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```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 ∞.  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.

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 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.

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	      ±∞
Divide by Zero	      ±∞
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 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 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‐
mals.
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.

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
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
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 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
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
they pertain to superseded drafts of the standard.

STANDARDS
IEEE Std 754-1985

BSD			       January 26, 2005				   BSD
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