# FreeBSD Manual Pages

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

NAME
ieee -- IEEE standard 754 for floating-point arithmetic

DESCRIPTION
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

Overflow and underflow:
Overflow goes by default to a signed	infinity.  Underflow is

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 infinity = 1/(x-y) !=
-1/(y-x) = -infinity.

Infinity is signed.
It persists when added to itself or to any finite number.  Its sign
transforms correctly	through	multiplication and division, and (fi-
nite)/+-infinity = +-0 (nonzero)/0 =	+-infinity.  But infinity-in-
finity, infinity*0 and infinity/infinity 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 op-
erations.  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 (+, -, *, /, <sqrt>) is rounded by	de-
fault 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 ev-
ery 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
+infinity or	towards	-infinity at the programmer's option.

Exceptions:
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	      +-infinity
Divide	by Zero	      +-infinity
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
Single-precision:
Type	name: float

Wordsize: 32	bits.

Precision: 24 significant bits, roughly like	7 significant deci-
mals.

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.

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 deci-
mals.

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.

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

Wordsize: 96	bits.

Precision: 64 significant bits, roughly like	19 significant deci-
mals.

If x	and x' are consecutive positive	extended-precision numbers
(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-
mals.

If x	and x' are consecutive positive	quad-extended-precision	num-
bers	(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.

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 in-
cident 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 op-
eration'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 ex-
ception 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 ex-
ception suspends execution as	near as	it can to the offending	opera-
tion so that the programmer can look around to see how it happened.
Quite	often the first	several	exceptions turn	out to be quite	unex-
ceptionable, so the programmer ought ideally to be able to resume
execution after each one as if execution had not been	stopped.

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-
dling.

The functions in libm are only approximately atomic.  They	signal no in-
appropriate 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 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
threshold.
Divide-by-Zero is signaled only when
a function takes exactly infinite values at finite oper-
ands.
Underflow is	signaled only when
the exact result would be nonzero but tinier	than the un-
derflow 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