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@node Mathematics, Arithmetic, Low-Level Terminal Interface, Top
@chapter Mathematics

This chapter contains information about functions for performing
mathematical computations, such as trigonometric functions.  Most of
these functions have prototypes declared in the header file
@file{math.h}.
@pindex math.h

All of the functions that operate on floating-point numbers accept
arguments and return results of type @code{double}.  In the future,
there may be additional functions that operate on @code{float} and
@code{long double} values.  For example, @code{cosf} and @code{cosl}
would be versions of the @code{cos} function that operate on
@code{float} and @code{long double} arguments, respectively.  In the
meantime, you should avoid using these names yourself.  @xref{Reserved
Names}.

@menu
* Domain and Range Errors::     Detecting overflow conditions and the like.
* Trig Functions::              Sine, cosine, and tangent.
* Inverse Trig Functions::      Arc sine, arc cosine, and arc tangent.
* Exponents and Logarithms::    Also includes square root.
* Hyperbolic Functions::        Hyperbolic sine and friends.
* Pseudo-Random Numbers::       Functions for generating pseudo-random
                                 numbers.
@end menu

@node Domain and Range Errors
@section Domain and Range Errors

@cindex domain error
Many of the functions listed in this chapter are defined mathematically
over a domain that is only a subset of real numbers.  For example, the
@code{acos} function is defined over the domain between @code{-1} and
@code{1}.  If you pass an argument to one of these functions that is
outside the domain over which it is defined, the function sets
@code{errno} to @code{EDOM} to indicate a @dfn{domain error}.  On
machines that support @w{IEEE 754} floating point, functions reporting
error @code{EDOM} also return a NaN.

Some of these functions are defined mathematically to result in a
complex value over parts of their domains.  The most familiar example of
this is taking the square root of a negative number.  The functions in
this chapter take only real arguments and return only real values;
therefore, if the value ought to be nonreal, this is treated as a domain
error.

@cindex range error
A related problem is that the mathematical result of a function may not
be representable as a floating point number.  If magnitude of the
correct result is too large to be represented, the function sets
@code{errno} to @code{ERANGE} to indicate a @dfn{range error}, and
returns a particular very large value (named by the macro
@code{HUGE_VAL}) or its negation (@w{@code{- HUGE_VAL}}).

If the magnitude of the result is too small, a value of zero is returned
instead.  In this case, @code{errno} might or might not be
set to @code{ERANGE}.

The only completely reliable way to check for domain and range errors is
to set @code{errno} to @code{0} before you call the mathematical function
and test @code{errno} afterward.  As a consequence of this use of
@code{errno}, use of the mathematical functions is not reentrant if you
check for errors.

@c !!! this isn't always true at the moment....
None of the mathematical functions ever generates signals as a result of
domain or range errors.  In particular, this means that you won't see
@code{SIGFPE} signals generated within these functions.  (@xref{Signal
Handling}, for more information about signals.)

@comment math.h
@comment ISO
@deftypevr Macro double HUGE_VAL
An expression representing a particular very large number.  On machines
that use @w{IEEE 754}/@w{IEEE 854} floating point format, the value is
``infinity''.  On other machines, it's typically the largest positive
number that can be represented.

The value of this macro is used as the return value from various
mathematical @code{double} returning functions in overflow situations.
@end deftypevr

@comment math.h
@comment GNU
@deftypevr Macro float HUGE_VALf
This macro is similar to the @code{HUGE_VAL} macro except that it is
used by functions returning @code{float} values.

This macro is a GNU extension.
@end deftypevr

@comment math.h
@comment GNU
@deftypevr Macro {long double} HUGE_VALl
This macro is similar to the @code{HUGE_VAL} macro except that it is
used by functions returning @code{long double} values.  The value is
only different from @code{HUGE_VAL} if the architecture really supports
@code{long double} values.

This macro is a GNU extension.
@end deftypevr


@comment

For more information about floating-point representations and limits,
see @ref{Floating Point Parameters}.  In particular, the macro
@code{DBL_MAX} might be more appropriate than @code{HUGE_VAL} for many
uses other than testing for an error in a mathematical function.

