@c We need some definitions here.
-@ifclear cdot
+@ifclear mult
@ifhtml
-@set cdot ยท
+@set mult @U{00B7}
+@set infty @U{221E}
+@set pie @U{03C0}
@end ifhtml
@iftex
-@set cdot @cdot
-@end iftex
-@ifclear cdot
-@set cdot x
-@end ifclear
-@macro mul
-@value{cdot}
-@end macro
-@end ifclear
-@iftex
+@set mult @cdot
@set infty @infty
@end iftex
-@ifclear infty
+@ifclear mult
+@set mult *
@set infty oo
+@set pie pi
@end ifclear
+@macro mul
+@value{mult}
+@end macro
@macro infinity
@value{infty}
@end macro
+@ifnottex
+@macro pi
+@value{pie}
+@end macro
+@end ifnottex
+@end ifclear
-@node Mathematics, Arithmetic, Low-Level Terminal Interface, Top
+@node Mathematics, Arithmetic, Syslog, Top
+@c %MENU% Math functions, useful constants, random numbers
@chapter Mathematics
This chapter contains information about functions for performing
mathematical computations, such as trigonometric functions. Most of
these functions have prototypes declared in the header file
-@file{math.h}.
+@file{math.h}. The complex-valued functions are defined in
+@file{complex.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.
-* FP Function Optimizations:: Fast code or small code.
-* 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 divisor 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.
+@pindex complex.h
+
+All mathematical functions which take a floating-point argument
+have three variants, one each for @code{double}, @code{float}, and
+@code{long double} arguments. The @code{double} versions are mostly
+defined in @w{ISO C89}. The @code{float} and @code{long double}
+versions are from the numeric extensions to C included in @w{ISO C99}.
+
+Which of the three versions of a function should be used depends on the
+situation. For most calculations, the @code{float} functions are the
+fastest. On the other hand, the @code{long double} functions have the
+highest precision. @code{double} is somewhere in between. It is
+usually wise to pick the narrowest type that can accommodate your data.
+Not all machines have a distinct @code{long double} type; it may be the
+same as @code{double}.
+
+@Theglibc{} also provides @code{_Float@var{N}} and
+@code{_Float@var{N}x} types. These types are defined in @w{ISO/IEC TS
+18661-3}, which extends @w{ISO C} and defines floating-point types that
+are not machine-dependent. When such a type, such as @code{_Float128},
+is supported by @theglibc{}, extra variants for most of the mathematical
+functions provided for @code{double}, @code{float}, and @code{long
+double} are also provided for the supported type. Throughout this
+manual, the @code{_Float@var{N}} and @code{_Float@var{N}x} variants of
+these functions are described along with the @code{double},
+@code{float}, and @code{long double} variants and they come from
+@w{ISO/IEC TS 18661-3}, unless explicitly stated otherwise.
+
+Support for @code{_Float@var{N}} or @code{_Float@var{N}x} types is
+provided for @code{_Float32}, @code{_Float64} and @code{_Float32x} on
+all platforms.
+It is also provided for @code{_Float128} and @code{_Float64x} on
+powerpc64le (PowerPC 64-bits little-endian), x86_64, x86, ia64,
+aarch64, alpha, mips64, riscv, s390 and sparc.
@menu
-* Status bit operations:: Manipulate the FP status word.
-* FPU environment:: Controlling the status of the FPU.
-* Rounding Modes:: Controlling the rounding mode.
+* Mathematical Constants:: Precise numeric values for often-used
+ constants.
+* Trig Functions:: Sine, cosine, tangent, and friends.
+* Inverse Trig Functions:: Arcsine, arccosine, etc.
+* Exponents and Logarithms:: Also pow and sqrt.
+* Hyperbolic Functions:: sinh, cosh, tanh, etc.
+* Special Functions:: Bessel, gamma, erf.
+* Errors in Math Functions:: Known Maximum Errors in Math Functions.
+* Pseudo-Random Numbers:: Functions for generating pseudo-random
+ numbers.
+* FP Function Optimizations:: Fast code or small code.
@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 saved.
-
-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:
+The header @file{math.h} defines several useful mathematical constants.
+All values are defined as preprocessor macros starting with @code{M_}.
+The values provided are:
@vtable @code
@item M_E
-The value is that of the base of the natural logarithm.
+The base of natural logarithms.
@item M_LOG2E
-The value is computed as the logarithm to base @code{2} of @code{M_E}.
+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}.
+The logarithm to base @code{10} of @code{M_E}.
@item M_LN2
-The value is computed as the natural logarithm of @code{2}.
+The natural logarithm of @code{2}.
@item M_LN10
-The value is computed as the natural logarithm of @code{10}.
+The natural logarithm of @code{10}.
@item M_PI
-The value is those of the number pi.
+Pi, the ratio of a circle's circumference to its diameter.
@item M_PI_2
-The value is those of the number pi divided by two.
+Pi divided by two.
@item M_PI_4
-The value is those of the number pi divided by four.
+Pi divided by four.
@item M_1_PI
-The value is the reziprocal of the value of the number pi.
+The reciprocal of pi (1/pi)
@item M_2_PI
-The value is two times the reziprocal of the value of the number pi.
+Two times the reciprocal of pi.
@item M_2_SQRTPI
-The value is two times the reziprocal of the square root of the number pi.
+Two times the reciprocal of the square root of pi.
@item M_SQRT2
-The value is the square root of the value of the number pi.
+The square root of two.
@item M_SQRT1_2
-The value is the reziprocal of the square root of the value of the number pi.
+The reciprocal of the square root of two (also the square root of 1/2).
@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}.
+These constants come from the Unix98 standard and were also available in
+4.4BSD; therefore they are only defined if
+@code{_XOPEN_SOURCE=500}, or a more general feature select macro, is
+defined. The default set of features includes these constants.
+@xref{Feature Test Macros}.
+
+All values are of type @code{double}. As an extension, @theglibc{}
+also defines these constants with type @code{long double} and
+@code{float}. The @code{long double} macros have a lowercase @samp{l}
+while the @code{float} macros have a lowercase @samp{f} appended to
+their names: @code{M_El}, @code{M_PIl}, and so forth. These are only
+available if @code{_GNU_SOURCE} is defined.
+
+Likewise, @theglibc{} also defines these constants with the types
+@code{_Float@var{N}} and @code{_Float@var{N}x} for the machines that
+have support for such types enabled (@pxref{Mathematics}) and if
+@code{_GNU_SOURCE} is defined. When available, the macros names are
+appended with @samp{f@var{N}} or @samp{f@var{N}x}, such as @samp{f128}
+for the type @code{_Float128}.
@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.
-
-@strong{Note:} There are no macros @code{isequal} or @code{isunequal}.
-These macros are not necessary since the @w{IEEE 754} standard requires
-that the comparison for equality and unequality do @emph{not} throw an
-exception if one of the arguments is an unordered value.
-
-
-@node FP Function Optimizations
-@section Is Fast Code or Small Code preferred?
-@cindex Optimization
-
-If an application uses many floating point function it is often the case
-that the costs for the function calls itselfs are not neglectable.
-Modern processor implementation often can execute the operation itself
-very fast but the call means a disturbance of the control flow.
-
-For this reason the GNU C Library provides optimizations for many of the
-frequently used math functions. When the GNU CC is used and the user
-activates the optimizer several new inline functions and macros get
-defined. These new functions and macros have the same names as the
-library function and so get used instead of the later. In case of
-inline functions the compiler will decide whether it is reasonable to
-use the inline function and this decision is usually correct.
-
-For the generated code this means that no calls to the library functions
-are necessary. This increases the speed significantly. But the
-drawback is that the code size increases and this increase is not always
-neglectable.
-
-In cases where the inline functions and macros are not wanted the symbol
-@code{__NO_MATH_INLINES} should be defined before any system header is
-included. This will make sure only library functions are used. Of
-course it can be determined for each single file in the project whether
-giving this option is preferred or not.
-
+same value as @code{M_PI}. This constant is not standard; it may have
+appeared in some old AT&T headers, and is mentioned in Stroustrup's book
+on C++. It infringes on the user's name space, so @theglibc{}
+does not define it. Fixing programs written to expect it is simple:
+replace @code{PI} with @code{M_PI} throughout, or put @samp{-DPI=M_PI}
+on the compiler command line.
@node Trig Functions
@section Trigonometric Functions
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:
+The math library normally defines @code{M_PI} to a @code{double}
+approximation of pi. If strict ISO and/or POSIX compliance
+are requested this constant is not defined, but you can easily define it
+yourself:
@smallexample
#define M_PI 3.14159265358979323846264338327
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})
+@deftypefunx _FloatN sinfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx sinfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{sinfN, TS 18661-3:2015, math.h}
+@standardsx{sinfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+@deftypefunx _FloatN cosfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx cosfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{cosfN, TS 18661-3:2015, math.h}
+@standardsx{cosfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+@deftypefunx _FloatN tanfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx tanfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{tanfN, TS 18661-3:2015, math.h}
+@standardsx{tanfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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
+singularities, @code{tan} will signal overflow.
