594 lines
20 KiB
Plaintext
594 lines
20 KiB
Plaintext
@node Representation Limits, System Configuration Limits, System Information, Top
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@chapter Representation Limits
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This chapter contains information about constants and parameters that
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characterize the representation of the various integer and
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floating-point types supported by the GNU C library.
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@menu
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* Integer Representation Limits:: Determining maximum and minimum
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representation values of
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various integer subtypes.
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* Floating-Point Limits :: Parameters which characterize
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supported floating-point
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representations on a particular
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system.
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@end menu
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@node Integer Representation Limits, Floating-Point Limits , , Representation Limits
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@section Integer Representation Limits
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@cindex integer representation limits
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@cindex representation limits, integer
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@cindex limits, integer representation
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Sometimes it is necessary for programs to know about the internal
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representation of various integer subtypes. For example, if you want
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your program to be careful not to overflow an @code{int} counter
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variable, you need to know what the largest representable value that
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fits in an @code{int} is. These kinds of parameters can vary from
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compiler to compiler and machine to machine. Another typical use of
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this kind of parameter is in conditionalizing data structure definitions
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with @samp{#ifdef} to select the most appropriate integer subtype that
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can represent the required range of values.
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Macros representing the minimum and maximum limits of the integer types
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are defined in the header file @file{limits.h}. The values of these
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macros are all integer constant expressions.
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@pindex limits.h
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int CHAR_BIT
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This is the number of bits in a @code{char}, usually eight.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int SCHAR_MIN
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This is the minimum value that can be represented by a @code{signed char}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int SCHAR_MAX
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This is the maximum value that can be represented by a @code{signed char}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int UCHAR_MAX
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This is the maximum value that can be represented by a @code{unsigned char}.
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(The minimum value of an @code{unsigned char} is zero.)
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int CHAR_MIN
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This is the minimum value that can be represented by a @code{char}.
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It's equal to @code{SCHAR_MIN} if @code{char} is signed, or zero
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otherwise.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int CHAR_MAX
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This is the maximum value that can be represented by a @code{char}.
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It's equal to @code{SCHAR_MAX} if @code{char} is signed, or
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@code{UCHAR_MAX} otherwise.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int SHRT_MIN
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This is the minimum value that can be represented by a @code{signed
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short int}. On most machines that the GNU C library runs on,
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@code{short} integers are 16-bit quantities.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int SHRT_MAX
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This is the maximum value that can be represented by a @code{signed
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short int}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int USHRT_MAX
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This is the maximum value that can be represented by an @code{unsigned
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short int}. (The minimum value of an @code{unsigned short int} is zero.)
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int INT_MIN
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This is the minimum value that can be represented by a @code{signed
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int}. On most machines that the GNU C system runs on, an @code{int} is
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a 32-bit quantity.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro int INT_MAX
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This is the maximum value that can be represented by a @code{signed
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int}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro {unsigned int} UINT_MAX
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This is the maximum value that can be represented by an @code{unsigned
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int}. (The minimum value of an @code{unsigned int} is zero.)
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro {long int} LONG_MIN
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This is the minimum value that can be represented by a @code{signed long
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int}. On most machines that the GNU C system runs on, @code{long}
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integers are 32-bit quantities, the same size as @code{int}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro {long int} LONG_MAX
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This is the maximum value that can be represented by a @code{signed long
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int}.
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@end deftypevr
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@comment limits.h
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@comment ANSI
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@deftypevr Macro {unsigned long int} ULONG_MAX
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This is the maximum value that can be represented by an @code{unsigned
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long int}. (The minimum value of an @code{unsigned long int} is zero.)
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@end deftypevr
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@strong{Incomplete:} There should be corresponding limits for the GNU
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C Compiler's @code{long long} type, too. (But they are not now present
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in the header file.)
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The header file @file{limits.h} also defines some additional constants
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that parameterize various operating system and file system limits. These
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constants are described in @ref{System Parameters} and @ref{File System
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Parameters}.
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@pindex limits.h
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@node Floating-Point Limits , , Integer Representation Limits, Representation Limits
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@section Floating-Point Limits
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@cindex floating-point number representation
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@cindex representation, floating-point number
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@cindex limits, floating-point representation
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Because floating-point numbers are represented internally as approximate
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quantities, algorithms for manipulating floating-point data often need
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to be parameterized in terms of the accuracy of the representation.