@node Trig Functions
@section Trigonometric Functions
@cindex trigonometric functions

These are the familiar @code{sin}, @code{cos}, and @code{tan} functions.
The arguments to all of these functions are in units of radians; recall
that pi radians equals 180 degrees.

@cindex pi (trigonometric constant)
The math library doesn't define a symbolic constant for pi, but you can
define your own if you need one:

@smallexample
#define PI 3.14159265358979323846264338327
@end smallexample

@noindent
You can also compute the value of pi with the expression @code{acos
(-1.0)}.


@comment math.h
@comment ISO
@deftypefun double sin (double @var{x})
This function returns the sine of @var{x}, where @var{x} is given in
radians.  The return value is in the range @code{-1} to @code{1}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double cos (double @var{x})
This function returns the cosine of @var{x}, where @var{x} is given in
radians.  The return value is in the range @code{-1} to @code{1}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double tan (double @var{x})
This function returns the tangent of @var{x}, where @var{x} is given in
radians.

The following @code{errno} error conditions are defined for this function:

@table @code
@item ERANGE
Mathematically, the tangent function has singularities at odd multiples
of pi/2.  If the argument @var{x} is too close to one of these
singularities, @code{tan} sets @code{errno} to @code{ERANGE} and returns
either positive or negative @code{HUGE_VAL}.
@end table
@end deftypefun


@node Inverse Trig Functions
@section Inverse Trigonometric Functions
@cindex inverse trigonometric functions

These are the usual arc sine, arc cosine and arc tangent functions,
which are the inverses of the sine, cosine and tangent functions,
respectively.

@comment math.h
@comment ISO
@deftypefun double asin (double @var{x})
This function computes the arc sine of @var{x}---that is, the value whose
sine is @var{x}.  The value is in units of radians.  Mathematically,
there are infinitely many such values; the one actually returned is the
one between @code{-pi/2} and @code{pi/2} (inclusive).

@code{asin} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is
out of range.  The arc sine function is defined mathematically only
over the domain @code{-1} to @code{1}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double acos (double @var{x})
This function computes the arc cosine of @var{x}---that is, the value
whose cosine is @var{x}.  The value is in units of radians.
Mathematically, there are infinitely many such values; the one actually
returned is the one between @code{0} and @code{pi} (inclusive).

@code{acos} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is
out of range.  The arc cosine function is defined mathematically only
over the domain @code{-1} to @code{1}.
@end deftypefun


@comment math.h
@comment ISO
@deftypefun double atan (double @var{x})
This function computes the arc tangent of @var{x}---that is, the value
whose tangent is @var{x}.  The value is in units of radians.
Mathematically, there are infinitely many such values; the one actually
returned is the one between @code{-pi/2} and @code{pi/2}
(inclusive).
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double atan2 (double @var{y}, double @var{x})
This is the two argument arc tangent function.  It is similar to computing
the arc tangent of @var{y}/@var{x}, except that the signs of both arguments
are used to determine the quadrant of the result, and @var{x} is
permitted to be zero.  The return value is given in radians and is in
the range @code{-pi} to @code{pi}, inclusive.

If @var{x} and @var{y} are coordinates of a point in the plane,
@code{atan2} returns the signed angle between the line from the origin
to that point and the x-axis.  Thus, @code{atan2} is useful for
converting Cartesian coordinates to polar coordinates.  (To compute the
radial coordinate, use @code{hypot}; see @ref{Exponents and
Logarithms}.)

The function @code{atan2} sets @code{errno} to @code{EDOM} if both
@var{x} and @var{y} are zero; the return value is not defined in this
case.
@end deftypefun


@node Exponents and Logarithms
@section Exponentiation and Logarithms
@cindex exponentiation functions
@cindex power functions
@cindex logarithm functions

@comment math.h
@comment ISO
@deftypefun double exp (double @var{x})
The @code{exp} function returns the value of e (the base of natural
logarithms) raised to power @var{x}.

