path: root/manual/math.texi

@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

For all functions which take a single floating-point argument and for
several other functions as well there are three different functions
available for the type @code{double}, @code{float}, and @code{long
double}.  The @code{double} versions of the functions are mostly defined
even in the @w{ISO C 89} standard.  The @code{float} and @code{long
double} variants are introduced in the numeric extensions for the C
language which are part of the @w{ISO C 9X} standard.

Which of the three versions of the function should be used depends on
the situation.  For most functions and implementation it is true that
speed and precision do not go together.  I.e., the @code{float} versions
are normally faster than the @code{double} and @code{long double}
versions.  On the other hand the @code{long double} version has the
highest precision.  One should always think about the actual needs and
in case of double using @code{double} is a good compromise.


@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 ISO
@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 introduced in @w{ISO C 9X}.
@end deftypevr

@comment math.h
@comment ISO
@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 introduced in @w{ISO C 9X}.
@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 does define a symbolic constant for pi in @file{math.h}
when BSD compliance is required (@pxref{Feature Test Macros}).  Beside
pi several other constants are defined.

@noindent
In case it is not possible to use this macro one easily can define it:

@smallexample
#define M_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})
@end deftypefun
@deftypefun float sinf (float @var{x})
@end deftypefun
@deftypefun {long double} sinl (long double @var{x})
These functions return 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})
@end deftypefun
@deftypefun float cosf (float @var{x})
@end deftypefun
@deftypefun {long double} cosl (long double @var{x})
These functions return 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})
@end deftypefun
@deftypefun float tanf (float @var{x})
@end deftypefun
@deftypefun {long double} tanl (long double @var{x})
These functions return 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

In many applications where @code{sin} and @code{cos} are used, the value
for the same argument of both of these functions is used at the same
time.  Since the algorithm to compute these values is very similar for
both functions there is an additional function with computes both values
at the same time.

@comment math.h
@comment GNU
@deftypefun void sincos (double @var{x}, double *@var{sinx}, double *@var{cosx})
@end deftypefun
@deftypefun void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx})
@end deftypefun
@deftypefun void sincosl (long double @var{x}, long double *@var{sinx}, long double *@var{cosx})
These functions return the sine of @var{x} in @code{*@var{sinx}} and the
cosine of @var{x} in @code{*@var{cos}}, where @var{x} is given in
radians.  Both values, @code{*@var{sinx}} and @code{*@var{cosx}}, are in
the range of @code{-1} to @code{1}.
@end deftypefun

@cindex complex trigonometric functions

The trigonometric functions are in mathematics not only on real numbers.
They can be extended to complex numbers and the @w{ISO C 9X} standard
introduces these variants in the standard math library.

@comment complex.h
@comment ISO
@deftypefun {complex double} csin (complex double @var{z})
@end deftypefun
@deftypefun {complex float} csinf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} csinl (complex long double @var{z})
These functions return the complex sine of the complex value in @var{z}.
The mathematical definition of the complex sine is

@smallexample
sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i))
@end smallexample
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ccos (complex double @var{z})
@end deftypefun
@deftypefun {complex float} ccosf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} ccosl (complex long double @var{z})
These functions return the complex cosine of the complex value in @var{z}.
The mathematical definition of the complex cosine is

@smallexample
cos (z) = 1/2 * (exp (z*i) + exp (-z*i))
@end smallexample
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ctan (complex double @var{z})
@end deftypefun
@deftypefun {complex float} ctanf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} ctanl (complex long double @var{z})
These functions return the complex tangent of the complex value in @var{z}.
The mathematical definition of the complex tangent is

@smallexample
tan (z) = 1/i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))
@end smallexample
@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})
@end deftypefun
@deftypefun float asinf (float @var{x})
@end deftypefun
@deftypefun {long double} asinl (long double @var{x})
These functions compute 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})
@end deftypefun
@deftypefun float acosf (float @var{x})
@end deftypefun
@deftypefun {long double} acosl (long double @var{x})
These functions compute 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})
@end deftypefun
@deftypefun float atanf (float @var{x})
@end deftypefun
@deftypefun {long double} atanl (long double @var{x})
These functions compute 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})
@end deftypefun
@deftypefun float atan2f (float @var{y}, float @var{x})
@end deftypefun
@deftypefun {long double} atan2l (long double @var{y}, long 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

@cindex inverse complex trigonometric functions

The inverse trigonometric functions also exist is separate versions
which are usable with complex numbers.

