 1xxxxxxxxxxxxxxx (payload in x’s). en.wikipedia

Payload Behavior

  • Propagation: Operations with NaN return NaN, typically copying the input NaN’s payload (or preferring one if multiple). discourse.llvm
  • Ordering: NaNs sort after numbers; among NaNs, by payload (lesser payload < greater in IEEE 754-2008). en.wikipedia
  • Uses: Diagnostic codes (e.g., error origins), NaN-boxing (type-tagged non-floats in payloads, as in JavaScript engines), or proprietary data. dan.corlan

Standard Operations (IEEE 754-2008/2019)

IEEE 754-2019 recommends getPayload, setPayload, setPayloadSignaling for payload access (required for new implementations). englishwiki.en-us.nina

Operation Behavior
qNaN op anything qNaN (payload preserved)
sNaN op non-NaN qNaN (if no exception)
sNaN op sNaN qNaN
Comparison with NaN False (except != self)

Payloads enhance NaN utility without altering core semantics, ensuring vast NaN space (~0.4% of values) isn’t wasted. Implementations vary (e.g., x86 may embed instruction addresses). stackoverflow

Edited: As for 0**0 result, according to englishwiki.en-us.nina link above, also found elsewhere, the standard (IEEE, I presume) provides several power functions with different expected results:

  • The standard pow function and the integer exponent pown function define 0**0, 1**(±∞), and (±∞)**0 as 1.
  • The powr function defines all three indeterminate forms as invalid operations and so returns NaN.
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@urbanjost is of course right about x0**x0. But x**y may not tend to 1 if x and y both tend to zero, but are not equal. If y=c/log(x) where c is a constant, the limit is exp(c). Then there’s y=c/sqrt(abs(log(x))) …