The function fails, and sets @code{errno} to @code{ERANGE}, if the
magnitude of the result is too large to be representable.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double log (double @var{x})
This function returns the natural logarithm of @var{x}.  @code{exp (log
(@var{x}))} equals @var{x}, exactly in mathematics and approximately in
C.

The following @code{errno} error conditions are defined for this function:

@table @code
@item EDOM
The argument @var{x} is negative.  The log function is defined
mathematically to return a real result only on positive arguments.

@item ERANGE
The argument is zero.  The log of zero is not defined.
@end table
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double log10 (double @var{x})
This function returns the base-10 logarithm of @var{x}.  Except for the
different base, it is similar to the @code{log} function.  In fact,
@code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double pow (double @var{base}, double @var{power})
This is a general exponentiation function, returning @var{base} raised
to @var{power}.

@need 250
The following @code{errno} error conditions are defined for this function:

@table @code
@item EDOM
The argument @var{base} is negative and @var{power} is not an integral
value.  Mathematically, the result would be a complex number in this case.

@item ERANGE
An underflow or overflow condition was detected in the result.
@end table
@end deftypefun

@cindex square root function
@comment math.h
@comment ISO
@deftypefun double sqrt (double @var{x})
This function returns the nonnegative square root of @var{x}.

The @code{sqrt} function fails, and sets @code{errno} to @code{EDOM}, if
@var{x} is negative.  Mathematically, the square root would be a complex
number.
@end deftypefun

@cindex cube root function
@comment math.h
@comment BSD
@deftypefun double cbrt (double @var{x})
This function returns the cube root of @var{x}.  This function cannot
fail; every representable real value has a representable real cube root.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double hypot (double @var{x}, double @var{y})
The @code{hypot} function returns @code{sqrt (@var{x}*@var{x} +
@var{y}*@var{y})}.  (This is the length of the hypotenuse of a right
triangle with sides of length @var{x} and @var{y}, or the distance
of the point (@var{x}, @var{y}) from the origin.)  See also the function
@code{cabs} in @ref{Absolute Value}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double expm1 (double @var{x})
This function returns a value equivalent to @code{exp (@var{x}) - 1}.
It is computed in a way that is accurate even if the value of @var{x} is
near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate due
to subtraction of two numbers that are nearly equal.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double log1p (double @var{x})
This function returns a value equivalent to @w{@code{log (1 + @var{x})}}.
It is computed in a way that is accurate even if the value of @var{x} is
near zero.
@end deftypefun

@node Hyperbolic Functions
@section Hyperbolic Functions
@cindex hyperbolic functions

The functions in this section are related to the exponential functions;
see @ref{Exponents and Logarithms}.

@comment math.h
@comment ISO
@deftypefun double sinh (double @var{x})
The @code{sinh} function returns the hyperbolic sine of @var{x}, defined
mathematically as @w{@code{exp (@var{x}) - exp (-@var{x}) / 2}}.  The
function fails, and sets @code{errno} to @code{ERANGE}, if the value of
@var{x} is too large; that is, if overflow occurs.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double cosh (double @var{x})
The @code{cosh} function returns the hyperbolic cosine of @var{x},
defined mathematically as @w{@code{exp (@var{x}) + exp (-@var{x}) / 2}}.
The function fails, and sets @code{errno} to @code{ERANGE}, if the value
of @var{x} is too large; that is, if overflow occurs.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double tanh (double @var{x})
This function returns the hyperbolic tangent of @var{x}, whose
mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
@end deftypefun

@cindex inverse hyperbolic functions

@comment math.h
@comment BSD
@deftypefun double asinh (double @var{x})
This function returns the inverse hyperbolic sine of @var{x}---the
value whose hyperbolic sine is @var{x}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double acosh (double @var{x})
This function returns the inverse hyperbolic cosine of @var{x}---the
value whose hyperbolic cosine is @var{x}.  If @var{x} is less than
@code{1}, @code{acosh} returns @code{HUGE_VAL}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double atanh (double @var{x})
This function returns the inverse hyperbolic tangent of @var{x}---the
value whose hyperbolic tangent is @var{x}.  If the absolute value of
@var{x} is greater than or equal to @code{1}, @code{atanh} returns
@code{HUGE_VAL}.
@end deftypefun

@node Pseudo-Random Numbers
@section Pseudo-Random Numbers
@cindex random numbers
@cindex pseudo-random numbers
@cindex seed (for random numbers)

This section describes the GNU facilities for generating a series of
pseudo-random numbers.  The numbers generated are not truly random;
typically, they form a sequence that repeats periodically, with a
period so large that you can ignore it for ordinary purposes.  The
random number generator works by remembering at all times a @dfn{seed}
value which it uses to compute the next random number and also to
compute a new seed.

