path: root/manual/math.texi

@c We need some definitions here.
@ifclear mult
@ifhtml
@set mult ·
@set infty ∞
@set pie π
@end ifhtml
@iftex
@set mult @cdot
@set infty @infty
@end iftex
@ifclear mult
@set mult *
@set infty oo
@set pie pi
@end ifclear
@macro mul
@value{mult}
@end macro
@macro infinity
@value{infty}
@end macro
@ifnottex
@macro pi
@value{pie}
@end macro
@end ifnottex
@end ifclear

@node Mathematics, Arithmetic, Low-Level Terminal Interface, Top
@c %MENU% Math functions, useful constants, random numbers
@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}.  The complex-valued functions are defined in
@file{complex.h}.
@pindex math.h
@pindex complex.h

All mathematical functions which take a floating-point argument
have three variants, one each for @code{double}, @code{float}, and
@code{long double} arguments.  The @code{double} versions are mostly
defined in @w{ISO C 89}.  The @code{float} and @code{long double}
versions are from the numeric extensions to C included in @w{ISO C 9X}.

Which of the three versions of a function should be used depends on the
situation.  For most calculations, the @code{float} functions are the
fastest.  On the other hand, the @code{long double} functions have the
highest precision.  @code{double} is somewhere in between.  It is
usually wise to pick the narrowest type that can accomodate your data.
Not all machines have a distinct @code{long double} type; it may be the
same as @code{double}.

@menu
* Mathematical Constants::      Precise numeric values for often-used
                                 constants.
* Trig Functions::              Sine, cosine, tangent, and friends.
* Inverse Trig Functions::      Arcsine, arccosine, etc.
* Exponents and Logarithms::    Also pow and sqrt.
* Hyperbolic Functions::        sinh, cosh, tanh, etc.
* Special Functions::           Bessel, gamma, erf.
* Pseudo-Random Numbers::       Functions for generating pseudo-random
				 numbers.
* FP Function Optimizations::   Fast code or small code.
@end menu

@node Mathematical Constants
@section Predefined Mathematical Constants
@cindex constants
@cindex mathematical constants

The header @file{math.h} defines several useful mathematical constants.
All values are defined as preprocessor macros starting with @code{M_}.
The values provided are:

@vtable @code
@item M_E
The base of natural logarithms.
@item M_LOG2E
The logarithm to base @code{2} of @code{M_E}.
@item M_LOG10E
The logarithm to base @code{10} of @code{M_E}.
@item M_LN2
The natural logarithm of @code{2}.
@item M_LN10
The natural logarithm of @code{10}.
@item M_PI
Pi, the ratio of a circle's circumrefence to its diameter.
@item M_PI_2
Pi divided by two.
@item M_PI_4
Pi divided by four.
@item M_1_PI
The reciprocal of pi (1/pi)
@item M_2_PI
Two times the reciprocal of pi.
@item M_2_SQRTPI
Two times the reciprocal of the square root of pi.
@item M_SQRT2
The square root of two.
@item M_SQRT1_2
The reciprocal of the square root of two (also the square root of 1/2).
@end vtable

These constants come from the Unix98 standard and were also available in
4.4BSD; therefore, they are only defined if @code{_BSD_SOURCE} or
@code{_XOPEN_SOURCE=500}, or a more general feature select macro, is
defined.  The default set of features includes these constants.
@xref{Feature Test Macros}.

All values are of type @code{double}.  As an extension, the GNU C
library also defines these constants with type @code{long double}.  The
@code{long double} macros have a lowercase @samp{l} appended to their
names: @code{M_El}, @code{M_PIl}, and so forth.  These are only
available if @code{_GNU_SOURCE} is defined.

@vindex PI
@emph{Note:} Some programs use a constant named @code{PI} which has the
same value as @code{M_PI}.  This constant is not standard; it may have
appeared in some old AT&T headers, and is mentioned in Stroustrup's book
on C++.  It infringes on the user's name space, so the GNU C library
does not define it.  Fixing programs written to expect it is simple:
replace @code{PI} with @code{M_PI} throughout, or put @samp{-DPI=M_PI}
on the compiler command line.

