path: root/manual/arith.texi

@node Arithmetic, Date and Time, Mathematics, Top
@chapter Low-Level Arithmetic Functions

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}.

@menu
* Not a Number::                Making NaNs and testing for NaNs.
* Predicates on Floats::        Testing for infinity and for NaNs.
* Absolute Value::              Absolute value functions.
* Normalization Functions::     Hacks for radix-2 representations.
* Rounding and Remainders::     Determinining the integer and
			         fractional parts of a float.
* Integer Division::            Functions for performing integer
				 division.
* Parsing of Numbers::          Functions for ``reading'' numbers
			         from strings.
@end menu

@node Not a Number
@section ``Not a Number'' Values
@cindex NaN
@cindex not a number
@cindex IEEE floating point

The IEEE floating point format used by most modern computers supports
values that are ``not a number''.  These values are called @dfn{NaNs}.
``Not a number'' values result from certain operations which have no
meaningful numeric result, such as zero divided by zero or infinity
divided by infinity.

One noteworthy property of NaNs is that they are not equal to
themselves.  Thus, @code{x == x} can be 0 if the value of @code{x} is a
NaN.  You can use this to test whether a value is a NaN or not: if it is
not equal to itself, then it is a NaN.  But the recommended way to test
for a NaN is with the @code{isnan} function (@pxref{Predicates on Floats}).

Almost any arithmetic operation in which one argument is a NaN returns
a NaN.

@comment math.h
@comment GNU
@deftypevr Macro double NAN
An expression representing a value which is ``not a number''.  This
macro is a GNU extension, available only on machines that support ``not
a number'' values---that is to say, on all machines that support IEEE
floating point.

You can use @samp{#ifdef NAN} to test whether the machine supports
NaNs.  (Of course, you must arrange for GNU extensions to be visible,
such as by defining @code{_GNU_SOURCE}, and then you must include
@file{math.h}.)
@end deftypevr

@node Predicates on Floats
@section Predicates on Floats

@pindex math.h
This section describes some miscellaneous test functions on doubles.
Prototypes for these functions appear in @file{math.h}.  These are BSD
functions, and thus are available if you define @code{_BSD_SOURCE} or
@code{_GNU_SOURCE}.

@comment math.h
@comment BSD
@deftypefun int isinf (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

@comment math.h
@comment BSD
@deftypefun int isnan (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).
@end deftypefun

@comment math.h
@comment BSD
@deftypefun int finite (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

@comment math.h
@comment BSD
@deftypefun double infnan (int @var{error})
This function is provided for compatibility with BSD.  The other
mathematical functions use @code{infnan} to decide what to return on
occasion of an error.  Its argument is an error code, @code{EDOM} or
@code{ERANGE}; @code{infnan} returns a suitable value to indicate this
with.  @code{-ERANGE} is also acceptable as an argument, and corresponds
to @code{-HUGE_VAL} as a value.

In the BSD library, on certain machines, @code{infnan} raises a fatal
signal in all cases.  The GNU library does not do likewise, because that
does not fit the ANSI C specification.
@end deftypefun

@strong{Portability Note:} The functions listed in this section are BSD
extensions.

@node Absolute Value
@section Absolute Value
@cindex absolute value functions

These functions are provided for obtaining the @dfn{absolute value} (or
@dfn{magnitude}) of a number.  The absolute value of a real number
@var{x} is @var{x} is @var{x} is positive, @minus{}@var{x} if @var{x} is
negative.  For a complex number @var{z}, whose real part is @var{x} and
whose imaginary part is @var{y}, the absolute value is @w{@code{sqrt
(@var{x}*@var{x} + @var{y}*@var{y})}}.

@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}.

@comment stdlib.h
@comment ANSI
@deftypefun int abs (int @var{number})
This function returns the absolute value of @var{number}.