@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.
+In many applications where @code{sin} and @code{cos} are used, the sine
+and cosine of the same angle are needed at the same time. It is more
+efficient to compute them simultaneously, so the library provides a
+function to do that.
-@comment math.h
-@comment GNU
@deftypefun void sincos (double @var{x}, double *@var{sinx}, double *@var{cosx})
@deftypefunx void sincosf (float @var{x}, float *@var{sinx}, float *@var{cosx})
@deftypefunx void sincosl (long double @var{x}, long double *@var{sinx}, long double *@var{cosx})
+@deftypefunx _FloatN sincosfN (_Float@var{N} @var{x}, _Float@var{N} *@var{sinx}, _Float@var{N} *@var{cosx})
+@deftypefunx _FloatNx sincosfNx (_Float@var{N}x @var{x}, _Float@var{N}x *@var{sinx}, _Float@var{N}x *@var{cosx})
+@standards{GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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
+cosine of @var{x} in @code{*@var{cosx}}, 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 portable applications there should be a
-fallback method for systems without this function.
+All these functions, including the @code{_Float@var{N}} and
+@code{_Float@var{N}x} variants, are GNU extensions. Portable programs
+should be prepared to cope with their absence.
@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.
+@w{ISO C99} defines variants of the trig functions which work on
+complex numbers. @Theglibc{} provides these functions, but they
+are only useful if your compiler supports the new complex types defined
+by the standard.
+@c XXX Change this when gcc is fixed. -zw
+(As of this writing GCC supports complex numbers, but there are bugs in
+the implementation.)
-@comment complex.h
-@comment ISO
@deftypefun {complex double} csin (complex double @var{z})
@deftypefunx {complex float} csinf (complex float @var{z})
@deftypefunx {complex long double} csinl (complex long double @var{z})
-These functions return the complex sine of the complex value in @var{z}.
+@deftypefunx {complex _FloatN} csinfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} csinfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{csinfN, TS 18661-3:2015, complex.h}
+@standardsx{csinfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@c There are calls to nan* that could trigger @mtslocale if they didn't get
+@c empty strings.
+These functions return the complex sine of @var{z}.
The mathematical definition of the complex sine is
-@ifinfo
+@ifnottex
@math{sin (z) = 1/(2*i) * (exp (z*i) - exp (-z*i))}.
-@end ifinfo
-@iftex
+@end ifnottex
@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}.
+@deftypefunx {complex _FloatN} ccosfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} ccosfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{ccosfN, TS 18661-3:2015, complex.h}
+@standardsx{ccosfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the complex cosine of @var{z}.
The mathematical definition of the complex cosine is
-@ifinfo
+@ifnottex
@math{cos (z) = 1/2 * (exp (z*i) + exp (-z*i))}
-@end ifinfo
-@iftex
+@end ifnottex
@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}.
+@deftypefunx {complex _FloatN} ctanfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} ctanfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{ctanfN, TS 18661-3:2015, complex.h}
+@standardsx{ctanfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the complex tangent of @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
+@ifnottex
+@math{tan (z) = -i * (exp (z*i) - exp (-z*i)) / (exp (z*i) + exp (-z*i))}
+@end ifnottex
@tex
-$$\tan(z) = {1\over i} {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$
+$$\tan(z) = -i \cdot {e^{zi} - e^{-zi}\over e^{zi} + e^{-zi}}$$
@end tex
-@end iftex
+
+@noindent
+The complex tangent has poles at @math{pi/2 + 2n}, where @math{n} is an
+integer. @code{ctan} may signal overflow if @var{z} is too close to a
+pole.
@end deftypefun
@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,
+These are the usual arcsine, arccosine and arctangent 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
+@deftypefunx _FloatN asinfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx asinfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{asinfN, TS 18661-3:2015, math.h}
+@standardsx{asinfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the arcsine 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}.
+The arcsine function is defined mathematically only
+over the domain @code{-1} to @code{1}. If @var{x} is outside the
+domain, @code{asin} signals a domain error.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double acos (double @var{x})
@deftypefunx float acosf (float @var{x})
@deftypefunx {long double} acosl (long double @var{x})
-These functions compute the arc cosine of @var{x}---that is, the value
+@deftypefunx _FloatN acosfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx acosfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{acosfN, TS 18661-3:2015, math.h}
+@standardsx{acosfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the arccosine 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}.
+The arccosine function is defined mathematically only
+over the domain @code{-1} to @code{1}. If @var{x} is outside the
+domain, @code{acos} signals a domain error.
@end deftypefun
-
-@comment math.h
-@comment ISO
@deftypefun double atan (double @var{x})
@deftypefunx float atanf (float @var{x})
@deftypefunx {long double} atanl (long double @var{x})
-These functions compute the arc tangent of @var{x}---that is, the value
+@deftypefunx _FloatN atanfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx atanfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{atanfN, TS 18661-3:2015, math.h}
+@standardsx{atanfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the arctangent 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).
+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.
+@deftypefunx _FloatN atan2fN (_Float@var{N} @var{y}, _Float@var{N} @var{x})
+@deftypefunx _FloatNx atan2fNx (_Float@var{N}x @var{y}, _Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{atan2fN, TS 18661-3:2015, math.h}
+@standardsx{atan2fNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+This function computes the arctangent of @var{y}/@var{x}, but the signs
+of both arguments are used to determine the quadrant of the result, and
+@var{x} is permitted to be zero. The return value is given in radians
+and is in the range @code{-pi} to @code{pi}, inclusive.
If @var{x} and @var{y} are coordinates of a point in the plane,
@code{atan2} returns the signed angle between the line from the origin
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.
+@c This is experimentally true. Should it be so? -zw
+If both @var{x} and @var{y} are zero, @code{atan2} returns zero.
@end deftypefun
@cindex inverse complex trigonometric functions
+@w{ISO C99} defines complex versions of the inverse trig 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.
+@deftypefunx {complex _FloatN} casinfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} casinfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{casinfN, TS 18661-3:2015, complex.h}
+@standardsx{casinfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the complex arcsine of @var{z}---that is, the
+value whose sine is @var{z}. The value returned is in radians.
-Unlike the real version of the arc sine function @code{casin} has no
-limitation on the argument @var{z}.
+Unlike the real-valued functions, @code{casin} is defined for all
+values of @var{z}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} cacos (complex double @var{z})
@deftypefunx {complex float} cacosf (complex float @var{z})
@deftypefunx {complex long double} cacosl (complex long double @var{z})
-These functions compute the complex arc cosine of @var{z}---that is, the
-value whose cosine is @var{z}. The value is in units of radians.
+@deftypefunx {complex _FloatN} cacosfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} cacosfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{cacosfN, TS 18661-3:2015, complex.h}
+@standardsx{cacosfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the complex arccosine of @var{z}---that is, the
+value whose cosine is @var{z}. The value returned is in radians.
-Unlike the real version of the arc cosine function @code{cacos} has no
-limitation on the argument @var{z}.
+Unlike the real-valued functions, @code{cacos} is defined for all
+values of @var{z}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} catan (complex double @var{z})
@deftypefunx {complex float} catanf (complex float @var{z})
@deftypefunx {complex long double} catanl (complex long double @var{z})
-These functions compute the complex arc tangent of @var{z}---that is,
+@deftypefunx {complex _FloatN} catanfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} catanfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{catanfN, TS 18661-3:2015, complex.h}
+@standardsx{catanfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute the complex arctangent of @var{z}---that is,
the value whose tangent is @var{z}. The value is in units of radians.
@end deftypefun
@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))}.
+@deftypefunx _FloatN expfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx expfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{expfN, TS 18661-3:2015, math.h}
+@standardsx{expfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute @code{e} (the base of natural logarithms) raised
+to the 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.
+If the magnitude of the result is too large to be representable,
+@code{exp} signals overflow.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double exp2 (double @var{x})
@deftypefunx float exp2f (float @var{x})
@deftypefunx {long double} exp2l (long double @var{x})
-These functions return the value of @code{2} raised to the power @var{x}.
+@deftypefunx _FloatN exp2fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx exp2fNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{exp2fN, TS 18661-3:2015, math.h}
+@standardsx{exp2fNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute @code{2} raised to the power @var{x}.
Mathematically, @code{exp2 (x)} is the same as @code{exp (x * log (2))}.
+@end deftypefun
+
+@deftypefun double exp10 (double @var{x})
+@deftypefunx float exp10f (float @var{x})
+@deftypefunx {long double} exp10l (long double @var{x})
+@deftypefunx _FloatN exp10fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx exp10fNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{exp10fN, TS 18661-4:2015, math.h}
+@standardsx{exp10fNx, TS 18661-4:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute @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.