@comment complex.h
@comment ISO
@deftypefun {complex double} casin (complex double @var{z})
@end deftypefun
@deftypefun {complex float} casinf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} casinl (complex long double @var{z})
These functions compute the complex arc sine of @var{z}---that is, the
value whose sine is @var{z}.  The value is in units of radians.

Unlike the real version of the arc sine function @code{casin} has no
limitation on the argument @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cacos (complex double @var{z})
@end deftypefun
@deftypefun {complex float} cacosf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} cacosl (complex long double @var{z})
These functions compute the complex arc cosine of @var{z}---that is, the
value whose cosine is @var{z}.  The value is in units of radians.

Unlike the real version of the arc cosine function @code{cacos} has no
limitation on the argument @var{z}.
@end deftypefun


@comment complex.h
@comment ISO
@deftypefun {complex double} catan (complex double @var{z})
@end deftypefun
@deftypefun {complex float} catanf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} catanl (complex long double @var{z})
These functions compute the complex arc tangent of @var{z}---that is,
the value whose tangent is @var{z}.  The value is in units of radians.
@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})
@end deftypefun
@deftypefun float expf (float @var{x})
@end deftypefun
@deftypefun {long double} expl (long double @var{x})
These functions return the value of @code{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 exp10 (double @var{x})
@end deftypefun
@deftypefun float exp10f (float @var{x})
@end deftypefun
@deftypefun {long double} exp10l (long double @var{x})
These functions return the value of @code{10} raised to the power @var{x}.
Mathematically, @code{exp10 (x)} is the same as @code{exp (x * log (10))}.

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 exp2 (double @var{x})
@end deftypefun
@deftypefun float exp2f (float @var{x})
@end deftypefun
@deftypefun {long double} exp2l (long double @var{x})
These functions return the value of @code{2} raised to the power @var{x}.
Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}.

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})
@end deftypefun
@deftypefun float logf (floatdouble @var{x})
@end deftypefun
@deftypefun {long double} logl (long double @var{x})
These functions return 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})
@end deftypefun
@deftypefun float log10f (float @var{x})
@end deftypefun
@deftypefun {long double} log10l (long double @var{x})
These functions return 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 log2 (double @var{x})
@end deftypefun
@deftypefun float log2f (float @var{x})
@end deftypefun
@deftypefun {long double} log2l (long double @var{x})
These functions return the base-2 logarithm of @var{x}.  Except for the
different base, it is similar to the @code{log} function.  In fact,
@code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double pow (double @var{base}, double @var{power})
@end deftypefun
@deftypefun float powf (float @var{base}, float @var{power})
@end deftypefun
@deftypefun {long double} powl (long double @var{base}, long double @var{power})
These are general exponentiation functions, 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})
@end deftypefun
@deftypefun float sqrtf (float @var{x})
@end deftypefun
@deftypefun {long double} sqrtl (long double @var{x})
These functions return 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.
@c (@pxref{csqrt})
@end deftypefun

@cindex cube root function
@comment math.h
@comment BSD
@deftypefun double cbrt (double @var{x})
@end deftypefun
@deftypefun float cbrtf (float @var{x})
@end deftypefun
@deftypefun {long double} cbrtl (long double @var{x})
These functions return the cube root of @var{x}.  They cannot
fail; every representable real value has a representable real cube root.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double hypot (double @var{x}, double @var{y})
@end deftypefun
@deftypefun float hypotf (float @var{x}, float @var{y})
@end deftypefun
@deftypefun {long double} hypotl (long double @var{x}, long double @var{y})
These functions return @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.)  Using this function
instead of the direct formula is highly appreciated since the error is
much smaller.  See also the function @code{cabs} in @ref{Absolute Value}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double expm1 (double @var{x})
@end deftypefun
@deftypefun float expm1f (float @var{x})
@end deftypefun
@deftypefun {long double} expm1l (long double @var{x})
These functions return 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 ISO
@deftypefun double log1p (double @var{x})
@end deftypefun
@deftypefun float log1pf (float @var{x})
@end deftypefun
@deftypefun {long double} log1pl (long 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