@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 normally defines @code{M_PI} to a @code{double}
approximation of pi.  If strict ISO and/or POSIX compliance
are requested this constant is not defined, but you can easily define it
yourself:

@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})
@deftypefunx float sinf (float @var{x})
@deftypefunx {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})
@deftypefunx float cosf (float @var{x})
@deftypefunx {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})
@deftypefunx float tanf (float @var{x})
@deftypefunx {long double} tanl (long double @var{x})
These functions return the tangent of @var{x}, where @var{x} is given in
radians.

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} will signal overflow.
@end deftypefun

In many applications where @code{sin} and @code{cos} are used, the sine
and cosine of the same angle are needed at the same time.  It is more
efficient to compute them simultaneously, so the library provides a
function to do that.

@comment math.h
@comment GNU
@deftypefun void sincos (double @var{x}, double *@var{sinx}, double *@var{cosx})
@deftypefunx void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx})
@deftypefunx 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}.

This function is a GNU extension.  Portable programs should be prepared
to cope with its absence.
@end deftypefun

@cindex complex trigonometric functions

@w{ISO C 9x} defines variants of the trig functions which work on
complex numbers.  The GNU C library provides these functions, but they
are only useful if your compiler supports the new complex types defined
by the standard.
@c Change this when gcc is fixed. -zw
(As of this writing GCC supports complex numbers, but there are bugs in
the implementation.)

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

@ifinfo
@math{sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i))}.
@end ifinfo
@tex
$$\sin(z) = {1\over 2i} (e^{zi} - e^{-zi})$$
@end tex
@end deftypefun

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

@ifinfo
@math{cos (z) = 1/2 * (exp (z*i) + exp (-z*i))}
@end ifinfo
@tex
$$\cos(z) = {1\over 2} (e^{zi} + e^{-zi})$$
@end tex
@end deftypefun

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

@ifinfo
@math{tan (z) = -i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))}
@end ifinfo
@tex
$$\tan(z) = -i \cdot {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$
@end tex

@noindent
The complex tangent has poles at @math{pi/2 + 2n}, where @math{n} is an
integer.  @code{ctan} may signal overflow if @var{z} is too close to a
pole.
@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})
@deftypefunx float asinf (float @var{x})
@deftypefunx {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).

The arc sine function is defined mathematically only
over the domain @code{-1} to @code{1}.  If @var{x} is outside the
domain, @code{asin} signals a domain error.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double acos (double @var{x})
@deftypefunx float acosf (float @var{x})
@deftypefunx {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).

The arc cosine function is defined mathematically only
over the domain @code{-1} to @code{1}.  If @var{x} is outside the
domain, @code{acos} signals a domain error.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double atan (double @var{x})
@deftypefunx float atanf (float @var{x})
@deftypefunx {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})
@deftypefunx float atan2f (float @var{y}, float @var{x})
@deftypefunx {long double} atan2l (long double @var{y}, long double @var{x})
This function computes the arc tangent of @var{y}/@var{x}, but 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}.)

@c This is experimentally true.  Should it be so? -zw
If both @var{x} and @var{y} are zero, @code{atan2} returns zero.
@end deftypefun

@cindex inverse complex trigonometric functions
@w{ISO C 9x} defines complex versions of the inverse trig functions.

@comment complex.h
@comment ISO
@deftypefun {complex double} casin (complex double @var{z})
@deftypefunx {complex float} casinf (complex float @var{z})
@deftypefunx {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 returned is in radians.

Unlike the real-valued functions, @code{casin} is defined for all
values of @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cacos (complex double @var{z})
@deftypefunx {complex float} cacosf (complex float @var{z})
@deftypefunx {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 returned is in radians.

