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NaN spec in ieee754-2008
8. Infinity, NaNs, and sign bit 8.0
8.1 Infinity arithmetic
Infinity arithmetic shall be construed as the limiting case of real arithmetic with operands of arbitrarily large
magnitude, when such a limit exists. Infinities shall be interpreted in the affine sense, that is, –∞< (every finite number) < +∞.
Operations on infinite operands are usually exact and therefore signal no exceptions. The exceptions that do
pertain to infinities are signaled only when:
∞ is an invalid operand (see 9.2),
∞ is created from finite operands by overflow (see 9.4) or division by zero (see 9.3), remainder(subnormal, ∞) signals underflow,
nextAfter(x, ∞) signals underflow and inexact if the result would be subnormal,
nextAfter(max normal, ∞) signals overflow and inexact if the result would be infinite.
8.2 Operations with NaNs
8.2.0
Two different kinds of NaN, signaling and quiet, shall be supported in all operations. Signaling NaNs afford
representations for uninitialized variables and arithmetic-like enhancements (such as complex-affine infinities
or extremely wide range) that are not the subject of the standard. Quiet NaNs should, by means left to the
implementer's discretion, afford retrospective diagnostic information inherited from invalid or unavailable
data and results. To facilitate propagation of diagnostic information contained in NaNs, as much of that
information as possible should be preserved in NaN results of computational operations.
Signaling NaNs shall be reserved operands that signal the invalid operation exception (see 9.1) for every
general-computational and signaling-computational operation.
Under default exception handling, any operation signaling an invalid exception for which a floating-point
result is to be delivered, shall deliver a quiet NaN.
Every general-computational and quiet-computational operation involving one or more input NaNs, none of
them signaling, shall signal no exception, except fusedMultiplyAdd may signal the invalid operation
exception (see 9.2) . For an operation with quiet NaN inputs other than max and min operations, if a floating-
point result is to be delivered, the result shall be a quiet NaN, which should be one of the input NaNs. If the
trailing significand field of a decimal input NaN is canonical then the bit pattern of that field shall be
preserved if that NaN is chosen as the result NaN. Note that format conversions, including conversions
between internal formats and external representations as character sequences, might be unable to deliver the
same NaN. Quiet NaNs signal exceptions on some operations that do not deliver a floating-point result; these
operations, namely comparison and conversion to a format that has no NaNs, are discussed in 7.6, 7.8, and
9.2.
8.2.1 NaN encodings in binary formats
8.2.1.0
This clause further specifies the encodings of NaNs as bit strings when they are the results of operations.
When encoded, all NaNs have a sign bit and a pattern of bits necessary to identify the encoding as a NaN and
which determines its kind (sNaN vs. qNaN). The remaining bits, which are in the trailing field, encode the
payload, which might be diagnostic information (see 8.2).
9.2 Invalid operation
9.2.0
The invalid operation exception is signaled if and only if there is no usefully definable result. In these cases
the operands are invalid for the operation to be performed.
For operations producing results in floating-point format, the default result of an invalid exception operation
shall be a quiet NaN that should provide some diagnostic information (see 8.2). Such invalid exception
operations in this standard are:
a) any general-computational or signaling-computational operation on a signaling NaN (see 8.2);
b) multiplication: multiplication(0,∞ ) or multiplication(∞ , 0);
c) fusedMultiplyAdd: fusedMultiplyAdd(0, ∞, c) or fusedMultiplyAdd(∞, 0, c) unless c is a quiet NaN;
if c is a quiet NaN then it is implementation defined whether the invalid operation exception is
signaled;
d) addition or subtraction or fusedMultiplyAdd: magnitude subtraction of infinities, such as: addition
(+∞ , –∞ );
e) division: division(0, 0) or division( ∞,∞ );
f) remainder: remainder(x,y), where y is zero or x is infinite and neither is NaN;
g) squareRoot if the operand is less than zero;
h) quantize when the result does not fit in the destination format or when one operand is finite and the
other is infinite.
For operations producing no result in floating-point format, the invalid exception operations are:
i) conversion of an internal floating-point number to an integer format, when the source is NaN,
infinity, or a value which would convert to an integer outside the range of the result format under the
prevailing rounding mode.
j) comparison by way of unordered-signaling predicates listed in Table 9, when the operands are
unordered;
k) logB(NaN), logB(∞), and logB(0) when logBFormat is an integer format (see 7.3.3) .
8.3 The sign bit 8.3.0
When either an input or result is NaN, this standard does not interpret the sign of a NaN. Note however that
operations on bitstrings – copy, negate, abs, copySign – specify the sign bit of a NaN result, sometimes based
upon the sign bit of a NaN operand. The logical predicate totalOrder is also affected by the sign bit of a NaN
operand. For all other operations, this standard does not specify the sign bit of a NaN result, even when there
is only one input NaN, or when the NaN is produced from an invalid operation.
When neither the inputs nor result are NaN, the sign of a product or quotient is the exclusive OR of the
operands' signs; the sign of a sum, or of a difference x–y regarded as a sum x+(–y), differs from at most one of
the addends' signs; and the sign of the result of the roundToIntegral operations and roundToIntegralExact (see
7.3.1) is the sign of the operand. These rules shall apply even when operands or results are zero or infinite.
When the sum of two operands with opposite signs (or the difference of two operands with like signs) is
exactly zero, the sign of that sum (or difference) shall be +0 in all rounding direction modes except
roundTowardNegative; in that mode, the sign of an exact zero sum (or difference) shall be –0. However, x+x
= x–(–x) retains the same sign as x even when x is zero.