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Some of the functions in the C library itself need this information; for
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example, the algorithms for printing and reading floating-point numbers
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(@pxref{I/O on Streams}) and for calculating trigonometric and
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irrational functions (@pxref{Mathematics}) use information about the
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underlying floating-point representation to avoid round-off error and
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loss of accuracy. User programs that implement numerical analysis
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techniques also often need to be parameterized in this way in order to
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minimize or compute error bounds.
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The specific representation of floating-point numbers varies from
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machine to machine. The GNU C library defines a set of parameters which
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characterize each of the supported floating-point representations on a
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particular system.
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@menu
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* Floating-Point Representation:: Definitions of terminology.
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* Floating-Point Parameters:: Descriptions of the library
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facilities.
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* IEEE Floating Point:: An example of a common
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representation.
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@end menu
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@node Floating-Point Representation, Floating-Point Parameters, , Floating-Point Limits
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@subsection Floating-Point Representation
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This section introduces the terminology used to characterize the
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representation of floating-point numbers.
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You are probably already familiar with most of these concepts in terms
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of scientific or exponential notation for floating-point numbers. For
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example, the number @code{123456.0} could be expressed in exponential
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notation as @code{1.23456e+05}, a shorthand notation indicating that the
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mantissa @code{1.23456} is multiplied by the base @code{10} raised to
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power @code{5}.
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More formally, the internal representation of a floating-point number
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can be characterized in terms of the following parameters:
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@itemize @bullet
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@item
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The @dfn{sign} is either @code{-1} or @code{1}.
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@cindex sign (of floating-point number)
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@item
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The @dfn{base} or @dfn{radix} for exponentiation; an integer greater
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than @code{1}. This is a constant for the particular representation.
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@cindex base (of floating-point number)
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@cindex radix (of floating-point number)
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@item
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The @dfn{exponent} to which the base is raised. The upper and lower
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bounds of the exponent value are constants for the particular
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representation.
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@cindex exponent (of floating-point number)
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Sometimes, in the actual bits representing the floating-point number,
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the exponent is @dfn{biased} by adding a constant to it, to make it
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always be represented as an unsigned quantity. This is only important
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if you have some reason to pick apart the bit fields making up the
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floating-point number by hand, which is something for which the GNU
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library provides no support. So this is ignored in the discussion that
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follows.
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@cindex bias (of floating-point number exponent)
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@item
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The value of the @dfn{mantissa} or @dfn{significand}, which is an
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unsigned integer.
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@cindex mantissa (of floating-point number)
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@cindex significand (of floating-point number)
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@item
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The @dfn{precision} of the mantissa. If the base of the representation
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is @var{b}, then the precision is the number of base-@var{b} digits in
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the mantissa. This is a constant for the particular representation.
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Many floating-point representations have an implicit @dfn{hidden bit} in
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the mantissa. Any such hidden bits are counted in the precision.
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Again, the GNU library provides no facilities for dealing with such low-level
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aspects of the representation.
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@cindex precision (of floating-point number)
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@cindex hidden bit (of floating-point number mantissa)
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@end itemize
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The mantissa of a floating-point number actually represents an implicit
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fraction whose denominator is the base raised to the power of the
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precision. Since the largest representable mantissa is one less than
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this denominator, the value of the fraction is always strictly less than
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@code{1}. The mathematical value of a floating-point number is then the
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product of this fraction; the sign; and the base raised to the exponent.
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If the floating-point number is @dfn{normalized}, the mantissa is also
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greater than or equal to the base raised to the power of one less
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than the precision (unless the number represents a floating-point zero,
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in which case the mantissa is zero). The fractional quantity is
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therefore greater than or equal to @code{1/@var{b}}, where @var{b} is
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the base.
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@cindex normalized floating-point number
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@node Floating-Point Parameters, IEEE Floating Point, Floating-Point Representation, Floating-Point Limits
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@subsection Floating-Point Parameters
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@strong{Incomplete:} This section needs some more concrete examples
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of what these parameters mean and how to use them in a program.
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These macro definitions can be accessed by including the header file
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@file{float.h} in your program.
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@pindex float.h
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Macro names starting with @samp{FLT_} refer to the @code{float} type,
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while names beginning with @samp{DBL_} refer to the @code{double} type
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and names beginning with @samp{LDBL_} refer to the @code{long double}
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type. (In implementations that do not support @code{long double} as
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a distinct data type, the values for those constants are the same
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as the corresponding constants for the @code{double} type.)@refill
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@cindex @code{float} representation limits
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@cindex @code{double} representation limits
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@cindex @code{long double} representation limits
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Of these macros, only @code{FLT_RADIX} is guaranteed to be a constant
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expression. The other macros listed here cannot be reliably used in
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places that require constant expressions, such as @samp{#if}
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preprocessing directives or array size specifications.
