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Overflow and underflow: Overflow goes by default to a signed ∞. Underflow is gradual.
Zero is represented ambiguously as +0 or -0. Its sign transforms correctly through multiplication or division, and is preserved by addition of zeros with like signs; but x-x yields +0 for every finite x. The only operations that reveal zero's sign are division by zero and copysign(x, ±0). In particular, comparison (x > y, x ≥ y, etc.)amp; cannot be affected by the sign of zero; but if finite x = y then ∞ = 1/(x-y) != -1/(y-x) = -∞.
Infinity is signed. It persists when added to itself or to any finite number. Its sign transforms correctly through multiplication and division, and (finite)/±∞ = ±0 (nonzero)/0 = ±∞. But ∞-∞, ∞*0 and ∞/∞ are, like 0/0 and sqrt(-3), invalid operations that produce NaN. ...
Reserved operands (NaNs): An NaN is ( Not a Number). Some NaNs, called Signaling NaNs, trap any floating-point operation performed upon them; they are used to mark missing or uninitialized values, or nonexistent elements of arrays. The rest are Quiet NaNs; they are the default results of Invalid Operations, and propagate through subsequent arithmetic operations. If x != x then x is NaN; every other predicate (x > y, x = y, x < y, ...) is FALSE if NaN is involved.
Rounding: Every algebraic operation (+, -, *, /, √) is rounded by default to within half an ulp, and when the rounding error is exactly half an ulp then the rounded value's least significant bit is zero. (An ulp is one Unit in the Last Place.) This kind of rounding is usually the best kind, sometimes provably so; for instance, for every x = 1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find (x/3.0)*3.0 == x and (x/10.0)*10.0 == x and ... despite that both the quotients and the products have been rounded. Only rounding like IEEE 754 can do that. But no single kind of rounding can be proved best for every circumstance, so IEEE 754 provides rounding towards zero or towards +∞ or towards -∞ at the programmer's option.
Exceptions: IEEE 754 recognizes five kinds of floating-point exceptions, listed below in declining order of probable importance.
| Default Result | |
| Invalid Operation | NaN, or FALSE |
| Overflow | ±∞ |
| Divide by Zero | ±∞ |
| Underflow | Gradual Underflow |
| Inexact | Rounded value |
NOTE: An Exception is not an Error unless handled badly. What makes a class of exceptions exceptional is that no single default response can be satisfactory in every instance. On the other hand, if a default response will serve most instances satisfactorily, the unsatisfactory instances cannot justify aborting computation every time the exception occurs.
Wordsize: 32 bits.
Precision: 24 significant bits, roughly like 7 significant decimals.
If x and x' are consecutive positive single-precision numbers (they differ by 1 ulp), then
| Range: | Overflow threshold = 2.0**128 = 3.4e38 |
| Underflow threshold = 0.5**126 = 1.2e-38 |
Underflowed results round to the nearest integer multiple of
Double-precision: Type name: double (On some architectures, long double is the same as double )
Wordsize: 64 bits.
Precision: 53 significant bits, roughly like 16 significant decimals.
If x and x' are consecutive positive double-precision numbers (they differ by 1 ulp), then
| Range: | Overflow threshold = 2.0**1024 = 1.8e308 |
| Underflow threshold = 0.5**1022 = 2.2e-308 |
Underflowed results round to the nearest integer multiple of
Extended-precision: Type name: long double (when supported by the hardware)
Wordsize: 96 bits.
Precision: 64 significant bits, roughly like 19 significant decimals.
If x and x' are consecutive positive extended-precision numbers (they differ by 1 ulp), then
| Range: | Overflow threshold = 2.0**16384 = 1.2e4932 |
| Underflow threshold = 0.5**16382 = 3.4e-4932 |
Underflowed results round to the nearest integer multiple of
Quad-extended-precision: Type name: long double (when supported by the hardware)
Wordsize: 128 bits.
Precision: 113 significant bits, roughly like 34 significant decimals.
If x and x' are consecutive positive quad-extended-precision numbers (they differ by 1 ulp), then
| Range: | Overflow threshold = 2.0**16384 = 1.2e4932 |
| Underflow threshold = 0.5**16382 = 3.4e-4932 |
Underflowed results round to the nearest integer multiple of
CAUTION: The only reliable ways to discover whether Underflow has occurred are to test whether products or quotients lie closer to zero than the underflow threshold, or to test the Underflow flag. (Sums and differences cannot underflow in IEEE 754; if x != y then x-y is correct to full precision and certainly nonzero regardless of how tiny it may be.) Products and quotients that underflow gradually can lose accuracy gradually without vanishing, so comparing them with zero (as one might on a VAX) will not reveal the loss. Fortunately, if a gradually underflowed value is destined to be added to something bigger than the underflow threshold, as is almost always the case, digits lost to gradual underflow will not be missed because they would have been rounded off anyway. So gradual underflows are usually provably ignorable. The same cannot be said of underflows flushed to 0.
At the option of an implementor conforming to IEEE 754, other ways to cope with exceptions may be provided:
| No means is provided to substitute a value for the offending operation's result and resume computation from what may be the middle of an expression. An exceptional result is abandoned. | |
| In a subprogram that lacks an error-handling statement, an exception causes the subprogram to abort within whatever program called it, and so on back up the chain of calling subprograms until an error-handling statement is encountered or the whole task is aborted and memory is dumped. | |
Ideally, each elementary function should act as if it were indivisible, or atomic, in the sense that ...
The functions in libm are only approximately atomic. They signal no inappropriate exception except possibly ...
| Over/Underflow when a result, if properly computed, might have lain barely within range, and | |
| Inexact in cabs(), cbrt(), hypot(), log10() and pow() when it happens to be exact, thanks to fortuitous cancellation of errors. | |
| Invalid Operation is signaled only when any result but NaN would probably be misleading. | |
| Overflow is signaled only when the exact result would be finite but beyond the overflow threshold. | |
| Divide-by-Zero is signaled only when a function takes exactly infinite values at finite operands. | |
| Underflow is signaled only when the exact result would be nonzero but tinier than the underflow threshold. | |
| Inexact is signaled only when greater range or precision would be needed to represent the exact result. | |
An explanation of IEEE 754 and its proposed extension p854 was published in the IEEE magazine MICRO in August 1984 under the title "A Proposed Radix- and Word-length-independent Standard for Floating-point Arithmetic" by W. J. Cody et al. The manuals for Pascal, C and BASIC on the Apple Macintosh document the features of IEEE 754 pretty well. Articles in the IEEE magazine COMPUTER vol.amp; 14 no.amp; 3 (Mar.amp; 1981), and in the ACM SIGNUM Newsletter Special Issue of Oct.amp; 1979, may be helpful although they pertain to superseded drafts of the standard.
| IEEE (3) | January 26, 2005 |
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