@cindex complex exponentiation functions
@cindex complex logarithm functions

@w{ISO C 9X} defines variants of some of the exponentiation and
logarithm functions.  As for the other functions handlung complex
numbers these functions are perhaps better optimized and provide better
error checking than a direct use of the formulas of the mathematical
definition.

@comment complex.h
@comment ISO
@deftypefun {complex double} cexp (complex double @var{z})
@end deftypefun
@deftypefun {complex float} cexpf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} cexpl (complex long double @var{z})
These functions return the value of @code{e} (the base of natural
logarithms) raised to power of the complex value @var{z}.

Mathematically this corresponds to the value

@smallexample
exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))
@end smallexample
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} clog (complex double @var{z})
@end deftypefun
@deftypefun {complex float} clogf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} clogl (complex long double @var{z})
These functions return the natural logarithm of the complex value
@var{z}.  Unlike the real value version @code{log} and its variants,
@code{clog} has no limit for the range of its argument @var{z}.

Mathematically this corresponds to the value

@smallexample
log (z) = log (cabs (z)) + I * carg (z)
@end smallexample
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} csqrt (complex double @var{z})
@end deftypefun
@deftypefun {complex float} csqrtf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} csqrtl (complex long double @var{z})
These functions return the complex root of the argument @var{z}.  Unlike
the @code{sqrt} function these functions do not have any restriction on
the value of the argument.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power})
@end deftypefun
@deftypefun {complex float} cpowf (complex float @var{base}, complex float @var{power})
@end deftypefun
@deftypefun {complex long double} cpowl (complex long double @var{base}, complex long double @var{power})
These functions return the complex value @var{BASE} raised to the power of
@var{power}.  This is computed as

@smallexample
cpow (x, y) = cexp (y * clog (x))
@end smallexample
@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})
@end deftypefun
@deftypefun float sinhf (float @var{x})
@end deftypefun
@deftypefun {long double} sinhl (long double @var{x})
These functions return 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})
@end deftypefun
@deftypefun float coshf (float @var{x})
@end deftypefun
@deftypefun {long double} coshl (long double @var{x})
These function return 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})
@end deftypefun
@deftypefun float tanhf (float @var{x})
@end deftypefun
@deftypefun {long double} tanhl (long double @var{x})
These functions return the hyperbolic tangent of @var{x}, whose
mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
@end deftypefun

@cindex hyperbolic functions

There are counterparts for these hyperbolic functions which work with
complex valued arguments.  They should always be used instead of the
obvious mathematical formula since the implementations in the math
library are optimized for accuracy and speed.

@comment complex.h
@comment ISO
@deftypefun {complex double} csinh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} csinhf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} csinhl (complex long double @var{z})
These functions return the complex hyperbolic sine of @var{z}, defined
mathematically as @w{@code{(exp (@var{z}) - exp (-@var{z})) / 2}}.  The
function fails, and sets @code{errno} to @code{ERANGE}, if the value of
result is too large.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ccosh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} ccoshf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} ccoshl (complex long double @var{z})
These functions return the complex hyperbolic cosine of @var{z}, defined
mathematically as @w{@code{(exp (@var{z}) + exp (-@var{z})) / 2}}.  The
function fails, and sets @code{errno} to @code{ERANGE}, if the value of
result is too large.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ctanh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} ctanhf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} ctanhl (complex long double @var{z})
These functions return the complex hyperbolic tangent of @var{z}, whose
mathematical definition is @w{@code{csinh (@var{z}) / ccosh (@var{z})}}.
@end deftypefun