Unlike the real-valued functions, @code{cacos} is defined for all
values of @var{z}.
@end deftypefun


@comment complex.h
@comment ISO
@deftypefun {complex double} catan (complex double @var{z})
@deftypefunx {complex float} catanf (complex float @var{z})
@deftypefunx {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})
@deftypefunx float expf (float @var{x})
@deftypefunx {long double} expl (long double @var{x})
These functions compute @code{e} (the base of natural logarithms) raised
to the power @var{x}.

If the magnitude of the result is too large to be representable,
@code{exp} signals overflow.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double exp2 (double @var{x})
@deftypefunx float exp2f (float @var{x})
@deftypefunx {long double} exp2l (long double @var{x})
These functions compute @code{2} raised to the power @var{x}.
Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}.
@end deftypefun

@comment math.h
@comment GNU
@deftypefun double exp10 (double @var{x})
@deftypefunx float exp10f (float @var{x})
@deftypefunx {long double} exp10l (long double @var{x})
@deftypefunx double pow10 (double @var{x})
@deftypefunx float pow10f (float @var{x})
@deftypefunx {long double} pow10l (long double @var{x})
These functions compute @code{10} raised to the power @var{x}.
Mathematically, @code{exp10 (x)} is the same as @code{exp (x * log (10))}.

These functions are GNU extensions.  The name @code{exp10} is
preferred, since it is analogous to @code{exp} and @code{exp2}.
@end deftypefun


@comment math.h
@comment ISO
@deftypefun double log (double @var{x})
@deftypefunx float logf (float @var{x})
@deftypefunx {long double} logl (long double @var{x})
These functions compute the natural logarithm of @var{x}.  @code{exp (log
(@var{x}))} equals @var{x}, exactly in mathematics and approximately in
C.

If @var{x} is negative, @code{log} signals a domain error.  If @var{x}
is zero, it returns negative infinity; if @var{x} is too close to zero,
it may signal overflow.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double log10 (double @var{x})
@deftypefunx float log10f (float @var{x})
@deftypefunx {long double} log10l (long double @var{x})
These functions return the base-10 logarithm of @var{x}.
@code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}.

@end deftypefun

@comment math.h
@comment ISO
@deftypefun double log2 (double @var{x})
@deftypefunx float log2f (float @var{x})
@deftypefunx {long double} log2l (long double @var{x})
These functions return the base-2 logarithm of @var{x}.
@code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double logb (double @var{x})
@deftypefunx float logbf (float @var{x})
@deftypefunx {long double} logbl (long double @var{x})
These functions extract the exponent of @var{x} and return it as a
floating-point value.  If @code{FLT_RADIX} is two, @code{logb} is equal
to @code{floor (log2 (x))}, except it's probably faster.

If @var{x} is denormalized, @code{logb} returns the exponent @var{x}
would have if it were normalized.  If @var{x} is infinity (positive or
negative), @code{logb} returns @math{@infinity{}}.  If @var{x} is zero,
@code{logb} returns @math{@infinity{}}.  It does not signal.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun int ilogb (double @var{x})
@deftypefunx int ilogbf (float @var{x})
@deftypefunx int ilogbl (long double @var{x})
These functions are equivalent to the corresponding @code{logb}
functions except that they return signed integer values.
@end deftypefun

@noindent
Since integers cannot represent infinity and NaN, @code{ilogb} instead
returns an integer that can't be the exponent of a normal floating-point
number.  @file{math.h} defines constants so you can check for this.

@comment math.h
@comment ISO
@deftypevr Macro int FP_ILOGB0
@code{ilogb} returns this value if its argument is @code{0}.  The
numeric value is either @code{INT_MIN} or @code{-INT_MAX}.

This macro is defined in @w{ISO C 9X}.
@end deftypevr

@comment math.h
@comment ISO
@deftypevr Macro int FP_ILOGBNAN
@code{ilogb} returns this value if its argument is @code{NaN}.  The
numeric value is either @code{INT_MIN} or @code{INT_MAX}.