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Although the ANSI C standard specifies minimum and maximum values for
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most of these parameters, the GNU C implementation uses whatever
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floating-point representations are supported by the underlying hardware.
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So whether GNU C actually satisfies the ANSI C requirements depends on
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what machine it is running on.
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_ROUNDS
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This value characterizes the rounding mode for floating-point addition.
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The following values indicate standard rounding modes:
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@table @code
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@item -1
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The mode is indeterminable.
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@item 0
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Rounding is towards zero.
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@item 1
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Rounding is to the nearest number.
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@item 2
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Rounding is towards positive infinity.
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@item 3
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Rounding is towards negative infinity.
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@end table
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@noindent
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Any other value represents a machine-dependent nonstandard rounding
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mode.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_RADIX
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This is the value of the base, or radix, of exponent representation.
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This is guaranteed to be a constant expression, unlike the other macros
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described in this section.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_MANT_DIG
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This is the number of base-@code{FLT_RADIX} digits in the floating-point
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mantissa for the @code{float} data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_MANT_DIG
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This is the number of base-@code{FLT_RADIX} digits in the floating-point
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mantissa for the @code{double} data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_MANT_DIG
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This is the number of base-@code{FLT_RADIX} digits in the floating-point
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mantissa for the @code{long double} data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_DIG
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This is the number of decimal digits of precision for the @code{float}
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data type. Technically, if @var{p} and @var{b} are the precision and
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base (respectively) for the representation, then the decimal precision
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@var{q} is the maximum number of decimal digits such that any floating
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point number with @var{q} base 10 digits can be rounded to a floating
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point number with @var{p} base @var{b} digits and back again, without
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change to the @var{q} decimal digits.
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The value of this macro is guaranteed to be at least @code{6}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_DIG
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This is similar to @code{FLT_DIG}, but is for the @code{double} data
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type. The value of this macro is guaranteed to be at least @code{10}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_DIG
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This is similar to @code{FLT_DIG}, but is for the @code{long double}
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data type. The value of this macro is guaranteed to be at least
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@code{10}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_MIN_EXP
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This is the minimum negative integer such that the mathematical value
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@code{FLT_RADIX} raised to this power minus 1 can be represented as a
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normalized floating-point number of type @code{float}. In terms of the
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actual implementation, this is just the smallest value that can be
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represented in the exponent field of the number.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_MIN_EXP
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This is similar to @code{FLT_MIN_EXP}, but is for the @code{double} data
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type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_MIN_EXP
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This is similar to @code{FLT_MIN_EXP}, but is for the @code{long double}
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data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_MIN_10_EXP
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This is the minimum negative integer such that the mathematical value
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@code{10} raised to this power minus 1 can be represented as a
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normalized floating-point number of type @code{float}. This is
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guaranteed to be no greater than @code{-37}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_MIN_10_EXP
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This is similar to @code{FLT_MIN_10_EXP}, but is for the @code{double}
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data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_MIN_10_EXP
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This is similar to @code{FLT_MIN_10_EXP}, but is for the @code{long
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double} data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_MAX_EXP
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This is the maximum negative integer such that the mathematical value
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@code{FLT_RADIX} raised to this power minus 1 can be represented as a
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floating-point number of type @code{float}. In terms of the actual
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implementation, this is just the largest value that can be represented
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in the exponent field of the number.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_MAX_EXP
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This is similar to @code{FLT_MAX_EXP}, but is for the @code{double} data
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type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_MAX_EXP
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This is similar to @code{FLT_MAX_EXP}, but is for the @code{long double}
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data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int FLT_MAX_10_EXP
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This is the maximum negative integer such that the mathematical value
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@code{10} raised to this power minus 1 can be represented as a
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normalized floating-point number of type @code{float}. This is
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guaranteed to be at least @code{37}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int DBL_MAX_10_EXP
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This is similar to @code{FLT_MAX_10_EXP}, but is for the @code{double}
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data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro int LDBL_MAX_10_EXP
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This is similar to @code{FLT_MAX_10_EXP}, but is for the @code{long
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double} data type.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro double FLT_MAX
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The value of this macro is the maximum representable floating-point
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number of type @code{float}, and is guaranteed to be at least
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@code{1E+37}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro double DBL_MAX
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The value of this macro is the maximum representable floating-point
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number of type @code{double}, and is guaranteed to be at least
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@code{1E+37}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro {long double} LDBL_MAX
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The value of this macro is the maximum representable floating-point
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number of type @code{long double}, and is guaranteed to be at least
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@code{1E+37}.
