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       This  manual  page  is part of the POSIX Programmer's Manual.  The Linux implementation of
       this interface may differ (consult the corresponding Linux  manual  page  for  details  of
       Linux behavior), or the interface may not be implemented on Linux.

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.1‐2017 defers to the ISO C standard.

The characteristics of floating types are defined in terms of a model that describes a
representation of floating-point numbers and values that provide information about an
implementation's floating-point arithmetic.

The following parameters are used to define the model for each floating-point 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 base-b digits in the significand).

f_k Non-negative integers less than b (the significand digits).

A floating-point number x is defined by the following model:

x = sb^e kpf_k b^ −k, e_min ≤ e ≤ e_max

In addition to normalized floating-point numbers (f_1>0 if x≠0), floating types may be
able to contain other kinds of floating-point numbers, such as subnormal floating-point
numbers (x≠0, e=e_min, f_1=0) and unnormalized floating-point numbers (x≠0, e>e_min,
f_1=0), and values that are not floating-point numbers, such as infinities and NaNs. A NaN
is an encoding signifying Not-a-Number. A quiet NaN propagates through almost every
arithmetic operation without raising a floating-point exception; a signaling NaN generally
raises a floating-point exception when occurring as an arithmetic operand.

An implementation may give zero and non-numeric values, such as infinities and NaNs, a
sign, or may leave them unsigned. Wherever such values are unsigned, any requirement in
POSIX.1‐2008 to retrieve the sign shall produce an unspecified sign and any requirement to
set the sign shall be ignored.

The accuracy of the floating-point operations ('+', '-', '*', '/') and of the functions in
<math.h> and <complex.h> that return floating-point results is implementation-defined, as
is the accuracy of the conversion between floating-point 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 floating-point types. The floating-point model
representation is provided for all values except FLT_EVAL_METHOD and FLT_ROUNDS.

The rounding mode for floating-point addition is characterized by the implementation-
defined 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 implementation-defined 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 implementation-defined 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 implementation-defined
behavior.

The <float.h> header shall define the following values as constant expressions with
implementation-defined 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 base-FLT_RADIX digits in the floating-point significand, p.

FLT_MANT_DIG

DBL_MANT_DIG

LDBL_MANT_DIG

* Number of decimal digits, n, such that any floating-point number in the widest
supported floating type with p_max radix b digits can be rounded to a floating-point
number with n decimal digits and back again without change to the value.

p_max log_10 b if b is a power of 10
⎡ 1 + p_max log_10 b⎤ otherwise
DECIMAL_DIG 10

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

p log_10 b if b is a power of 10
⎣ (p − 1) log_10 b ⎦ 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 floating-point 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 floating-point numbers.

⎡ log_10 b^ e_min ^ −1 ⎤

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 floating-point 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 floating-point numbers.

⎣ log_10 ((1 − b^ −p) b^e _max ) ⎦

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
implementation-defined values that are greater than or equal to those shown:

* Maximum representable finite floating-point 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
implementation-defined (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 floating-point type, b^ 1 − p.

FLT_EPSILON 1E-5

DBL_EPSILON 1E-9

LDBL_EPSILON 1E-9

* Minimum normalized positive floating-point number, b^ e_min ^ −1.

FLT_MIN 1E-37

DBL_MIN 1E-37

LDBL_MIN 1E-37

The following sections are informative.

APPLICATION USAGE

       None.

RATIONALE

       All known hardware floating-point formats satisfy the property that the exponent range  is
       larger  than  the  number  of mantissa digits. The ISO C standard permits a floating-point
       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
       floating-point format, and it introduces boundary cases that portable programs should  not
       have  to  be  concerned  with (for example, a non-integral 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.

COPYRIGHT

       Portions  of  this  text  are  reprinted  and  reproduced in electronic form from IEEE Std
       1003.1-2017, Standard for Information Technology -- Portable  Operating  System  Interface
       (POSIX),  The  Open Group Base Specifications Issue 7, 2018 Edition, Copyright (C) 2018 by
       the Institute of Electrical and Electronics Engineers, Inc and The  Open  Group.   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.opengroup.org/unix/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/man-pages/reporting_bugs.html .