@c We need some definitions here. @ifhtml @set cdot À @end ifhtml @iftex @set cdot @cdot @end iftex @ifclear cdot @set cdot x @end ifclear @iftex @set infty @infty @end iftex @ifclear infty @set infty oo @end ifclear @macro mul @value{cdot} @end macro @macro infinity @value{infty} @end macro @node Mathematics, Arithmetic, Low-Level Terminal Interface, Top @chapter Mathematics This chapter contains information about functions for performing mathematical computations, such as trigonometric functions. Most of these functions have prototypes declared in the header file @file{math.h}. @pindex math.h For all functions which take a single floating-point argument and for several other functions as well there are three different functions available for the type @code{double}, @code{float}, and @code{long double}. The @code{double} versions of the functions are mostly defined even in the @w{ISO C 89} standard. The @code{float} and @code{long double} variants are introduced in the numeric extensions for the C language which are part of the @w{ISO C 9X} standard. Which of the three versions of the function should be used depends on the situation. For most functions and implementation it is true that speed and precision do not go together. I.e., the @code{float} versions are normally faster than the @code{double} and @code{long double} versions. On the other hand the @code{long double} version has the highest precision. One should always think about the actual needs and in case of double using @code{double} is a good compromise. @menu * Domain and Range Errors:: Detecting overflow conditions and the like. * Exceptions in Math Functions:: Signalling exception in math functions. * Mathematical Constants:: Precise numeric values for often used constant. * FP Comparison Functions:: Special functions to compare floating-point numbers. * Trig Functions:: Sine, cosine, and tangent. * Inverse Trig Functions:: Arc sine, arc cosine, and arc tangent. * Exponents and Logarithms:: Also includes square root. * Hyperbolic Functions:: Hyperbolic sine and friends. * Pseudo-Random Numbers:: Functions for generating pseudo-random numbers. @end menu @node Domain and Range Errors @section Domain and Range Errors @cindex domain error Many of the functions listed in this chapter are defined mathematically over a domain that is only a subset of real numbers. For example, the @code{acos} function is defined over the domain between @code{@minus{}1} and @code{1}. If you pass an argument to one of these functions that is outside the domain over which it is defined, the function sets @code{errno} to @code{EDOM} to indicate a @dfn{domain error}. On machines that support @w{IEEE 754} floating point, functions reporting error @code{EDOM} also return a NaN. Some of these functions are defined mathematically to result in a complex value over parts of their domains. The most familiar example of this is taking the square root of a negative number. The functions in this chapter take only real arguments and return only real values; therefore, if the value ought to be nonreal, this is treated as a domain error. @cindex range error A related problem is that the mathematical result of a function may not be representable as a floating point number. If magnitude of the correct result is too large to be represented, the function sets @code{errno} to @code{ERANGE} to indicate a @dfn{range error}, and returns a particular very large value (named by the macro @code{HUGE_VAL}) or its negation (@code{@minus{}HUGE_VAL}). If the magnitude of the result is too small, a value of zero is returned instead. In this case, @code{errno} might or might not be set to @code{ERANGE}. The only completely reliable way to check for domain and range errors is to set @code{errno} to @code{0} before you call the mathematical function and test @code{errno} afterward. As a consequence of this use of @code{errno}, use of the mathematical functions is not reentrant if you check for errors. @c ### This is no longer true. --drepper @c None of the mathematical functions ever generates signals as a result of @c domain or range errors. In particular, this means that you won't see @c @code{SIGFPE} signals generated within these functions. (@xref{Signal @c Handling}, for more information about signals.) @comment math.h @comment ISO @deftypevr Macro double HUGE_VAL An expression representing a particular very large number. On machines that use @w{IEEE 754}/@w{IEEE 854} floating point format, the value is infinity''. On other machines, it's typically the largest positive number that can be represented. The value of this macro is used as the return value from various mathematical @code{double} returning functions in overflow situations. @end deftypevr @comment math.h @comment ISO @deftypevr Macro float HUGE_VALF This macro is similar to the @code{HUGE_VAL} macro except that it is used by functions returning @code{float} values. This macro is introduced in @w{ISO C 9X}. @end deftypevr @comment math.h @comment ISO @deftypevr Macro {long double} HUGE_VALL This macro is similar to the @code{HUGE_VAL} macro except that it is used by functions returning @code{long double} values. The value is only different from @code{HUGE_VAL} if the architecture really supports @code{long double} values. This macro is introduced in @w{ISO C 9X}. @end deftypevr A special case is the @code{ilogb} function @pxref{Exponents and Logarithms}. Since the return value is an integer value, one cannot compare with @code{HUGE_VAL} etc. Therefore two further values are defined. @comment math.h @comment ISO @deftypevr Macro int FP_ILOGB0 This value is returned by @code{ilogb} if the argument is @code{0}. The numeric value is either @code{INT_MIN} or @code{-INT_MAX}. This macro is introduced in @w{ISO C 9X}. @end deftypevr @comment math.h @comment ISO @deftypevr Macro int FP_ILOGBNAN This value is returned by @code{ilogb} if the argument is @code{NaN}. The numeric value is either @code{INT_MIN} or @code{INT_MAX}. This macro is introduced in @w{ISO C 9X}. @end deftypevr For more information about floating-point representations and limits, see @ref{Floating Point Parameters}. In particular, the macro @code{DBL_MAX} might be more appropriate than @code{HUGE_VAL} for many uses other than testing for an error in a mathematical function. @node Exceptions in Math Functions @section Exceptions in Math Functions @cindex exception @cindex signal Due to the restrictions in the size of the floating-point number representation or the limitation of the input range of certain functions some of the mathematical operations and functions have to signal exceptional situations. The @w{IEEE 754} standard specifies which exceptions have to be supported and how they can be handled. @w{IEEE 