506 lines
18 KiB
Plaintext
506 lines
18 KiB
Plaintext
@node Mathematics, Arithmetic, Low-Level Terminal Interface, Top
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@chapter Mathematics
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This chapter contains information about functions for performing
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mathematical computations, such as trigonometric functions. Most of
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these functions have prototypes declared in the header file
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@file{math.h}.
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@pindex math.h
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All of the functions that operate on floating-point numbers accept
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arguments and return results of type @code{double}. In the future,
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there may be additional functions that operate on @code{float} and
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@code{long double} values. For example, @code{cosf} and @code{cosl}
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would be versions of the @code{cos} function that operate on
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@code{float} and @code{long double} arguments, respectively. In the
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meantime, you should avoid using these names yourself. @xref{Reserved
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Names}.
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@menu
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* Domain and Range Errors:: Detecting overflow conditions and the like.
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* Trig Functions:: Sine, cosine, and tangent.
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* Inverse Trig Functions:: Arc sine, arc cosine, and arc tangent.
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* Exponents and Logarithms:: Also includes square root.
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* Hyperbolic Functions:: Hyperbolic sine and friends.
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* Pseudo-Random Numbers:: Functions for generating pseudo-random
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numbers.
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@end menu
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@node Domain and Range Errors
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@section Domain and Range Errors
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@cindex domain error
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Many of the functions listed in this chapter are defined mathematically
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over a domain that is only a subset of real numbers. For example, the
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@code{acos} function is defined over the domain between @code{-1} and
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@code{1}. If you pass an argument to one of these functions that is
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outside the domain over which it is defined, the function sets
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@code{errno} to @code{EDOM} to indicate a @dfn{domain error}. On
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machines that support IEEE floating point, functions reporting error
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@code{EDOM} also return a NaN.
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Some of these functions are defined mathematically to result in a
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complex value over parts of their domains. The most familiar example of
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this is taking the square root of a negative number. The functions in
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this chapter take only real arguments and return only real values;
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therefore, if the value ought to be nonreal, this is treated as a domain
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error.
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@cindex range error
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A related problem is that the mathematical result of a function may not
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be representable as a floating point number. If magnitude of the
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correct result is too large to be represented, the function sets
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@code{errno} to @code{ERANGE} to indicate a @dfn{range error}, and
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returns a particular very large value (named by the macro
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@code{HUGE_VAL}) or its negation (@w{@code{- HUGE_VAL}}).
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If the magnitude of the result is too small, a value of zero is returned
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instead. In this case, @code{errno} might or might not be
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set to @code{ERANGE}.
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The only completely reliable way to check for domain and range errors is
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to set @code{errno} to @code{0} before you call the mathematical function
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and test @code{errno} afterward. As a consequence of this use of
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@code{errno}, use of the mathematical functions is not reentrant if you
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check for errors.
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@c !!! this isn't always true at the moment....
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None of the mathematical functions ever generates signals as a result of
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domain or range errors. In particular, this means that you won't see
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@code{SIGFPE} signals generated within these functions. (@xref{Signal
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Handling}, for more information about signals.)
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@comment math.h
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@comment ANSI
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@deftypevr Macro double HUGE_VAL
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An expression representing a particular very large number. On machines
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that use IEEE floating point format, the value is ``infinity''. On
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other machines, it's typically the largest positive number that can be
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represented.
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The value of this macro is used as the return value from various
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mathematical functions in overflow situations.
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@end deftypevr
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For more information about floating-point representations and limits,
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see @ref{Floating Point Parameters}. In particular, the macro
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@code{DBL_MAX} might be more appropriate than @code{HUGE_VAL} for many
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uses other than testing for an error in a mathematical function.
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@node Trig Functions
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@section Trigonometric Functions
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@cindex trigonometric functions
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These are the familiar @code{sin}, @code{cos}, and @code{tan} functions.
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The arguments to all of these functions are in units of radians; recall
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that pi radians equals 180 degrees.
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@cindex pi (trigonometric constant)
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The math library doesn't define a symbolic constant for pi, but you can
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define your own if you need one:
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@smallexample
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#define PI 3.14159265358979323846264338327
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@end smallexample
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@noindent
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You can also compute the value of pi with the expression @code{acos
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(-1.0)}.
