# 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 `math.h'.

## "Not a Number" Values

The IEEE floating point format used by most modern computers supports values that are "not a number". These values are called 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, `x == x` can be 0 if the value of `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 `isnan` function (see section Predicates on Floats).

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

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 `#ifdef NAN' to test whether the machine supports NaNs. (Of course, you must arrange for GNU extensions to be visible, such as by defining `_GNU_SOURCE`, and then you must include `math.h'.)

## Predicates on Floats

This section describes some miscellaneous test functions on doubles. Prototypes for these functions appear in `math.h'. These are BSD functions, and thus are available if you define `_BSD_SOURCE` or `_GNU_SOURCE`.

Function: int isinf (double x)
This function returns `-1` if x represents negative infinity, `1` if x represents positive infinity, and `0` otherwise.

Function: int isnan (double x)
This function returns a nonzero value if x is a "not a number" value, and zero otherwise. (You can just as well use ```x != x``` to get the same result).

Function: int finite (double x)
This function returns a nonzero value if x is finite or a "not a number" value, and zero otherwise.

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

In the BSD library, on certain machines, `infnan` raises a fatal signal in all cases. The GNU library does not do likewise, because that does not fit the ISO C specification.

Portability Note: The functions listed in this section are BSD extensions.

## Absolute Value

These functions are provided for obtaining the absolute value (or magnitude) of a number. The absolute value of a real number x is x is x is positive, -x if x is negative. For a complex number z, whose real part is x and whose imaginary part is y, the absolute value is ```sqrt (x*x + y*y)```.

Prototypes for `abs` and `labs` are in `stdlib.h'; `fabs` and `cabs` are declared in `math.h'.

Function: int abs (int number)
This function returns the absolute value of number.

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

Function: long int labs (long int number)
This is similar to `abs`, except that both the argument and result are of type `long int` rather than `int`.

Function: double fabs (double number)
This function returns the absolute value of the floating-point number number.

Function: double cabs (struct { double real, imag; } z)
The `cabs` function returns the absolute value of the complex number z, whose real part is `z.real` and whose imaginary part is `z.imag`. (See also the function `hypot` in section Exponentiation and Logarithms.) The value is:

```
sqrt (z.real*z.real + z.imag*z.imag)

```

## Normalization Functions

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 section Floating Point Representation 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.

All these functions are declared in `math.h'.

Function: double frexp (double value, int *exponent)
The `frexp` function is used to split the number value into a normalized fraction and an exponent.

If the argument value is not zero, the return value is value times a power of two, and is always in the range 1/2 (inclusive) to 1 (exclusive). The corresponding exponent is stored in `*exponent`; the return value multiplied by 2 raised to this exponent equals the original number value.

For example, `frexp (12.8, &exponent)` returns `0.8` and stores `4` in `exponent`.

If value is zero, then the return value is zero and zero is stored in `*exponent`.

Function: double ldexp (double value, int exponent)
This function returns the result of multiplying the floating-point number value by 2 raised to the power exponent. (It can be used to reassemble floating-point numbers that were taken apart by `frexp`.)

For example, `ldexp (0.8, 4)` returns `12.8`.

The following functions which come from BSD provide facilities equivalent to those of `ldexp` and `frexp`:

Function: double scalb (double value, int exponent)
The `scalb` function is the BSD name for `ldexp`.

Function: double logb (double x)
This BSD function returns the integer part of the base-2 logarithm of x, an integer value represented in type `double`. This is the highest integer power of `2` contained in x. The sign of x is ignored. For example, `logb (3.5)` is `1.0` and `logb (4.0)` is `2.0`.

When `2` raised to this power is divided into x, it gives a quotient between `1` (inclusive) and `2` (exclusive).

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

The value returned by `logb` is one less than the value that `frexp` would store into `*exponent`.

Function: double copysign (double value, double sign)
The `copysign` function returns a value whose absolute value is the same as that of value, and whose sign matches that of sign. This is a BSD function.

## Rounding and Remainder Functions

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 `math.h'.

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

Function: double ceil (double x)
The `ceil` function rounds x upwards to the nearest integer, returning that value as a `double`. Thus, `ceil (1.5)` is `2.0`.

Function: double floor (double x)
The `ceil` function rounds x downwards to the nearest integer, returning that value as a `double`. Thus, ```floor (1.5)``` is `1.0` and `floor (-1.5)` is `-2.0`.

Function: double rint (double x)
This function rounds x to an integer value according to the current rounding mode. See section 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.

Function: double modf (double value, double *integer-part)
This function breaks the argument value into an integer part and a fractional part (between `-1` and `1`, exclusive). Their sum equals value. Each of the parts has the same sign as value, so the rounding of the integer part is towards zero.

