@cindex elementary functions
@cindex mathematical functions, elementary

This chapter describes basic mathematical functions.  Some of these
functions are present in system libraries, but the alternative versions
given here can be used as a substitute when the system functions are not
available.

The functions and macros described in this chapter are defined in the
header file @file{gsl_math.h}.

@menu
* Mathematical Constants::      
* Infinities and Not-a-number::  
* Elementary Functions::        
* Small integer powers::        
* Testing the Sign of Numbers::  
* Testing for Odd and Even Numbers::  
* Maximum and Minimum functions::  
* Approximate Comparison of Floating Point Numbers::  
@end menu

@node Mathematical Constants
@section Mathematical Constants
@cindex mathematical constants, defined as macros
@cindex numerical constants, defined as macros
@cindex constants, mathematical---defined as macros
@cindex macros for mathematical constants
The library ensures that the standard @sc{bsd} mathematical constants
are defined. For reference, here is a list of the constants:

@table @code
@item M_E
@cindex e, defined as a macro
The base of exponentials, @math{e}

@item M_LOG2E
The base-2 logarithm of @math{e}, @math{\log_2 (e)}

@item M_LOG10E
The base-10 logarithm of @math{e}, @c{$\log_{10}(e)$}
@math{\log_10 (e)}

@item M_SQRT2
The square root of two, @math{\sqrt 2}

@item M_SQRT1_2
The square root of one-half, @c{$\sqrt{1/2}$}
@math{\sqrt@{1/2@}}

@item M_SQRT3
The square root of three, @math{\sqrt 3}

@item M_PI
@cindex pi, defined as a macro
The constant pi, @math{\pi}

@item M_PI_2
Pi divided by two, @math{\pi/2}

@item M_PI_4
Pi divided by four, @math{\pi/4}

@item M_SQRTPI
The square root of pi, @math{\sqrt\pi}

@item M_2_SQRTPI
Two divided by the square root of pi, @math{2/\sqrt\pi}

@item M_1_PI
The reciprocal of pi, @math{1/\pi}

@item M_2_PI
Twice the reciprocal of pi, @math{2/\pi}

@item M_LN10
The natural logarithm of ten, @math{\ln(10)}

@item M_LN2
The natural logarithm of two, @math{\ln(2)}

@item M_LNPI
The natural logarithm of pi, @math{\ln(\pi)}

@item M_EULER
@cindex Euler's constant, defined as a macro
Euler's constant, @math{\gamma}

@end table

@node Infinities and Not-a-number
@section Infinities and Not-a-number

@cindex infinity, defined as a macro
@cindex IEEE infinity, defined as a macro
@cindex NaN, defined as a macro
@cindex Not-a-number, defined as a macro
@cindex IEEE NaN, defined as a macro

@defvr {Macro} GSL_POSINF
This macro contains the IEEE representation of positive infinity,
@math{+\infty}. It is computed from the expression @code{+1.0/0.0}.
@end defvr

@defvr {Macro} GSL_NEGINF
This macro contains the IEEE representation of negative infinity,
@math{-\infty}. It is computed from the expression @code{-1.0/0.0}.
@end defvr

@defvr {Macro} GSL_NAN
This macro contains the IEEE representation of the Not-a-Number symbol,
@code{NaN}. It is computed from the ratio @code{0.0/0.0}.
@end defvr

@deftypefun int gsl_isnan (const double @var{x})
This function returns 1 if @var{x} is not-a-number.
@end deftypefun

@deftypefun int gsl_isinf (const double @var{x})
This function returns @math{+1} if @var{x} is positive infinity,
@math{-1} if @var{x} is negative infinity and 0
otherwise.@footnote{Note that the C99 standard only requires the
system @code{isinf} function to return a non-zero value, without the
sign of the infinity.  The implementation in some earlier versions of
GSL used the system @code{isinf} function and may have this behavior
on some platforms.  Therefore, it is advisable to test the sign of
@var{x} separately, if needed, rather than relying the sign of the
return value from @code{gsl_isinf()}.}
@end deftypefun

@deftypefun int gsl_finite (const double @var{x})
This function returns 1 if @var{x} is a real number, and 0 if it is
infinite or not-a-number.
@end deftypefun


@node Elementary Functions
@section Elementary Functions

The following routines provide portable implementations of functions
found in the BSD math library.  When native versions are not available
the functions described here can be used instead.  The substitution can
be made automatically if you use @code{autoconf} to compile your
application (@pxref{Portability functions}).

