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diff --git a/manual/arith.texi b/manual/arith.texi index d8703ea..86fb266 100644 --- a/manual/arith.texi +++ b/manual/arith.texi @@ -3,12 +3,17 @@ This chapter contains information about functions for doing basic arithmetic operations, such as splitting a float into its integer and -fractional parts. These functions are declared in the header file -@file{math.h}. +fractional parts or retrieving the imaginary part of a complex value. +These functions are declared in the header files @file{math.h} and +@file{complex.h}. @menu +* Infinity:: What is Infinity and how to test for it. * Not a Number:: Making NaNs and testing for NaNs. +* Imaginary Unit:: Constructing complex Numbers. * Predicates on Floats:: Testing for infinity and for NaNs. +* Floating-Point Classes:: Classifiy floating-point numbers. +* Operations on Complex:: Projections, Conjugates, and Decomposing. * Absolute Value:: Absolute value functions. * Normalization Functions:: Hacks for radix-2 representations. * Rounding and Remainders:: Determining the integer and @@ -19,6 +24,44 @@ fractional parts. These functions are declared in the header file from strings. @end menu +@node Infinity +@section Infinity Values +@cindex Infinity +@cindex IEEE floating point + +Mathematical operations easily can produce as the result values which +are not representable by the floating-point format. The functions in +the mathematics library also have this problem. The situation is +generally solved by raising an overflow exception and by returning a +huge value. + +The @w{IEEE 754} floating-point defines a special value to be used in +these situations. There is a special value for infinity. + +@comment math.h +@comment ISO +@deftypevr Macro float_t INFINITY +A expression representing the inifite value. @code{INFINITY} values are +produce by mathematical operations like @code{1.0 / 0.0}. It is +possible to continue the computations with this value since the basic +operations as well as the mathematical library functions are prepared to +handle values like this. + +Beside @code{INFINITY} also the value @code{-INIFITY} is representable +and it is handled differently if needed. It is possible to test a +variables for infinite value using a simple comparison but the +recommended way is to use the the @code{isinf} function. + +This macro was introduced in the @w{ISO C 9X} standard. +@end deftypevr + +@vindex HUGE_VAL +The macros @code{HUGE_VAL}, @code{HUGE_VALF} and @code{HUGE_VALL} are +defined in a similar way but they are not required to represent the +infinite value, only a very large value (@pxref{Domain and Range Errors}). +If actually infinity is wanted, @code{INFINITY} should be used. + + @node Not a Number @section ``Not a Number'' Values @cindex NaN @@ -54,6 +97,46 @@ such as by defining @code{_GNU_SOURCE}, and then you must include @file{math.h}.) @end deftypevr +@node Imaginary Unit +@section Constructing complex Numbers + +@pindex complex.h +To construct complex numbers it is necessary have a way to express the +imaginary part of the numbers. In mathematics one uses the symbol ``i'' +to mark a number as imaginary. For convenienve the @file{complex.h} +header defines two macros which allow to use a similar easy notation. + +@deftypevr Macro float_t _Imaginary_I +This macro is a (compiler specific) representation of the value ``1i''. +I.e., it is the value for which + +@smallexample +_Imaginary_I * _Imaginary_I = -1 +@end smallexample + +@noindent +One can use it to easily construct complex number like in + +@smallexample +3.0 - _Imaginary_I * 4.0 +@end smallexample + +@noindent +which results in the complex number with a real part of 3.0 and a +imaginary part -4.0. +@end deftypevr + +@noindent +A more intuitive approach is to use the following macro. + +@deftypevr Macro float_t I +This macro has exactly the same value as @code{_Imaginary_I}. The +problem is that the name @code{I} very easily can clash with macros or +variables in programs and so it might be a good idea to avoid this name +and stay at the safe side by using @code{_Imaginary_I}. +@end deftypevr + + @node Predicates on Floats @section Predicates on Floats @@ -66,6 +149,10 @@ functions, and thus are available if you define @code{_BSD_SOURCE} or @comment math.h @comment BSD @deftypefun int isinf (double @var{x}) +@end