417 lines
14 KiB
Plaintext
417 lines
14 KiB
Plaintext
@node Floating-Point Limits
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@chapter Floating-Point Limits
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@pindex <float.h>
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@cindex floating-point number representation
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@cindex representation of floating-point numbers
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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 facilities.
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* IEEE Floating-Point:: An example of a common representation.
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@end menu
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@node Floating-Point Representation
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@section 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, in exponent (of floating-point number)
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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 quantity.
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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, in mantissa (of floating-point number)
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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
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@section 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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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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Note that only @code{FLT_RADIX} is guaranteed to be a constant
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expression, so 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 and 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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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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 defvr
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@comment float.h
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@comment ANSI
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@defvr Macro 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
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guaranteed to be no more than @code{1E-37}.
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@end defvr
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@comment float.h
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@comment ANSI
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@defvr Macro DBL_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{double}, and
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is guaranteed to be no more than @code{1E-37}.
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@end defvr
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@comment float.h
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@comment ANSI
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@defvr Macro LDBL_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{long double},
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and is guaranteed to be no more than @code{1E-37}.
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@end defvr
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@comment float.h
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@comment ANSI
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@defvr Macro FLT_EPSILON
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This is the minimum positive floating-point number of type @code{float}
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such that @code{1.0 + FLT_EPSILON != 1.0} is true. It's guaranteed to
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be no greater than @code{1E-5}.
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@end defvr
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@comment float.h
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@comment ANSI
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@defvr Macro DBL_EPSILON
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This is similar to @code{FLT_EPSILON}, but is for the @code{double}
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type. The maximum value is @code{1E-9}.
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@end defvr
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@comment float.h
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@comment ANSI
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@defvr Macro LDBL_EPSILON
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This is similar to @code{FLT_EPSILON}, but is for the @code{long double}
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type. The maximum value is @code{1E-9}.
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@end defvr
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@node IEEE Floating Point
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@section IEEE Floating Point
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Here is an example showing how these parameters work for a common
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floating point representation, specified by the @cite{IEEE Standard for
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Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985)}.
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The IEEE single-precision float representation uses a base of 2. There
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is a sign bit, a mantissa with 23 bits plus one hidden bit (so the total
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precision is 24 base-2 digits), and an 8-bit exponent that can represent
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values in the range -125 to 128, inclusive.
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So, for an implementation that uses this representation for the
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@code{float} data type, appropriate values for the corresponding
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parameters are:
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@example
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FLT_RADIX 2
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FLT_MANT_DIG 24
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FLT_DIG 6
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FLT_MIN_EXP -125
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FLT_MIN_10_EXP -37
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FLT_MAX_EXP 128
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FLT_MAX_10_EXP +38
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FLT_MIN 1.17549435E-38F
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FLT_MAX 3.40282347E+38F
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FLT_EPSILON 1.19209290E-07F
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@end example
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