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@comment math.h
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@comment ANSI
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@deftypefun double sin (double @var{x})
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This function returns the sine of @var{x}, where @var{x} is given in
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radians. The return value is in the range @code{-1} to @code{1}.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double cos (double @var{x})
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This function returns the cosine of @var{x}, where @var{x} is given in
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radians. The return value is in the range @code{-1} to @code{1}.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double tan (double @var{x})
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This function returns the tangent of @var{x}, where @var{x} is given in
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radians.
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The following @code{errno} error conditions are defined for this function:
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@table @code
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@item ERANGE
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Mathematically, the tangent function has singularities at odd multiples
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of pi/2. If the argument @var{x} is too close to one of these
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singularities, @code{tan} sets @code{errno} to @code{ERANGE} and returns
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either positive or negative @code{HUGE_VAL}.
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@end table
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@end deftypefun
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@node Inverse Trig Functions
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@section Inverse Trigonometric Functions
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@cindex inverse trigonmetric functions
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These are the usual arc sine, arc cosine and arc tangent functions,
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which are the inverses of the sine, cosine and tangent functions,
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respectively.
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@comment math.h
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@comment ANSI
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@deftypefun double asin (double @var{x})
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This function computes the arc sine of @var{x}---that is, the value whose
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sine is @var{x}. The value is in units of radians. Mathematically,
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there are infinitely many such values; the one actually returned is the
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one between @code{-pi/2} and @code{pi/2} (inclusive).
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@code{asin} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is
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out of range. The arc sine function is defined mathematically only
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over the domain @code{-1} to @code{1}.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double acos (double @var{x})
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This function computes the arc cosine of @var{x}---that is, the value
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whose cosine is @var{x}. The value is in units of radians.
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Mathematically, there are infinitely many such values; the one actually
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returned is the one between @code{0} and @code{pi} (inclusive).
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@code{acos} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is
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out of range. The arc cosine function is defined mathematically only
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over the domain @code{-1} to @code{1}.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double atan (double @var{x})
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This function computes the arc tangent of @var{x}---that is, the value
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whose tangent is @var{x}. The value is in units of radians.
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Mathematically, there are infinitely many such values; the one actually
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returned is the one between @code{-pi/2} and @code{pi/2}
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(inclusive).
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double atan2 (double @var{y}, double @var{x})
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This is the two argument arc tangent function. It is similar to computing
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the arc tangent of @var{y}/@var{x}, except that the signs of both arguments
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are used to determine the quadrant of the result, and @var{x} is
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permitted to be zero. The return value is given in radians and is in
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the range @code{-pi} to @code{pi}, inclusive.
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If @var{x} and @var{y} are coordinates of a point in the plane,
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@code{atan2} returns the signed angle between the line from the origin
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to that point and the x-axis. Thus, @code{atan2} is useful for
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converting Cartesian coordinates to polar coordinates. (To compute the
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radial coordinate, use @code{hypot}; see @ref{Exponents and
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Logarithms}.)
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The function @code{atan2} sets @code{errno} to @code{EDOM} if both
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@var{x} and @var{y} are zero; the return value is not defined in this
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case.
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@end deftypefun
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@node Exponents and Logarithms
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@section Exponentiation and Logarithms
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@cindex exponentiation functions
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@cindex power functions
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@cindex logarithm functions
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@comment math.h
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@comment ANSI
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@deftypefun double exp (double @var{x})
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The @code{exp} function returns the value of e (the base of natural
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logarithms) raised to power @var{x}.
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The function fails, and sets @code{errno} to @code{ERANGE}, if the
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magnitude of the result is too large to be representable.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double log (double @var{x})
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This function returns the natural logarithm of @var{x}. @code{exp (log
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(@var{x}))} equals @var{x}, exactly in mathematics and approximately in
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C.
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The following @code{errno} error conditions are defined for this function:
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@table @code
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@item EDOM
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The argument @var{x} is negative. The log function is defined
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mathematically to return a real result only on positive arguments.
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@item ERANGE
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The argument is zero. The log of zero is not defined.
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@end table
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double log10 (double @var{x})
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This function returns the base-10 logarithm of @var{x}. Except for the
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different base, it is similar to the @code{log} function. In fact,
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@code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double pow (double @var{base}, double @var{power})
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This is a general exponentiation function, returning @var{base} raised
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to @var{power}.