754} specifies two actions for floating-point exception: taking a trap or continuing without doing so. If the trap is taken a (possibly) user defined trap handler is called and this function can correct the argument or react somehow else on the call. If the trap handler returns, its return value is taken as the result of the operation. If no trap handler is called each of the known exceptions has a default action. This consists of setting a corresponding bit in the floating-point status word to indicate which kind of exception was raised and to return a default value, which depends on the exception (see the table below). @noindent The exceptions defined in @w{IEEE 754} are: @table @samp @item Invalid Operation This exception is raised if the given operands are invalid for the operation to be performed. Examples are (see @w{IEEE 754}, @w{section 7}): @enumerate @item Any operation on a signalling NaN. @item Addition or subtraction; magnitude subtraction of infinities such as @math{(+@infinity{}) + (-@infinity{})}. @item Multiplication: @math{0 @mul{} @infinity{}}. @item Division: @math{0/0} or @math{@infinity{}/@infinity{}}. @item Remainder: @math{x} REM @math{y}, where @math{y} is zero or @math{x} is infinite. @item Square root if the operand is less then zero. @item Conversion of an internal floating-point number to an integer or to a decimal string when overflow, infinity, or NaN precludes a faithful representation in that format and this cannot otherwise be signaled. @item Conversion of an unrecognizable input string. @item Comparison via predicates involving @math{<} or @math{>}, without @code{?}, when the operands are @dfn{unordered}. (@math{?>} means the unordered greater relation, @xref{FP Comparison Functions}). @end enumerate If the exception does not cause a trap handler to be called the result of the operation is taken as a quiet NaN. @item Division by Zero This exception is raised if the devisor is zero and the dividend is a finite nonzero number. If no trap occurs the result is either @math{+@infinity{}} or @math{-@infinity{}}, depending on the signs of the operands. @item Overflow This exception is signalled whenever the result cannot be represented as a finite value in the precision format of the destination. If no trap occurs the result depends on the sign of the intermediate result and the current rounding mode (@w{IEEE 754}, @w{section 7.3}): @enumerate @item Round to nearest carries all overflows to @math{@infinity{}} with the sign of the intermediate result. @item Round toward @math{0} carries all overflows to the precision's largest finite number with the sign of the intermediate result. @item Round toward @math{-@infinity{}} carries positive overflows to the precision's largest finite number and carries negative overflows to @math{-@infinity{}}. @item Round toward @math{@infinity{}} carries negative overflows to the precision's most negative finite number and carries positive overflows to @math{@infinity{}}. @end enumerate @item Underflow The underflow exception is created when an intermediate result is too small for the operation or if the operations result rounded to the destination precision causes a loss of accuracy by approximating the result by denormalized numbers. When no trap is installed for the underflow exception, underflow shall be signaled (via the underflow flag) only when both tininess and loss of accuracy have been detected. If no trap handler is installed the operation continues with an inprecise small value or zero if the destination precision cannot hold the small exact result. @item Inexact This exception is signalled if the rounded result is not exact (such as computing the square root of two) or the result overflows without an overflow trap. @end table To control whether an exception causes a trap to occur all @w{IEEE 754} conformant floating-point implementations (either hardware or software) have a control word. By setting specific bits for each exception in this control word the programmer can decide whether a trap is wanted or not. @w{ISO C 9X} introduces a set of function which can be used to control exceptions. There are functions to manipulate the control word, to query the status word or to save and restore the whole state of the floating-point unit. There are also functions to control the rounding mode used. @menu * Status bit operations:: Manipulate the FP status word. * FPU environment:: Controlling the status of the FPU. * Rounding Modes:: Controlling the rounding mode. @end menu @node Status bit operations @subsection Controlling the FPU status word To control the five types of exceptions defined in @w{IEEE 754} some functions are defined which abstract the interface to the FPU. The actual implementation can be very different, depending on the underlying hardware or software. To address the single exception the @file{fenv.h} headers defines a number of macros: @vtable @code @comment fenv.h @comment ISO @item FE_INEXACT Represents the inexact exception iff the FPU supports this exception. @comment fenv.h @comment ISO @item FE_DIVBYZERO Represents the divide by zero exception iff the FPU supports this exception. @comment fenv.h @comment ISO @item FE_UNDERFLOW Represents the underflow exception iff the FPU supports this exception. @comment fenv.h @comment ISO @item FE_OVERFLOW Represents the overflow exception iff the FPU supports this exception. @comment fenv.h @comment ISO @item FE_INVALID Represents the invalid exception iff the FPU supports this exception. @end vtable The macro @code{FE_ALL_EXCEPT} is the bitwise OR of all exception macros which are supported by the FP implementation. Each of the supported exception flags can either be set or unset. The @w{ISO C 9X} standard defines functions to set, unset and test the status of the flags. @comment fenv.h @comment ISO @deftypefun void feclearexcept (int @var{excepts}) This function clears all of the supported exception flags denoted by @var{excepts} in the status word. @end deftypefun To safe the current status of the flags in the status word @file{fenv.h} defines the type @code{fexcept_t} which can hold all the information. The following function can be used to retrieve the current setting. @comment fenv.h @comment ISO @deftypefun void fegetexceptflag (fexcept_t *@var{flagp}, int @var{excepts}) Store in the variable pointed to by @var{flagp} an implementation-defined value representing the current setting of the exception flags indicated by the parameter @var{excepts}. @end deftypefun @noindent To restore the previously saved values one can use this function: @comment fenv.h @comment ISO @deftypefun void fesetexceptflag (const fexcept_t *@var{flagp}, int @var{excepts}) Restore from the variable pointed to by @var{flagp} the setting of the flags for the exceptions denoted by the value of the parameter @var{excepts}. @end deftypefun The last function allows to query the current status of the flags. The flags can be set either explicitely (using @code{fesetexceptflag} or @code{feclearexcept}) or by a floating-point operation which happened before. Since the flags are accumulative, the flags must be explicitely reset using @code{feclearexcept} if one wants to test for a certain exceptions raised by a specific piece of code. @comment fenv.h @comment ISO @deftypefun int fetestexcept (int @var{excepts}) Test whether a subset of the flags indicated by the parameter @var{except} is currently set. If yes, a nonzero value is returned which specifies which exceptions are set. Otherwise the result is zero. @end deftypefun @noindent Code which uses the @code{fetestexcept} function could look like this: @smallexample @{ double f; int raised; feclearexcept (FE_ALL_EXCEPT); f = compute (); raised = fetestexcept (FE_OVERFLOW | FE_INVALID); if (raised & FE_OVERFLOW) @{ /* ... */ @} if (raised & FE_INVALID) @{ /* ... */ @} /* ... */ @} @end smallexample Please note that the return value of @code{fetestexcept} is @code{int} but this does not mean that the @code{fexcept_t} type is generally representable as an integer. These are completely independent types. @node FPU environment @subsection Controlling the Floating-Point environment It is sometimes necessary so save the complete status of the floating-point unit for a certain time to perform some completely different actions. Beside the status of the exception flags, the control word for the exceptions and the rounding mode can be safed. The file @file{fenv.h} defines the type @code{fenv_t}. The layout of a variable of this type is implementation defined but the variable is able to contain the complete status information. To fill a variable of this type one can use this function: @comment fenv.h @comment ISO @deftypefun void fegetenv (fenv_t *@var{envp}) Store the current floating-point environment in the object pointed to by @var{envp}. @end deftypefun @noindent Another possibility which is useful in several situations is @comment fenv.h @comment ISO @deftypefun int feholdexcept (fenv_t *@var{envp}) Store the current floating-point environment in the object pointed to by @var{envp}. Afterwards, all exception flags are cleared and if available a mode is installed which continues on all exception and does not cause a trap to occur. In this case a nonzero value is returned. If the floating-point implementation does not support such a non-stop mode, the return value is zero. @end deftypefun The functions which allow a state of the floating-point unit to be restored can take two kinds of arguments: @itemize @bullet @item Pointed to objects which previously were initialized by a call to @code{fegetenv} or @code{feholdexcept}. @item @vindex FE_DFL_ENV The special macro @code{FE_DFL_ENV} which represents the floating-point environment as it was available at program start. @item Implementation defined macros with names starting with @code{FE_}. @vindex FE_NOMASK_ENV If possible, the GNU C Library defines a macro @code{FE_NOMASK_ENV} which represents an environment where no exception is masked and so each raised exception causes a trap to occur. Whether this macro is available can easily be tested using @code{#ifdef}. Some platforms might define further predefined environments. @end itemize @noindent To set any of the environments there are two functions defined. @deftypefun void fesetenv (const fenv_t *@var{envp}) Establish the floating-point environment described in the object pointed to by @var{envp}. Even if one or more exceptions flags in the restored environment are set no exception is raised. @end deftypefun In some situations the previous status of the exception flags must not simply be discarded and so this function is useful: @deftypefun void feupdateenv (const fenv_t *@var{envp}) The current status of the floating-point unit is preserved in some automatic storage before the environment described by the object pointed to by @var{envp} is installed. Once this is finished all exceptions set in the original environment which is saved in the automatic storage, is raised. @end deftypefun This function can be used to execute a part of the program with an environment which masks all exceptions and before switching back remove unwanted exception and raise the remaining exceptions. @node Rounding Modes @subsection Rounding modes of the Floating-Point Unit @w{IEEE 754} defines four different rounding modes. If the rounding mode is supported by the floating-point implementation the corresponding of the following macros is defined: @table @code @comment fenv.h @comment ISO @vindex FE_TONEAREST @item FE_TONEAREST Round to nearest. This is the default mode and should always be used except when a different mode is explicitely required. Only rounding to nearest guarantees numeric stability of the computations. @comment fenv.h @comment ISO @vindex FE_UPWARD @item FE_UPWARD Round toward @math{+@infinity{}}. @comment fenv.h @comment ISO @vindex FE_DOWNWARD @item FE_DOWNWARD Round toward @math{-@infinity{}}. @comment fenv.h @comment ISO @vindex FE_TOWARDZERO @item FE_TOWARDZERO Round toward zero. @end table At any time one of the above four rounding modes is selected. To get information about the currently selected mode one can use this function: @comment fenv.h @comment ISO @deftypefun int fegetround (void) Return the currently selected rounding mode, represented by one of the values of the defined rounding mode macros. @end deftypefun @noindent To set a specific rounding mode the next function can be used. @comment fenv.h @comment ISO @deftypefun int fesetround (int @var{round}) Change the currently selected rounding mode to the mode described by the parameter @var{round}. If @var{round} does not correspond to one of the supported rounding modes nothing is changed. The function returns a nonzero value iff the requested rounding mode can be established. Otherwise zero is returned. @end deftypefun Changing the rounding mode might be necessary for various reasons. But changing the mode only to round a given number normally is no good idea. The standard defines a set of functions which can be used to round an argument according to some rules and for all of the rounding modes there is a corresponding function. If a large set of number has to be rounded it might be good to change the rounding mode and to not use the function the library provides. So the perhaps necessary switching of the rounding mode in the library function can be avoided. But since not all rounding modes are guaranteed to exist on any platform this possible