+The @code{exp10} functions are from TS 18661-4:2015.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double log (double @var{x})
@deftypefunx float logf (float @var{x})
@deftypefunx {long double} logl (long double @var{x})
-These functions return the natural logarithm of @var{x}. @code{exp (log
+@deftypefunx _FloatN logfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx logfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{logfN, TS 18661-3:2015, math.h}
+@standardsx{logfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions compute 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
+If @var{x} is negative, @code{log} signals a domain error. If @var{x}
+is zero, it returns negative infinity; if @var{x} is too close to zero,
+it may signal overflow.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double log10 (double @var{x})
@deftypefunx float log10f (float @var{x})
@deftypefunx {long double} log10l (long double @var{x})
-These functions return the base-10 logarithm of @var{x}. Except for the
-different base, it is similar to the @code{log} function. In fact,
+@deftypefunx _FloatN log10fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx log10fNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{log10fN, TS 18661-3:2015, math.h}
+@standardsx{log10fNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the base-10 logarithm of @var{x}.
@code{log10 (@var{x})} equals @code{log (@var{x}) / log (10)}.
+
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double log2 (double @var{x})
@deftypefunx float log2f (float @var{x})
@deftypefunx {long double} log2l (long double @var{x})
-These functions return the base-2 logarithm of @var{x}. Except for the
-different base, it is similar to the @code{log} function. In fact,
+@deftypefunx _FloatN log2fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx log2fNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{log2fN, TS 18661-3:2015, math.h}
+@standardsx{log2fNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the base-2 logarithm of @var{x}.
@code{log2 (@var{x})} equals @code{log (@var{x}) / log (2)}.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double logb (double @var{x})
@deftypefunx float logbf (float @var{x})
@deftypefunx {long double} logbl (long double @var{x})
+@deftypefunx _FloatN logbfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx logbfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{logbfN, TS 18661-3:2015, math.h}
+@standardsx{logbfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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.
+floating-point value. If @code{FLT_RADIX} is two, @code{logb} is equal
+to @code{floor (log2 (x))}, except it's probably faster.
-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.
+If @var{x} is de-normalized, @code{logb} returns the exponent @var{x}
+would have if it were normalized. If @var{x} is infinity (positive or
+negative), @code{logb} returns @math{@infinity{}}. If @var{x} is zero,
+@code{logb} returns @math{@infinity{}}. It does not signal.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun int ilogb (double @var{x})
@deftypefunx int ilogbf (float @var{x})
@deftypefunx int ilogbl (long double @var{x})
+@deftypefunx int ilogbfN (_Float@var{N} @var{x})
+@deftypefunx int ilogbfNx (_Float@var{N}x @var{x})
+@deftypefunx {long int} llogb (double @var{x})
+@deftypefunx {long int} llogbf (float @var{x})
+@deftypefunx {long int} llogbl (long double @var{x})
+@deftypefunx {long int} llogbfN (_Float@var{N} @var{x})
+@deftypefunx {long int} llogbfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{ilogbfN, TS 18661-3:2015, math.h}
+@standardsx{ilogbfNx, TS 18661-3:2015, math.h}
+@standardsx{llogbfN, TS 18661-3:2015, math.h}
+@standardsx{llogbfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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:
+functions except that they return signed integer values. The
+@code{ilogb}, @code{ilogbf}, and @code{ilogbl} functions are from ISO
+C99; the @code{llogb}, @code{llogbf}, @code{llogbl} functions are from
+TS 18661-1:2014; the @code{ilogbfN}, @code{ilogbfNx}, @code{llogbfN},
+and @code{llogbfNx} functions are from TS 18661-3:2015.
+@end deftypefun
+
+@noindent
+Since integers cannot represent infinity and NaN, @code{ilogb} instead
+returns an integer that can't be the exponent of a normal floating-point
+number. @file{math.h} defines constants so you can check for this.
+
+@deftypevr Macro int FP_ILOGB0
+@standards{ISO, math.h}
+@code{ilogb} returns this value if its argument is @code{0}. The
+numeric value is either @code{INT_MIN} or @code{-INT_MAX}.
+
+This macro is defined in @w{ISO C99}.
+@end deftypevr
+
+@deftypevr Macro {long int} FP_LLOGB0
+@standards{ISO, math.h}
+@code{llogb} returns this value if its argument is @code{0}. The
+numeric value is either @code{LONG_MIN} or @code{-LONG_MAX}.
+
+This macro is defined in TS 18661-1:2014.
+@end deftypevr
+
+@deftypevr Macro int FP_ILOGBNAN
+@standards{ISO, math.h}
+@code{ilogb} returns this value if its argument is @code{NaN}. The
+numeric value is either @code{INT_MIN} or @code{INT_MAX}.
+
+This macro is defined in @w{ISO C99}.
+@end deftypevr
+
+@deftypevr Macro {long int} FP_LLOGBNAN
+@standards{ISO, math.h}
+@code{llogb} returns this value if its argument is @code{NaN}. The
+numeric value is either @code{LONG_MIN} or @code{LONG_MAX}.
+
+This macro is defined in TS 18661-1:2014.
+@end deftypevr
+
+These values are system specific. They might even be the same. The
+proper way to test the result of @code{ilogb} is as follows:
@smallexample
i = ilogb (f);
@}
@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})
+@deftypefunx _FloatN powfN (_Float@var{N} @var{base}, _Float@var{N} @var{power})
+@deftypefunx _FloatNx powfNx (_Float@var{N}x @var{base}, _Float@var{N}x @var{power})
+@standards{ISO, math.h}
+@standardsx{powfN, TS 18661-3:2015, math.h}
+@standardsx{powfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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
+Mathematically, @code{pow} would return a complex number when @var{base}
+is negative and @var{power} is not an integral value. @code{pow} can't
+do that, so instead it signals a domain error. @code{pow} may also
+underflow or overflow the destination type.
@end deftypefun
@cindex square root function
-@comment math.h
-@comment ISO
@deftypefun double sqrt (double @var{x})
@deftypefunx float sqrtf (float @var{x})
@deftypefunx {long double} sqrtl (long double @var{x})
+@deftypefunx _FloatN sqrtfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx sqrtfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{sqrtfN, TS 18661-3:2015, math.h}
+@standardsx{sqrtfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+If @var{x} is negative, @code{sqrt} signals a domain error.
+Mathematically, it should return a complex number.
@end deftypefun
@cindex cube root function
-@comment math.h
-@comment BSD
@deftypefun double cbrt (double @var{x})
@deftypefunx float cbrtf (float @var{x})
@deftypefunx {long double} cbrtl (long double @var{x})
+@deftypefunx _FloatN cbrtfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx cbrtfNx (_Float@var{N}x @var{x})
+@standards{BSD, math.h}
+@standardsx{cbrtfN, TS 18661-3:2015, math.h}
+@standardsx{cbrtfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+@deftypefunx _FloatN hypotfN (_Float@var{N} @var{x}, _Float@var{N} @var{y})
+@deftypefunx _FloatNx hypotfNx (_Float@var{N}x @var{x}, _Float@var{N}x @var{y})
+@standards{ISO, math.h}
+@standardsx{hypotfN, TS 18661-3:2015, math.h}
+@standardsx{hypotfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
These functions return @code{sqrt (@var{x}*@var{x} +
-@var{y}*@var{y})}. (This is the length of the hypotenuse of a right
+@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
+of the point (@var{x}, @var{y}) from the origin. Using this function
+instead of the direct formula is wise, since the error is
much smaller. See also the function @code{cabs} in @ref{Absolute Value}.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double expm1 (double @var{x})
@deftypefunx float expm1f (float @var{x})
@deftypefunx {long double} expm1l (long double @var{x})
+@deftypefunx _FloatN expm1fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx expm1fNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{expm1fN, TS 18661-3:2015, math.h}
+@standardsx{expm1fNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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
+They are computed in a way that is accurate even if @var{x} is
+near zero---a case where @code{exp (@var{x}) - 1} would be inaccurate owing
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
+@deftypefunx _FloatN log1pfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx log1pfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{log1pfN, TS 18661-3:2015, math.h}
+@standardsx{log1pfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return a value equivalent to @w{@code{log (1 + @var{x})}}.
+They are computed in a way that is accurate even if @var{x} is
near zero.
@end deftypefun
@cindex complex exponentiation functions
@cindex complex logarithm functions
-@w{ISO C 9X} defines variants of some of the exponentiation and
-logarithm functions. As for the other functions handling complex
-numbers these functions are perhaps better optimized and provide better
-error checking than a direct use of the formulas of the mathematical
-definition.
+@w{ISO C99} defines complex variants of some of the exponentiation and
+logarithm functions.
-@comment complex.h
-@comment ISO
@deftypefun {complex double} cexp (complex double @var{z})
@deftypefunx {complex float} cexpf (complex float @var{z})
@deftypefunx {complex long double} cexpl (complex long double @var{z})
-These functions return 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
+@deftypefunx {complex _FloatN} cexpfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} cexpfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{cexpfN, TS 18661-3:2015, complex.h}
+@standardsx{cexpfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return @code{e} (the base of natural
+logarithms) raised to the power of @var{z}.
