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IEEE(3)			 BSD Library Functions Manual		       IEEE(3)

     ieee — IEEE standard 754 for floating-point arithmetic

     The IEEE Standard 754 for Binary Floating-Point Arithmetic defines repre‐
     sentations of floating-point numbers and abstract properties of arith‐
     metic operations relating to precision, rounding, and exceptional cases,
     as described below.

   IEEE STANDARD 754 Floating-Point Arithmetic
     Radix: Binary.

     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 particu‐
	   lar, comparison (x > y, x ≥ y, etc.) 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.

	   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 usu‐
	   ally 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.

	   IEEE 754 recognizes five kinds of floating-point exceptions, listed
	   below in declining order of probable importance.

		 Exception	      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.

   Data Formats
	   Type name: float

	   Wordsize: 32 bits.

	   Precision: 24 significant bits, roughly like 7 significant deci‐
		 If x and x' are consecutive positive single-precision numbers
		 (they differ by 1 ulp), then
		 5.9e-08 < 0.5**24 < (x'-x)/x ≤ 0.5**23 < 1.2e-07.

	   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
		 0.5**149 = 1.4e-45.

	   Type name: double
		 On some architectures, long double is the the same as double.

	   Wordsize: 64 bits.

	   Precision: 53 significant bits, roughly like 16 significant deci‐
		 If x and x' are consecutive positive double-precision numbers
		 (they differ by 1 ulp), then
		 1.1e-16 < 0.5**53 < (x'-x)/x ≤ 0.5**52 < 2.3e-16.

	   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
		 0.5**1074 = 4.9e-324.

	   Type name: long double (when supported by the hardware)

	   Wordsize: 96 bits.

	   Precision: 64 significant bits, roughly like 19 significant deci‐
		 If x and x' are consecutive positive extended-precision num‐
		 bers (they differ by 1 ulp), then
		 1.0e-19 < 0.5**63 < (x'-x)/x ≤ 0.5**62 < 2.2e-19.

	   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
		 0.5**16445 = 5.7e-4953.

	   Type name: long double (when supported by the hardware)

	   Wordsize: 128 bits.

	   Precision: 113 significant bits, roughly like 34 significant deci‐
		 If x and x' are consecutive positive quad-extended-precision
		 numbers (they differ by 1 ulp), then
		 9.6e-35 < 0.5**113 < (x'-x)/x ≤ 0.5**112 < 2.0e-34.

	   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
		 0.5**16494 = 6.5e-4966.

   Additional Information Regarding Exceptions
     For each kind of floating-point exception, IEEE 754 provides a Flag that
     is raised each time its exception is signaled, and stays raised until the
     program resets it.	 Programs may also test, save and restore a flag.
     Thus, IEEE 754 provides three ways by which programs may cope with excep‐
     tions for which the default result might be unsatisfactory:

     1.	  Test for a condition that might cause an exception later, and branch
	  to avoid the exception.

     2.	  Test a flag to see whether an exception has occurred since the pro‐
	  gram last reset its flag.

     3.	  Test a result to see whether it is a value that only an exception
	  could have produced.

	  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 gradu‐
	  ally 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 under‐
	  flows 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:

     1.	  ABORT.  This mechanism classifies an exception in advance as an
	  incident to be handled by means traditionally associated with error-
	  handling statements like "ON ERROR GO TO ...".  Different languages
	  offer different forms of this statement, but most share the follow‐
	  ing characteristics:

	  -   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.

     2.	  STOP.	 This mechanism, requiring an interactive debugging environ‐
	  ment, is more for the programmer than the program.  It classifies an
	  exception in advance as a symptom of a programmer's error; the
	  exception suspends execution as near as it can to the offending
	  operation so that the programmer can look around to see how it hap‐
	  pened.  Quite often the first several exceptions turn out to be
	  quite unexceptionable, so the programmer ought ideally to be able to
	  resume execution after each one as if execution had not been

     3.	  ... Other ways lie beyond the scope of this document.

     Ideally, each elementary function should act as if it were indivisible,
     or atomic, in the sense that ...

     1.	  No exception should be signaled that is not deserved by the data
	  supplied to that function.

     2.	  Any exception signaled should be identified with that function
	  rather than with one of its subroutines.

     3.	  The internal behavior of an atomic function should not be disrupted
	  when a calling program changes from one to another of the five or so
	  ways of handling exceptions listed above, although the definition of
	  the function may be correlated intentionally with exception han‐

     The functions in libm are only approximately atomic.  They signal no
     inappropriate exception except possibly ...
		   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 cancella‐
		   tion of errors.
     Otherwise, ...
	   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
	   Divide-by-Zero is signaled only when
		   a function takes exactly infinite values at finite oper‐
	   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.

     fenv(3), ieee_test(3), math(3)

     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 Arith‐
     metic" 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. 14 no. 3 (Mar. 1981), and in the ACM
     SIGNUM Newsletter Special Issue of Oct. 1979, may be helpful although
     they pertain to superseded drafts of the standard.

     IEEE Std 754-1985

BSD			       January 26, 2005				   BSD

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