implementation cannot be portably used. A default method has to be implemented as well. @node Mathematical Constants @section Predefined Mathematical Constants @cindex constants @cindex mathematical constants The header @file{math.h} defines a series of mathematical constants if @code{_BSD_SOURCE} or a more general feature select macro is defined before including this file. All values are defined as preprocessor macros starting with @code{M_}. The collection includes: @vtable @code @item M_E The value is that of the base of the natural logarithm. @item M_LOG2E The value is computed as the logarithm to base @code{2} of @code{M_E}. @item M_LOG10E The value is computed as the logarithm to base @code{10} of @code{M_E}. @item M_LN2 The value is computed as the natural logarithm of @code{2}. @item M_LN10 The value is computed as the natural logarithm of @code{10}. @item M_PI The value is those of the number pi. @item M_PI_2 The value is those of the number pi divided by two. @item M_PI_4 The value is those of the number pi divided by four. @item M_1_PI The value is the reziprocal of the value of the number pi. @item M_2_PI The value is two times the reziprocal of the value of the number pi. @item M_2_SQRTPI The value is two times the reziprocal of the square root of the number pi. @item M_SQRT2 The value is the square root of the value of the number pi. @item M_SQRT1_2 The value is the reziprocal of the square root of the value of the number pi. @end vtable All values are defined as @code{long double} values unless the compiler does not support this type or @code{__STDC__} is not defined (both is unlikely). Historically the numbers were @code{double} values and some old code still relies on this so you might want to add explicit casts if the extra precision of the @code{long double} value is not needed. One critical case are functions with a variable number of arguments, such as @code{printf}. @vindex PI @emph{Note:} Some programs use a constant named @code{PI} which has the same value as @code{M_PI}. This probably derives from Stroustroup's book about his C++ programming language where this value is used in examples (and perhaps some AT&T headers contain this value). But due to possible name space problems (@code{PI} is a quite frequently used name) this value is not added to @file{math.h}. Every program should use @code{M_PI} instead or add on the compiler command line @code{-DPI=M_PI}. @node FP Comparison Functions @section Floating-Point Comparison Functions @cindex unordered comparison The @w{IEEE 754} standards defines a set of functions which allows to compare even those numbers which normally would cause an exception to be raised since they are unordered. E.g., the expression @smallexample int v = a < 1.0; @end smallexample @noindent would raise an exception if @var{a} would be a NaN. Functions to compare unordered numbers are part of the FORTRAN language for a long time and the extensions in @w{ISO C 9X} finally introduce them as well for the C programming language. All of the operations are implemented as macros which allow their arguments to be of either @code{float}, @code{double}, or @code{long double} type. @comment math.h @comment ISO @deftypefn {Macro} int isgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether the argument @var{x} is greater than @var{y}. This is equivalent to @code{(@var{x}) > (@var{y})} but no exception is raised if @var{x} or @var{y} are unordered. @end deftypefn @comment math.h @comment ISO @deftypefn {Macro} int isgreaterequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether the argument @var{x} is greater than or equal to @var{y}. This is equivalent to @code{(@var{x}) >= (@var{y})} but no exception is raised if @var{x} or @var{y} are unordered. @end deftypefn @comment math.h @comment ISO @deftypefn {Macro} int isless (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether the argument @var{x} is less than @var{y}. This is equivalent @code{(@var{x}) < (@var{y})} but no exception is raised if @var{x} or @var{y} are unordered. @end deftypefn @comment math.h @comment ISO @deftypefn {Macro} int islessequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether the argument @var{x} is less than or equal to @var{y}. This is equivalent to @code{(@var{x}) <= (@var{y})} but no exception is raised if @var{x} or @var{y} are unordered. @end deftypefn @comment math.h @comment ISO @deftypefn {Macro} int islessgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether the argument @var{x} is less or greater than @var{y}. This is equivalent to @code{(@var{x}) < (@var{y}) || (@var{x}) > (@var{y})} (except that @var{x} and @var{y} are only evaluated once) but no exception is raised if @var{x} or @var{y} are unordered. @end deftypefn @comment math.h @comment ISO @deftypefn {Macro} int isunordered (@emph{real-floating} @var{x}, @emph{real-floating} @var{y}) This macro determines whether its arguments are unordered. @end deftypefn All the macros are defined in a way to ensure that both arguments are evaluated exactly once and so they can be used exactly like the builtin operators. On several platform these macros are mapped to efficient instructions the processor understands. But on machines missing these functions, the macros above might be rather slow. So it is best to use the builtin operators unless it is necessary to use unordered comparisons. @node Trig Functions @section Trigonometric Functions @cindex trigonometric functions These are the familiar @code{sin}, @code{cos}, and @code{tan} functions. The arguments to all of these functions are in units of radians; recall that pi radians equals 180 degrees. @cindex pi (trigonometric constant) The math library does define a symbolic constant for pi in @file{math.h} (@pxref{Mathematical Constants}) when BSD compliance is required (@pxref{Feature Test Macros}). In case it is not possible to use this predefined macro one easily can define it: @smallexample #define M_PI 3.14159265358979323846264338327 @end smallexample @noindent You can also compute the value of pi with the expression @code{acos (-1.0)}. @comment math.h @comment ISO @deftypefun double sin (double @var{x}) @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. The following @code{errno} error conditions are defined for this function: @table @code @item ERANGE Mathematically, the tangent function has singularities at odd multiples of pi/2. If the argument @var{x} is too close to one of these singularities, @code{tan} sets @code{errno} to @code{ERANGE} and returns either positive or negative @code{HUGE_VAL}. @end table @end deftypefun In many applications where @code{sin} and @code{cos} are used, the value for the same argument of both of these functions is used at the same time. Since the