This macro is defined in @w{ISO C 9X}.
@end deftypevr

These values are system specific.  They might even be the same.  The
proper way to test the result of @code{ilogb} is as follows:

@smallexample
i = ilogb (f);
if (i == FP_ILOGB0 || i == FP_ILOGBNAN)
  @{
    if (isnan (f))
      @{
        /* @r{Handle NaN.}  */
      @}
    else if (f  == 0.0)
      @{
        /* @r{Handle 0.0.}  */
      @}
    else
      @{
        /* @r{Some other value with large exponent,}
           @r{perhaps +Inf.}  */
      @}
  @}
@end smallexample

@comment math.h
@comment ISO
@deftypefun double pow (double @var{base}, double @var{power})
@deftypefunx float powf (float @var{base}, float @var{power})
@deftypefunx {long double} powl (long double @var{base}, long double @var{power})
These are general exponentiation functions, returning @var{base} raised
to @var{power}.

Mathematically, @code{pow} would return a complex number when @var{base}
is negative and @var{power} is not an integral value.  @code{pow} can't
do that, so instead it signals a domain error. @code{pow} may also
underflow or overflow the destination type.
@end deftypefun

@cindex square root function
@comment math.h
@comment ISO
@deftypefun double sqrt (double @var{x})
@deftypefunx float sqrtf (float @var{x})
@deftypefunx {long double} sqrtl (long double @var{x})
These functions return the nonnegative square root of @var{x}.

If @var{x} is negative, @code{sqrt} signals a domain error.
Mathematically, it should return a complex number.
@end deftypefun

@cindex cube root function
@comment math.h
@comment BSD
@deftypefun double cbrt (double @var{x})
@deftypefunx float cbrtf (float @var{x})
@deftypefunx {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})
@deftypefunx float hypotf (float @var{x}, float @var{y})
@deftypefunx {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 wise, 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})
@deftypefunx float expm1f (float @var{x})
@deftypefunx {long double} expm1l (long double @var{x})
These functions return a value equivalent to @code{exp (@var{x}) - 1}.
They are computed in a way that is accurate even if @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})
@deftypefunx float log1pf (float @var{x})
@deftypefunx {long double} log1pl (long double @var{x})
These functions returns a value equivalent to @w{@code{log (1 + @var{x})}}.
They are computed in a way that is accurate even if @var{x} is
near zero.
@end deftypefun

@cindex complex exponentiation functions
@cindex complex logarithm functions

@w{ISO C 9X} defines complex variants of some of the exponentiation and
logarithm functions.

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

@ifinfo
@math{exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))}
@end ifinfo
@tex
$$\exp(z) = e^z = e^{{\rm Re}\,z} (\cos ({\rm Im}\,z) + i \sin ({\rm Im}\,z))$$
@end tex
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} clog (complex double @var{z})
@deftypefunx {complex float} clogf (complex float @var{z})
@deftypefunx {complex long double} clogl (complex long double @var{z})
These functions return the natural logarithm of @var{z}.
Mathematically this corresponds to the value

@ifinfo
@math{log (z) = log (cabs (z)) + I * carg (z)}
@end ifinfo
@tex
$$\log(z) = \log |z| + i \arg z$$
@end tex

@noindent
@code{clog} has a pole at 0, and will signal overflow if @var{z} equals
or is very close to 0.  It is well-defined for all other values of
@var{z}.
@end deftypefun


@comment complex.h
@comment GNU
@deftypefun {complex double} clog10 (complex double @var{z})
@deftypefunx {complex float} clog10f (complex float @var{z})
@deftypefunx {complex long double} clog10l (complex long double @var{z})
These functions return the base 10 logarithm of the complex value
@var{z}. Mathematically this corresponds to the value

@ifinfo
@math{log (z) = log10 (cabs (z)) + I * carg (z)}
@end ifinfo
@tex
$$\log_{10}(z) = \log_{10}|z| + i \arg z$$
@end tex

