path: root/manual/arith.texi

@c We need some definitions here.
@ifclear cdot
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@iftex
@set cdot @cdot
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@ifclear cdot
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@macro mul
@value{cdot}
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@node Arithmetic, Date and Time, Mathematics, Top
@chapter Low-Level Arithmetic Functions

This chapter contains information about functions for doing basic
arithmetic operations, such as splitting a float into its integer and
fractional parts or retrieving the imaginary part of a complex value.
These functions are declared in the header files @file{math.h} and
@file{complex.h}.

@menu
* Infinity::                    What is Infinity and how to test for it.
* Not a Number::                Making NaNs and testing for NaNs.
* Imaginary Unit::              Constructing complex Numbers.
* Predicates on Floats::        Testing for infinity and for NaNs.
* Floating-Point Classes::      Classify floating-point numbers.
* Operations on Complex::       Projections, Conjugates, and Decomposing.
* Absolute Value::              Absolute value functions.
* Normalization Functions::     Hacks for radix-2 representations.
* Rounding and Remainders::     Determining the integer and
			         fractional parts of a float.
* Arithmetic on FP Values::     Setting and Modifying Single Bits of FP Values.
* Special arithmetic on FPs::   Special Arithmetic on FPs.
* Integer Division::            Functions for performing integer
				 division.
* Parsing of Numbers::          Functions for ``reading'' numbers
			         from strings.
@end menu

@node Infinity
@section Infinity Values
@cindex Infinity
@cindex IEEE floating point

Mathematical operations easily can produce as the result values which
are not representable by the floating-point format.  The functions in
the mathematics library also have this problem.  The situation is
generally solved by raising an overflow exception and by returning a
huge value.

The @w{IEEE 754} floating-point defines a special value to be used in
these situations.  There is a special value for infinity.

@comment math.h
@comment ISO
@deftypevr Macro float INFINITY
An expression representing the infinite value.  @code{INFINITY} values are
produced by mathematical operations like @code{1.0 / 0.0}.  It is
possible to continue the computations with this value since the basic
operations as well as the mathematical library functions are prepared to
handle values like this.

Beside @code{INFINITY} also the value @code{-INFINITY} is representable
and it is handled differently if needed.  It is possible to test a
value for infiniteness using a simple comparison but the
recommended way is to use the the @code{isinf} function.

This macro was introduced in the @w{ISO C 9X} standard.
@end deftypevr

@vindex HUGE_VAL
The macros @code{HUGE_VAL}, @code{HUGE_VALF} and @code{HUGE_VALL} are
defined in a similar way but they are not required to represent the
infinite value, only a very large value (@pxref{Domain and Range Errors}).
If actually infinity is wanted, @code{INFINITY} should be used.


@node Not a Number
@section ``Not a Number'' Values
@cindex NaN
@cindex not a number
@cindex IEEE floating point

The IEEE floating point format used by most modern computers supports
values that are ``not a number''.  These values are called @dfn{NaNs}.
``Not a number'' values result from certain operations which have no
meaningful numeric result, such as zero divided by zero or infinity
divided by infinity.

One noteworthy property of NaNs is that they are not equal to
themselves.  Thus, @code{x == x} can be 0 if the value of @code{x} is a
NaN.  You can use this to test whether a value is a NaN or not: if it is
not equal to itself, then it is a NaN.  But the recommended way to test
for a NaN is with the @code{isnan} function (@pxref{Predicates on Floats}).

Almost any arithmetic operation in which one argument is a NaN returns
a NaN.

@comment math.h
@comment GNU
@deftypevr Macro float NAN
An expression representing a value which is ``not a number''.  This
macro is a GNU extension, available only on machines that support ``not
a number'' values---that is to say, on all machines that support IEEE
floating point.

You can use @samp{#ifdef NAN} to test whether the machine supports
NaNs.  (Of course, you must arrange for GNU extensions to be visible,
such as by defining @code{_GNU_SOURCE}, and then you must include
@file{math.h}.)
@end deftypevr

@node Imaginary Unit
@section Constructing complex Numbers

@pindex complex.h
To construct complex numbers it is necessary have a way to express the
imaginary part of the numbers.  In mathematics one uses the symbol ``i''
to mark a number as imaginary.  For convenience the @file{complex.h}
header defines two macros which allow to use a similar easy notation.

@deftypevr Macro {const float complex} _Complex_I
This macro is a representation of the complex number ``@math{0+1i}''.
Computing

@smallexample
_Complex_I * _Complex_I = -1
@end smallexample

@noindent
leads to a real-valued result.  If no @code{imaginary} types are
available it is easiest to use this value to construct complex numbers
from real values:

@smallexample
3.0 - _Complex_I * 4.0
@end smallexample
@end deftypevr

@noindent
Without an optimizing compiler this is more expensive than the use of
@code{_Imaginary_I} but with is better than nothing.  You can avoid all
the hassles if you use the @code{I} macro below if the name is not
problem.

@deftypevr Macro {const float imaginary} _Imaginary_I
This macro is a representation of the value ``@math{1i}''.  I.e., it is
the value for which

@smallexample
_Imaginary_I * _Imaginary_I = -1
@end smallexample

@noindent
The result is not of type @code{float imaginary} but instead @code{float}.
One can use it to easily construct complex number like in

@smallexample
3.0 - _Imaginary_I * 4.0
@end smallexample
@end deftypevr

@noindent
which results in the complex number with a real part of 3.0 and a
imaginary part -4.0.