algorithm to compute these values is very similar for both functions there is an additional function which computes both values at the same time. @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. It should be used whenever both sine and cosine are needed but in protable applications there should be a fallback method for systems without this function. @end deftypefun @cindex complex trigonometric functions The trigonometric functions are in mathematics not only defined on real numbers. They can be extended to complex numbers and the @w{ISO C 9X} standard introduces these variants in the standard math library. @comment complex.h @comment ISO @deftypefun {complex double} csin (complex double @var{z}) @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 the complex value in @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 @iftex @tex $$\sin(z) = {1\over 2i} (e^{zi} - e^{-zi})$$ @end tex @end iftex @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 the complex value in @var{z}. The mathematical definition of the complex cosine is @ifinfo @math{cos (z) = 1/2 * (exp (z*i) + exp (-z*i))} @end ifinfo @iftex @tex $$\cos(z) = {1\over 2} (e^{zi} + e^{-zi})$$ @end tex @end iftex @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 the complex value in @var{z}. The mathematical definition of the complex tangent is @ifinfo @math{tan (z) = 1/i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))} @end ifinfo @iftex @tex $$\tan(z) = {1\over i} {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$ @end tex @end iftex @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). @code{asin} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is out of range. The arc sine function is defined mathematically only over the domain @code{-1} to @code{1}. @end deftypefun @comment math.h @comment ISO @deftypefun double acos (double @var{x}) @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). @code{acos} fails, and sets @code{errno} to @code{EDOM}, if @var{x} is out of range. The arc cosine function is defined mathematically only over the domain @code{-1} to @code{1}. @end deftypefun @comment math.h @comment ISO @deftypefun double atan (double @var{x}) @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 is the two argument arc tangent function. It is similar to computing the arc tangent of @var{y}/@var{x}, except that the signs of both arguments are used to determine the quadrant of the result, and @var{x} is permitted to be zero. The return value is given in radians and is in the range @code{-pi} to @code{pi}, inclusive. If @var{x} and @var{y} are coordinates of a point in the plane, @code{atan2} returns the signed angle between the line from the origin to that point and the x-axis. Thus, @code{atan2} is useful for converting Cartesian coordinates to polar coordinates. (To compute the radial coordinate, use @code{hypot}; see @ref{Exponents and Logarithms}.) The function @code{atan2} sets @code{errno} to @code{EDOM} if both @var{x} and @var{y} are zero; the return value is not defined in this case. @end deftypefun @cindex inverse complex trigonometric functions The inverse trigonometric functions also exist is separate versions which are usable with complex numbers. @comment complex.h @comment ISO @deftypefun {complex double} casin (complex double @var{z}) @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 is in units of radians. Unlike the real version of the arc sine function @code{casin} has no limitation on the argument @var{z}. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} cacos (complex double @var{z}) @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 is in units of radians. Unlike the real version of the arc cosine function @code{cacos} has no limitation on the argument @var{z}. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} catan (complex double @var{z}) @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 return the value of @code{e} (the base of natural logarithms) raised to power @var{x}. The function fails, and sets @code{errno} to @code{ERANGE}, if the magnitude of the result is too large to be representable. @end deftypefun @comment math.h @comment ISO @deftypefun double exp10 (double @var{x}) @deftypefunx float exp10f (float @var{x}) @deftypefunx {long double} exp10l (long double @var{x}) These functions return the value of @code{10} raised to the power @var{x}. Mathematically, @code{exp10 (x)} is the same as @code{exp (x * log (10))}. The function fails, and sets @code{errno} to @code{ERANGE}, if the magnitude of the result is too large to be representable. @end deftypefun @comment math.h @comment ISO @deftypefun double exp2 (double @var{x}) @deftypefunx float exp2f (float @var{x}) @deftypefunx {long double} exp2l (long double @var{x}) These functions return the value of @code{2} raised to the power @var{x}. Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}. The function fails, and sets @code{errno} to @code{ERANGE}, if the magnitude of the result is too large to be representable. @end deftypefun @comment math.h @comment ISO @deftypefun double log (double @var{x}) @deftypefunx float logf (floatdouble @var{x}) @deftypefunx {long double} logl (long double @var{x}) These functions return the natural logarithm of @var{x}. @code{exp (log (@var{x}))} equals @var{x}, exactly in mathematics and approximately in C. The following @code{errno} error conditions are defined for this function: @table @code @item EDOM The argument @var{x} is negative. The log function is defined mathematically to return a real result only on positive arguments. @item ERANGE The argument is zero. The log of zero is not defined. @end table @end deftypefun @comment math.h @comment ISO @deftypefun double log10 (double @var{x}) @deftypefunx float log10f (float @var{x}) @deftypefunx {long double} log10l (long double @var{x}) These functions return the base-10 logarithm of @var{x}. Except for the different base, it is similar to the @code{log} function. In fact, @code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}. @end deftypefun @comment math.h @comment ISO @deftypefun double log2 (double @var{x}) @deftypefunx float log2f (float @var{x}) @deftypefunx {long double} log2l (long double @var{x}) These functions return the base-2 logarithm of @var{x}. Except for the different base, it is similar to the @code{log} function. In fact, @code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}. @end deftypefun @comment math.h @comment ISO @deftypefun double 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 signed integer value. If @var{x} is zero, a range error may occur. A special case are subnormal numbers (if supported by the floating-point format). The exponent returned is not the actual value from @var{x}. Instead the