When (a×b)+c is exactly zero, the sign of fusedMultiplyAdd(a, b, c) shall be determined by the rules above
for a sum of operands. When the exact result of (a×b)+c is nonzero yet the result of fusedMultiplyAdd is
zero because of rounding, the zero result takes the sign of the exact result.
Except that squareRoot(–0) shall be –0, every valid squareRoot shall have a positive sign. !!!
The representation r of the floating-point datum, and value v of the floating-point datum represented, are
inferred from the constituent fields, thus:
a) If G0 through G4 are 11111, then v is NaN regardless of S. Furthermore, if G5 is 1, then r is sNaN;
otherwise r is qNaN. The remaining bits of G are ignored, and T constitutes the NaN's payload,
which can be used to distinguish various NaNs.
The NaN payload is encoded similarly to finite numbers described below, with G treated as though
all bits were zero. The payload corresponds to the significand of finite numbers, interpreted as an
integer with a maximum value of 10^(3×J) − 1, and the exponent field is ignored (it is treated as if it
were zero). A NaN is in its preferred (canonical) representation if the bits G6 through G_(w+4) are zero
and the encoding of the payload is canonical.
class(x) tells which of the following ten classes x falls into:
signalingNaN
quietNaN
...
The bit pattern in a NaN significand can affect how the
NaN is propagated
When a NaN operand cannot be represented in the destination format and this cannot otherwise be indicated,
the invalid exception shall be signaled. When a numeric operand would convert to an integer outside the
range of the destination format, the invalid exception shall be signaled if this situation cannot otherwise be
indicated.
7.10 Details of totalOrder predicate
7.10.0
For each supported non-storage floating-point format, an implementation shall provide certain predicates that
define orderings among all operands in a particular format.
totalOrder(x,y) imposes a total ordering on canonical members of the format of x and y;
a) if x < y, totalOrder(x, y) is true
b) if x > y, totalOrder(x, y) is false
c) if x = y:
1) totalOrder(−0, +0) is true
2) totalOrder(+0, −0) is false
3) if x and y represent the same floating-point datum:
i) if x and y have negative sign,
totalOrder(x, y) is true if and only if the exponent of x ≥ the exponent of y
ii) otherwise
totalOrder(x, y) is true if and only if the exponent of x ≤ the exponent of y
Note that totalOrder does not impose a total ordering on all encodings in a format. In particular
it does not distinguish among different encodings of the same floating-point representation, as
when one or both encodings are non-canonical.
d) if x and y are unordered numerically because x or y is NaN:
1) totalOrder(−NaN, y) is true where −NaN represents a NaN with negative sign bit and y is a
floating-point number.
2) totalOrder(x, +NaN) is true where +NaN represents a NaN with positive sign bit and x is a
floating-point number.
3) if x and y are both NaNs, then totalOrder reflects a total ordering based on
i) negative sign bit < positive sign bit
ii) signaling < quiet for +NaN, reverse for −NaN
iii) lesser payload < greater payload for +NaN, reverse for −NaN
Neither signaling nor quiet NaNs signal an exception.
For canonical x and y, totalOrder(x,y) and totalOrder(y,x) are both true only if x and y are bitwise identical.
7.11 Details of comparison predicates
7.11.0
For every supported non-storage floating-point format, it shall be possible to compare one floating-point
datum to another in that format. Additionally, floating-point data represented in different formats shall be
comparable as long as the operands' formats have the same radix.
Comparisons are exact and never overflow or underflow. Four mutually exclusive relations are possible: less
than, equal, greater than, and unordered. The last case arises when at least one operand is NaN. Every NaN
shall compare unordered with everything, including itself. Comparisons shall ignore the sign of zero
(so +0 = −0). Infinite operands of the same sign shall compare equal.
Languages define how the result of a comparison shall be delivered, in one of two ways: either as a condition
7.12.1 External character sequences representing zeros, infinities, and NaNs 7.12.1.0
The conversions (described in 7.12) from internal formats to external character sequences and back that
recover the original floating-point representation, recover zeros, infinities, and quiet NaNs, as well as nonzero
finite numbers. In particular, signs of zeros and infinities are preserved.
Conversion of an infinity in internal format to an external character sequence shall produce a language-
defined one of “inf” or “infinity” or a sequence that is equivalent except for case (e.g., “Infinity” or “INF”),
with a preceding minus sign if the input is negative. Whether the conversion produces a preceding plus sign if
the input is positive is language defined.
Conversion of external character sequences “inf” and “infinity”, regardless of case, with an optional
preceding sign, to an internal floating-point format shall produce an infinity (with the same sign as the input).
Conversion of a quiet NaN in internal format to an external character sequence shall produce a language-
defined one of “nan” or a sequence that is equivalent except for case (e.g., “NaN”), with an optional
preceding sign.
Conversion of a signaling NaN in internal format to an external character sequence should produce a
language-defined one of "snan" or "nan" or a sequence that is equivalent except for case, with an optional
preceding sign. If the conversion of a signaling NaN produces "nan" or a sequence that is equivalent except
for case, with an optional preceding sign, then the invalid exception should be signaled.
Conversion of external character sequences “nan”, regardless of case, with an optional preceding sign, to an
internal floating-point format shall produce a quiet NaN.
Conversion of an external character sequence "snan", regardless of case, with an optional preceding sign, to
an internal format should either produce a signaling NaN or else produce a quiet NaN and signal the invalid
exception.
Languages should provide an optional conversion of NaNs in internal format to external character sequences
that appends to the basic NaN character sequences a suffix that can represent the NaN payload (see 8.2). The
form and interpretation of the payload suffix is language defined. The language should require that any such
optional output sequences be recognized as input in conversion of external character sequences to internal
formats.
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