@noindent
A more intuitive approach is to use the following macro.

@deftypevr Macro {const float imaginary} I
This macro has exactly the same value as @code{_Imaginary_I}.  The
problem is that the name @code{I} very easily can clash with macros or
variables in programs and so it might be a good idea to avoid this name
and stay at the safe side by using @code{_Imaginary_I}.

If the implementation does not support the @code{imaginary} types
@code{I} is defined as @code{_Complex_I} which is the second best
solution.  It still can be used in the same way but requires a most
clever compiler to get the same results.
@end deftypevr


@node Predicates on Floats
@section Predicates on Floats

@pindex math.h
This section describes some miscellaneous test functions on doubles.
Prototypes for these functions appear in @file{math.h}.  These are BSD
functions, and thus are available if you define @code{_BSD_SOURCE} or
@code{_GNU_SOURCE}.

@comment math.h
@comment BSD
@deftypefun int isinf (double @var{x})
@deftypefunx int isinff (float @var{x})
@deftypefunx int isinfl (long double @var{x})
This function returns @code{-1} if @var{x} represents negative infinity,
@code{1} if @var{x} represents positive infinity, and @code{0} otherwise.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun int isnan (double @var{x})
@deftypefunx int isnanf (float @var{x})
@deftypefunx int isnanl (long double @var{x})
This function returns a nonzero value if @var{x} is a ``not a number''
value, and zero otherwise.  (You can just as well use @code{@var{x} !=
@var{x}} to get the same result).
@end deftypefun

@comment math.h
@comment BSD
@deftypefun int finite (double @var{x})
@deftypefunx int finitef (float @var{x})
@deftypefunx int finitel (long double @var{x})
This function returns a nonzero value if @var{x} is finite or a ``not a
number'' value, and zero otherwise.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double infnan (int @var{error})
This function is provided for compatibility with BSD.  The other
mathematical functions use @code{infnan} to decide what to return on
occasion of an error.  Its argument is an error code, @code{EDOM} or
@code{ERANGE}; @code{infnan} returns a suitable value to indicate this
with.  @code{-ERANGE} is also acceptable as an argument, and corresponds
to @code{-HUGE_VAL} as a value.

In the BSD library, on certain machines, @code{infnan} raises a fatal
signal in all cases.  The GNU library does not do likewise, because that
does not fit the @w{ISO C} specification.
@end deftypefun

@strong{Portability Note:} The functions listed in this section are BSD
extensions.

@node Floating-Point Classes
@section Floating-Point Number Classification Functions

Instead of using the BSD specific functions from the last section it is
better to use those in this section which are introduced in the @w{ISO C
9X} standard and are therefore widely available.

@comment math.h
@comment ISO
@deftypefn {Macro} int fpclassify (@emph{float-type} @var{x})
This is a generic macro which works on all floating-point types and
which returns a value of type @code{int}.  The possible values are:

@vtable @code
@item FP_NAN
The floating-point number @var{x} is ``Not a Number'' (@pxref{Not a Number})
@item FP_INFINITE
The value of @var{x} is either plus or minus infinity (@pxref{Infinity})
@item FP_ZERO
The value of @var{x} is zero.  In floating-point formats like @w{IEEE
754} where the zero value can be signed this value is also returned if
@var{x} is minus zero.
@item FP_SUBNORMAL
Some floating-point formats (such as @w{IEEE 754}) allow floating-point
numbers to be represented in a denormalized format.  This happens if the
absolute value of the number is too small to be represented in the
normal format.  @code{FP_SUBNORMAL} is returned for such values of @var{x}.
@item FP_NORMAL
This value is returned for all other cases which means the number is a
plain floating-point number without special meaning.
@end vtable

This macro is useful if more than property of a number must be
tested.  If one only has to test for, e.g., a NaN value, there are
function which are faster.
@end deftypefn

The remainder of this section introduces some more specific functions.
They might be implemented faster than the call to @code{fpclassify} and
if the actual need in the program is covered be these functions they
should be used (and not @code{fpclassify}).

@comment math.h
@comment ISO
@deftypefn {Macro} int isfinite (@emph{float-type} @var{x})
The value returned by this macro is nonzero if the value of @var{x} is
not plus or minus infinity and not NaN.  I.e., it could be implemented as

@smallexample
(fpclassify (x) != FP_NAN && fpclassify (x) != FP_INFINITE)
@end smallexample

@code{isfinite} is also implemented as a macro which can handle all
floating-point types.  Programs should use this function instead of
@var{finite} (@pxref{Predicates on Floats}).
@end deftypefn

@comment math.h
@comment ISO
@deftypefn {Macro} int isnormal (@emph{float-type} @var{x})
If @code{isnormal} returns a nonzero value the value or @var{x} is
neither a NaN, infinity, zero, nor a denormalized number.  I.e., it
could be implemented as

@smallexample
(fpclassify (x) == FP_NORMAL)
@end smallexample
@end deftypefn

@comment math.h
@comment ISO
@deftypefn {Macro} int isnan (@emph{float-type} @var{x})
The situation with this macro is a bit complicated.  Here @code{isnan}
is a macro which can handle all kinds of floating-point types.  It
returns a nonzero value is @var{x} does not represent a NaN value and
could be written like this

@smallexample
(fpclassify (x) == FP_NAN)
@end smallexample

The complication is that there is a function of the same name and the
same semantic defined for compatibility with BSD (@pxref{Predicates on
Floats}).  Fortunately this should not yield to problems in most cases
since the macro and the function have the same semantic.  Should in a
situation the function be absolutely necessary one can use

@smallexample
(isnan) (x)
@end smallexample

@noindent
to avoid the macro expansion.  Using the macro has two big advantages:
it is more portable and one does not have to choose the right function
among @code{isnan}, @code{isnanf}, and @code{isnanl}.
@end deftypefn


@node Operations on Complex
@section Projections, Conjugates, and Decomposing of Complex Numbers
@cindex project complex numbers
@cindex conjugate complex numbers
@cindex decompose complex numbers

This section lists functions performing some of the simple mathematical
operations on complex numbers.  Using any of the function requires that
the C compiler understands the @code{complex} keyword, introduced to the
C language in the @w{ISO C 9X} standard.