+Mathematically, this corresponds to the value
+
+@ifnottex
@math{exp (z) = exp (creal (z)) * (cos (cimag (z)) + I * sin (cimag (z)))}
-@end ifinfo
-@iftex
+@end ifnottex
@tex
-$$\exp(z) = e^z = e^{{\rm Re} z} (\cos ({\rm Im} z) + i \sin ({\rm Im} z))$$
+$$\exp(z) = e^z = e^{{\rm Re}\,z} (\cos ({\rm Im}\,z) + i \sin ({\rm Im}\,z))$$
@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
+@deftypefunx {complex _FloatN} clogfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} clogfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{clogfN, TS 18661-3:2015, complex.h}
+@standardsx{clogfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the natural logarithm of @var{z}.
+Mathematically, this corresponds to the value
+
+@ifnottex
@math{log (z) = log (cabs (z)) + I * carg (z)}
-@end ifinfo
-@iftex
+@end ifnottex
@tex
-$$\log(z) = \log(|z|) + i \arg(z)$$
+$$\log(z) = \log |z| + i \arg z$$
@end tex
-@end iftex
+
+@noindent
+@code{clog} has a pole at 0, and will signal overflow if @var{z} equals
+or is very close to 0. It is well-defined for all other values of
+@var{z}.
@end deftypefun
-@comment complex.h
-@comment GNU
@deftypefun {complex double} clog10 (complex double @var{z})
@deftypefunx {complex float} clog10f (complex float @var{z})
@deftypefunx {complex long double} clog10l (complex long double @var{z})
+@deftypefunx {complex _FloatN} clog10fN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} clog10fNx (complex _Float@var{N}x @var{z})
+@standards{GNU, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
These functions return the base 10 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
+@var{z}. Mathematically, this corresponds to the value
-@ifinfo
-@math{log (z) = log10 (cabs (z)) + I * carg (z)}
-@end ifinfo
-@iftex
+@ifnottex
+@math{log10 (z) = log10 (cabs (z)) + I * carg (z) / log (10)}
+@end ifnottex
@tex
-$$\log_{10}(z) = \log_{10}(|z|) + i \arg(z)$$
+$$\log_{10}(z) = \log_{10}|z| + i \arg z / \log (10)$$
@end tex
-@end iftex
-This function is a GNU extension.
+All these functions, including the @code{_Float@var{N}} and
+@code{_Float@var{N}x} variants, are GNU extensions.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} csqrt (complex double @var{z})
@deftypefunx {complex float} csqrtf (complex float @var{z})
@deftypefunx {complex long double} csqrtl (complex long double @var{z})
-These functions return the complex root of the argument @var{z}. Unlike
-the @code{sqrt} function these functions do not have any restriction on
-the value of the argument.
+@deftypefunx {complex _FloatN} csqrtfN (_Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} csqrtfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{csqrtfN, TS 18661-3:2015, complex.h}
+@standardsx{csqrtfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the complex square root of the argument @var{z}. Unlike
+the real-valued functions, they are defined for all values of @var{z}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} cpow (complex double @var{base}, complex double @var{power})
@deftypefunx {complex float} cpowf (complex float @var{base}, complex float @var{power})
@deftypefunx {complex long double} cpowl (complex long double @var{base}, complex long double @var{power})
-These functions return 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
+@deftypefunx {complex _FloatN} cpowfN (complex _Float@var{N} @var{base}, complex _Float@var{N} @var{power})
+@deftypefunx {complex _FloatNx} cpowfNx (complex _Float@var{N}x @var{base}, complex _Float@var{N}x @var{power})
+@standards{ISO, complex.h}
+@standardsx{cpowfN, TS 18661-3:2015, complex.h}
+@standardsx{cpowfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return @var{base} raised to the power of
+@var{power}. This is equivalent to @w{@code{cexp (y * clog (x))}}
@end deftypefun
-
@node Hyperbolic Functions
@section Hyperbolic Functions
@cindex hyperbolic functions
The functions in this section are related to the exponential functions;
see @ref{Exponents and Logarithms}.
-@comment math.h
-@comment ISO
@deftypefun double sinh (double @var{x})
@deftypefunx float sinhf (float @var{x})
@deftypefunx {long double} sinhl (long double @var{x})
+@deftypefunx _FloatN sinhfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx sinhfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{sinhfN, TS 18661-3:2015, math.h}
+@standardsx{sinhfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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.
+mathematically as @w{@code{(exp (@var{x}) - exp (-@var{x})) / 2}}. They
+may signal overflow if @var{x} is too large.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double cosh (double @var{x})
@deftypefunx float coshf (float @var{x})
@deftypefunx {long double} coshl (long double @var{x})
-These function return the hyperbolic cosine of @var{x},
+@deftypefunx _FloatN coshfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx coshfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{coshfN, TS 18661-3:2015, math.h}
+@standardsx{coshfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions 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.
+They may signal overflow if @var{x} is too large.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double tanh (double @var{x})
@deftypefunx float tanhf (float @var{x})
@deftypefunx {long double} tanhl (long double @var{x})
-These functions return the hyperbolic tangent of @var{x}, whose
-mathematical definition is @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
+@deftypefunx _FloatN tanhfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx tanhfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{tanhfN, TS 18661-3:2015, math.h}
+@standardsx{tanhfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the hyperbolic tangent of @var{x},
+defined mathematically as @w{@code{sinh (@var{x}) / cosh (@var{x})}}.
+They may signal overflow if @var{x} is too large.
@end deftypefun
@cindex hyperbolic functions
-There are counterparts for 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.
+There are counterparts for the hyperbolic functions which take
+complex arguments.
-@comment complex.h
-@comment ISO
@deftypefun {complex double} csinh (complex double @var{z})
@deftypefunx {complex float} csinhf (complex float @var{z})
@deftypefunx {complex long double} csinhl (complex long double @var{z})
+@deftypefunx {complex _FloatN} csinhfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} csinhfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{csinhfN, TS 18661-3:2015, complex.h}
+@standardsx{csinhfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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.
+mathematically as @w{@code{(exp (@var{z}) - exp (-@var{z})) / 2}}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} ccosh (complex double @var{z})
@deftypefunx {complex float} ccoshf (complex float @var{z})
@deftypefunx {complex long double} ccoshl (complex long double @var{z})
+@deftypefunx {complex _FloatN} ccoshfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} ccoshfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{ccoshfN, TS 18661-3:2015, complex.h}
+@standardsx{ccoshfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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.
+mathematically as @w{@code{(exp (@var{z}) + exp (-@var{z})) / 2}}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} ctanh (complex double @var{z})
@deftypefunx {complex float} ctanhf (complex float @var{z})
@deftypefunx {complex long double} ctanhl (complex long double @var{z})
-These functions return the complex hyperbolic tangent of @var{z}, whose
-mathematical definition is @w{@code{csinh (@var{z}) / ccosh (@var{z})}}.
+@deftypefunx {complex _FloatN} ctanhfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} ctanhfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{ctanhfN, TS 18661-3:2015, complex.h}
+@standardsx{ctanhfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+These functions return the complex hyperbolic tangent of @var{z},
+defined mathematically as @w{@code{csinh (@var{z}) / ccosh (@var{z})}}.
@end deftypefun
@cindex inverse hyperbolic functions
-@comment math.h
-@comment ISO
@deftypefun double asinh (double @var{x})
@deftypefunx float asinhf (float @var{x})
@deftypefunx {long double} asinhl (long double @var{x})
+@deftypefunx _FloatN asinhfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx asinhfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{asinhfN, TS 18661-3:2015, math.h}
+@standardsx{asinhfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+@deftypefunx _FloatN acoshfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx acoshfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{acoshfN, TS 18661-3:2015, math.h}
+@standardsx{acoshfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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}.
+@code{1}, @code{acosh} signals a domain error.
@end deftypefun
-@comment math.h
-@comment ISO
@deftypefun double atanh (double @var{x})
@deftypefunx float atanhf (float @var{x})
@deftypefunx {long double} atanhl (long double @var{x})
+@deftypefunx _FloatN atanhfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx atanhfNx (_Float@var{N}x @var{x})
+@standards{ISO, math.h}
+@standardsx{atanhfN, TS 18661-3:2015, math.h}
+@standardsx{atanhfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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}.
+@var{x} is greater than @code{1}, @code{atanh} signals a domain error;
+if it is equal to 1, @code{atanh} returns infinity.
@end deftypefun
@cindex inverse complex hyperbolic functions
-@comment complex.h
-@comment ISO
@deftypefun {complex double} casinh (complex double @var{z})
@deftypefunx {complex float} casinhf (complex float @var{z})
@deftypefunx {complex long double} casinhl (complex long double @var{z})
+@deftypefunx {complex _FloatN} casinhfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} casinhfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{casinhfN, TS 18661-3:2015, complex.h}
+@standardsx{casinhfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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})
+@deftypefunx {complex _FloatN} cacoshfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} cacoshfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{cacoshfN, TS 18661-3:2015, complex.h}
+@standardsx{cacoshfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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.