@deftypefun double gsl_log1p (const double @var{x})
@cindex log1p
@cindex logarithm, computed accurately near 1
This function computes the value of @math{\log(1+x)} in a way that is
accurate for small @var{x}. It provides an alternative to the BSD math
function @code{log1p(x)}.
@end deftypefun

@deftypefun double gsl_expm1 (const double @var{x})
@cindex expm1
@cindex exponential, difference from 1 computed accurately
This function computes the value of @math{\exp(x)-1} in a way that is
accurate for small @var{x}. It provides an alternative to the BSD math
function @code{expm1(x)}.
@end deftypefun

@deftypefun double gsl_hypot (const double @var{x}, const double @var{y})
@cindex hypot
@cindex euclidean distance function, hypot
@cindex length, computed accurately using hypot
This function computes the value of
@c{$\sqrt{x^2 + y^2}$}
@math{\sqrt@{x^2 + y^2@}} in a way that avoids overflow. It provides an
alternative to the BSD math function @code{hypot(x,y)}.
@end deftypefun

@deftypefun double gsl_hypot3 (const double @var{x}, const double @var{y}, const double @var{z})
@cindex euclidean distance function, hypot
@cindex length, computed accurately using hypot
This function computes the value of
@c{$\sqrt{x^2 + y^2 + z^2}$}
@math{\sqrt@{x^2 + y^2 + z^2@}} in a way that avoids overflow.
@end deftypefun

@deftypefun double gsl_acosh (const double @var{x})
@cindex acosh
@cindex hyperbolic cosine, inverse
@cindex inverse hyperbolic cosine
This function computes the value of @math{\arccosh(x)}. It provides an
alternative to the standard math function @code{acosh(x)}.
@end deftypefun

@deftypefun double gsl_asinh (const double @var{x})
@cindex asinh
@cindex hyperbolic sine, inverse
@cindex inverse hyperbolic sine
This function computes the value of @math{\arcsinh(x)}. It provides an
alternative to the standard math function @code{asinh(x)}.
@end deftypefun

@deftypefun double gsl_atanh (const double @var{x})
@cindex atanh
@cindex hyperbolic tangent, inverse
@cindex inverse hyperbolic tangent
This function computes the value of @math{\arctanh(x)}. It provides an
alternative to the standard math function @code{atanh(x)}.
@end deftypefun


@deftypefun double gsl_ldexp (double @var{x}, int @var{e})
@cindex ldexp
This function computes the value of @math{x * 2^e}. It provides an
alternative to the standard math function @code{ldexp(x,e)}.
@end deftypefun

@deftypefun double gsl_frexp (double @var{x}, int * @var{e})
@cindex frexp
This function splits the number @math{x} into its normalized fraction
@math{f} and exponent @math{e}, such that @math{x = f * 2^e} and
@c{$0.5 \le f < 1$}
@math{0.5 <= f < 1}. The function returns @math{f} and stores the
exponent in @math{e}. If @math{x} is zero, both @math{f} and @math{e}
are set to zero. This function provides an alternative to the standard
math function @code{frexp(x, e)}.
@end deftypefun

@node Small integer powers
@section Small integer powers

A common complaint about the standard C library is its lack of a
function for calculating (small) integer powers.  GSL provides some simple
functions to fill this gap.  For reasons of efficiency, these functions
do not check for overflow or underflow conditions. 

@deftypefun double gsl_pow_int (double @var{x}, int @var{n})
@deftypefunx double gsl_pow_uint (double @var{x}, unsigned int @var{n})
These routines computes the power @math{x^n} for integer @var{n}.  The
power is computed efficiently---for example, @math{x^8} is computed as
@math{((x^2)^2)^2}, requiring only 3 multiplications.  A version of this
function which also computes the numerical error in the result is
available as @code{gsl_sf_pow_int_e}.
@end deftypefun

@deftypefun double gsl_pow_2 (const double @var{x})
@deftypefunx double gsl_pow_3 (const double @var{x})
@deftypefunx double gsl_pow_4 (const double @var{x})
@deftypefunx double gsl_pow_5 (const double @var{x})
@deftypefunx double gsl_pow_6 (const double @var{x})
@deftypefunx double gsl_pow_7 (const double @var{x})
@deftypefunx double gsl_pow_8 (const double @var{x})
@deftypefunx double gsl_pow_9 (const double @var{x})
These functions can be used to compute small integer powers @math{x^2},
@math{x^3}, etc. efficiently. The functions will be inlined when 
@code{HAVE_INLINE} is defined, so that use of these functions 
should be as efficient as explicitly writing the corresponding 
product expression.
@end deftypefun