@pindex complex.h
The prototypes for all functions in this section can be found in
@file{complex.h}.  All functions are available in three variants, one
for each of the three floating-point types.

The easiest operation on complex numbers is the decomposition in the
real part and the imaginary part.  This is done by the next two
functions.

@comment complex.h
@comment ISO
@deftypefun double creal (complex double @var{z})
@deftypefunx float crealf (complex float @var{z})
@deftypefunx {long double} creall (complex long double @var{z})
These functions return the real part of the complex number @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double cimag (complex double @var{z})
@deftypefunx float cimagf (complex float @var{z})
@deftypefunx {long double} cimagl (complex long double @var{z})
These functions return the imaginary part of the complex number @var{z}.
@end deftypefun


The conjugate complex value of a given complex number has the same value
for the real part but the complex part is negated.

@comment complex.h
@comment ISO
@deftypefun {complex double} conj (complex double @var{z})
@deftypefunx {complex float} conjf (complex float @var{z})
@deftypefunx {complex long double} conjl (complex long double @var{z})
These functions return the conjugate complex value of the complex number
@var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double carg (complex double @var{z})
@deftypefunx float cargf (complex float @var{z})
@deftypefunx {long double} cargl (complex long double @var{z})
These functions return argument of the complex number @var{z}.

Mathematically, the argument is the phase angle of @var{z} with a branch
cut along the negative real axis.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cproj (complex double @var{z})
@deftypefunx {complex float} cprojf (complex float @var{z})
@deftypefunx {complex long double} cprojl (complex long double @var{z})
Return the projection of the complex value @var{z} on the Riemann
sphere.  Values with a infinite complex part (even if the real part
is NaN) are projected to positive infinite on the real axis.  If the
real part is infinite, the result is equivalent to

@smallexample
INFINITY + I * copysign (0.0, cimag (z))
@end smallexample
@end deftypefun


@node Absolute Value
@section Absolute Value
@cindex absolute value functions

These functions are provided for obtaining the @dfn{absolute value} (or
@dfn{magnitude}) of a number.  The absolute value of a real number
@var{x} is @var{x} is @var{x} is positive, @minus{}@var{x} if @var{x} is
negative.  For a complex number @var{z}, whose real part is @var{x} and
whose imaginary part is @var{y}, the absolute value is @w{@code{sqrt
(@var{x}*@var{x} + @var{y}*@var{y})}}.

@pindex math.h
@pindex stdlib.h
Prototypes for @code{abs}, @code{labs} and @code{llabs} are in @file{stdlib.h};
@code{fabs}, @code{fabsf} and @code{fabsl} are declared in @file{math.h};
@code{cabs}, @code{cabsf} and @code{cabsl} are declared in @file{complex.h}.

@comment stdlib.h
@comment ISO
@deftypefun int abs (int @var{number})
This function returns the absolute value of @var{number}.

Most computers use a two's complement integer representation, in which
the absolute value of @code{INT_MIN} (the smallest possible @code{int})
cannot be represented; thus, @w{@code{abs (INT_MIN)}} is not defined.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long int} labs (long int @var{number})
This is similar to @code{abs}, except that both the argument and result
are of type @code{long int} rather than @code{int}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long long int} llabs (long long int @var{number})
This is similar to @code{abs}, except that both the argument and result
are of type @code{long long int} rather than @code{int}.

This function is defined in @w{ISO C 9X}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fabs (double @var{number})
@deftypefunx float fabsf (float @var{number})
@deftypefunx {long double} fabsl (long double @var{number})
This function returns the absolute value of the floating-point number
@var{number}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun double cabs (complex double @var{z})
@deftypefunx float cabsf (complex float @var{z})
@deftypefunx {long double} cabsl (complex long double @var{z})
These functions return the absolute value of the complex number @var{z}.
The compiler must support complex numbers to use these functions.  The
value is:

@smallexample
sqrt (creal (@var{z}) * creal (@var{z}) + cimag (@var{z}) * cimag (@var{z}))
@end smallexample

This function should always be used instead of the direct formula since
using the simple straight-forward method can mean to lose accuracy.  If
one of the squared values is neglectable in size compared to the other
value the result should be the same as the larger value.  But squaring
the value and afterwards using the square root function leads to
inaccuracy.  See @code{hypot} in @xref{Exponents and Logarithms}.
@end deftypefun

@node Normalization Functions
@section Normalization Functions
@cindex normalization functions (floating-point)

The functions described in this section are primarily provided as a way
to efficiently perform certain low-level manipulations on floating point
numbers that are represented internally using a binary radix;
see @ref{Floating Point Concepts}.  These functions are required to
have equivalent behavior even if the representation does not use a radix
of 2, but of course they are unlikely to be particularly efficient in
those cases.