@cindex inverse hyperbolic functions

@comment math.h
@comment ISO
@deftypefun double asinh (double @var{x})
@end deftypefun
@deftypefun float asinhf (float @var{x})
@end deftypefun
@deftypefun {long double} asinhl (long double @var{x})
These functions return the inverse hyperbolic sine of @var{x}---the
value whose hyperbolic sine is @var{x}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double acosh (double @var{x})
@end deftypefun
@deftypefun float acoshf (float @var{x})
@end deftypefun
@deftypefun {long double} acoshl (long double @var{x})
These functions return 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 ISO
@deftypefun double atanh (double @var{x})
@end deftypefun
@deftypefun float atanhf (float @var{x})
@end deftypefun
@deftypefun {long double} atanhl (long double @var{x})
These functions return 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

@cindex inverse complex hyperbolic functions

@comment complex.h
@comment ISO
@deftypefun {complex double} casinh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} casinhf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} casinhl (complex long double @var{z})
These functions return the inverse complex hyperbolic sine of
@var{z}---the value whose complex hyperbolic sine is @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cacosh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} cacoshf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} cacoshl (complex long double @var{z})
These functions return the inverse complex hyperbolic cosine of
@var{z}---the value whose complex hyperbolic cosine is @var{z}.  Unlike
the real valued function @code{acosh} there is not limit for the range
of the argument.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} catanh (complex double @var{z})
@end deftypefun
@deftypefun {complex float} catanhf (complex float @var{z})
@end deftypefun
@deftypefun {complex long double} catanhl (complex long double @var{z})
These functions return the inverse complex hyperbolic tangent of
@var{z}---the value whose complex hyperbolic tangent is @var{z}.  Unlike
the real valued function @code{atanh} there is not limit for the range
of the argument.
@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.
* SVID Random::      @code{drand48} 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


@node SVID Random
@subsection SVID Random Number Function

The C library on SVID systems contains yet another kind of random number
generator functions.  They use a state of 48 bits of data.  The user can
choose among a collection of functions which all return the random bits
in different forms.

Generally there are two kinds of functions: those which use a state of
the random number generator which is shared among several functions and
by all threads of the process.  The second group of functions require
the user to handle the state.

All functions have in common that they use the same congruential
formula with the same constants.  The formula is

@smallexample
Y = (a * X + c) mod m
@end smallexample

@noindent
where @var{X} is the state of the generator at the beginning and
@var{Y} the state at the end.  @code{a} and @code{c} are constants
determining the way the generator work.  By default they are

@smallexample
a = 0x5DEECE66D = 25214903917
c = 0xb = 11
@end smallexample

@noindent
but they can also be changed by the user.  @code{m} is of course 2^48
since the state consists of a 48 bit array.


@comment stdlib.h
@comment SVID
@deftypefun double drand48 ()
This function returns a @code{double} value in the range of @code{0.0}
to @code{1.0} (exclusive).  The random bits are determined by the global
state of the random number generator in the C library.

Since the @code{double} type according to @w{IEEE 754} has a 52 bit
mantissa this means 4 bits are not initialized by the random number
generator.  These are (of course) chosen to be the least significant
bits and they are initialized to @code{0}.
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun double erand48 (unsigned short int @var{xsubi}[3])
This function returns a @code{double} value in the range of @code{0.0}
to @code{1.0} (exclusive), similar to @code{drand48}.  The argument is
an array describing the state of the random number generator.

This function can be called subsequently since it updates the array to
guarantee random numbers.  The array should have been initialized before
using to get reproducible results.
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun {long int} lrand48 ()
The @code{lrand48} functions return an integer value in the range of
@code{0} to @code{2^31} (exclusive).  Even if the size of the @code{long
int} type can take more than 32 bits no higher numbers are returned.
The random bits are determined by the global state of the random number
generator in the C library.
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun {long int} nrand48 (unsigned short int @var{xsubi}[3])
This function is similar to the @code{lrand48} function in that it
returns a number in the range of @code{0} to @code{2^31} (exclusive) but
the state of the random number generator used to produce the random bits
is determined by the array provided as the parameter to the function.