Most computers use a two's complement integer representation, in which
the absolute value of @code{INT_MIN} (the smallest possible @code{int})
cannot be represented; thus, @w{@code{abs (INT_MIN)}} is not defined.
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftypefun {long int} labs (long int @var{number})
This is similar to @code{abs}, except that both the argument and result
are of type @code{long int} rather than @code{int}.
@end deftypefun

@comment math.h
@comment ANSI
@deftypefun double fabs (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:

@smallexample
sqrt (@var{z}.real*@var{z}.real + @var{z}.imag*@var{z}.imag)
@end smallexample
@end deftypefun

@node Normalization Functions
@section Normalization Functions
@cindex normalization functions (floating-point)

The functions described in this section are primarily provided as a way
to efficiently perform certain low-level manipulations on floating point
numbers that are represented internally using a binary radix;
see @ref{Floating Point Concepts}.  These functions are required to
have equivalent behavior even if the representation does not use a radix
of 2, but of course they are unlikely to be particularly efficient in
those cases.

@pindex math.h
All these functions are declared in @file{math.h}.

@comment math.h
@comment ANSI
@deftypefun double frexp (double @var{value}, int *@var{exponent})
The @code{frexp} function is 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}
times a power of two, and is always in the range 1/2 (inclusive) to 1
(exclusive).  The corresponding exponent is stored in
@code{*@var{exponent}}; the return value multiplied by 2 raised to this
exponent equals the original number @var{value}.

For example, @code{frexp (12.8, &exponent)} returns @code{0.8} and
stores @code{4} in @code{exponent}.

If @var{value} is zero, then the return value is zero and
zero is stored in @code{*@var{exponent}}.
@end deftypefun

@comment math.h
@comment ANSI
@deftypefun double ldexp (double @var{value}, int @var{exponent})
This function returns 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}.)

For example, @code{ldexp (0.8, 4)} returns @code{12.8}.
@end deftypefun

The following functions which come from BSD provide facilities
equivalent to those of @code{ldexp} and @code{frexp}:

@comment math.h
@comment BSD
@deftypefun double scalb (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
@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
@code{logb (4.0)} is @code{2.0}.

When @code{2} raised to this power is divided into @var{x}, it gives a
quotient between @code{1} (inclusive) and @code{2} (exclusive).

If @var{x} is zero, the value is minus infinity (if the machine supports
such a value), or else a very small number.  If @var{x} is infinity, the
value is infinity.

The value returned by @code{logb} is one less than the value that
@code{frexp} would store into @code{*@var{exponent}}.
@end deftypefun

@comment math.h
@comment BSD
@deftypefun double copysign (double @var{value}, double @var{sign})
The @code{copysign} function returns 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.
@end deftypefun

@node Rounding and Remainders
@section Rounding and Remainder Functions
@cindex rounding functions
@cindex remainder functions
@cindex converting floats to integers

@pindex math.h
The functions listed here perform operations such as rounding,
truncation, and remainder in division of floating point numbers.  Some
of these functions convert floating point numbers to integer values.
They are all declared in @file{math.h}.

You can also convert floating-point numbers to integers simply by
casting them to @code{int}.  This discards the fractional part,
effectively rounding towards zero.  However, this only works if the
result can actually be represented as an @code{int}---for very large
numbers, this is impossible.  The functions listed here return the
result as a @code{double} instead to get around this problem.

@comment math.h
@comment ANSI
@deftypefun double ceil (double @var{x})
The @code{ceil} function rounds @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

@comment math.h
@comment ANSI
@deftypefun double floor (double @var{x})
The @code{ceil} function rounds @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
@deftypefun double rint (double @var{x})
This function rounds @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
support other modes, but round-to-nearest is always used unless
you explicit select another.
@end deftypefun

@comment math.h
@comment ANSI
@deftypefun double modf (double @var{value}, double *@var{integer-part})
This function breaks 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.

@code{modf} stores the integer part in @code{*@var{integer-part}}, and
returns the fractional part.  For example, @code{modf (2.5, &intpart)}
returns @code{0.5} and stores @code{2.0} into @code{intpart}.
@end deftypefun

@comment math.h
@comment ANSI
@deftypefun double fmod (double @var{numerator}, double @var{denominator})
This function computes the remainder from the division of
@var{numerator} by @var{denominator}.  Specifically, the return value is
@w{@code{@var{numerator} - @var{n} * @var{denominator}}}, where @var{n}
is the quotient of @var{numerator} divided by @var{denominator}, rounded
towards zero to an integer.  Thus, @w{@code{fmod (6.5, 2.3)}} returns
@code{1.9}, which is @code{6.5} minus @code{4.6}.

The result has the same sign as the @var{numerator} and has magnitude
less than the magnitude of the @var{denominator}.