@pindex math.h
All these functions are declared in @file{math.h}.

@comment math.h
@comment ISO
@deftypefun double frexp (double @var{value}, int *@var{exponent})
@deftypefunx float frexpf (float @var{value}, int *@var{exponent})
@deftypefunx {long double} frexpl (long double @var{value}, int *@var{exponent})
These functions are used to split the number @var{value}
into a normalized fraction and an exponent.

If the argument @var{value} is not zero, the return value is @var{value}
times a power of two, and is always in the range 1/2 (inclusive) to 1
(exclusive).  The corresponding exponent is stored in
@code{*@var{exponent}}; the return value multiplied by 2 raised to this
exponent equals the original number @var{value}.

For example, @code{frexp (12.8, &exponent)} returns @code{0.8} and
stores @code{4} in @code{exponent}.

If @var{value} is zero, then the return value is zero and
zero is stored in @code{*@var{exponent}}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double ldexp (double @var{value}, int @var{exponent})
@deftypefunx float ldexpf (float @var{value}, int @var{exponent})
@deftypefunx {long double} ldexpl (long double @var{value}, int @var{exponent})
These functions return the result of multiplying the floating-point
number @var{value} by 2 raised to the power @var{exponent}.  (It can
be used to reassemble floating-point numbers that were taken apart
by @code{frexp}.)

For example, @code{ldexp (0.8, 4)} returns @code{12.8}.
@end deftypefun

The following functions which come from BSD provide facilities
equivalent to those of @code{ldexp} and @code{frexp}:

@comment math.h
@comment BSD
@deftypefun double scalb (double @var{value}, int @var{exponent})
@deftypefunx float scalbf (float @var{value}, int @var{exponent})
@deftypefunx {long double} scalbl (long double @var{value}, int @var{exponent})
The @code{scalb} function is the BSD name for @code{ldexp}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double logb (double @var{x})
@deftypefunx float logbf (float @var{x})
@deftypefunx {long double} logbl (long double @var{x})
These BSD functions return the integer part of the base-2 logarithm of
@var{x}, an integer value represented in type @code{double}.  This is
the highest integer power of @code{2} contained in @var{x}.  The sign of
@var{x} is ignored.  For example, @code{logb (3.5)} is @code{1.0} and
@code{logb (4.0)} is @code{2.0}.

When @code{2} raised to this power is divided into @var{x}, it gives a
quotient between @code{1} (inclusive) and @code{2} (exclusive).

If @var{x} is zero, the value is minus infinity (if the machine supports
such a value), or else a very small number.  If @var{x} is infinity, the
value is infinity.

The value returned by @code{logb} is one less than the value that
@code{frexp} would store into @code{*@var{exponent}}.
@end deftypefun

@node Rounding and Remainders
@section Rounding and Remainder Functions
@cindex rounding functions
@cindex remainder functions
@cindex converting floats to integers

@pindex math.h
The functions listed here perform operations such as rounding,
truncation, and remainder in division of floating point numbers.  Some
of these functions convert floating point numbers to integer values.
They are all declared in @file{math.h}.

You can also convert floating-point numbers to integers simply by
casting them to @code{int}.  This discards the fractional part,
effectively rounding towards zero.  However, this only works if the
result can actually be represented as an @code{int}---for very large
numbers, this is impossible.  The functions listed here return the
result as a @code{double} instead to get around this problem.

@comment math.h
@comment ISO
@deftypefun double ceil (double @var{x})
@deftypefunx float ceilf (float @var{x})
@deftypefunx {long double} ceill (long double @var{x})
These functions round @var{x} upwards to the nearest integer,
returning that value as a @code{double}.  Thus, @code{ceil (1.5)}
is @code{2.0}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double floor (double @var{x})
@deftypefunx float floorf (float @var{x})
@deftypefunx {long double} floorl (long double @var{x})
These functions round @var{x} downwards to the nearest
integer, returning that value as a @code{double}.  Thus, @code{floor
(1.5)} is @code{1.0} and @code{floor (-1.5)} is @code{-2.0}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double rint (double @var{x})
@deftypefunx float rintf (float @var{x})
@deftypefunx {long double} rintl (long double @var{x})
These functions round @var{x} to an integer value according to the
current rounding mode.  @xref{Floating Point Parameters}, for
information about the various rounding modes.  The default
rounding mode is to round to the nearest integer; some machines
support other modes, but round-to-nearest is always used unless
you explicit select another.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double nearbyint (double @var{x})
@deftypefunx float nearbyintf (float @var{x})
@deftypefunx {long double} nearbyintl (long double @var{x})
These functions return the same value as the @code{rint} functions but
even some rounding actually takes place @code{nearbyint} does @emph{not}
raise the inexact exception.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double modf (double @var{value}, double *@var{integer-part})
@deftypefunx float modff (float @var{value}, float *@var{integer-part})
@deftypefunx {long double} modfl (long double @var{value}, long double *@var{integer-part})
These functions break the argument @var{value} into an integer part and a
fractional part (between @code{-1} and @code{1}, exclusive).  Their sum
equals @var{value}.  Each of the parts has the same sign as @var{value},
so the rounding of the integer part is towards zero.

