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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.1‐2008
       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	float‐
       ing-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	 kΣ1 f_k  b^ −k, e_min	≤ e ≤ e_max

       In addition to normalized floating-point numbers (f_1>0 if x≠0), float‐
       ing 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 infini‐
       ties 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 representa‐
       tions  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 DECI‐
       MAL_DIG, FLT_EVAL_METHOD, FLT_RADIX, and FLT_ROUNDS have separate names
       for  all three floating-point types. The floating-point model represen‐
       tation  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	evalu‐
       ated 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 pre‐
	     cision 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  implementa‐
       tion-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 deci‐
	   mal 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;  how‐
       ever,  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-inte‐
       gral 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 Electri‐
       cal and Electronics Engineers,  Inc  and	 The  Open  Group.   (This  is
       POSIX.1-2008  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.ker‐
       nel.org/doc/man-pages/reporting_bugs.html .

IEEE/The Open Group		     2013			   float.h(0P)
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