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@need 250
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The following @code{errno} error conditions are defined for this function:
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@table @code
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@item EDOM
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The argument @var{base} is negative and @var{power} is not an integral
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value. Mathematically, the result would be a complex number in this case.
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@item ERANGE
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An underflow or overflow condition was detected in the result.
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@end table
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@end deftypefun
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@cindex square root function
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@comment math.h
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@comment ANSI
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@deftypefun double sqrt (double @var{x})
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This function returns the nonnegative square root of @var{x}.
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The @code{sqrt} function fails, and sets @code{errno} to @code{EDOM}, if
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@var{x} is negative. Mathematically, the square root would be a complex
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number.
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@end deftypefun
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@cindex cube root function
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@comment math.h
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@comment BSD
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@deftypefun double cbrt (double @var{x})
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This function returns the cube root of @var{x}. This function cannot
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fail; every representable real value has a representable real cube root.
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@end deftypefun
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@comment math.h
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@comment BSD
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@deftypefun double hypot (double @var{x}, double @var{y})
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The @code{hypot} function returns @code{sqrt (@var{x}*@var{x} +
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@var{y}*@var{y})}. (This is the length of the hypotenuse of a right
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triangle with sides of length @var{x} and @var{y}, or the distance
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of the point (@var{x}, @var{y}) from the origin.) See also the function
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@code{cabs} in @ref{Absolute Value}.
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@end deftypefun
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@comment math.h
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@comment BSD
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@deftypefun double expm1 (double @var{x})
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This function returns a value equivalent to @code{exp (@var{x}) - 1}.
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It is computed in a way that is accurate even if the value of @var{x} is
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near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate due
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to subtraction of two numbers that are nearly equal.
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@end deftypefun
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@comment math.h
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@comment BSD
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@deftypefun double log1p (double @var{x})
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This function returns a value equivalent to @w{@code{log (1 + @var{x})}}.
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It is computed in a way that is accurate even if the value of @var{x} is
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near zero.
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@end deftypefun
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@node Hyperbolic Functions
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@section Hyperbolic Functions
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@cindex hyperbolic functions
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The functions in this section are related to the exponential functions;
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see @ref{Exponents and Logarithms}.
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@comment math.h
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@comment ANSI
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@deftypefun double sinh (double @var{x})
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The @code{sinh} function returns the hyperbolic sine of @var{x}, defined
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mathematically as @w{@code{exp (@var{x}) - exp (-@var{x}) / 2}}. The
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function fails, and sets @code{errno} to @code{ERANGE}, if the value of
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@var{x} is too large; that is, if overflow occurs.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double cosh (double @var{x})
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The @code{cosh} function returns the hyperbolic cosine of @var{x},
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defined mathematically as @w{@code{exp (@var{x}) + exp (-@var{x}) / 2}}.
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The function fails, and sets @code{errno} to @code{ERANGE}, if the value
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of @var{x} is too large; that is, if overflow occurs.
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@end deftypefun
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@comment math.h
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@comment ANSI
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@deftypefun double tanh (double @var{x})
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This function returns the hyperbolic tangent of @var{x}, whose
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mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
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@end deftypefun
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@cindex inverse hyperbolic functions
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@comment math.h
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@comment BSD
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@deftypefun double asinh (double @var{x})
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This function returns the inverse hyperbolic sine of @var{x}---the
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value whose hyperbolic sine is @var{x}.
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@end deftypefun
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@comment math.h
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@comment BSD
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@deftypefun double acosh (double @var{x})
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This function returns the inverse hyperbolic cosine of @var{x}---the
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value whose hyperbolic cosine is @var{x}. If @var{x} is less than
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@code{1}, @code{acosh} returns @code{HUGE_VAL}.
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@end deftypefun
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@comment math.h
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@comment BSD
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@deftypefun double atanh (double @var{x})
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This function returns the inverse hyperbolic tangent of @var{x}---the
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value whose hyperbolic tangent is @var{x}. If the absolute value of
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@var{x} is greater than or equal to @code{1}, @code{atanh} returns
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@code{HUGE_VAL}.
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@end deftypefun
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@node Pseudo-Random Numbers
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@section Pseudo-Random Numbers
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@cindex random numbers
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@cindex pseudo-random numbers
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@cindex seed (for random numbers)
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This section describes the GNU facilities for generating a series of
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pseudo-random numbers. The numbers generated are not truly random;
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typically, they form a sequence that repeats periodically, with a
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period so large that you can ignore it for ordinary purposes. The
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random number generator works by remembering at all times a @dfn{seed}
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value which it uses to compute the next random number and also to
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compute a new seed.