@code{modf} stores the integer part in @code{*@var{integer-part}}, and
returns the fractional part.  For example, @code{modf (2.5, &intpart)}
returns @code{0.5} and stores @code{2.0} into @code{intpart}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fmod (double @var{numerator}, double @var{denominator})
@deftypefunx float fmodf (float @var{numerator}, float @var{denominator})
@deftypefunx {long double} fmodl (long double @var{numerator}, long double @var{denominator})
These functions compute the remainder from the division of
@var{numerator} by @var{denominator}.  Specifically, the return value is
@code{@var{numerator} - @w{@var{n} * @var{denominator}}}, where @var{n}
is the quotient of @var{numerator} divided by @var{denominator}, rounded
towards zero to an integer.  Thus, @w{@code{fmod (6.5, 2.3)}} returns
@code{1.9}, which is @code{6.5} minus @code{4.6}.

The result has the same sign as the @var{numerator} and has magnitude
less than the magnitude of the @var{denominator}.

If @var{denominator} is zero, @code{fmod} fails and sets @code{errno} to
@code{EDOM}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double drem (double @var{numerator}, double @var{denominator})
@deftypefunx float dremf (float @var{numerator}, float @var{denominator})
@deftypefunx {long double} dreml (long double @var{numerator}, long double @var{denominator})
These functions are like @code{fmod} etc except that it rounds the
internal quotient @var{n} to the nearest integer instead of towards zero
to an integer.  For example, @code{drem (6.5, 2.3)} returns @code{-0.4},
which is @code{6.5} minus @code{6.9}.

The absolute value of the result is less than or equal to half the
absolute value of the @var{denominator}.  The difference between
@code{fmod (@var{numerator}, @var{denominator})} and @code{drem
(@var{numerator}, @var{denominator})} is always either
@var{denominator}, minus @var{denominator}, or zero.

If @var{denominator} is zero, @code{drem} fails and sets @code{errno} to
@code{EDOM}.
@end deftypefun


@node Arithmetic on FP Values
@section Setting and modifying Single Bits of FP Values
@cindex FP arithmetic

In certain situations it is too complicated (or expensive) to modify a
floating-point value by the normal operations.  For a few operations
@w{ISO C 9X} defines functions to modify the floating-point value
directly.

@comment math.h
@comment ISO
@deftypefun double copysign (double @var{x}, double @var{y})
@deftypefunx float copysignf (float @var{x}, float @var{y})
@deftypefunx {long double} copysignl (long double @var{x}, long double @var{y})
The @code{copysign} function allows to specifiy the sign of the
floating-point value given in the parameter @var{x} by discarding the
prior content and replacing it with the sign of the value @var{y}.
The so found value is returned.

This function also works and throws no exception if the parameter
@var{x} is a @code{NaN}.  If the platform supports the signed zero
representation @var{x} might also be zero.

This function is defined in @w{IEC 559} (and the appendix with
recommended functions in @w{IEEE 754}/@w{IEEE 854}).
@end deftypefun

@comment math.h
@comment ISO
@deftypefun int signbit (@emph{float-type} @var{x})
@code{signbit} is a generic macro which can work on all floating-point
types.  It returns a nonzero value if the value of @var{x} has its sign
bit set.

This is not the same as @code{x < 0.0} since in some floating-point
formats (e.g., @w{IEEE 754}) the zero value is optionally signed.  The
comparison @code{-0.0 < 0.0} will not be true while @code{signbit
(-0.0)} will return a nonzero value.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double nextafter (double @var{x}, double @var{y})
@deftypefunx float nextafterf (float @var{x}, float @var{y})
@deftypefunx {long double} nextafterl (long double @var{x}, long double @var{y})
The @code{nextafter} function returns the next representable neighbor of
@var{x} in the direction towards @var{y}.  Depending on the used data
type the steps make have a different size.  If @math{@var{x} = @var{y}}
the function simply returns @var{x}.  If either value is a @code{NaN}
one the @code{NaN} values is returned.  Otherwise a value corresponding
to the value of the least significant bit in the mantissa is
added/subtracted (depending on the direction).  If the resulting value
is not finite but @var{x} is, overflow is signaled.  Underflow is
signaled if the resulting value is a denormalized number (if the @w{IEEE
754}/@w{IEEE 854} representation is used).

This function is defined in @w{IEC 559} (and the appendix with
recommended functions in @w{IEEE 754}/@w{IEEE 854}).
@end deftypefun

@cindex NaN
@comment math.h
@comment ISO
@deftypefun double nan (const char *@var{tagp})
@deftypefunx float nanf (const char *@var{tagp})
@deftypefunx {long double} nanl (const char *@var{tagp})
The @code{nan} function returns a representation of the NaN value.  If
quiet NaNs are supported by the platform a call like @code{nan
("@var{n-char-sequence}")} is equivalent to @code{strtod
("NAN(@var{n-char-sequence})")}.  The exact implementation is left
unspecified but on systems using IEEE arithmethic the
@var{n-char-sequence} specifies the bits of the mantissa for the NaN
value.
@end deftypefun


@node Special arithmetic on FPs
@section Special Arithmetic on FPs
@cindex positive difference
@cindex minimum
@cindex maximum

A frequent operation of numbers is the determination of mimuma, maxima,
or the difference between numbers.  The @w{ISO C 9X} standard introduces
three functions which implement this efficiently while also providing
some useful functions which is not so efficient to implement.  Machine
specific implementation might perform this very efficient.

@comment math.h
@comment ISO
@deftypefun double fmin (double @var{x}, double @var{y})
@deftypefunx float fminf (float @var{x}, float @var{y})
@deftypefunx {long double} fminl (long double @var{x}, long double @var{y})
The @code{fmin} function determine the minimum of the two values @var{x}
and @var{y} and returns it.