`modf` stores the integer part in `*integer-part`, and returns the fractional part. For example, `modf (2.5, &intpart)` returns `0.5` and stores `2.0` into `intpart`.

Function: double fmod (double numerator, double denominator)
This function computes the remainder from the division of numerator by denominator. Specifically, the return value is `numerator - n * denominator`, where n is the quotient of numerator divided by denominator, rounded towards zero to an integer. Thus, `fmod (6.5, 2.3)` returns `1.9`, which is `6.5` minus `4.6`.

The result has the same sign as the numerator and has magnitude less than the magnitude of the denominator.

If denominator is zero, `fmod` fails and sets `errno` to `EDOM`.

Function: double drem (double numerator, double denominator)
The function `drem` is like `fmod` except that it rounds the internal quotient n to the nearest integer instead of towards zero to an integer. For example, `drem (6.5, 2.3)` returns `-0.4`, which is `6.5` minus `6.9`.

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

If denominator is zero, `drem` fails and sets `errno` to `EDOM`.

## Integer Division

This section describes functions for performing integer division. These functions are redundant in the GNU C library, since in GNU C the `/' operator always rounds towards zero. But in other C implementations, `/' may round differently with negative arguments. `div` and `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 r such that the value `r.quot*denominator + r.rem` equals numerator.

To use these facilities, you should include the header file `stdlib.h' in your program.

Data Type: div_t
This is a structure type used to hold the result returned by the `div` function. It has the following members:

`int quot`
The quotient from the division.
`int rem`
The remainder from the division.

Function: div_t div (int numerator, int denominator)
This function `div` computes the quotient and remainder from the division of numerator by denominator, returning the result in a structure of type `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.

```
div_t result;

result = div (20, -6);

```

Now `result.quot` is `-3` and `result.rem` is `2`.

Data Type: ldiv_t
This is a structure type used to hold the result returned by the `ldiv` function. It has the following members:

`long int quot`
The quotient from the division.
`long int rem`
The remainder from the division.

(This is identical to `div_t` except that the components are of type `long int` rather than `int`.)

Function: ldiv_t ldiv (long int numerator, long int denominator)
The `ldiv` function is similar to `div`, except that the arguments are of type `long int` and the result is returned as a structure of type `ldiv`.

## Parsing of 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 `sscanf` or one of the related functions; see section 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.

### Parsing of Integers

These functions are declared in `stdlib.h'.

Function: long int strtol (const char *string, char **tailptr, int base)
The `strtol` ("string-to-long") function converts the initial part of string to a signed integer, which is returned as a value of type `long int`.

This function attempts to decompose string as follows:

• A (possibly empty) sequence of whitespace characters. Which characters are whitespace is determined by the `isspace` function (see section Classification of Characters). These are discarded.
• An optional plus or minus sign (`+' or `-').
• A nonempty sequence of digits in the radix specified by base. If base is zero, decimal radix is assumed unless the series of digits begins with `0' (specifying octal radix), or `0x' or `0X' (specifying hexadecimal radix); in other words, the same syntax used for integer constants in C. Otherwise base must have a value between `2` and `35`. If base is `16`, the digits may optionally be preceded by `0x' or `0X'. If base has no legal value the value returned is `0l` and the global variable `errno` is set to `EINVAL`.
• Any remaining characters in the string. If tailptr is not a null pointer, `strtol` stores a pointer to this tail in `*tailptr`.

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 base, no conversion is performed. In this case, `strtol` returns a value of zero and the value stored in `*tailptr` is the value of string.

In a locale other than the standard `"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, `strtol` returns either `LONG_MAX` or `LONG_MIN` (see section Range of an Integer Type), as appropriate for the sign of the value. It also sets `errno` to `ERANGE` to indicate there was overflow.

Because the value `0l` is a correct result for `strtol` the user who is interested in handling errors should set the global variable `errno` to `0` before calling this function, so that the program can later test whether an error occurred.

There is an example at the end of this section.

Function: unsigned long int strtoul (const char *string, char **tailptr, int base)
The `strtoul` ("string-to-unsigned-long") function is like `strtol` except it deals with unsigned numbers, and returns its value with type `unsigned long int`. No `+' or `-' sign may appear before the number, but the syntax is otherwise the same as described above for `strtol`. The value returned in case of overflow is `ULONG_MAX` (see section Range of an Integer Type).

Like `strtol` this function sets `errno` and returns the value `0ul` in case the value for base is not in the legal range. For `strtoul` this can happen in another situation. In case the number to be converted is negative `strtoul` also sets `errno` to `EINVAL` and returns `0ul`.