If @var{denominator} is zero, @code{fmod} fails and sets @code{errno} to
@code{EDOM}.
@end deftypefun

@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
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}.

The absolute value of the result is less than or equal to half the
absolute value of the @var{denominator}.  The difference between
@code{fmod (@var{numerator}, @var{denominator})} and @code{drem
(@var{numerator}, @var{denominator})} is always either
@var{denominator}, minus @var{denominator}, or zero.

If @var{denominator} is zero, @code{drem} fails and sets @code{errno} to
@code{EDOM}.
@end deftypefun


@node Integer Division
@section Integer Division
@cindex integer division functions

This section describes functions for performing integer division.  These
functions are redundant in the GNU C library, since in GNU C the @samp{/}
operator always rounds towards zero.  But in other C implementations,
@samp{/} may round differently with negative arguments.  @code{div} and
@code{ldiv} are useful because they specify how to round the quotient:
towards zero.  The remainder has the same sign as the numerator.

These functions are specified to return a result @var{r} such that the value
@code{@var{r}.quot*@var{denominator} + @var{r}.rem} equals
@var{numerator}.

@pindex stdlib.h
To use these facilities, you should include the header file
@file{stdlib.h} in your program.

@comment stdlib.h
@comment ANSI
@deftp {Data Type} div_t
This is a structure type used to hold the result returned by the @code{div}
function.  It has the following members:

@table @code
@item int quot
The quotient from the division.

@item int rem
The remainder from the division.
@end table
@end deftp

@comment stdlib.h
@comment ANSI
@deftypefun div_t div (int @var{numerator}, int @var{denominator})
This function @code{div} computes the quotient and remainder from
the division of @var{numerator} by @var{denominator}, returning the
result in a structure of type @code{div_t}.

If the result cannot be represented (as in a division by zero), the
behavior is undefined.

Here is an example, albeit not a very useful one.

@smallexample
div_t result;
result = div (20, -6);
@end smallexample

@noindent
Now @code{result.quot} is @code{-3} and @code{result.rem} is @code{2}.
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftp {Data Type} ldiv_t
This is a structure type used to hold the result returned by the @code{ldiv}
function.  It has the following members:

@table @code
@item long int quot
The quotient from the division.

@item long int rem
The remainder from the division.
@end table

(This is identical to @code{div_t} except that the components are of
type @code{long int} rather than @code{int}.)
@end deftp

@comment stdlib.h
@comment ANSI
@deftypefun ldiv_t ldiv (long int @var{numerator}, long int @var{denominator})
The @code{ldiv} function is similar to @code{div}, except that the
arguments are of type @code{long int} and the result is returned as a
structure of type @code{ldiv}.
@end deftypefun


@node Parsing of Numbers
@section Parsing of Numbers
@cindex parsing numbers (in formatted input)
@cindex converting strings to numbers
@cindex number syntax, parsing
@cindex syntax, for reading numbers

This section describes functions for ``reading'' integer and
floating-point numbers from a string.  It may be more convenient in some
cases to use @code{sscanf} or one of the related functions; see
@ref{Formatted Input}.  But often you can make a program more robust by
finding the tokens in the string by hand, then converting the numbers
one by one.

@menu
* Parsing of Integers::         Functions for conversion of integer values.
* Parsing of Floats::           Functions for conversion of floating-point
				 values.
@end menu

@node Parsing of Integers
@subsection Parsing of Integers

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ANSI
@deftypefun {long int} strtol (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtol} (``string-to-long'') function converts the initial
part of @var{string} to a signed integer, which is returned as a value
of type @code{long int}.  

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item 
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item 
An optional plus or minus sign (@samp{+} or @samp{-}).

@item 
A nonempty sequence of digits in the radix specified by @var{base}.

If @var{base} is zero, decimal radix is assumed unless the series of
digits begins with @samp{0} (specifying octal radix), or @samp{0x} or
@samp{0X} (specifying hexadecimal radix); in other words, the same
syntax used for integer constants in C.

Otherwise @var{base} must have a value between @code{2} and @code{35}.
If @var{base} is @code{16}, the digits may optionally be preceded by
@samp{0x} or @samp{0X}.

@item 
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, @code{strtol} stores a pointer to this tail in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for an integer in the
specified @var{base}, no conversion is performed.  In this case,
@code{strtol} returns a value of zero and the value stored in
@code{*@var{tailptr}} is the value of @var{string}.