If an argument is NaN it as treated as missing and the other value is
returned.  If both values are NaN one of the values is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fmax (double @var{x}, double @var{y})
@deftypefunx float fmaxf (float @var{x}, float @var{y})
@deftypefunx {long double} fmaxl (long double @var{x}, long double @var{y})
The @code{fmax} function determine the maximum of the two values @var{x}
and @var{y} and returns it.

If an argument is NaN it as treated as missing and the other value is
returned.  If both values are NaN one of the values is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fdim (double @var{x}, double @var{y})
@deftypefunx float fdimf (float @var{x}, float @var{y})
@deftypefunx {long double} fdiml (long double @var{x}, long double @var{y})
The @code{fdim} function computes the positive difference between
@var{x} and @var{y} and returns this value.  @dfn{Positive difference}
means that if @var{x} is greater than @var{y} the value @math{@var{x} -
@var{y}} is returned.  Otherwise the return value is @math{+0}.

If any of the arguments is NaN this value is returned.  If both values
are NaN, one of the values is returned.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double fma (double @var{x}, double @var{y}, double @var{z})
@deftypefunx float fmaf (float @var{x}, float @var{y}, float @var{z})
@deftypefunx {long double} fmal (long double @var{x}, long double @var{y}, long double @var{z})
@cindex butterfly
The name of the function @code{fma} means floating-point multiply-add.
I.e., the operation performed is @math{(@var{x} @mul{} @var{y}) +
@var{z}}.  The speciality of this function is that the intermediate
result is not rounded and the addition is performed with the full
precision of the multiplcation.

This function was introduced because some processors provide such a
function in their FPU implementation.  Since compilers cannot optimize
code which performs the operation in single steps using this opcode
because of rounding differences the operation is available separately so
the programmer can select when the rounding of the intermediate result
is not important.

@vindex FP_FAST_FMA
If the @file{math.h} header defines the symbol @code{FP_FAST_FMA} (or
@code{FP_FAST_FMAF} and @code{FP_FAST_FMAL} for @code{float} and
@code{long double} respectively) the processor typically defines the
operation in hardware.  The symbols might also be defined if the
software implementation is as fast as a multiply and an add but in the
GNU C Library the macros indicate hardware support.
@end deftypefun


@node Integer Division
@section Integer Division
@cindex integer division functions

This section describes functions for performing integer division.  These
functions are redundant in the GNU C library, since in GNU C the @samp{/}
operator always rounds towards zero.  But in other C implementations,
@samp{/} may round differently with negative arguments.  @code{div} and
@code{ldiv} are useful because they specify how to round the quotient:
towards zero.  The remainder has the same sign as the numerator.

These functions are specified to return a result @var{r} such that the value
@code{@var{r}.quot*@var{denominator} + @var{r}.rem} equals
@var{numerator}.

@pindex stdlib.h
To use these facilities, you should include the header file
@file{stdlib.h} in your program.

@comment stdlib.h
@comment ISO
@deftp {Data Type} div_t
This is a structure type used to hold the result returned by the @code{div}
function.  It has the following members:

@table @code
@item int quot
The quotient from the division.

@item int rem
The remainder from the division.
@end table
@end deftp

@comment stdlib.h
@comment ISO
@deftypefun div_t div (int @var{numerator}, int @var{denominator})
This function @code{div} computes the quotient and remainder from
the division of @var{numerator} by @var{denominator}, returning the
result in a structure of type @code{div_t}.

If the result cannot be represented (as in a division by zero), the
behavior is undefined.

Here is an example, albeit not a very useful one.

@smallexample
div_t result;
result = div (20, -6);
@end smallexample

@noindent
Now @code{result.quot} is @code{-3} and @code{result.rem} is @code{2}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftp {Data Type} ldiv_t
This is a structure type used to hold the result returned by the @code{ldiv}
function.  It has the following members:

@table @code
@item long int quot
The quotient from the division.

@item long int rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{long int} rather than @code{int}.)
@end deftp

@comment stdlib.h
@comment ISO
@deftypefun ldiv_t ldiv (long int @var{numerator}, long int @var{denominator})
The @code{ldiv} function is similar to @code{div}, except that the
arguments are of type @code{long int} and the result is returned as a
structure of type @code{ldiv_t}.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftp {Data Type} lldiv_t
This is a structure type used to hold the result returned by the @code{lldiv}
function.  It has the following members:

@table @code
@item long long int quot
The quotient from the division.

@item long long int rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{long long int} rather than @code{int}.)
@end deftp

@comment stdlib.h
@comment GNU
@deftypefun lldiv_t lldiv (long long int @var{numerator}, long long int @var{denominator})
The @code{lldiv} function is like the @code{div} function, but the
arguments are of type @code{long long int} and the result is returned as
a structure of type @code{lldiv_t}.

The @code{lldiv} function is a GNU extension but it will eventually be
part of the next ISO C standard.
@end deftypefun


@node Parsing of Numbers
@section Parsing of Numbers
@cindex parsing numbers (in formatted input)
@cindex converting strings to numbers
@cindex number syntax, parsing
@cindex syntax, for reading numbers

This section describes functions for ``reading'' integer and
floating-point numbers from a string.  It may be more convenient in some
cases to use @code{sscanf} or one of the related functions; see
@ref{Formatted Input}.  But often you can make a program more robust by
finding the tokens in the string by hand, then converting the numbers
one by one.