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@end deftypevr
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@comment float.h
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@comment ANSI
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@deftypevr Macro double FLT_MIN
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The value of this macro is the minimum normalized positive
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floating-point number that is representable by type @code{float}, and is
|
|
guaranteed to be no more than @code{1E-37}.
|
|
@end deftypevr
|
|
|
|
@comment float.h
|
|
@comment ANSI
|
|
@deftypevr Macro double DBL_MIN
|
|
The value of this macro is the minimum normalized positive
|
|
floating-point number that is representable by type @code{double}, and
|
|
is guaranteed to be no more than @code{1E-37}.
|
|
@end deftypevr
|
|
|
|
@comment float.h
|
|
@comment ANSI
|
|
@deftypevr Macro {long double} LDBL_MIN
|
|
The value of this macro is the minimum normalized positive
|
|
floating-point number that is representable by type @code{long double},
|
|
and is guaranteed to be no more than @code{1E-37}.
|
|
@end deftypevr
|
|
|
|
|
|
@comment float.h
|
|
@comment ANSI
|
|
@deftypevr Macro double FLT_EPSILON
|
|
This is the minimum positive floating-point number of type @code{float}
|
|
such that @code{1.0 + FLT_EPSILON != 1.0} is true. It's guaranteed to
|
|
be no greater than @code{1E-5}.
|
|
@end deftypevr
|
|
|
|
@comment float.h
|
|
@comment ANSI
|
|
@deftypevr Macro double DBL_EPSILON
|
|
This is similar to @code{FLT_EPSILON}, but is for the @code{double}
|
|
type. The maximum value is @code{1E-9}.
|
|
@end deftypevr
|
|
|
|
@comment float.h
|
|
@comment ANSI
|
|
@deftypevr Macro {long double} LDBL_EPSILON
|
|
This is similar to @code{FLT_EPSILON}, but is for the @code{long double}
|
|
type. The maximum value is @code{1E-9}.
|
|
@end deftypevr
|
|
|
|
|
|
@node IEEE Floating Point, , Floating-Point Parameters, Floating-Point Limits
|
|
@subsection IEEE Floating Point
|
|
@cindex IEEE floating-point representation
|
|
@cindex floating-point, IEEE
|
|
@cindex IEEE Std 754
|
|
|
|
|
|
Here is an example showing how these parameters work for a common
|
|
floating point representation, specified by the @cite{IEEE Standard for
|
|
Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985)}. Nearly
|
|
all computers today use this format.
|
|
|
|
The IEEE single-precision float representation uses a base of 2. There
|
|
is a sign bit, a mantissa with 23 bits plus one hidden bit (so the total
|
|
precision is 24 base-2 digits), and an 8-bit exponent that can represent
|
|
values in the range -125 to 128, inclusive.
|
|
|
|
So, for an implementation that uses this representation for the
|
|
@code{float} data type, appropriate values for the corresponding
|
|
parameters are:
|
|
|
|
@example
|
|
FLT_RADIX 2
|
|
FLT_MANT_DIG 24
|
|
FLT_DIG 6
|
|
FLT_MIN_EXP -125
|
|
FLT_MIN_10_EXP -37
|
|
FLT_MAX_EXP 128
|
|
FLT_MAX_10_EXP +38
|
|
FLT_MIN 1.17549435E-38F
|
|
FLT_MAX 3.40282347E+38F
|
|
FLT_EPSILON 1.19209290E-07F
|
|
@end example
|
|
|
|
Here are the values for the @code{double} data type:
|
|
|
|
@example
|
|
DBL_MANT_DIG 53
|
|
DBL_DIG 15
|
|
DBL_MIN_EXP -1021
|
|
DBL_MIN_10_EXP -307
|
|
DBL_MAX_EXP 1024
|
|
DBL_MAX_10_EXP 308
|
|
DBL_MAX 1.7976931348623157E+308
|
|
DBL_MIN 2.2250738585072014E-308
|
|
DBL_EPSILON 2.2204460492503131E-016
|
|
@end example
|