deftypefun +@deftypefun int isinff (float @var{x}) +@end deftypefun +@deftypefun int isinfl (long double @var{x}) This function returns @code{-1} if @var{x} represents negative infinity, @code{1} if @var{x} represents positive infinity, and @code{0} otherwise. @end deftypefun @@ -73,6 +160,10 @@ This function returns @code{-1} if @var{x} represents negative infinity, @comment math.h @comment BSD @deftypefun int isnan (double @var{x}) +@end deftypefun +@deftypefun int isnanf (float @var{x}) +@end deftypefun +@deftypefun int isnanl (long double @var{x}) This function returns a nonzero value if @var{x} is a ``not a number'' value, and zero otherwise. (You can just as well use @code{@var{x} != @var{x}} to get the same result). @@ -81,6 +172,10 @@ value, and zero otherwise. (You can just as well use @code{@var{x} != @comment math.h @comment BSD @deftypefun int finite (double @var{x}) +@end deftypefun +@deftypefun int finitef (float @var{x}) +@end deftypefun +@deftypefun int finitel (long double @var{x}) This function returns a nonzero value if @var{x} is finite or a ``not a number'' value, and zero otherwise. @end deftypefun @@ -103,6 +198,189 @@ does not fit the @w{ISO C} specification. @strong{Portability Note:} The functions listed in this section are BSD extensions. +@node Floating-Point Classes +@section Floating-Point Number Classification Functions + +Instead of using the BSD specific functions from the last section it is +better to use those in this section will are introduced in the @w{ISO C +9X} standard and are therefore widely available. + +@comment math.h +@comment ISO +@deftypefun int fpclassify (@emph{float-type} @var{x}) +This is a generic macro which works on all floating-point types and +which returns a value of type @code{int}. The possible values are: + +@vtable @code +@item FP_NAN + The floating-point number @var{x} is ``Not a Number'' (@pxref{Not a Number}) +@item FP_INFINITE + The value of @var{x} is either plus or minus infinity (@pxref{Infinity}) +@item FP_ZERO + The value of @var{x} is zero. In floating-point formats like @w{IEEE + 754} where the zero value can be signed this value is also returned if + @var{x} is minus zero. +@item FP_SUBNORMAL + Some floating-point formats (such as @w{IEEE 754}) allow floating-point + numbers to be represented in a denormalized format. This happens if the + absolute value of the number is too small to be represented in the + normal format. @code{FP_SUBNORMAL} is returned for such values of @var{x}. +@item FP_NORMAL + This value is returned for all other cases which means the number is a + plain floating-point number without special meaning. +@end vtable + +This macro is useful if more than property of a number must be +tested. If one only has to test for, e.g., a NaN value, there are +function which are faster. +@end deftypefun + +The remainder of this section introduces some more specific functions. +They might be implemented faster than the call to @code{fpclassify} and +if the actual need in the program is covered be these functions they +should be used (and not @code{fpclassify}). + +@comment math.h +@comment ISO +@deftypefun int isfinite (@emph{float-type} @var{x}) +The value returned by this macro is nonzero if the value of @var{x} is +not plus or minus infinity and not NaN. I.e., it could be implemented as + +@smallexample +(fpclassify (x) != FP_NAN && fpclassify (x) != FP_INFINITE) +@end smallexample + +@code{isfinite} is also implemented as a macro which can handle all +floating-point types. Programs should use this function instead of +@var{finite} (@pxref{Predicates on Floats}). +@end deftypefun + +@comment math.h +@comment ISO +@deftypefun int isnormal (@emph{float-type} @var{x}) +If @code{isnormal} returns a nonzero value the value or @var{x} is +neither a NaN, infinity, zero, nor a denormalized number. I.e., it +could be implemented as + +@smallexample +(fpclassify (x) == FP_NORMAL) +@end smallexample +@end deftypefun + +@comment math.h +@comment ISO +@deftypefun int isnan (@emph{float-type} @var{x}) +The situation with this macro is a bit complicated. Here @code{isnan} +is a macro which can handle all kinds of floating-point types. It +returns a nonzero value is @var{x} does not represent a NaN value and +could be written like this + +@smallexample +(fpclassify (x) == FP_NAN) +@end smallexample + +The