@menu
* Parsing of Integers::         Functions for conversion of integer values.
* Parsing of Floats::           Functions for conversion of floating-point
				 values.
@end menu

@node Parsing of Integers
@subsection Parsing of Integers

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ISO
@deftypefun {long int} strtol (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtol} (``string-to-long'') function converts the initial
part of @var{string} to a signed integer, which is returned as a value
of type @code{long int}.

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item
An optional plus or minus sign (@samp{+} or @samp{-}).

@item
A nonempty sequence of digits in the radix specified by @var{base}.

If @var{base} is zero, decimal radix is assumed unless the series of
digits begins with @samp{0} (specifying octal radix), or @samp{0x} or
@samp{0X} (specifying hexadecimal radix); in other words, the same
syntax used for integer constants in C.

Otherwise @var{base} must have a value between @code{2} and @code{35}.
If @var{base} is @code{16}, the digits may optionally be preceded by
@samp{0x} or @samp{0X}.  If base has no legal value the value returned
is @code{0l} and the global variable @code{errno} is set to @code{EINVAL}.

@item
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, @code{strtol} stores a pointer to this tail in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for an integer in the
specified @var{base}, no conversion is performed.  In this case,
@code{strtol} returns a value of zero and the value stored in
@code{*@var{tailptr}} is the value of @var{string}.

In a locale other than the standard @code{"C"} locale, this function
may recognize additional implementation-dependent syntax.

If the string has valid syntax for an integer but the value is not
representable because of overflow, @code{strtol} returns either
@code{LONG_MAX} or @code{LONG_MIN} (@pxref{Range of Type}), as
appropriate for the sign of the value.  It also sets @code{errno}
to @code{ERANGE} to indicate there was overflow.

Because the value @code{0l} is a correct result for @code{strtol} the
user who is interested in handling errors should set the global variable
@code{errno} to @code{0} before calling this function, so that the program
can later test whether an error occurred.

There is an example at the end of this section.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {unsigned long int} strtoul (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoul} (``string-to-unsigned-long'') function is like
@code{strtol} except it deals with unsigned numbers, and returns its
value with type @code{unsigned long int}.  No @samp{+} or @samp{-} sign
may appear before the number, but the syntax is otherwise the same as
described above for @code{strtol}.  The value returned in case of
overflow is @code{ULONG_MAX} (@pxref{Range of Type}).

Like @code{strtol} this function sets @code{errno} and returns the value
@code{0ul} in case the value for @var{base} is not in the legal range.
For @code{strtoul} this can happen in another situation.  In case the
number to be converted is negative @code{strtoul} also sets @code{errno}
to @code{EINVAL} and returns @code{0ul}.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {long long int} strtoll (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoll} function is like @code{strtol} except that is deals
with extra long numbers and it returns its value with type @code{long
long int}.

If the string has valid syntax for an integer but the value is not
representable because of overflow, @code{strtoll} returns either
@code{LONG_LONG_MAX} or @code{LONG_LONG_MIN} (@pxref{Range of Type}), as
appropriate for the sign of the value.  It also sets @code{errno} to
@code{ERANGE} to indicate there was overflow.

The @code{strtoll} function is a GNU extension but it will eventually be
part of the next ISO C standard.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {long long int} strtoq (const char *@var{string}, char **@var{tailptr}, int @var{base})
@code{strtoq} (``string-to-quad-word'') is only an commonly used other
name for the @code{strtoll} function.  Everything said for
@code{strtoll} applies to @code{strtoq} as well.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {unsigned long long int} strtoull (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoull} function is like @code{strtoul} except that is deals
with extra long numbers and it returns its value with type
@code{unsigned long long int}.  The value returned in case of overflow
is @code{ULONG_LONG_MAX} (@pxref{Range of Type}).

The @code{strtoull} function is a GNU extension but it will eventually be
part of the next ISO C standard.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {unsigned long long int} strtouq (const char *@var{string}, char **@var{tailptr}, int @var{base})
@code{strtouq} (``string-to-unsigned-quad-word'') is only an commonly
used other name for the @code{strtoull} function.  Everything said for
@code{strtoull} applies to @code{strtouq} as well.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun {long int} atol (const char *@var{string})
This function is similar to the @code{strtol} function with a @var{base}
argument of @code{10}, except that it need not detect overflow errors.
The @code{atol} function is provided mostly for compatibility with
existing code; using @code{strtol} is more robust.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun int atoi (const char *@var{string})
This function is like @code{atol}, except that it returns an @code{int}
value rather than @code{long int}.  The @code{atoi} function is also
considered obsolete; use @code{strtol} instead.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {long long int} atoll (const char *@var{string})
This function is similar to @code{atol}, except it returns a @code{long
long int} value rather than @code{long int}.

The @code{atoll} function is a GNU extension but it will eventually be
part of the next ISO C standard.
@end deftypefun

The POSIX locales contain some information about how to format numbers
(@pxref{General Numeric}).  This mainly deals with representing numbers
for better readability for humans.  The functions present so far in this
section cannot handle numbers in this form.

If this functionality is needed in a program one can use the functions
from the @code{scanf} family which know about the flag @samp{'} for
parsing numeric input (@pxref{Numeric Input Conversions}).  Sometimes it
is more desirable to have finer control.