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Although the generated numbers look unpredictable within one run of a
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program, the sequence of numbers is @emph{exactly the same} from one run
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to the next. This is because the initial seed is always the same. This
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is convenient when you are debugging a program, but it is unhelpful if
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you want the program to behave unpredictably. If you want truly random
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numbers, not just pseudo-random, specify a seed based on the current
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time.
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You can get repeatable sequences of numbers on a particular machine type
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by specifying the same initial seed value for the random number
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generator. There is no standard meaning for a particular seed value;
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the same seed, used in different C libraries or on different CPU types,
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will give you different random numbers.
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The GNU library supports the standard ANSI C random number functions
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plus another set derived from BSD. We recommend you use the standard
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ones, @code{rand} and @code{srand}.
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@menu
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* ANSI Random:: @code{rand} and friends.
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* BSD Random:: @code{random} and friends.
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@end menu
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@node ANSI Random
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@subsection ANSI C Random Number Functions
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This section describes the random number functions that are part of
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the ANSI C standard.
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To use these facilities, you should include the header file
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@file{stdlib.h} in your program.
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@pindex stdlib.h
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@comment stdlib.h
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@comment ANSI
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@deftypevr Macro int RAND_MAX
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The value of this macro is an integer constant expression that
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represents the maximum possible value returned by the @code{rand}
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function. In the GNU library, it is @code{037777777}, which is the
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largest signed integer representable in 32 bits. In other libraries, it
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may be as low as @code{32767}.
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@end deftypevr
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@comment stdlib.h
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@comment ANSI
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@deftypefun int rand ()
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The @code{rand} function returns the next pseudo-random number in the
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series. The value is in the range from @code{0} to @code{RAND_MAX}.
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@end deftypefun
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@comment stdlib.h
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@comment ANSI
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@deftypefun void srand (unsigned int @var{seed})
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This function establishes @var{seed} as the seed for a new series of
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pseudo-random numbers. If you call @code{rand} before a seed has been
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established with @code{srand}, it uses the value @code{1} as a default
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seed.
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To produce truly random numbers (not just pseudo-random), do @code{srand
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(time (0))}.
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@end deftypefun
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@node BSD Random
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@subsection BSD Random Number Functions
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This section describes a set of random number generation functions that
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are derived from BSD. There is no advantage to using these functions
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with the GNU C library; we support them for BSD compatibility only.
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The prototypes for these functions are in @file{stdlib.h}.
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@pindex stdlib.h
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@comment stdlib.h
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@comment BSD
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@deftypefun {long int} random ()
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This function returns the next pseudo-random number in the sequence.
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The range of values returned is from @code{0} to @code{RAND_MAX}.
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@end deftypefun
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@comment stdlib.h
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@comment BSD
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@deftypefun void srandom (unsigned int @var{seed})
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The @code{srandom} function sets the seed for the current random number
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state based on the integer @var{seed}. If you supply a @var{seed} value
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of @code{1}, this will cause @code{random} to reproduce the default set
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of random numbers.
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To produce truly random numbers (not just pseudo-random), do
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@code{srandom (time (0))}.
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@end deftypefun
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@comment stdlib.h
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@comment BSD
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@deftypefun {void *} initstate (unsigned int @var{seed}, void *@var{state}, size_t @var{size})
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The @code{initstate} function is used to initialize the random number
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generator state. The argument @var{state} is an array of @var{size}
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bytes, used to hold the state information. The size must be at least 8
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bytes, and optimal sizes are 8, 16, 32, 64, 128, and 256. The bigger
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the @var{state} array, the better.
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The return value is the previous value of the state information array.
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You can use this value later as an argument to @code{setstate} to
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restore that state.
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@end deftypefun
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@comment stdlib.h
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@comment BSD
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@deftypefun {void *} setstate (void *@var{state})
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The @code{setstate} function restores the random number state
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information @var{state}. The argument must have been the result of
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a previous call to @var{initstate} or @var{setstate}.
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The return value is the previous value of the state information array.
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You can use thise value later as an argument to @code{setstate} to
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restore that state.
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@end deftypefun
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