These functions are GNU extensions.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} csqrt (complex double @var{z})
@deftypefunx {complex float} csqrtf (complex float @var{z})
@deftypefunx {complex long double} csqrtl (complex long double @var{z})
These functions return the complex square root of the argument @var{z}.  Unlike
the real-valued functions, they are defined for all values of @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power})
@deftypefunx {complex float} cpowf (complex float @var{base}, complex float @var{power})
@deftypefunx {complex long double} cpowl (complex long double @var{base}, complex long double @var{power})
These functions return @var{base} raised to the power of
@var{power}.  This is equivalent to @w{@code{cexp (y * clog (x))}}
@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})
@deftypefunx float sinhf (float @var{x})
@deftypefunx {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}}.  They
may signal overflow if @var{x} is too large.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double cosh (double @var{x})
@deftypefunx float coshf (float @var{x})
@deftypefunx {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}}.
They may signal overflow if @var{x} is too large.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double tanh (double @var{x})
@deftypefunx float tanhf (float @var{x})
@deftypefunx {long double} tanhl (long double @var{x})
These functions return the hyperbolic tangent of @var{x},
defined mathematically as @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
They may signal overflow if @var{x} is too large.
@end deftypefun

@cindex hyperbolic functions

There are counterparts for the hyperbolic functions which take
complex arguments.

@comment complex.h
@comment ISO
@deftypefun {complex double} csinh (complex double @var{z})
@deftypefunx {complex float} csinhf (complex float @var{z})
@deftypefunx {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}}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ccosh (complex double @var{z})
@deftypefunx {complex float} ccoshf (complex float @var{z})
@deftypefunx {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}}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} ctanh (complex double @var{z})
@deftypefunx {complex float} ctanhf (complex float @var{z})
@deftypefunx {complex long double} ctanhl (complex long double @var{z})
These functions return the complex hyperbolic tangent of @var{z},
defined mathematically as @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})
@deftypefunx float asinhf (float @var{x})
@deftypefunx {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})
@deftypefunx float acoshf (float @var{x})
@deftypefunx {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} signals a domain error.
@end deftypefun

@comment math.h
@comment ISO
@deftypefun double atanh (double @var{x})
@deftypefunx float atanhf (float @var{x})
@deftypefunx {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 @code{1}, @code{atanh} signals a domain error;
if it is equal to 1, @code{atanh} returns infinity.
@end deftypefun

@cindex inverse complex hyperbolic functions

@comment complex.h
@comment ISO
@deftypefun {complex double} casinh (complex double @var{z})
@deftypefunx {complex float} casinhf (complex float @var{z})
@deftypefunx {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})
@deftypefunx {complex float} cacoshf (complex float @var{z})
@deftypefunx {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 functions, there are no restrictions on the value of @var{z}.
@end deftypefun

@comment complex.h
@comment ISO
@deftypefun {complex double} catanh (complex double @var{z})
@deftypefunx {complex float} catanhf (complex float @var{z})
@deftypefunx {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 functions, there are no restrictions on the value of
@var{z}.
@end deftypefun

@node Special Functions
@section Special Functions
@cindex special functions
@cindex Bessel functions
@cindex gamma function

These are some more exotic mathematical functions, which are sometimes
useful.  Currently they only have real-valued versions.

@comment math.h
@comment SVID
@deftypefun double erf (double @var{x})
@deftypefunx float erff (float @var{x})
@deftypefunx {long double} erfl (long double @var{x})
@code{erf} returns the error function of @var{x}.  The error
function is defined as
@tex
$$\hbox{erf}(x) = {2\over\sqrt{\pi}}\cdot\int_0^x e^{-t^2} \hbox{d}t$$
@end tex
@ifnottex
@smallexample
erf (x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt
@end smallexample
@end ifnottex
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double erfc (double @var{x})
@deftypefunx float erfcf (float @var{x})
@deftypefunx {long double} erfcl (long double @var{x})
@code{erfc} returns @code{1.0 - erf(@var{x})}, but computed in a
fashion that avoids round-off error when @var{x} is large.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double lgamma (double @var{x})
@deftypefunx float lgammaf (float @var{x})
@deftypefunx {long double} lgammal (long double @var{x})
@code{lgamma} returns the natural logarithm of the absolute value of
the gamma function of @var{x}.  The gamma function is defined as
@tex
$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$
@end tex
@ifnottex
@smallexample
gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt
@end smallexample
@end ifnottex

@vindex signgam
The sign of the gamma function is stored in the global variable
@var{signgam}, which is declared in @file{math.h}.  It is @code{1} if
the intermediate result was positive or zero, and, @code{-1} if it was
negative.