@example
#include <gsl/gsl_math.h>
double y = gsl_pow_4 (3.141)  /* compute 3.141**4 */
@end example

@node Testing the Sign of Numbers
@section Testing the Sign of Numbers

@defmac GSL_SIGN (x)
This macro returns the sign of @var{x}. It is defined as @code{((x) >= 0
? 1 : -1)}. Note that with this definition the sign of zero is positive
(regardless of its @sc{ieee} sign bit).
@end defmac

@node Testing for Odd and Even Numbers
@section Testing for Odd and Even Numbers

@defmac GSL_IS_ODD (n)
This macro evaluates to 1 if @var{n} is odd and 0 if @var{n} is
even. The argument @var{n} must be of integer type.
@end defmac

@defmac GSL_IS_EVEN (n)
This macro is the opposite of @code{GSL_IS_ODD(n)}. It evaluates to 1 if
@var{n} is even and 0 if @var{n} is odd. The argument @var{n} must be of
integer type.
@end defmac

@node Maximum and Minimum functions
@section Maximum and Minimum functions

Note that the following macros perform multiple evaluations of their
arguments, so they should not be used with arguments that have side
effects (such as a call to a random number generator).

@defmac GSL_MAX (a, b)
@cindex maximum of two numbers
This macro returns the maximum of @var{a} and @var{b}. It is defined
as @code{((a) > (b) ? (a):(b))}.  
@end defmac

@defmac GSL_MIN (a, b)
@cindex minimum of two numbers
This macro returns the minimum of @var{a} and @var{b}. It is defined as 
@code{((a) < (b) ? (a):(b))}.
@end defmac

@deftypefun {extern inline double} GSL_MAX_DBL (double @var{a}, double @var{b})
This function returns the maximum of the double precision numbers
@var{a} and @var{b} using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
@code{GSL_MAX} will be automatically substituted.
@end deftypefun

@deftypefun {extern inline double} GSL_MIN_DBL (double @var{a}, double @var{b})
This function returns the minimum of the double precision numbers
@var{a} and @var{b} using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
@code{GSL_MIN} will be automatically substituted.
@end deftypefun

@deftypefun {extern inline int} GSL_MAX_INT (int @var{a}, int @var{b})
@deftypefunx {extern inline int} GSL_MIN_INT (int @var{a}, int @var{b})
These functions return the maximum or minimum of the integers @var{a}
and @var{b} using an inline function.  On platforms where inline
functions are not available the macros @code{GSL_MAX} or @code{GSL_MIN}
will be automatically substituted.
@end deftypefun

@deftypefun {extern inline long double} GSL_MAX_LDBL (long double @var{a}, long double @var{b})
@deftypefunx {extern inline long double} GSL_MIN_LDBL (long double @var{a}, long double @var{b})
These functions return the maximum or minimum of the long doubles @var{a}
and @var{b} using an inline function.  On platforms where inline
functions are not available the macros @code{GSL_MAX} or @code{GSL_MIN}
will be automatically substituted.
@end deftypefun

@node Approximate Comparison of Floating Point Numbers
@section Approximate Comparison of Floating Point Numbers

It is sometimes useful to be able to compare two floating point numbers
approximately, to allow for rounding and truncation errors.  The following
function implements the approximate floating-point comparison algorithm
proposed by D.E. Knuth in Section 4.2.2 of @cite{Seminumerical
Algorithms} (3rd edition).

@deftypefun int gsl_fcmp (double @var{x}, double @var{y}, double @var{epsilon})
@cindex approximate comparison of floating point numbers
@cindex safe comparison of floating point numbers
@cindex floating point numbers, approximate comparison
This function determines whether @math{x} and @math{y} are approximately
equal to a relative accuracy @var{epsilon}.

The relative accuracy is measured using an interval of size @math{2
\delta}, where @math{\delta = 2^k \epsilon} and @math{k} is the
maximum base-2 exponent of @math{x} and @math{y} as computed by the
function @code{frexp}.  

If @math{x} and @math{y} lie within this interval, they are considered
approximately equal and the function returns 0. Otherwise if @math{x <
y}, the function returns @math{-1}, or if @math{x > y}, the function returns
@math{+1}.

Note that @math{x} and @math{y} are compared to relative accuracy, so
this function is not suitable for testing whether a value is
approximately zero. 

The implementation is based on the package @code{fcmp} by T.C. Belding.
@end deftypefun