+the real-valued functions, there are no restrictions on the value of @var{z}.
@end deftypefun
-@comment complex.h
-@comment ISO
@deftypefun {complex double} catanh (complex double @var{z})
@deftypefunx {complex float} catanhf (complex float @var{z})
@deftypefunx {complex long double} catanhl (complex long double @var{z})
+@deftypefunx {complex _FloatN} catanhfN (complex _Float@var{N} @var{z})
+@deftypefunx {complex _FloatNx} catanhfNx (complex _Float@var{N}x @var{z})
+@standards{ISO, complex.h}
+@standardsx{catanhfN, TS 18661-3:2015, complex.h}
+@standardsx{catanhfNx, TS 18661-3:2015, complex.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
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
+the real-valued functions, there are no restrictions on the value of
+@var{z}.
+@end deftypefun
+
+@node Special Functions
+@section Special Functions
+@cindex special functions
+@cindex Bessel functions
+@cindex gamma function
+
+These are some more exotic mathematical functions which are sometimes
+useful. Currently they only have real-valued versions.
+
+@deftypefun double erf (double @var{x})
+@deftypefunx float erff (float @var{x})
+@deftypefunx {long double} erfl (long double @var{x})
+@deftypefunx _FloatN erffN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx erffNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{erffN, TS 18661-3:2015, math.h}
+@standardsx{erffNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{erf} returns the error function of @var{x}. The error
+function is defined as
+@tex
+$$\hbox{erf}(x) = {2\over\sqrt{\pi}}\cdot\int_0^x e^{-t^2} \hbox{d}t$$
+@end tex
+@ifnottex
+@smallexample
+erf (x) = 2/sqrt(pi) * integral from 0 to x of exp(-t^2) dt
+@end smallexample
+@end ifnottex
+@end deftypefun
+
+@deftypefun double erfc (double @var{x})
+@deftypefunx float erfcf (float @var{x})
+@deftypefunx {long double} erfcl (long double @var{x})
+@deftypefunx _FloatN erfcfN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx erfcfNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{erfcfN, TS 18661-3:2015, math.h}
+@standardsx{erfcfNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{erfc} returns @code{1.0 - erf(@var{x})}, but computed in a
+fashion that avoids round-off error when @var{x} is large.
+@end deftypefun
+
+@deftypefun double lgamma (double @var{x})
+@deftypefunx float lgammaf (float @var{x})
+@deftypefunx {long double} lgammal (long double @var{x})
+@deftypefunx _FloatN lgammafN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx lgammafNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{lgammafN, TS 18661-3:2015, math.h}
+@standardsx{lgammafNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:signgam}}@asunsafe{}@acsafe{}}
+@code{lgamma} returns the natural logarithm of the absolute value of
+the gamma function of @var{x}. The gamma function is defined as
+@tex
+$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$
+@end tex
+@ifnottex
+@smallexample
+gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt
+@end smallexample
+@end ifnottex
+
+@vindex signgam
+The sign of the gamma function is stored in the global variable
+@var{signgam}, which is declared in @file{math.h}. It is @code{1} if
+the intermediate result was positive or zero, or @code{-1} if it was
+negative.
+
+To compute the real gamma function you can use the @code{tgamma}
+function or you can compute the values as follows:
+@smallexample
+lgam = lgamma(x);
+gam = signgam*exp(lgam);
+@end smallexample
+
+The gamma function has singularities at the non-positive integers.
+@code{lgamma} will raise the zero divide exception if evaluated at a
+singularity.
+@end deftypefun
+
+@deftypefun double lgamma_r (double @var{x}, int *@var{signp})
+@deftypefunx float lgammaf_r (float @var{x}, int *@var{signp})
+@deftypefunx {long double} lgammal_r (long double @var{x}, int *@var{signp})
+@deftypefunx _FloatN lgammafN_r (_Float@var{N} @var{x}, int *@var{signp})
+@deftypefunx _FloatNx lgammafNx_r (_Float@var{N}x @var{x}, int *@var{signp})
+@standards{XPG, math.h}
+@standardsx{lgammafN_r, GNU, math.h}
+@standardsx{lgammafNx_r, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{lgamma_r} is just like @code{lgamma}, but it stores the sign of
+the intermediate result in the variable pointed to by @var{signp}
+instead of in the @var{signgam} global. This means it is reentrant.
+
+The @code{lgammaf@var{N}_r} and @code{lgammaf@var{N}x_r} functions are
+GNU extensions.
+@end deftypefun
+
+@deftypefun double gamma (double @var{x})
+@deftypefunx float gammaf (float @var{x})
+@deftypefunx {long double} gammal (long double @var{x})
+@standards{SVID, math.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:signgam}}@asunsafe{}@acsafe{}}
+These functions exist for compatibility reasons. They are equivalent to
+@code{lgamma} etc. It is better to use @code{lgamma} since for one the
+name reflects better the actual computation, and moreover @code{lgamma} is
+standardized in @w{ISO C99} while @code{gamma} is not.
+@end deftypefun
+
+@deftypefun double tgamma (double @var{x})
+@deftypefunx float tgammaf (float @var{x})
+@deftypefunx {long double} tgammal (long double @var{x})
+@deftypefunx _FloatN tgammafN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx tgammafNx (_Float@var{N}x @var{x})
+@standardsx{tgamma, XPG, math.h}
+@standardsx{tgamma, ISO, math.h}
+@standardsx{tgammaf, XPG, math.h}
+@standardsx{tgammaf, ISO, math.h}
+@standardsx{tgammal, XPG, math.h}
+@standardsx{tgammal, ISO, math.h}
+@standardsx{tgammafN, TS 18661-3:2015, math.h}
+@standardsx{tgammafNx, TS 18661-3:2015, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{tgamma} applies the gamma function to @var{x}. The gamma
+function is defined as
+@tex
+$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \hbox{d}t$$
+@end tex
+@ifnottex
+@smallexample
+gamma (x) = integral from 0 to @infinity{} of t^(x-1) e^-t dt
+@end smallexample
+@end ifnottex
+
+This function was introduced in @w{ISO C99}. The @code{_Float@var{N}}
+and @code{_Float@var{N}x} variants were introduced in @w{ISO/IEC TS
+18661-3}.
+@end deftypefun
+
+@deftypefun double j0 (double @var{x})
+@deftypefunx float j0f (float @var{x})
+@deftypefunx {long double} j0l (long double @var{x})
+@deftypefunx _FloatN j0fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx j0fNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{j0fN, GNU, math.h}
+@standardsx{j0fNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{j0} returns the Bessel function of the first kind of order 0 of
+@var{x}. It may signal underflow if @var{x} is too large.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@deftypefun double j1 (double @var{x})
+@deftypefunx float j1f (float @var{x})
+@deftypefunx {long double} j1l (long double @var{x})
+@deftypefunx _FloatN j1fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx j1fNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{j1fN, GNU, math.h}
+@standardsx{j1fNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{j1} returns the Bessel function of the first kind of order 1 of
+@var{x}. It may signal underflow if @var{x} is too large.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@deftypefun double jn (int @var{n}, double @var{x})
+@deftypefunx float jnf (int @var{n}, float @var{x})
+@deftypefunx {long double} jnl (int @var{n}, long double @var{x})
+@deftypefunx _FloatN jnfN (int @var{n}, _Float@var{N} @var{x})
+@deftypefunx _FloatNx jnfNx (int @var{n}, _Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{jnfN, GNU, math.h}
+@standardsx{jnfNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{jn} returns the Bessel function of the first kind of order
+@var{n} of @var{x}. It may signal underflow if @var{x} is too large.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@deftypefun double y0 (double @var{x})
+@deftypefunx float y0f (float @var{x})
+@deftypefunx {long double} y0l (long double @var{x})
+@deftypefunx _FloatN y0fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx y0fNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{y0fN, GNU, math.h}
+@standardsx{y0fNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{y0} returns the Bessel function of the second kind of order 0 of
+@var{x}. It may signal underflow if @var{x} is too large. If @var{x}
+is negative, @code{y0} signals a domain error; if it is zero,
+@code{y0} signals overflow and returns @math{-@infinity}.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@deftypefun double y1 (double @var{x})
+@deftypefunx float y1f (float @var{x})
+@deftypefunx {long double} y1l (long double @var{x})
+@deftypefunx _FloatN y1fN (_Float@var{N} @var{x})
+@deftypefunx _FloatNx y1fNx (_Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{y1fN, GNU, math.h}
+@standardsx{y1fNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{y1} returns the Bessel function of the second kind of order 1 of
+@var{x}. It may signal underflow if @var{x} is too large. If @var{x}
+is negative, @code{y1} signals a domain error; if it is zero,
+@code{y1} signals overflow and returns @math{-@infinity}.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@deftypefun double yn (int @var{n}, double @var{x})
+@deftypefunx float ynf (int @var{n}, float @var{x})
+@deftypefunx {long double} ynl (int @var{n}, long double @var{x})
+@deftypefunx _FloatN ynfN (int @var{n}, _Float@var{N} @var{x})
+@deftypefunx _FloatNx ynfNx (int @var{n}, _Float@var{N}x @var{x})
+@standards{SVID, math.h}
+@standardsx{ynfN, GNU, math.h}
+@standardsx{ynfNx, GNU, math.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
+@code{yn} returns the Bessel function of the second kind of order @var{n} of
+@var{x}. It may signal underflow if @var{x} is too large. If @var{x}
+is negative, @code{yn} signals a domain error; if it is zero,
+@code{yn} signals overflow and returns @math{-@infinity}.