In these situation one could use the function
@code{__strto@var{XXX}_internal}.  @var{XXX} here stands for any of the
above forms.  All numeric conversion functions (including the functions
to process floating-point numbers) have such a counterpart.  The
difference to the normal form is the extra argument at the end of the
parameter list.  If this value has an non-zero value the handling of
number grouping is enabled.  The advantage of using these functions is
that the @var{tailptr} parameters allow to determine which part of the
input is processed.  The @code{scanf} functions don't provide this
information.  The drawback of using these functions is that they are not
portable.  They only exist in the GNU C library.


Here is a function which parses a string as a sequence of integers and
returns the sum of them:

@smallexample
int
sum_ints_from_string (char *string)
@{
  int sum = 0;

  while (1) @{
    char *tail;
    int next;

    /* @r{Skip whitespace by hand, to detect the end.}  */
    while (isspace (*string)) string++;
    if (*string == 0)
      break;

    /* @r{There is more nonwhitespace,}  */
    /* @r{so it ought to be another number.}  */
    errno = 0;
    /* @r{Parse it.}  */
    next = strtol (string, &tail, 0);
    /* @r{Add it in, if not overflow.}  */
    if (errno)
      printf ("Overflow\n");
    else
      sum += next;
    /* @r{Advance past it.}  */
    string = tail;
  @}

  return sum;
@}
@end smallexample

@node Parsing of Floats
@subsection Parsing of Floats

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ISO
@deftypefun double strtod (const char *@var{string}, char **@var{tailptr})
The @code{strtod} (``string-to-double'') function converts the initial
part of @var{string} to a floating-point number, which is returned as a
value of type @code{double}.

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item
An optional plus or minus sign (@samp{+} or @samp{-}).

@item
A nonempty sequence of digits optionally containing a decimal-point
character---normally @samp{.}, but it depends on the locale
(@pxref{Numeric Formatting}).

@item
An optional exponent part, consisting of a character @samp{e} or
@samp{E}, an optional sign, and a sequence of digits.

@item
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, a pointer to this tail of the string is stored in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for a floating-point
number, no conversion is performed.  In this case, @code{strtod} returns
a value of zero and the value returned in @code{*@var{tailptr}} is the
value of @var{string}.

In a locale other than the standard @code{"C"} or @code{"POSIX"} locales,
this function may recognize additional locale-dependent syntax.

If the string has valid syntax for a floating-point number but the value
is not representable because of overflow, @code{strtod} returns either
positive or negative @code{HUGE_VAL} (@pxref{Mathematics}), depending on
the sign of the value.  Similarly, if the value is not representable
because of underflow, @code{strtod} returns zero.  It also sets @code{errno}
to @code{ERANGE} if there was overflow or underflow.

There are two more special inputs which are recognized by @code{strtod}.
The string @code{"inf"} or @code{"infinity"} (without consideration of
case and optionally preceded by a @code{"+"} or @code{"-"} sign) is
changed to the floating-point value for infinity if the floating-point
format supports this; and to the largest representable value otherwise.

If the input string is @code{"nan"} or
@code{"nan(@var{n-char-sequence})"} the return value of @code{strtod} is
the representation of the NaN (not a number) value (if the
floating-point format supports this).  In the second form the part
@var{n-char-sequence} allows to specify the form of the NaN value in an
implementation specific way.  When using the @w{IEEE 754}
floating-point format, the NaN value can have a lot of forms since only
at least one bit in the mantissa must be set.  In the GNU C library
implementation of @code{strtod} the @var{n-char-sequence} is interpreted
as a number (as recognized by @code{strtol}, @pxref{Parsing of Integers}).
The mantissa of the return value corresponds to this given number.

Since the value zero which is returned in the error case is also a valid
result the user should set the global variable @code{errno} to zero
before calling this function.  So one can test for failures after the
call since all failures set @code{errno} to a non-zero value.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun float strtof (const char *@var{string}, char **@var{tailptr})
This function is similar to the @code{strtod} function but it returns a
@code{float} value instead of a @code{double} value.  If the precision
of a @code{float} value is sufficient this function should be used since
it is much faster than @code{strtod} on some architectures.  The reasons
are obvious: @w{IEEE 754} defines @code{float} to have a mantissa of 23
bits while @code{double} has 53 bits and every additional bit of
precision can require additional computation.

If the string has valid syntax for a floating-point number but the value
is not representable because of overflow, @code{strtof} returns either
positive or negative @code{HUGE_VALF} (@pxref{Mathematics}), depending on
the sign of the value.

This function is a GNU extension.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun {long double} strtold (const char *@var{string}, char **@var{tailptr})
This function is similar to the @code{strtod} function but it returns a
@code{long double} value instead of a @code{double} value.  It should be
used when high precision is needed.  On systems which define a @code{long
double} type (i.e., on which it is not the same as @code{double})
running this function might take significantly more time since more bits
of precision are required.

If the string has valid syntax for a floating-point number but the value
is not representable because of overflow, @code{strtold} returns either
positive or negative @code{HUGE_VALL} (@pxref{Mathematics}), depending on
the sign of the value.

This function is a GNU extension.
@end deftypefun

As for the integer parsing functions there are additional functions
which will handle numbers represented using the grouping scheme of the
current locale (@pxref{Parsing of Integers}).

@comment stdlib.h
@comment ISO
@deftypefun double atof (const char *@var{string})
This function is similar to the @code{strtod} function, except that it
need not detect overflow and underflow errors.  The @code{atof} function
is provided mostly for compatibility with existing code; using
@code{strtod} is more robust.
@end deftypefun