In a locale other than the standard @code{"C"} locale, this function
may recognize additional implementation-dependent syntax.

If the string has valid syntax for an integer but the value is not
representable because of overflow, @code{strtol} returns either
@code{LONG_MAX} or @code{LONG_MIN} (@pxref{Range of Type}), as
appropriate for the sign of the value.  It also sets @code{errno}
to @code{ERANGE} to indicate there was overflow.

There is an example at the end of this section.
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftypefun {unsigned long int} strtoul (const char *@var{string}, char **@var{tailptr}, int @var{base})
The @code{strtoul} (``string-to-unsigned-long'') function is like
@code{strtol} except that it returns its value with type @code{unsigned
long int}.  The value returned in case of overflow is @code{ULONG_MAX}
(@pxref{Range of Type}).
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftypefun {long int} atol (const char *@var{string})
This function is similar to the @code{strtol} function with a @var{base}
argument of @code{10}, except that it need not detect overflow errors.
The @code{atol} function is provided mostly for compatibility with
existing code; using @code{strtol} is more robust.
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftypefun int atoi (const char *@var{string})
This function is like @code{atol}, except that it returns an @code{int}
value rather than @code{long int}.  The @code{atoi} function is also
considered obsolete; use @code{strtol} instead.
@end deftypefun

Here is a function which parses a string as a sequence of integers and
returns the sum of them:

@smallexample
int
sum_ints_from_string (char *string)
@{
  int sum = 0;

  while (1) @{
    char *tail;
    int next;

    /* @r{Skip whitespace by hand, to detect the end.}  */
    while (isspace (*string)) string++;
    if (*string == 0)
      break;

    /* @r{There is more nonwhitespace,}  */
    /* @r{so it ought to be another number.}  */
    errno = 0;
    /* @r{Parse it.}  */
    next = strtol (string, &tail, 0);
    /* @r{Add it in, if not overflow.}  */
    if (errno)
      printf ("Overflow\n");
    else
      sum += next;
    /* @r{Advance past it.}  */
    string = tail;
  @}

  return sum;
@}
@end smallexample

@node Parsing of Floats
@subsection Parsing of Floats

@pindex stdlib.h
These functions are declared in @file{stdlib.h}.

@comment stdlib.h
@comment ANSI
@deftypefun double strtod (const char *@var{string}, char **@var{tailptr})
The @code{strtod} (``string-to-double'') function converts the initial
part of @var{string} to a floating-point number, which is returned as a
value of type @code{double}.  

This function attempts to decompose @var{string} as follows:

@itemize @bullet
@item 
A (possibly empty) sequence of whitespace characters.  Which characters
are whitespace is determined by the @code{isspace} function
(@pxref{Classification of Characters}).  These are discarded.

@item
An optional plus or minus sign (@samp{+} or @samp{-}).

@item
A nonempty sequence of digits optionally containing a decimal-point
character---normally @samp{.}, but it depends on the locale
(@pxref{Numeric Formatting}).

@item
An optional exponent part, consisting of a character @samp{e} or
@samp{E}, an optional sign, and a sequence of digits.

@item
Any remaining characters in the string.  If @var{tailptr} is not a null
pointer, a pointer to this tail of the string is stored in
@code{*@var{tailptr}}.
@end itemize

If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for a floating-point
number, no conversion is performed.  In this case, @code{strtod} returns
a value of zero and the value returned in @code{*@var{tailptr}} is the
value of @var{string}.

In a locale other than the standard @code{"C"} locale, this function may
recognize additional locale-dependent syntax.

If the string has valid syntax for a floating-point number but the value
is not representable because of overflow, @code{strtod} returns either
positive or negative @code{HUGE_VAL} (@pxref{Mathematics}), depending on
the sign of the value.  Similarly, if the value is not representable
because of underflow, @code{strtod} returns zero.  It also sets @code{errno}
to @code{ERANGE} if there was overflow or underflow.
@end deftypefun

@comment stdlib.h
@comment ANSI
@deftypefun double atof (const char *@var{string})
This function is similar to the @code{strtod} function, except that it
need not detect overflow and underflow errors.  The @code{atof} function
is provided mostly for compatibility with existing code; using
@code{strtod} is more robust.
@end deftypefun