+
+The @code{_Float@var{N}} and @code{_Float@var{N}x} variants are GNU
+extensions.
+@end deftypefun
+
+@node Errors in Math Functions
+@section Known Maximum Errors in Math Functions
+@cindex math errors
+@cindex ulps
+
+This section lists the known errors of the functions in the math
+library. Errors are measured in ``units of the last place''. This is a
+measure for the relative error. For a number @math{z} with the
+representation @math{d.d@dots{}d@mul{}2^e} (we assume IEEE
+floating-point numbers with base 2) the ULP is represented by
+
+@tex
+$${|d.d\dots d - (z/2^e)|}\over {2^{p-1}}$$
+@end tex
+@ifnottex
+@smallexample
+|d.d...d - (z / 2^e)| / 2^(p - 1)
+@end smallexample
+@end ifnottex
+
+@noindent
+where @math{p} is the number of bits in the mantissa of the
+floating-point number representation. Ideally the error for all
+functions is always less than 0.5ulps in round-to-nearest mode. Using
+rounding bits this is also
+possible and normally implemented for the basic operations. Except
+for certain functions such as @code{sqrt}, @code{fma} and @code{rint}
+whose results are fully specified by reference to corresponding IEEE
+754 floating-point operations, and conversions between strings and
+floating point, @theglibc{} does not aim for correctly rounded results
+for functions in the math library, and does not aim for correctness in
+whether ``inexact'' exceptions are raised. Instead, the goals for
+accuracy of functions without fully specified results are as follows;
+some functions have bugs meaning they do not meet these goals in all
+cases. In the future, @theglibc{} may provide some other correctly
+rounding functions under the names such as @code{crsin} proposed for
+an extension to ISO C.
+
+@itemize @bullet
+
+@item
+Each function with a floating-point result behaves as if it computes
+an infinite-precision result that is within a few ulp (in both real
+and complex parts, for functions with complex results) of the
+mathematically correct value of the function (interpreted together
+with ISO C or POSIX semantics for the function in question) at the
+exact value passed as the input. Exceptions are raised appropriately
+for this value and in accordance with IEEE 754 / ISO C / POSIX
+semantics, and it is then rounded according to the current rounding
+direction to the result that is returned to the user. @code{errno}
+may also be set (@pxref{Math Error Reporting}). (The ``inexact''
+exception may be raised, or not raised, even if this is inconsistent
+with the infinite-precision value.)
+
+@item
+For the IBM @code{long double} format, as used on PowerPC GNU/Linux,
+the accuracy goal is weaker for input values not exactly representable
+in 106 bits of precision; it is as if the input value is some value
+within 0.5ulp of the value actually passed, where ``ulp'' is
+interpreted in terms of a fixed-precision 106-bit mantissa, but not
+necessarily the exact value actually passed with discontiguous
+mantissa bits.
+
+@item
+For the IBM @code{long double} format, functions whose results are
+fully specified by reference to corresponding IEEE 754 floating-point
+operations have the same accuracy goals as other functions, but with
+the error bound being the same as that for division (3ulp).
+Furthermore, ``inexact'' and ``underflow'' exceptions may be raised
+for all functions for any inputs, even where such exceptions are
+inconsistent with the returned value, since the underlying
+floating-point arithmetic has that property.
+
+@item
+Functions behave as if the infinite-precision result computed is zero,
+infinity or NaN if and only if that is the mathematically correct
+infinite-precision result. They behave as if the infinite-precision
+result computed always has the same sign as the mathematically correct
+result.
+
+@item
+If the mathematical result is more than a few ulp above the overflow
+threshold for the current rounding direction, the value returned is
+the appropriate overflow value for the current rounding direction,
+with the overflow exception raised.
+
+@item
+If the mathematical result has magnitude well below half the least
+subnormal magnitude, the returned value is either zero or the least
+subnormal (in each case, with the correct sign), according to the
+current rounding direction and with the underflow exception raised.
+@item
+Where the mathematical result underflows (before rounding) and is not
+exactly representable as a floating-point value, the function does not
+behave as if the computed infinite-precision result is an exact value
+in the subnormal range. This means that the underflow exception is
+raised other than possibly for cases where the mathematical result is
+very close to the underflow threshold and the function behaves as if
+it computes an infinite-precision result that does not underflow. (So
+there may be spurious underflow exceptions in cases where the
+underflowing result is exact, but not missing underflow exceptions in
+cases where it is inexact.)
+
+@item
+@Theglibc{} does not aim for functions to satisfy other properties of
+the underlying mathematical function, such as monotonicity, where not
+implied by the above goals.
+
+@item
+All the above applies to both real and complex parts, for complex
+functions.
+
+@end itemize
+
+Therefore many of the functions in the math library have errors. The
+table lists the maximum error for each function which is exposed by one
+of the existing tests in the test suite. The table tries to cover as much
+as possible and list the actual maximum error (or at least a ballpark
+figure) but this is often not achieved due to the large search space.
+
+The table lists the ULP values for different architectures. Different
+architectures have different results since their hardware support for
+floating-point operations varies and also the existing hardware support
+is different. Only the round-to-nearest rounding mode is covered by
+this table. Functions not listed do not have known errors. Vector
+versions of functions in the x86_64 libmvec library have a maximum error
+of 4 ulps.
+
+@page
+@c This multitable does not fit on a single page
+@include libm-err.texi
@node Pseudo-Random Numbers
@section Pseudo-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.
+typically, they form a sequence that repeats periodically, with a period
+so large that you can ignore it for ordinary purposes. The random
+number generator works by remembering a @dfn{seed} value which it uses
+to compute the next random number and also to compute a new seed.
Although the generated numbers look unpredictable within one run of a
program, the sequence of numbers is @emph{exactly the same} from one run
to the next. This is because the initial seed is always the same. This
is convenient when you are debugging a program, but it is unhelpful if
-you want the program to behave unpredictably. If you want truly random
-numbers, not just pseudo-random, specify a seed based on the current
-time.
+you want the program to behave unpredictably. If you want a different
+pseudo-random series each time your program runs, you must specify a
+different seed each time. For ordinary purposes, basing the seed on the
+current time works well. For random numbers in cryptography,
+@pxref{Unpredictable Bytes}.
-You can get repeatable sequences of numbers on a particular machine type
+You can obtain repeatable sequences of numbers on a particular machine type
by specifying the same initial seed value for the random number
generator. There is no standard meaning for a particular seed value;
the same seed, used in different C libraries or on different CPU types,
will give you different random numbers.
-The GNU library supports the standard @w{ISO C} random number functions
-plus two other sets derived from BSD and SVID. We recommend you use the
-standard ones, @code{rand} and @code{srand} if only a small number of
-random bits are required. The SVID functions provide an interface which
-allows better random number generator algorithms and they return up to
-48 random bits in one calls and they also return random floating-point
-numbers if wanted. The SVID function might not be available on some BSD
-derived systems but since they are required in the XPG they are
-available on all Unix-conformant systems.
+@Theglibc{} supports the standard @w{ISO C} random number functions
+plus two other sets derived from BSD and SVID. The BSD and @w{ISO C}
+functions provide identical, somewhat limited functionality. If only a
+small number of random bits are required, we recommend you use the
+@w{ISO C} interface, @code{rand} and @code{srand}. The SVID functions
+provide a more flexible interface, which allows better random number
+generator algorithms, provides more random bits (up to 48) per call, and
+can provide random floating-point numbers. These functions are required
+by the XPG standard and therefore will be present in all modern Unix
+systems.
@menu
-* ISO Random:: @code{rand} and friends.
-* BSD Random:: @code{random} and friends.
-* SVID Random:: @code{drand48} and friends.
+* ISO Random:: @code{rand} and friends.
+* BSD Random:: @code{random} and friends.
+* SVID Random:: @code{drand48} and friends.
@end menu
@node ISO Random
@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}.
+@standards{ISO, stdlib.h}
+The value of this macro is an integer constant representing the largest
+value the @code{rand} function can return. In @theglibc{}, it is
+@code{2147483647}, which is the largest signed integer representable in
+32 bits. In other libraries, it may be as low as @code{32767}.