To compute the real gamma function you can use the @code{tgamma}
function or you can compute the values as follows:
@smallexample
lgam = lgamma(x);
gam  = signgam*exp(lgam);
@end smallexample

The gamma function has singularities at the nonpositive integers.
@code{lgamma} will raise the zero divide exception if evaluated at a
singularity.
@end deftypefun

@comment math.h
@comment XPG
@deftypefun double lgamma_r (double @var{x}, int *@var{signp})
@deftypefunx float lgammaf_r (float @var{x}, int *@var{signp})
@deftypefunx {long double} lgammal_r (long double @var{x}, int *@var{signp})
@code{lgamma_r} is just like @code{lgamma}, but it stores the sign of
the intermediate result in the variable pointed to by @var{signp}
instead of in the @var{signgam} global.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double gamma (double @var{x})
@deftypefunx float gammaf (float @var{x})
@deftypefunx {long double} gammal (long double @var{x})
These functions exist for compatibility reasons.  They are equivalent to
@code{lgamma} etc.  It is better to use @code{lgamma} since for one the
name reflects better the actual computation and @code{lgamma} is also
standardized in @w{ISO C 9x} while @code{gamma} is not.
@end deftypefun

@comment math.h
@comment XPG
@deftypefun double tgamma (double @var{x})
@deftypefunx float tgammaf (float @var{x})
@deftypefunx {long double} tgammal (long double @var{x})
@code{tgamma} applies the gamma function to @var{x}.  The gamma
function is defined as
@tex
$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$
@end tex
@ifnottex
@smallexample
gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt
@end smallexample
@end ifnottex

This function was introduced in @w{ISO C 9x}.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double j0 (double @var{x})
@deftypefunx float j0f (float @var{x})
@deftypefunx {long double} j0l (long double @var{x})
@code{j0} returns the Bessel function of the first kind of order 0 of
@var{x}.  It may signal underflow if @var{x} is too large.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double j1 (double @var{x})
@deftypefunx float j1f (float @var{x})
@deftypefunx {long double} j1l (long double @var{x})
@code{j1} returns the Bessel function of the first kind of order 1 of
@var{x}.  It may signal underflow if @var{x} is too large.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double jn (int n, double @var{x})
@deftypefunx float jnf (int n, float @var{x})
@deftypefunx {long double} jnl (int n, long double @var{x})
@code{jn} returns the Bessel function of the first kind of order
@var{n} of @var{x}.  It may signal underflow if @var{x} is too large.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double y0 (double @var{x})
@deftypefunx float y0f (float @var{x})
@deftypefunx {long double} y0l (long double @var{x})
@code{y0} returns the Bessel function of the second kind of order 0 of
@var{x}.  It may signal underflow if @var{x} is too large.  If @var{x}
is negative, @code{y0} signals a domain error; if it is zero,
@code{y0} signals overflow and returns @math{-@infinity}.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double y1 (double @var{x})
@deftypefunx float y1f (float @var{x})
@deftypefunx {long double} y1l (long double @var{x})
@code{y1} returns the Bessel function of the second kind of order 1 of
@var{x}.  It may signal underflow if @var{x} is too large.  If @var{x}
is negative, @code{y1} signals a domain error; if it is zero,
@code{y1} signals overflow and returns @math{-@infinity}.
@end deftypefun

@comment math.h
@comment SVID
@deftypefun double yn (int n, double @var{x})
@deftypefunx float ynf (int n, float @var{x})
@deftypefunx {long double} ynl (int n, long double @var{x})
@code{yn} returns the Bessel function of the second kind of order @var{n} of
@var{x}.  It may signal underflow if @var{x} is too large.  If @var{x}
is negative, @code{yn} signals a domain error; if it is zero,
@code{yn} signals overflow and returns @math{-@infinity}.
@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 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 a different
pseudo-random series each time your program runs, you must specify a
different seed each time.  For ordinary purposes, basing the seed on the
current time works well.

