float.h
Section: POSIX Programmer's Manual (0P)
Updated: 2013
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NAME
float.h
 floating types
SYNOPSIS
#include <float.h>
DESCRIPTION
The functionality described on this reference page is aligned with the
ISO C standard. Any conflict between the requirements described here and the
ISO C standard is unintentional. This volume of POSIX.12008 defers to the ISO C standard.
The characteristics of floating types are defined in terms of a model
that describes a representation of floatingpoint numbers and values
that provide information about an implementation's floatingpoint
arithmetic.
The following parameters are used to define the model for each
floatingpoint type:
 s

Sign (±1).
 b

Base or radix of exponent representation (an integer >1).
 e

Exponent (an integer between a minimum $e_ min$ and a maximum
$e_ max$).
 p

Precision (the number of baseb digits in the significand).
 $f_ k$

Nonnegative integers less than
b
(the significand digits).
A floatingpoint number
x
is defined by the following model:
x " " = " " sb"^" e" " " " sum from k=1 to p^ " " f_ k" " " " b"^" " "k ,
" " e_ min" " " " <= " " e " " <= " " e_ max" "
In addition to normalized floatingpoint numbers ($f_ 1$>0 if
x!=0),
floating types may be able to contain other kinds of floatingpoint
numbers, such as subnormal floatingpoint numbers (x!=0,
e=$e_ min$, $f_ 1$=0) and unnormalized floatingpoint numbers (x!=0,
e>$e_ min$, $f_ 1$=0), and values that are not floatingpoint
numbers, such as infinities and NaNs. A
NaN
is an encoding signifying NotaNumber. A
quiet NaN
propagates through almost every arithmetic operation without raising a
floatingpoint exception; a
signaling NaN
generally raises a floatingpoint exception when occurring as an
arithmetic operand.
An implementation may give zero and nonnumeric values, such as
infinities and NaNs, a sign, or may leave them unsigned. Wherever such
values are unsigned, any requirement in POSIX.12008 to retrieve the
sign shall produce an unspecified sign and any requirement to set the
sign shall be ignored.
The accuracy of the floatingpoint operations ('+',
'',
'*',
'/')
and of the functions in
<math.h>
and
<complex.h>
that return floatingpoint results is implementationdefined, as is the
accuracy of the conversion between floatingpoint internal
representations and string representations performed by the functions
in
<stdio.h>,
<stdlib.h>,
and
<wchar.h>.
The implementation may state that the accuracy is unknown.
All integer values in the
<float.h>
header, except FLT_ROUNDS, shall be constant expressions suitable for
use in
#if
preprocessing directives; all floating values shall be constant
expressions. All except DECIMAL_DIG, FLT_EVAL_METHOD, FLT_RADIX, and
FLT_ROUNDS have separate names for all three floatingpoint types. The
floatingpoint model representation is provided for all values except
FLT_EVAL_METHOD and FLT_ROUNDS.
The rounding mode for floatingpoint addition is characterized by the
implementationdefined value of FLT_ROUNDS:
 1

Indeterminable.
 0

Toward zero.
 1

To nearest.
 2

Toward positive infinity.
 3

Toward negative infinity.
All other values for FLT_ROUNDS characterize implementationdefined
rounding behavior.
The values of operations with floating operands and values subject to
the usual arithmetic conversions and of floating constants are
evaluated to a format whose range and precision may be greater than
required by the type. The use of evaluation formats is characterized by
the implementationdefined value of FLT_EVAL_METHOD:
 1

Indeterminable.
 0

Evaluate all operations and constants just to the range and
precision of the type.
 1

Evaluate operations and constants of type
float
and
double
to the range and precision of the
double
type; evaluate
long double
operations and constants to the range and precision of the
long double
type.
 2

Evaluate all operations and constants to the range and precision of the
long double
type.
All other negative values for FLT_EVAL_METHOD characterize
implementationdefined behavior.
The
<float.h>
header shall define the following values as constant expressions with
implementationdefined values that are greater or equal in magnitude
(absolute value) to those shown, with the same sign.
 *

Radix of exponent representation,
b.

 FLT_RADIX

2
 *

Number of baseFLT_RADIX digits in the floatingpoint significand,
p.

 FLT_MANT_DIG

 DBL_MANT_DIG

 LDBL_MANT_DIG

 *

Number of decimal digits,
n,
such that any floatingpoint number in the widest supported floating
type with $p_ max$ radix
b
digits can be rounded to a floatingpoint number with
n
decimal digits and back again without change to the value.

lpile { p_ max" " " " log_ 10" " " " b above
left ceiling " " 1 " " + " " p_ max" " " " log_ 10" " " " b right ceiling }
" " " " lpile {if " " b " " is " " a " " power " " of " " 10 above otherwise}
 DECIMAL_DIG

10
 *

Number of decimal digits,
q,
such that any floatingpoint number with
q
decimal digits can be rounded into a floatingpoint number with
p
radix
b
digits and back again without change to the
q
decimal digits.

lpile { p " " log_ 10" " " " b above
left floor " " (p " "  " " 1) " " log_ 10" " " " b " " right floor }
" " " " lpile {if " " b " " is " " a " " power " " of " " 10 above otherwise}
 FLT_DIG

6
 DBL_DIG

10
 LDBL_DIG

10
 *

Minimum negative integer such that FLT_RADIX raised to that power
minus 1 is a normalized floatingpoint number, $e_ min$.