Although the generated numbers look unpredictable within one run of a
program, the sequence of numbers is @emph{exactly the same} from one run
to the next.  This is because the initial seed is always the same.  This
is convenient when you are debugging a program, but it is unhelpful if
you want the program to behave unpredictably.  If you want truly random
numbers, not just pseudo-random, specify a seed based on the current
time.

You can get repeatable sequences of numbers on a particular machine type
by specifying the same initial seed value for the random number
generator.  There is no standard meaning for a particular seed value;
the same seed, used in different C libraries or on different CPU types,
will give you different random numbers.

The GNU library supports the standard @w{ISO C} random number functions
plus another set derived from BSD.  We recommend you use the standard
ones, @code{rand} and @code{srand}.

@menu
* ISO Random::       @code{rand} and friends.
* BSD Random::       @code{random} and friends.
@end menu

@node ISO Random
@subsection ISO C Random Number Functions

This section describes the random number functions that are part of
the @w{ISO C} standard.

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

@comment stdlib.h
@comment ISO
@deftypevr Macro int RAND_MAX
The value of this macro is an integer constant expression that
represents the maximum possible value returned by the @code{rand}
function.  In the GNU library, it is @code{037777777}, which is the
largest signed integer representable in 32 bits.  In other libraries, it
may be as low as @code{32767}.
@end deftypevr

@comment stdlib.h
@comment ISO
@deftypefun int rand ()
The @code{rand} function returns the next pseudo-random number in the
series.  The value is in the range from @code{0} to @code{RAND_MAX}.
@end deftypefun

@comment stdlib.h
@comment ISO
@deftypefun void srand (unsigned int @var{seed})
This function establishes @var{seed} as the seed for a new series of
pseudo-random numbers.  If you call @code{rand} before a seed has been
established with @code{srand}, it uses the value @code{1} as a default
seed.

To produce truly random numbers (not just pseudo-random), do @code{srand
(time (0))}.
@end deftypefun

@node BSD Random
@subsection BSD Random Number Functions

This section describes a set of random number generation functions that
are derived from BSD.  There is no advantage to using these functions
with the GNU C library; we support them for BSD compatibility only.

The prototypes for these functions are in @file{stdlib.h}.
@pindex stdlib.h

@comment stdlib.h
@comment BSD
@deftypefun {long int} random ()
This function returns the next pseudo-random number in the sequence.
The range of values returned is from @code{0} to @code{RAND_MAX}.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun void srandom (unsigned int @var{seed})
The @code{srandom} function sets the seed for the current random number
state based on the integer @var{seed}.  If you supply a @var{seed} value
of @code{1}, this will cause @code{random} to reproduce the default set
of random numbers.

To produce truly random numbers (not just pseudo-random), do
@code{srandom (time (0))}.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {void *} initstate (unsigned int @var{seed}, void *@var{state}, size_t @var{size})
The @code{initstate} function is used to initialize the random number
generator state.  The argument @var{state} is an array of @var{size}
bytes, used to hold the state information.  The size must be at least 8
bytes, and optimal sizes are 8, 16, 32, 64, 128, and 256.  The bigger
the @var{state} array, the better.

The return value is the previous value of the state information array.
You can use this value later as an argument to @code{setstate} to
restore that state.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun {void *} setstate (void *@var{state})
The @code{setstate} function restores the random number state
information @var{state}.  The argument must have been the result of
a previous call to @var{initstate} or @var{setstate}.

The return value is the previous value of the state information array.
You can use thise value later as an argument to @code{setstate} to
restore that state.
@end deftypefun