complication is that there is a function of the same name and the +same semantic defined for compatibility with BSD (@pxref{Predicates on +Floats}). Fortunately this should not yield to problems in most cases +since the macro and the function have the same semantic. Should in a +situation the function be absolutely necessary one can use + +@smallexample +(isnan) (x) +@end smallexample + +@noindent +to avoid the macro expansion. Using the macro has two big adavantages: +it is more portable and one does not have to choose the right function +among @code{isnan}, @code{isnanf}, and @code{isnanl}. +@end deftypefun + + +@node Operations on Complex +@section Projections, Conjugates, and Decomposing of Complex Numbers +@cindex project complex numbers +@cindex conjugate complex numbers +@cindex decompose complex numbers + +This section lists functions performing some of the simple mathematical +operations on complex numbers. Using any of the function requries that +the C compiler understands the @code{complex} keyword, introduced to the +C language in the @w{ISO C 9X} standard. + +@pindex complex.h +The prototypes for all functions in this section can be found in +@file{complex.h}. All functions are available in three variants, one +for each of the three floating-point types. + +The easiest operation on complex numbers is the decomposition in the +real part and the imaginary part. This is done by the next two +functions. + +@comment complex.h +@comment ISO +@deftypefun double creal (complex double @var{z}) +@end deftypefun +@deftypefun float crealf (complex float @var{z}) +@end deftypefun +@deftypefun {long double} creall (complex long double @var{z}) +These functions return the real part of the complex number @var{z}. +@end deftypefun + +@comment complex.h +@comment ISO +@deftypefun double cimag (complex double @var{z}) +@end deftypefun +@deftypefun float cimagf (complex float @var{z}) +@end deftypefun +@deftypefun {long double} cimagl (complex long double @var{z}) +These functions return the imaginary part of the complex number @var{z}. +@end deftypefun + + +The conjugate complex value of a given complex number has the same value +for the real part but the complex part is negated. + +@comment complex.h +@comment ISO +@deftypefun {complex double} conj (complex double @var{z}) +@end deftypefun +@deftypefun {complex float} conjf (complex float @var{z}) +@end deftypefun +@deftypefun {complex long double} conjl (complex long double @var{z}) +These functions return the conjugate complex value of the complex number +@var{z}. +@end deftypefun + +@comment complex.h +@comment ISO +@deftypefun double carg (complex double @var{z}) +@end deftypefun +@deftypefun float cargf (complex float @var{z}) +@end deftypefun +@deftypefun {long double} cargl (complex long double @var{z}) +These functions return argument of the complex number @var{z}. + +Mathematically, the argument is the phase angle of @var{z} with a branch +cut along the negative real axis. +@end deftypefun + +@comment complex.h +@comment ISO +@deftypefun {complex double} cproj (complex double @var{z}) +@end deftypefun +@deftypefun {complex float} cprojf (complex float @var{z}) +@end deftypefun +@deftypefun {complex long double} cprojl (complex long double @var{z}) +Return the projection of the complex value @var{z} on the Riemann +sphere. Values with a infinite complex part (even if the real part +is NaN) are projected to positive infinte on the real axis. If the real part is infinite, the result is equivalent to + +@smallexample +INFINITY + I * copysign (0.0, cimag (z)) +@end smallexample +@end deftypefun + + @node Absolute Value @section Absolute Value @cindex absolute value functions @@ -117,7 +395,8 @@ whose imaginary part is @var{y}, the absolute value is @w{@code{sqrt @pindex math.h @pindex stdlib.h Prototypes for @code{abs} and @code{labs} are in @file{stdlib.h}; -@code{fabs} and @code{cabs} are declared in @file{math.h}. +@code{fabs}, @code{fabsf} and @code{fabsl} are declared in @file{math.h}; +@code{cabs}, @code{cabsf} and @code{cabsl} are declared in @file{complex.h}. @comment stdlib.h @comment ISO @@ -139,20 +418,28 @@ are of type @code{long int} rather than @code{int}. @comment math.h @comment ISO @deftypefun double fabs (double @var{number}) +@end deftypefun +@deftypefun float fabsf (float @var{number}) +@end deftypefun +@deftypefun {long double} fabsl (long double @var{number}) This function returns the absolute value of the floating-point number @var{number}. @end deftypefun -@comment math.h -@comment BSD -@deftypefun double cabs (struct @{ double real, imag; @} @var{z}) -The @code{cabs} function returns the absolute value of the complex -number @var{z}, whose real part is @code{@var{z}.real} and whose -imaginary part is @code{@var{z}.imag}. (See also the function -@code{hypot} in @ref{Exponents and Logarithms}.) The value is: +@comment complex.h +@comment ISO +@deftypefun double cabs (complex double @var{z}) +@end deftypefun +@deftypefun float cabsf (complex float @var{z}) +@end deftypefun +@deftypefun {long double} cabsl (complex long double @var{z}) +These functions return the absolute value of the complex number @var{z}. +The compiler must support complex numbers to use these functions. (See +also the function @code{hypot} in @ref{Exponents and Logarithms}.) The +value is: @smallexample -sqrt (@var{z}.real*@var{z}.real + @var{z}.imag*@var{z}.imag) +sqrt (creal (@var{z}) * creal (@var{z}) + cimag (@var{z}) * cimag (@var{z})) @end smallexample @end deftypefun @@ -174,7 +461,11 @@ All these functions are declared in @file{math.h}. @comment math.h @comment ISO @deftypefun double frexp (double @var{value}, int *@var{exponent}) -The @code{frexp} function is used to split the number @var{value} +@end deftypefun +@deftypefun float frexpf (float @var{value}, int *@var{exponent}) +@end deftypefun +@deftypefun {long double} frexpl (long double @var{value}, int *@var{exponent}) +These functions are used to split the number @var{value} into a normalized fraction and an exponent. If the argument @var{value} is not zero, the return value is @var{value} @@ -193,7 +484,11 @@ zero is stored in @code{*@var{exponent}}. @comment math.h @comment ISO @deftypefun double ldexp (double @var{value}, int @var{exponent}) -This function returns the result of multiplying the floating-point +@end deftypefun +@deftypefun float ldexpf (float @var{value}, int @var{exponent}) +@end deftypefun +@deftypefun {long double} ldexpl (long double @var{value}, int @var{exponent}) +These functions return the result of multiplying the floating-point number @var{value} by 2 raised to the power @var{exponent}. (It can be used to reassemble floating-point numbers that were taken apart by @code{frexp}.) @@ -207,13 +502,21 @@ equivalent to those of @code{ldexp} and @code{frexp}: @comment math.h @comment BSD @deftypefun double scalb (double @var{value}, int @var{exponent}) +@end deftypefun +@deftypefun float scalbf (float @var{value}, int @var{exponent}) +@end deftypefun +@deftypefun {long double} scalbl (long double @var{value}, int @var{exponent}) The @code{scalb} function is the BSD name for @code{ldexp}. @end deftypefun @comment math.h @comment BSD @deftypefun double logb (double @var{x}) -This BSD function returns the integer part of the base-2 logarithm of +@end deftypefun +@deftypefun float logbf (float @var{x}) +@end deftypefun +@deftypefun {long double} logbl (long double @var{x}) +These BSD functions return the integer part of the base-2 logarithm of @var{x}, an integer value represented in type @code{double}. This is the highest integer power of @code{2} contained in @var{x}. The sign of @var{x} is ignored. For example, @code{logb (3.5)} is @code{1.0} and @@ -231,11 +534,28 @@ The value returned by @code{logb} is one less than the value that @end deftypefun @comment math.h -@comment BSD +@comment ISO @deftypefun double copysign (double @var{value}, double @var{sign}) -The @code{copysign} function returns a value whose absolute value is the +@end deftypefun +@deftypefun float copysignf (float @var{value}, float @var{sign}) +@end deftypefun +@deftypefun {long double} copysignl (long double @var{value}, long double @var{sign}) +These functions return a value whose absolute value is the same as that of @var{value}, and whose sign matches that of @var{sign}. -This is a BSD function. +This function appears in BSD and was standardized in @w{ISO C 9X}. +@end deftypefun + +@comment math.h +@comment ISO +@deftypefun int signbit (@emph{float-type} @var{x}) +@code{signbit} is a generic macro which can work on all floating-point +types. It returns a nonzero value if the value of @var{x} has its sign +bit set. + +This is not the same as @code{x < 0.0} since