 FLT_MIN_EXP

 DBL_MIN_EXP

 LDBL_MIN_EXP

 *

Minimum negative integer such that 10 raised to that power is in
the range of normalized floatingpoint numbers.

left ceiling " " log_ 10" " " " b"^" " "{ e_ min" " " " "^" " "1 } ^ " " right ceiling
 FLT_MIN_10_EXP

37
 DBL_MIN_10_EXP

37
 LDBL_MIN_10_EXP

37
 *

Maximum integer such that FLT_RADIX raised to that power
minus 1 is a representable finite floatingpoint number, $e_ max$.

 FLT_MAX_EXP

 DBL_MAX_EXP

 LDBL_MAX_EXP

Additionally, FLT_MAX_EXP shall be at least as large as FLT_MANT_DIG,
DBL_MAX_EXP shall be at least as large as DBL_MANT_DIG, and LDBL_MAX_EXP
shall be at least as large as LDBL_MANT_DIG; which has the effect that
FLT_MAX, DBL_MAX, and LDBL_MAX are integral.
 *

Maximum integer such that 10 raised to that power is in the range
of representable finite floatingpoint numbers.

left floor " " log_ 10" " ( ( 1 " "  " " b"^" " "p ) " "
b"^" e" "_ max" "^ ) " " right floor
 FLT_MAX_10_EXP

+37
 DBL_MAX_10_EXP

+37
 LDBL_MAX_10_EXP

+37
The
<float.h>
header shall define the following values as constant expressions with
implementationdefined values that are greater than or equal to those
shown:
 *

Maximum representable finite floatingpoint number.

(1 " "  " " b"^" " "p^) " " b"^" e" "_ max" "
 FLT_MAX

1E+37
 DBL_MAX

1E+37
 LDBL_MAX

1E+37
The
<float.h>
header shall define the following values as constant expressions with
implementationdefined (positive) values that are less than or equal to
those shown:
 *

The difference between 1 and the least value greater than 1
that is representable in the given floatingpoint type, $b"^" " "{1 " "  " " p}$.

 FLT_EPSILON

1E5
 DBL_EPSILON

1E9
 LDBL_EPSILON

1E9
 *

Minimum normalized positive floatingpoint number,
$b"^" " "{ e_ min" " " " "^" " "1 }$.

 FLT_MIN

1E37
 DBL_MIN

1E37
 LDBL_MIN

1E37
The following sections are informative.
APPLICATION USAGE
None.
RATIONALE
All known hardware floatingpoint formats satisfy the property that the
exponent range is larger than the number of mantissa digits. The ISO C standard
permits a floatingpoint format where this property is not true, such that
the largest finite value would not be integral; however, it is unlikely
that there will ever be hardware support for such a floatingpoint format,
and it introduces boundary cases that portable programs should not have
to be concerned with (for example, a nonintegral DBL_MAX means that
ceil()
would have to worry about overflow). Therefore, this standard imposes
an additional requirement that the largest representable finite value
is integral.
FUTURE DIRECTIONS
None.
SEE ALSO
<complex.h>,
<math.h>,
<stdio.h>,
<stdlib.h>,
<wchar.h>
COPYRIGHT
Portions of this text are reprinted and reproduced in electronic form
from IEEE Std 1003.1, 2013 Edition, Standard for Information Technology
 Portable Operating System Interface (POSIX), The Open Group Base
Specifications Issue 7, Copyright (C) 2013 by the Institute of
Electrical and Electronics Engineers, Inc and The Open Group.
(This is POSIX.12008 with the 2013 Technical Corrigendum 1 applied.) In the
event of any discrepancy between this version and the original IEEE and
The Open Group Standard, the original IEEE and The Open Group Standard
is the referee document. The original Standard can be obtained online at
http://www.unix.org/online.html .
Any typographical or formatting errors that appear
in this page are most likely
to have been introduced during the conversion of the source files to
man page format. To report such errors, see
https://www.kernel.org/doc/manpages/reporting_bugs.html .
Index
 PROLOG

 NAME

 SYNOPSIS

 DESCRIPTION

 APPLICATION USAGE

 RATIONALE

 FUTURE DIRECTIONS

 SEE ALSO

 COPYRIGHT