The numbers in the array are afterwards updated so that subsequent calls
to this function yield to different results (as it is expected by a
random number generator).  The array should have been initialized before
the first call to get reproducible results.
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun {long int} mrand48 ()
The @code{mrand48} function is similar to @code{lrand48}.  The only
difference is that the numbers returned are in the range @code{-2^31} to
@code{2^31} (exclusive).
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun {long int} jrand48 (unsigned short int @var{xsubi}[3])
The @code{jrand48} function is similar to @code{nrand48}.  The only
difference is that the numbers returned are in the range @code{-2^31} to
@code{2^31} (exclusive).  For the @code{xsubi} parameter the same
requirements are necessary.
@end deftypefun

The internal state of the random number generator can be initialized in
several ways.  The functions differ in the completeness of the
information provided.

@comment stdlib.h
@comment SVID
@deftypefun void srand48 (long int @var{seedval}))
The @code{srand48} function sets the most significant 32 bits of the
state internal state of the random number generator to the least
significant 32 bits of the @var{seedval} parameter.  The lower 16 bts
are initilialized to the value @code{0x330E}.  Even if the @code{long
int} type contains more the 32 bits only the lower 32 bits are used.

Due to this limitation the initialization of the state using this
function of not very useful.  But it makes it easy to use a constrcut
like @code{srand48 (time (0))}.

A side-effect of this function is that the values @code{a} and @code{c}
from the internal state, which are used in the congruential formula,
are reset to the default values given above.  This is of importance once
the user called the @code{lcong48} function (see below).
@end deftypefun

@comment stdlib.h
@comment SVID
@deftypefun {unsigned short int *} seed48 (unsigned short int @var{seed16v}[3])
The @code{seed48} function initializes all 48 bits of the state of the
internal random number generator from the content of the parameter
@var{seed16v}.  Here the lower 16 bits of the first element of
@var{see16v} initialize the least significant 16 bits of the internal
state, the lower 16 bits of @code{@var{seed16v}[1]} initialize the mid-order
16 bits of the state and the 16 lower bits of @code{@var{seed16v}[2]}
initialize the most significant 16 bits of the state.

Unlike @code{srand48} this function lets the user initialize all 48 bits
of the state.

The value returned by @code{seed48} is a pointer to an array containing
the values of the internal state before the change.  This might be
useful to restart the random number generator at a certain state.
Otherwise, the value can simply be ignored.

As for @code{srand48}, the values @code{a} and @code{c} from the
congruential formula are reset to the default values.
@end deftypefun

There is one more function to initialize the random number generator
which allows to specify even more information by allowing to change the
parameters in the congruential formula.

@comment stdlib.h
@comment SVID
@deftypefun void lcong48 (unsigned short int @var{param}[7])
The @code{lcong48} function allows the user to change the complete state
of the random number generator.  Unlike @code{srand48} and
@code{seed48}, this function also changes the constants in the
congruential formula.

From the seven elements in the array @var{param} the least significant
16 bits of the entries @code{@var{param}[0]} to @code{@var{param}[2]}
determine the the initial state, the least 16 bits of
@code{@var{param}[3]} to @code{@var{param}[5]} determine the 48 bit
constant @code{a} and @code{@var{param}[6]} determines the 16 bit value
@code{c}.
@end deftypefun

All the above functions have in common that they use the global
parameters for the congruential formula.  In multi-threaded programs it
might sometimes be useful to have different parameters in different
threads.  For this reason all the above functions have a counterpart
which works on a description of the random number generator in the
user-supplied buffer instead of the global state.

Please note that it is no problem if several threads use the global
state if all threads use the functions which take a pointer to an array
containing the state.  The random numbers are computed following the
same loop but if the state in the array is different all threads will
get an individuual random number generator.

The user supplied buffer must be of type @code{struct drand48_data}.
This type should be regarded as opaque and no member should be used
directly.