@end deftypevr
-@comment stdlib.h
-@comment ISO
@deftypefun int rand (void)
+@standards{ISO, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
+@c Just calls random.
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}.
+series. The value ranges from @code{0} to @code{RAND_MAX}.
@end deftypefun
-@comment stdlib.h
-@comment ISO
@deftypefun void srand (unsigned int @var{seed})
+@standards{ISO, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
+@c Alias to srandom.
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))}.
+To produce a different pseudo-random series each time your program is
+run, do @code{srand (time (0))}.
@end deftypefun
-A completely broken interface was designed by the POSIX.1 committee to
-support reproducible random numbers in multi-threaded programs.
+POSIX.1 extended the C standard functions to support reproducible random
+numbers in multi-threaded programs. However, the extension is badly
+designed and unsuitable for serious work.
-@comment stdlib.h
-@comment POSIX.1
@deftypefun int rand_r (unsigned int *@var{seed})
+@standards{POSIX.1, stdlib.h}
+@safety{@prelim{}@mtsafe{}@assafe{}@acsafe{}}
This function returns a random number in the range 0 to @code{RAND_MAX}
-just as @code{rand} does. But this function does not keep an internal
-state for the RNG. Instead the @code{unsigned int} variable pointed to
-by the argument @var{seed} is the only state. Before the value is
-returned the state will be updated so that the next call will return a
-new number.
+just as @code{rand} does. However, all its state is stored in the
+@var{seed} argument. This means the RNG's state can only have as many
+bits as the type @code{unsigned int} has. This is far too few to
+provide a good RNG.
-I.e., the state of the RNG can only have as much bits as the type
-@code{unsigned int} has. This is far too few to provide a good RNG.
-This interface is broken by design.
-
-If the program requires reproducible random numbers in multi-threaded
-programs the reentrant SVID functions are probably a better choice. But
-these functions are GNU extensions and therefore @code{rand_r}, as being
-standardized in POSIX.1, should always be kept as a default method.
+If your program requires a reentrant RNG, we recommend you use the
+reentrant GNU extensions to the SVID random number generator. The
+POSIX.1 interface should only be used when the GNU extensions are not
+available.
@end deftypefun
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.
+with @theglibc{}; 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)
+@deftypefun {long int} random (void)
+@standards{BSD, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
+@c Takes a lock and calls random_r with an automatic variable and the
+@c global state, while holding a lock.
This function returns the next pseudo-random number in the sequence.
-The range of values returned is from @code{0} to @code{RAND_MAX}.
+The value returned ranges from @code{0} to @code{2147483647}.
-@strong{Please note:} Historically this function returned a @code{long
-int} value. But with the appearance of 64bit machines this could lead
-to severe compatibility problems and therefore the type now explicitly
-limits the return value to 32bit.
+@strong{NB:} Temporarily this function was defined to return a
+@code{int32_t} value to indicate that the return value always contains
+32 bits even if @code{long int} is wider. The standard demands it
+differently. Users must always be aware of the 32-bit limitation,
+though.
@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
+@standards{BSD, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
+@c Takes a lock and calls srandom_r with an automatic variable and a
+@c static buffer. There's no MT-safety issue because the static buffer
+@c is internally protected by a lock, although other threads may modify
+@c the set state before it is used.
+The @code{srandom} function sets the state of the random number
+generator based on the integer @var{seed}. If you supply a @var{seed} value
of @code{1}, this will cause @code{random} to reproduce the default set
of random numbers.
-To produce truly random numbers (not just pseudo-random), do
-@code{srandom (time (0))}.
+To produce a different set of pseudo-random numbers each time your
+program runs, do @code{srandom (time (0))}.
@end deftypefun
-@comment stdlib.h
-@comment BSD
-@deftypefun {void *} initstate (unsigned int @var{seed}, void *@var{state}, size_t @var{size})
+@deftypefun {char *} initstate (unsigned int @var{seed}, char *@var{state}, size_t @var{size})
+@standards{BSD, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
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.
+bytes, used to hold the state information. It is initialized based on
+@var{seed}. The size must be between 8 and 256 bytes, and should be a
+power of two. The bigger the @var{state} array, the better.
The return value is the previous value of the state information array.
You can use this value later as an argument to @code{setstate} to
restore that state.
@end deftypefun
-@comment stdlib.h
-@comment BSD
-@deftypefun {void *} setstate (void *@var{state})
+@deftypefun {char *} setstate (char *@var{state})
+@standards{BSD, stdlib.h}
+@safety{@prelim{}@mtsafe{}@asunsafe{@asulock{}}@acunsafe{@aculock{}}}
The @code{setstate} function restores the random number state
information @var{state}. The argument must have been the result of
a previous call to @var{initstate} or @var{setstate}.
The return value is the previous value of the state information array.
You can use this value later as an argument to @code{setstate} to
restore that state.
+
+If the function fails the return value is @code{NULL}.
+@end deftypefun
+
+The four functions described so far in this section all work on a state
+which is shared by all threads. The state is not directly accessible to
+the user and can only be modified by these functions. This makes it
+hard to deal with situations where each thread should have its own
+pseudo-random number generator.
+
+@Theglibc{} contains four additional functions which contain the
+state as an explicit parameter and therefore make it possible to handle
+thread-local PRNGs. Besides this there is no difference. In fact, the
+four functions already discussed are implemented internally using the
+following interfaces.
+
+The @file{stdlib.h} header contains a definition of the following type:
+
+@deftp {Data Type} {struct random_data}
+@standards{GNU, stdlib.h}
+
+Objects of type @code{struct random_data} contain the information
+necessary to represent the state of the PRNG. Although a complete
+definition of the type is present the type should be treated as opaque.
+@end deftp
+
+The functions modifying the state follow exactly the already described
+functions.
+
+@deftypefun int random_r (struct random_data *restrict @var{buf}, int32_t *restrict @var{result})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buf}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{random_r} function behaves exactly like the @code{random}
+function except that it uses and modifies the state in the object
+pointed to by the first parameter instead of the global state.
+@end deftypefun
+
+@deftypefun int srandom_r (unsigned int @var{seed}, struct random_data *@var{buf})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buf}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{srandom_r} function behaves exactly like the @code{srandom}
+function except that it uses and modifies the state in the object
+pointed to by the second parameter instead of the global state.
+@end deftypefun
+
+@deftypefun int initstate_r (unsigned int @var{seed}, char *restrict @var{statebuf}, size_t @var{statelen}, struct random_data *restrict @var{buf})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buf}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{initstate_r} function behaves exactly like the @code{initstate}
+function except that it uses and modifies the state in the object
+pointed to by the fourth parameter instead of the global state.
@end deftypefun
+@deftypefun int setstate_r (char *restrict @var{statebuf}, struct random_data *restrict @var{buf})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buf}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{setstate_r} function behaves exactly like the @code{setstate}
+function except that it uses and modifies the state in the object
+pointed to by the first parameter instead of the global 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
+choose among a collection of functions which return the random bits
in different forms.
-Generally there are two kinds of functions: those which use a state of
+Generally there are two kinds of function. The first uses 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.
+by all threads of the process. The second requires 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
@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
+determining the way the generator works. By default they are
@smallexample
a = 0x5DEECE66D = 25214903917
@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.
+since the state consists of a 48-bit array.
+
+The prototypes for these functions are in @file{stdlib.h}.
+@pindex stdlib.h
-@comment stdlib.h
-@comment SVID
@deftypefun double drand48 (void)
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
+@c Uses of the static state buffer are not guarded by a lock (thus
+@c @mtasurace:drand48), so they may be found or left at a
+@c partially-updated state in case of calls from within signal handlers
+@c or cancellation. None of this will break safety rules or invoke
+@c undefined behavior, but it may affect randomness.
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
+Since the @code{double} type according to @w{IEEE 754} has a 52-bit
mantissa this means 4 bits are not initialized by the random number
generator. These are (of course) chosen to be the least significant
bits and they are initialized to @code{0}.
@end deftypefun
-@comment stdlib.h
-@comment SVID
@deftypefun double erand48 (unsigned short int @var{xsubi}[3])
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
+@c The static buffer is just initialized with default parameters, which
+@c are later read to advance the state held in 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
+to @code{1.0} (exclusive), similarly 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.
+initial use to obtain 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
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
+The @code{lrand48} function returns 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.
+int} type can take more than 32 bits, no higher numbers are returned.
The random bits are determined by the global state of the random number
generator in the C library.
@end deftypefun
-@comment stdlib.h
-@comment SVID
@deftypefun {long int} nrand48 (unsigned short int @var{xsubi}[3])
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
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.
+The numbers in the array are updated afterwards so that subsequent calls
+to this function yield different results (as is expected of a random
+number generator). The array should have been initialized before the
+first call to obtain reproducible results.
@end deftypefun
-@comment stdlib.h
-@comment SVID
@deftypefun {long int} mrand48 (void)
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
The @code{mrand48} function is similar to @code{lrand48}. The only
difference is that the numbers returned are in the range @code{-2^31} to
@code{2^31} (exclusive).