number is first normalized as if the range of the exponent field is large enough. @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 the values are returned as signed integer values. Since integer values cannot represent infinity and NaN, there are some special symbols defined to help detect these situations. @vindex FP_ILOGB0 @vindex FP_ILOGBNAN @code{ilogb} returns @code{FP_ILOGB0} if @var{x} is @code{0} and it returns @code{FP_ILOGBNAN} if @var{x} is @code{NaN}. These values are system specific and no fixed value is assigned. More concrete, these values might even have the same value. So a piece of code handling the result of @code{ilogb} could look like this: @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 @end deftypefun @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}. @need 250 The following @code{errno} error conditions are defined for this function: @table @code @item EDOM The argument @var{base} is negative and @var{power} is not an integral value. Mathematically, the result would be a complex number in this case. @item ERANGE An underflow or overflow condition was detected in the result. @end table @end deftypefun @cindex square root function @comment math.h @comment ISO @deftypefun double sqrt (double @var{x}) @deftypefunx float sqrtf (float @var{x}) @deftypefunx {long double} sqrtl (long double @var{x}) These functions return the nonnegative square root of @var{x}. The @code{sqrt} function fails, and sets @code{errno} to @code{EDOM}, if @var{x} is negative. Mathematically, the square root would be a complex number. @c (@pxref{csqrt}) @end deftypefun @cindex cube root function @comment math.h @comment BSD @deftypefun double cbrt (double @var{x}) @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 highly appreciated since the error is much smaller. See also the function @code{cabs} in @ref{Absolute Value}. @end deftypefun @comment math.h @comment ISO @deftypefun double expm1 (double @var{x}) @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}. It is computed in a way that is accurate even if the value of @var{x} is near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate due to subtraction of two numbers that are nearly equal. @end deftypefun @comment math.h @comment ISO @deftypefun double log1p (double @var{x}) @deftypefunx float log1pf (float @var{x}) @deftypefunx {long double} log1pl (long double @var{x}) This function returns a value equivalent to @w{@code{log (1 + @var{x})}}. It is computed in a way that is accurate even if the value of @var{x} is near zero. @end deftypefun @cindex complex exponentiation functions @cindex complex logarithm functions @w{ISO C 9X} defines variants of some of the exponentiation and logarithm functions. As for the other functions handlung complex numbers these functions are perhaps better optimized and provide better error checking than a direct use of the formulas of the mathematical definition. @comment complex.h @comment ISO @deftypefun {complex double} cexp (complex double @var{z}) @deftypefunx {complex float} cexpf (complex float @var{z}) @deftypefunx {complex long double} cexpl (complex long double @var{z}) These functions return the value of @code{e} (the base of natural logarithms) raised to power of the complex value @var{z}. @noindent Mathematically this corresponds to the value @ifinfo @math{exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))} @end ifinfo @iftex @tex $$\exp(z) = e^z = e^{{\rm Re} z} (\cos ({\rm Im} z) + i \sin ({\rm Im} z))$$ @end tex @end iftex @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 the complex value @var{z}. Unlike the real value version @code{log} and its variants, @code{clog} has no limit for the range of its argument @var{z}. @noindent Mathematically this corresponds to the value @ifinfo @math{log (z) = log (cabs (z)) + I * carg (z)} @end ifinfo @iftex @tex $$\log(z) = \log(|z|) + i \arg(z)$$ @end tex @end iftex @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 root of the argument @var{z}. Unlike the @code{sqrt} function these functions do not have any restriction on the value of the argument. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power}) @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 the complex value @var{BASE} raised to the power of @var{power}. This is computed as @ifinfo @math{cpow (x, y) = cexp (y * clog (x))} @end ifinfo @iftex @tex $${\rm cpow}(x, y) = e^{y \log(x)}$$ @end tex @end iftex @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}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of @var{x} is too large; that is, if overflow occurs. @end deftypefun @comment math.h @comment ISO @deftypefun double cosh (double @var{x}) @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}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of @var{x} is too large; that is, if overflow occurs. @end deftypefun @comment math.h @comment ISO @deftypefun double tanh (double @var{x}) @deftypefunx float tanhf (float @var{x}) @deftypefunx {long double} tanhl (long double @var{x}) These functions return the hyperbolic tangent of @var{x}, whose mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}. @end deftypefun @cindex hyperbolic functions There are counterparts for these hyperbolic functions which work with complex valued arguments. They should always be used instead of the obvious mathematical formula since the implementations in the math library are optimized for accuracy and speed. @comment complex.h @comment ISO @deftypefun {complex double} csinh (complex double @var{z}) @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}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of result is too large. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} ccosh (complex double @var{z}) @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}}. The function fails, and sets @code{errno} to @code{ERANGE}, if the value of result is too large. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} ctanh (complex double @var{z}) @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}, whose mathematical definition is @w{@code{csinh (@var{z}) / ccosh (@var{z})}}. @end deftypefun @cindex inverse hyperbolic functions @comment math.h @comment ISO @deftypefun double asinh (double @var{x}) @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} returns @code{HUGE_VAL}. @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 or equal to @code{1}, @code{atanh} returns @code{HUGE_VAL}. @end deftypefun @cindex inverse complex hyperbolic functions @comment complex.h @comment ISO @deftypefun {complex double} casinh (complex double @var{z}) @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 function @code{acosh} there is not limit for the range of the argument. @end deftypefun @comment complex.h @comment ISO @deftypefun {complex double} catanh (complex double @var{z}) @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 function @code{atanh} there is not limit for the range of the argument. @end deftypefun @node Pseudo-Random Numbers @section Pseudo-Random Numbers @cindex random numbers @cindex pseudo-random numbers @cindex seed (for random numbers) This section describes the GNU facilities for generating a series of pseudo-random numbers. The numbers generated are not truly random; typically, they form a sequence that repeats periodically, with a period so large that you can ignore it for ordinary purposes. The random number generator works by remembering at all times a @dfn{seed} value which it uses to compute the next random number and also to compute a new seed. Although the generated numbers look unpredictable within one run of a program, the sequence of numbers is @emph{exactly the same} from one run to the next. This is because the initial seed is always the same. This is convenient when you are debugging a program, but it is unhelpful if you want the program to behave unpredictably. If you want truly random numbers, not just pseudo-random, specify a seed based on the current time. You can get repeatable sequences of numbers on a particular machine type by specifying the same initial seed value for the random number generator. There is no standard meaning for a particular seed value; the same seed, used in different C libraries or on different CPU types, will give you different random numbers. The GNU library supports the standard @w{ISO C} random number functions plus another set derived from BSD. We recommend you use the standard ones, @code{rand} and @code{srand}. @menu * ISO Random:: @code{rand} and friends. * BSD Random:: @code{random} and friends. * SVID Random:: @code{drand48} and friends. @end menu @node ISO Random @subsection ISO C Random Number Functions This section describes the random number functions that are part of the @w{ISO C} standard. To use these facilities, you should include the header file @file{stdlib.h} in your program. @pindex stdlib.h @comment stdlib.h @comment ISO @deftypevr Macro int RAND_MAX The value of this macro is an integer constant expression that represents the maximum possible value returned by the @code{rand} function. In the GNU library, it is @code{037777777}, which is the largest signed integer representable in 32 bits. In other libraries, it may be as low as @code{32767}. @end deftypevr @comment stdlib.h @comment ISO @deftypefun int rand (void) The @code{rand} function returns the next pseudo-random number in the series. The value is in the range from @code{0} to @code{RAND_MAX}. @end deftypefun @comment stdlib.h @comment ISO @deftypefun void srand (unsigned int @var{seed}) This function establishes @var{seed} as the seed for a new series of pseudo-random numbers. If you call @code{rand} before a seed has been established with @code{srand}, it uses the value @code{1} as a default seed. To produce truly random numbers (not just pseudo-random), do @code{srand (time (0))}. @end deftypefun @node BSD Random @subsection BSD Random Number Functions This section describes a set of random number generation functions that are derived from BSD. There is no advantage to using these functions with the GNU C library; we support them for BSD compatibility only. The prototypes for these functions are in @file{stdlib.h}. @pindex stdlib.h @comment stdlib.h @comment BSD @deftypefun {int32_t} random (void) This function returns the next pseudo-random number in the sequence. The range of values returned is from @code{0} to @code{RAND_MAX}. @strong{Please note:} Historically this function returned a @code{long int} value. But with the appearence of 64bit machines this could lead to severe compatibility problems and therefore the type now explicitly limits the return value to 32bit. @end deftypefun @comment stdlib.h @comment BSD @deftypefun void srandom (unsigned int @var{seed}) The @code{srandom} function sets the seed for the current random number state based on the integer @var{seed}. If you supply a @var{seed} value of @code{1}, this will cause @code{random} to reproduce the default set of random numbers. To produce truly random numbers (not just pseudo-random), do @code{srandom (time (0))}. @end deftypefun @comment stdlib.h @comment BSD @deftypefun {void *} initstate (unsigned int @var{seed}, void *@var{state}, size_t @var{size}) The @code{initstate} function is used to initialize the random number generator state. The argument @var{state} is an array of @var{size} bytes, used to hold the state information. The size must be at least 8 bytes, and optimal sizes are 8, 16, 32, 64, 128, and 256. The bigger the @var{state} array, the better. The return value is the previous value of the state information array. You can use this value later as an argument to @code{setstate} to restore that state. @end deftypefun @comment stdlib.h @comment BSD @deftypefun {void *} setstate (void *@var{state}) The @code{setstate} function restores the random number state information @var{state}. The argument must have been the result of a previous call to @var{initstate} or @var{setstate}. The return value is the previous value of the state information array. You can use thise value later as an argument to @code{setstate} to restore that state. @end deftypefun @node SVID Random @subsection SVID Random Number Function The C library on SVID systems contains yet another kind of random number generator functions. They use a state of 48 bits of data. The user can choose among a collection of functions which all return the random bits in different forms. Generally there are two kinds of functions: those which use a state of the random number generator which is shared among several functions and by all threads of the process. The second group of functions require the user to handle the state. All functions have in common that they use the same congruential formula with the same constants. The formula is @smallexample Y = (a * X + c) mod m @end smallexample @noindent where @var{X} is the state of the generator at the beginning and @var{Y} the state at the end. @code{a} and @code{c} are constants determining the way the generator work. By default they are @smallexample a = 0x5DEECE66D = 25214903917 c = 0xb = 11 @end smallexample @noindent but they can also be changed by the user. @code{m} is of course 2^48 since the state consists of a 48 bit array. @comment stdlib.h @comment SVID @deftypefun double drand48 (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}) 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}) 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}) The @code{jrand48} function is similar to @code{nrand48}. The only difference is that the numbers returned are in the range @code{-2^31} to @code{2^31} (exclusive). For the @code{xsubi} parameter the same requirements are necessary. @end deftypefun The internal state of the random number generator can be initialized in several ways. The functions differ in the completeness of the information provided. @comment stdlib.h @comment SVID @deftypefun void srand48 (long int @var{seedval})) The @code{srand48} function sets the most significant 32 bits of the state internal state of the random number generator to the least significant 32 bits of the @var{seedval} parameter. The lower 16 bts are initilialized to the value @code{0x330E}. Even if the @code{long int} type contains more the 32 bits only the lower 32 bits are used. Due to this limitation the initialization of the state using this function of not very useful. But it makes it easy to use a constrcut like @code{srand48 (time (0))}. A side-effect of this function is that the values @code{a} and @code{c} from the internal state, which are used in the congruential formula, are reset to the default values given above. This is of importance once the user called the @code{lcong48} function (see below). @end deftypefun @comment stdlib.h @comment SVID @deftypefun {unsigned short int *} seed48 (unsigned short int @var{seed16v}) 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}} initialize the mid-order 16 bits of the state and the 16 lower bits of @code{@var{seed16v}} 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}) 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}} to @code{@var{param}} determine the the initial state, the least 16 bits of @code{@var{param}} to @code{@var{param}} determine the 48 bit constant @code{a} and @code{@var{param}} determines the 16 bit value @code{c}. @end deftypefun All the above functions have in common that they use the global parameters for the congruential formula. In multi-threaded programs it might sometimes be useful to have different parameters in different threads. For this reason all the above functions have a counterpart which works on a description of the random number generator in the user-supplied buffer instead of the global state. Please note that it is no problem if several threads use the global state if all threads use the functions which take a pointer to an array containing the state. The random numbers are computed following the same loop but if the state in the array is different all threads will get an individuual random number generator. The user supplied buffer must be of type @code{struct drand48_data}. This type should be regarded as opaque and no member should be used directly. @comment stdlib.h @comment GNU @deftypefun int drand48_r (struct drand48_data *@var{buffer}, double *@var{result}) This function is equivalent to the @code{drand48} function with the difference it does not modify the global random number generator parameters but instead the parameters is the buffer supplied by the buffer through the pointer @var{buffer}. The random number is return in the variable pointed to by @var{result}. The return value of the function indicate whether the call succeeded. If the value is less than @code{0} an error occurred and @var{errno} is set to indicate the problem. This function is a GNU extension and should not be used in portable programs. @end deftypefun @comment stdlib.h @comment GNU @deftypefun int erand48_r (unsigned short int @var{xsubi}, struct drand48_data *@var{buffer}, double *@var{result}) The @code{erand48_r} function works like the @code{erand48} and it takes an argument @var{buffer} which describes the random number generator. The state of the random number genertor is taken from the @code{xsubi} array, the parameters for the congruential formula from the global random number generator data. The random number is return in the variable pointed to by @var{result}. The return value is non-negative is the call succeeded. This function is a GNU extension and should not be used in portable programs. @end deftypefun @comment stdlib.h @comment GNU @deftypefun int lrand48_r (struct drand48_data *@var{buffer}, double *@var{result}) This function is similar to @code{lrand48} and it takes a pointer to a buffer describing the state of the random number generator as a parameter just like @code{drand48}. If the return value of the function is non-negative the variable pointed to by @var{result} contains the result. Otherwise an error occurred. This function is a GNU extension and should not be used in portable programs. @end deftypefun @comment stdlib.h @comment GNU @deftypefun int nrand48_r (unsigned short int @var{xsubi}, 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}, struct drand48_data *@var{buffer}, long int *@var{result}) The @code{jrand48_r} function is similar to @code{jrand48}. But as the other reentrant functions of this function family it uses the congruential formula parameters from the buffer pointed to by @var{buffer}. If the return value is non-negative the variable pointed to by @var{result} contains the result. This function is a GNU extension and should not be used in portable programs. @end deftypefun Before any of the above functions should be used the buffer of type @code{struct drand48_data} should initialized. The easiest way is to fill the whole buffer with null bytes, e.g., using @smallexample memset (buffer, '\0', sizeof (struct drand48_data)); @end smallexample @noindent Using any of the reetrant functions of this family now will automatically initialize the random number generator to the default values for the state and the parameters of the congruential formula. The other possibility is too use any of the functions which explicitely initialize the buffer. Though it might be obvious how to initialize the buffer from the data given as parameter from the function it is highly recommended to use these functions since the result might not always be what you expect. @comment stdlib.h @comment GNU @deftypefun int srand48_r (long int @var{seedval}, struct drand48_data *@var{buffer}) The description of the random number generator represented by the information in @var{buffer} is initialized similar to what the function @code{srand48} does. The state is initialized from the paramter @var{seedval} and the paameters for the congruential formula are initialized to the default values. If the return value is non-negative the function call succeeded. This function is a GNU extension and should not be used in portable programs. @end deftypefun @comment stdlib.h @comment GNU @deftypefun int seed48_r (unsigned short int @var{seed16v}, 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}, 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