@comment stdlib.h
@comment GNU
@deftypefun int drand48_r (struct drand48_data *@var{buffer}, double *@var{result})
This function is equivalent to the @code{drand48} function with the
difference it does not modify the global random number generator
parameters but instead the parameters is the buffer supplied by the
buffer through the pointer @var{buffer}.  The random number is return in
the variable pointed to by @var{result}.

The return value of the function indicate whether the call succeeded.
If the value is less than @code{0} an error occurred and @var{errno} is
set to indicate the problem.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int erand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, double *@var{result})
The @code{erand48_r} function works like the @code{erand48} and it takes
an argument @var{buffer} which describes the random number generator.
The state of the random number genertor is taken from the @code{xsubi}
array, the parameters for the congruential formula from the global
random number generator data.  The random number is return in the
variable pointed to by @var{result}.

The return value is non-negative is the call succeeded.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int lrand48_r (struct drand48_data *@var{buffer}, double *@var{result})
This function is similar to @code{lrand48} and it takes a pointer to a
buffer describing the state of the random number generator as a
parameter just like @code{drand48}.

If the return value of the function is non-negative the variable pointed
to by @var{result} contains the result.  Otherwise an error occurred.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int nrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result})
The @code{nrand48_r} function works like @code{nrand48} in that it
produces a random number in range @code{0} to @code{2^31}.  But instead
of using the global parameters for the congruential formula it uses the
information from the buffer pointed to by @var{buffer}.  The state is
described by the values in @var{xsubi}.

If the return value is non-negative the variable pointed to by
@var{result} contains the result.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int mrand48_r (struct drand48_data *@var{buffer}, double *@var{result})
This function is similar to @code{mrand48} but as the other reentrant
function it uses the random number generator described by the value in
the buffer pointed to by @var{buffer}.

If the return value is non-negative the variable pointed to by
@var{result} contains the result.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int jrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result})
The @code{jrand48_r} function is similar to @code{jrand48}.  But as the
other reentrant functions of this function family it uses the
congruential formula parameters from the buffer pointed to by
@var{buffer}.

If the return value is non-negative the variable pointed to by
@var{result} contains the result.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

Before any of the above functions should be used the buffer of type
@code{struct drand48_data} should initialized.  The easiest way is to
fill the whole buffer with null bytes, e.g., using

@smallexample
memset (buffer, '\0', sizeof (struct drand48_data));
@end smallexample

@noindent
Using any of the reetrant functions of this family now will
automatically initialize the random number generator to the default
values for the state and the parameters of the congruential formula.

The other possibility is too use any of the functions which explicitely
initialize the buffer.  Though it might be obvious how to initialize the
buffer from the data given as parameter from the function it is highly
recommended to use these functions since the result might not always be
what you expect.

@comment stdlib.h
@comment GNU
@deftypefun int srand48_r (long int @var{seedval}, struct drand48_data *@var{buffer})
The description of the random number generator represented by the
information in @var{buffer} is initialized similar to what the function
@code{srand48} does.  The state is initialized from the paramter
@var{seedval} and the paameters for the congruential formula are
initialized to the default values.

If the return value is non-negative the function call succeeded.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int seed48_r (unsigned short int @var{seed16v}[3], struct drand48_data *@var{buffer})
This function is similar to @code{srand48_r} but like @code{seed48} it
initializes all 48 bits of the state from the parameter @var{seed16v}.

If the return value is non-negative the function call succeeded.  It
does not return a pointer to the previous state of the random number
generator like the @code{seed48} function does.  if the user wants to
preserve the state for a later rerun s/he can copy the whole buffer
pointed to by @var{buffer}.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun

@comment stdlib.h
@comment GNU
@deftypefun int lcong48_r (unsigned short int @var{param}[7], struct drand48_data *@var{buffer})
This function initializes all aspects of the random number generator
described in @var{buffer} by the data in @var{param}.  Here it is
especially true the function does more than just copying the contents of
@var{param} of @var{buffer}.  Some more actions are required and
therefore it is important to use this function and not initialized the
random number generator directly.

If the return value is non-negative the function call succeeded.

This function is a GNU extension and should not be used in portable
programs.
@end deftypefun