@end deftypefun
-@comment stdlib.h
-@comment SVID
@deftypefun {long int} jrand48 (unsigned short int @var{xsubi}[3])
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
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
@end deftypefun
The internal state of the random number generator can be initialized in
-several ways. The functions differ in the completeness of the
+several ways. The methods differ in the completeness of the
information provided.
-@comment stdlib.h
-@comment SVID
-@deftypefun void srand48 (long int @var{seedval}))
+@deftypefun void srand48 (long int @var{seedval})
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
The @code{srand48} function sets the most significant 32 bits of the
-state internal state of the random number generator to the least
+internal state of the random number generator to the least
significant 32 bits of the @var{seedval} parameter. The lower 16 bits
are initialized to the value @code{0x330E}. Even if the @code{long
-int} type contains more the 32 bits only the lower 32 bits are used.
+int} type contains more than 32 bits only the lower 32 bits are used.
-Due to this limitation the initialization of the state using this
-function of not very useful. But it makes it easy to use a construct
+Owing to this limitation, initialization of the state of this
+function is not very useful. But it makes it easy to use a construct
like @code{srand48 (time (0))}.
A side-effect of this function is that the values @code{a} and @code{c}
from the internal state, which are used in the congruential formula,
are reset to the default values given above. This is of importance once
-the user called the @code{lcong48} function (see below).
+the user has called the @code{lcong48} function (see below).
@end deftypefun
-@comment stdlib.h
-@comment SVID
@deftypefun {unsigned short int *} seed48 (unsigned short int @var{seed16v}[3])
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
The @code{seed48} function initializes all 48 bits of the state of the
-internal random number generator from the content of the parameter
+internal random number generator from the contents 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
+@var{seed16v} initialize the least significant 16 bits of the internal
state, the lower 16 bits of @code{@var{seed16v}[1]} initialize the mid-order
16 bits of the state and the 16 lower bits of @code{@var{seed16v}[2]}
initialize the most significant 16 bits of the state.
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.
+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.
+which enables you to specify even more information by allowing you to
+change the parameters in the congruential formula.
-@comment stdlib.h
-@comment SVID
@deftypefun void lcong48 (unsigned short int @var{param}[7])
+@standards{SVID, stdlib.h}
+@safety{@prelim{}@mtunsafe{@mtasurace{:drand48}}@asunsafe{}@acunsafe{@acucorrupt{}}}
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
From the seven elements in the array @var{param} the least significant
16 bits of the entries @code{@var{param}[0]} to @code{@var{param}[2]}
-determine the the initial state, the least 16 bits of
+determine the initial state, the least significant 16 bits of
@code{@var{param}[3]} to @code{@var{param}[5]} determine the 48 bit
-constant @code{a} and @code{@var{param}[6]} determines the 16 bit value
+constant @code{a} and @code{@var{param}[6]} determines the 16-bit value
@code{c}.
@end deftypefun
state if all threads use the functions which take a pointer to an array
containing the state. The random numbers are computed following the
same loop but if the state in the array is different all threads will
-get an individual random number generator.
+obtain an individual random number generator.
-The user supplied buffer must be of type @code{struct drand48_data}.
-This type should be regarded as opaque and no member should be used
-directly.
+The user-supplied buffer must be of type @code{struct drand48_data}.
+This type should be regarded as opaque and not manipulated directly.
-@comment stdlib.h
-@comment GNU
@deftypefun int drand48_r (struct drand48_data *@var{buffer}, double *@var{result})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
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}.
+difference that it does not modify the global random number generator
+parameters but instead the parameters in the buffer supplied through the
+pointer @var{buffer}. The random number is returned 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
+The return value of the function indicates whether the call succeeded.
+If the value is less than @code{0} an error occurred and @code{errno} is
set to indicate the problem.
This function is a GNU extension and should not be used in portable
programs.
@end deftypefun
-@comment stdlib.h
-@comment GNU
@deftypefun int erand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, double *@var{result})
-The @code{erand48_r} function works like the @code{erand48} and it takes
-an argument @var{buffer} which describes the random number generator.
-The state of the random number generator is taken from the @code{xsubi}
-array, the parameters for the congruential formula from the global
-random number generator data. The random number is return in the
-variable pointed to by @var{result}.
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{erand48_r} function works like @code{erand48}, but in addition
+it takes an argument @var{buffer} which describes the random number
+generator. The state of the random number generator is taken from the
+@code{xsubi} array, the parameters for the congruential formula from the
+global random number generator data. The random number is returned in
+the variable pointed to by @var{result}.
-The return value is non-negative is the call succeeded.
+The return value is non-negative if 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}.
+@deftypefun int lrand48_r (struct drand48_data *@var{buffer}, long int *@var{result})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
+This function is similar to @code{lrand48}, but in addition it takes a
+pointer to a buffer describing the state of the random number generator
+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.
programs.
@end deftypefun
-@comment stdlib.h
-@comment GNU
@deftypefun int nrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
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
+produces a random number in the 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}.
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
+@deftypefun int mrand48_r (struct drand48_data *@var{buffer}, long int *@var{result})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
+This function is similar to @code{mrand48} but like the other reentrant
+functions 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
programs.
@end deftypefun
-@comment stdlib.h
-@comment GNU
@deftypefun int jrand48_r (unsigned short int @var{xsubi}[3], struct drand48_data *@var{buffer}, long int *@var{result})
-The @code{jrand48_r} function is similar to @code{jrand48}. But as the
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
+The @code{jrand48_r} function is similar to @code{jrand48}. Like the
other reentrant functions of this function family it uses the
congruential formula parameters from the buffer pointed to by
@var{buffer}.
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
+Before any of the above functions are used the buffer of type
+@code{struct drand48_data} should be initialized. The easiest way to do
+this is to fill the whole buffer with null bytes, e.g. by
@smallexample
memset (buffer, '\0', sizeof (struct drand48_data));
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
+The other possibility is to use any of the functions which explicitly
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
+buffer from looking at the parameter to 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})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
The description of the random number generator represented by the
-information in @var{buffer} is initialized similar to what the function
+information in @var{buffer} is initialized similarly to what the function
@code{srand48} does. The state is initialized from the parameter
@var{seedval} and the parameters for the congruential formula are
-initialized to the default values.
+initialized to their default values.
If the return value is non-negative the function call succeeded.
programs.
@end deftypefun
-@comment stdlib.h
-@comment GNU
@deftypefun int seed48_r (unsigned short int @var{seed16v}[3], struct drand48_data *@var{buffer})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
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
+generator like the @code{seed48} function does. If the user wants to
+preserve the state for a later re-run s/he can copy the whole buffer
pointed to by @var{buffer}.
This function is a GNU extension and should not be used in portable
programs.
@end deftypefun
-@comment stdlib.h
-@comment GNU
@deftypefun int lcong48_r (unsigned short int @var{param}[7], struct drand48_data *@var{buffer})
+@standards{GNU, stdlib.h}
+@safety{@prelim{}@mtsafe{@mtsrace{:buffer}}@assafe{}@acunsafe{@acucorrupt{}}}
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.
+described in @var{buffer} with the data in @var{param}. Here it is
+especially true that the function does more than just copying the
+contents of @var{param} and @var{buffer}. More work is required and
+therefore it is important to use this function rather than initializing
+the random number generator directly.
If the return value is non-negative the function call succeeded.
This function is a GNU extension and should not be used in portable
programs.
@end deftypefun
+
+@node FP Function Optimizations
+@section Is Fast Code or Small Code preferred?
+@cindex Optimization
+
+If an application uses many floating point functions it is often the case
+that the cost of the function calls themselves is not negligible.
+Modern processors can often execute the operations themselves
+very fast, but the function call disrupts the instruction pipeline.
+
+For this reason @theglibc{} provides optimizations for many of the
+frequently-used math functions. When GNU CC is used and the user
+activates the optimizer, several new inline functions and macros are
+defined. These new functions and macros have the same names as the
+library functions and so are used instead of the latter. In the case of
+inline functions the compiler will decide whether it is reasonable to
+use them, and this decision is usually correct.
+
+This means that no calls to the library functions may be necessary, and
+can increase the speed of generated code significantly. The drawback is
+that code size will increase, and the increase is not always negligible.
+
+There are two kinds of inline functions: those that give the same result
+as the library functions and others that might not set @code{errno} and
+might have a reduced precision and/or argument range in comparison with
+the library functions. The latter inline functions are only available
+if the flag @code{-ffast-math} is given to GNU CC.
+
+Not all hardware implements the entire @w{IEEE 754} standard, and even
+if it does there may be a substantial performance penalty for using some
+of its features. For example, enabling traps on some processors forces
+the FPU to run un-pipelined, which can more than double calculation time.
+@c ***Add explanation of -lieee, -mieee.
projects / thirdparty / glibc.git / blobdiff