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 two other sets derived from BSD and SVID.  The BSD and @w{ISO C}
functions provide identical, somewhat limited functionality.  If only a
small number of random bits are required, we recommend you use the
@w{ISO C} interface, @code{rand} and @code{srand}.  The SVID functions
provide a more flexible interface, which allows better random number
generator algorithms, provides more random bits (up to 48) per call, and
can provide random floating-point numbers.  These functions are required
by the XPG standard and therefore will be present in all modern Unix
systems.

@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 representing the largest
value the @code{rand} function can return.  In the GNU library, it is
@code{2147483647}, 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 (void)
The @code{rand} function returns the next pseudo-random number in the
series.  The value ranges 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 a different pseudo-random series each time your program is
run, do @code{srand (time (0))}.
@end deftypefun

POSIX.1 extended the C standard functions to support reproducible random
numbers in multi-threaded programs.  However, the extension is badly
designed and unsuitable for serious work.

@comment stdlib.h
@comment POSIX.1
@deftypefun int rand_r (unsigned int *@var{seed})
This function returns a random number in the range 0 to @code{RAND_MAX}
just as @code{rand} does.  However, all its state is stored in the
@var{seed} argument.  This means the RNG's state can only have as many
bits as the type @code{unsigned int} has.  This is far too few to
provide a good RNG.

If your program requires a reentrant RNG, we recommend you use the
reentrant GNU extensions to the SVID random number generator.  The
POSIX.1 interface should only be used when the GNU extensions are not
available.
@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 {int32_t} random (void)
This function returns the next pseudo-random number in the sequence.
The value returned ranges from @code{0} to @code{RAND_MAX}.

@strong{Note:} Historically this function returned a @code{long
int} value.  On 64bit systems @code{long int} would have been larger
than programs expected, so @code{random} is now defined to return
exactly 32 bits.
@end deftypefun

@comment stdlib.h
@comment BSD
@deftypefun void srandom (unsigned int @var{seed})
The @code{srandom} function sets the state of the random number
generator 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 a different set of pseudo-random numbers each time your
program runs, 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.  It is initialized based on
@var{seed}.  The size must be between 8 and 256 bytes, and should be a
power of two.  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 this 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 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 (void)
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 (void)
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 (void)
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 bits
are initialized 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 construct
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 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 individual 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 generator 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 reentrant 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 parameter
@var{seedval} and the parameters 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

@node FP Function Optimizations
@section Is Fast Code or Small Code preferred?
@cindex Optimization

If an application uses many floating point function it is often the case
that the costs for the function calls itselfs are not neglectable.
Modern processor implementation often can execute the operation itself
very fast but the call means a disturbance of the control flow.

For this reason the GNU C Library provides optimizations for many of the
frequently used math functions.  When the GNU CC is used and the user
activates the optimizer several new inline functions and macros get
defined.  These new functions and macros have the same names as the
library function and so get used instead of the later.  In case of
inline functions the compiler will decide whether it is reasonable to
use the inline function and this decision is usually correct.

For the generated code this means that no calls to the library functions
are necessary.  This increases the speed significantly.  But the
drawback is that the code size increases and this increase is not always
neglectable.

In cases where the inline functions and macros are not wanted the symbol
@code{__NO_MATH_INLINES} should be defined before any system header is
included.  This will make sure only library functions are used.  Of
course it can be determined for each single file in the project whether
giving this option is preferred or not.

Not all hardware implements the entire @w{IEEE 754} standard, or if it
does, there may be a substantial performance penalty for using some of
its features.  For example, enabling traps on some processors forces
the FPU to run unpipelined, which more than doubles calculation time.
@c ***Add explanation of -lieee, -mieee.