float.h(0P)		   POSIX Programmer's Manual		   float.h(0P)

PROLOG
       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" 6 Non-negative integers less than b (the significand
	     digits).

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

       x   =   sb ^ e	  ∑_(k = 1)^p	f _ 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) ⌈  1	+   p _ max	log _ 10
	   b⌉)	   ((if	  b   is   a   power   of   10) 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) ⌊  (p	 −   1)	  log _ 10     b  ⌋)	 ((if
	   b   is   a	power	of   10) 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.

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 1003.1-2017, 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 .

IEEE/The Open Group		     2017			   float.h(0P)