in some floating-point +formats (e.g., @w{IEEE 754}) the zero value is optionally signed. The +comparison @code{-0.0 < 0.0} will not be true while @code{signbit +(-0.0)} will return a nonzeri value. @end deftypefun @node Rounding and Remainders @@ -260,7 +580,11 @@ result as a @code{double} instead to get around this problem. @comment math.h @comment ISO @deftypefun double ceil (double @var{x}) -The @code{ceil} function rounds @var{x} upwards to the nearest integer, +@end deftypefun +@deftypefun float ceilf (float @var{x}) +@end deftypefun +@deftypefun {long double} ceill (long double @var{x}) +These functions round @var{x} upwards to the nearest integer, returning that value as a @code{double}. Thus, @code{ceil (1.5)} is @code{2.0}. @end deftypefun @@ -268,15 +592,23 @@ is @code{2.0}. @comment math.h @comment ISO @deftypefun double floor (double @var{x}) -The @code{ceil} function rounds @var{x} downwards to the nearest +@end deftypefun +@deftypefun float floorf (float @var{x}) +@end deftypefun +@deftypefun {long double} floorl (long double @var{x}) +These functions round @var{x} downwards to the nearest integer, returning that value as a @code{double}. Thus, @code{floor (1.5)} is @code{1.0} and @code{floor (-1.5)} is @code{-2.0}. @end deftypefun @comment math.h -@comment BSD +@comment ISO @deftypefun double rint (double @var{x}) -This function rounds @var{x} to an integer value according to the +@end deftypefun +@deftypefun float rintf (float @var{x}) +@end deftypefun +@deftypefun {long double} rintl (long double @var{x}) +These functions round @var{x} to an integer value according to the current rounding mode. @xref{Floating Point Parameters}, for information about the various rounding modes. The default rounding mode is to round to the nearest integer; some machines @@ -286,8 +618,24 @@ you explicit select another. @comment math.h @comment ISO +@deftypefun double nearbyint (double @var{x}) +@end deftypefun +@deftypefun float nearbyintf (float @var{x}) +@end deftypefun +@deftypefun {long double} nearbyintl (long double @var{x}) +These functions return the same value as the @code{rint} functions but +even some rounding actually takes place @code{nearbyint} does @emph{not} +raise the inexact exception. +@end deftypefun + +@comment math.h +@comment ISO @deftypefun double modf (double @var{value}, double *@var{integer-part}) -This function breaks the argument @var{value} into an integer part and a +@end deftypefun +@deftypefun float modff (flaot @var{value}, float *@var{integer-part}) +@end deftypefun +@deftypefun {long double} modfl (long double @var{value}, long double *@var{integer-part}) +These functions break the argument @var{value} into an integer part and a fractional part (between @code{-1} and @code{1}, exclusive). Their sum equals @var{value}. Each of the parts has the same sign as @var{value}, so the rounding of the integer part is towards zero. @@ -300,7 +648,11 @@ returns @code{0.5} and stores @code{2.0} into @code{intpart}. @comment math.h @comment ISO @deftypefun double fmod (double @var{numerator}, double @var{denominator}) -This function computes the remainder from the division of +@end deftypefun +@deftypefun float fmodf (float @var{numerator}, float @var{denominator}) +@end deftypefun +@deftypefun {long double} fmodl (long double @var{numerator}, long double @var{denominator}) +These functions compute the remainder from the division of @var{numerator} by @var{denominator}. Specifically, the return value is @code{@var{numerator} - @w{@var{n} * @var{denominator}}}, where @var{n} is the quotient of @var{numerator} divided by @var{denominator}, rounded @@ -317,7 +669,11 @@ If @var{denominator} is zero, @code{fmod} fails and sets @code{errno} to @comment math.h @comment BSD @deftypefun double drem (double @var{numerator}, double @var{denominator}) -The function @code{drem} is like @code{fmod} except that it rounds the +@end deftypefun +@deftypefun float dremf (float @var{numerator}, float @var{denominator}) +@end deftypefun +@deftypefun {long double} dreml (long double @var{numerator}, long double @var{denominator}) +These functions are like @code{fmod} etc except that it rounds the internal quotient @var{n} to the nearest integer instead of towards zero to an integer. For example, @code{drem (6.5, 2.3)} returns @code{-0.4}, which is @code{6.5} minus @code{6.9}. |