Function: long long int strtoq (const char *string, char **tailptr, int base)
The `strtoq` ("string-to-quad-word") function is like `strtol` except that is deals with extra long numbers and it returns its value with type `long long int`.

If the string has valid syntax for an integer but the value is not representable because of overflow, `strtoq` returns either `LONG_LONG_MAX` or `LONG_LONG_MIN` (see section Range of an Integer Type), as appropriate for the sign of the value. It also sets `errno` to `ERANGE` to indicate there was overflow.

Function: long long int strtoll (const char *string, char **tailptr, int base)
`strtoll` is only an commonly used other name for the `strtoq` function. Everything said for `strtoq` applies to `strtoll` as well.

Function: unsigned long long int strtouq (const char *string, char **tailptr, int base)
The `strtouq` ("string-to-unsigned-quad-word") function is like `strtoul` except that is deals with extra long numbers and it returns its value with type `unsigned long long int`. The value returned in case of overflow is `ULONG_LONG_MAX` (see section Range of an Integer Type).

Function: unsigned long long int strtoull (const char *string, char **tailptr, int base)
`strtoull` is only an commonly used other name for the `strtouq` function. Everything said for `strtouq` applies to `strtoull` as well.

Function: long int atol (const char *string)
This function is similar to the `strtol` function with a base argument of `10`, except that it need not detect overflow errors. The `atol` function is provided mostly for compatibility with existing code; using `strtol` is more robust.

Function: int atoi (const char *string)
This function is like `atol`, except that it returns an `int` value rather than `long int`. The `atoi` function is also considered obsolete; use `strtol` instead.

The POSIX locales contain some information about how to format numbers (see section Generic Numeric Formatting Parameters). This mainly deals with representing numbers for better readability for humans. The functions present so far in this section cannot handle numbers in this form.

If this functionality is needed in a program one can use the functions from the `scanf` family which know about the flag `'' for parsing numeric input (see section Numeric Input Conversions). Sometimes it is more desirable to have finer control.

In these situation one could use the function `__strtoXXX_internal`. XXX here stands for any of the above forms. All numeric conversion functions (including the functions to process floating-point numbers) have such a counterpart. The difference to the normal form is the extra argument at the end of the parameter list. If this value has an non-zero value the handling of number grouping is enabled. The advantage of using these functions is that the tailptr parameters allow to determine which part of the input is processed. The `scanf` functions don't provide this information. The drawback of using these functions is that they are not portable. They only exist in the GNU C library.

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

```
int

sum_ints_from_string (char *string)

{

int sum = 0;

while (1) {

char *tail;

int next;

/* Skip whitespace by hand, to detect the end.  */

while (isspace (*string)) string++;

if (*string == 0)

break;

/* There is more nonwhitespace,  */

/* so it ought to be another number.  */

errno = 0;

/* Parse it.  */

next = strtol (string, &tail, 0);

/* Add it in, if not overflow.  */

if (errno)

printf ("Overflow\n");

else

sum += next;

string = tail;

}

return sum;

}

```

### Parsing of Floats

These functions are declared in `stdlib.h'.

Function: double strtod (const char *string, char **tailptr)
The `strtod` ("string-to-double") function converts the initial part of string to a floating-point number, which is returned as a value of type `double`.

This function attempts to decompose string as follows:

• A (possibly empty) sequence of whitespace characters. Which characters are whitespace is determined by the `isspace` function (see section Classification of Characters). These are discarded.
• An optional plus or minus sign (`+' or `-').
• A nonempty sequence of digits optionally containing a decimal-point character--normally `.', but it depends on the locale (see section Numeric Formatting).
• An optional exponent part, consisting of a character `e' or `E', an optional sign, and a sequence of digits.
• Any remaining characters in the string. If tailptr is not a null pointer, a pointer to this tail of the string is stored in `*tailptr`.

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, `strtod` returns a value of zero and the value returned in `*tailptr` is the value of string.

In a locale other than the standard `"C"` or `"POSIX"` locales, 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, `strtod` returns either positive or negative `HUGE_VAL` (see section Mathematics), depending on the sign of the value. Similarly, if the value is not representable because of underflow, `strtod` returns zero. It also sets `errno` to `ERANGE` if there was overflow or underflow.

Since the value zero which is returned in the error case is also a valid result the user should set the global variable `errno` to zero before calling this function. So one can test for failures after the call since all failures set `errno` to a non-zero value.

Function: float strtof (const char *string, char **tailptr)
This function is similar to the `strtod` function but it returns a `float` value instead of a `double` value. If the precision of a `float` value is sufficient this function should be used since it is much faster than `strtod` on some architectures. The reasons are obvious: IEEE 754 defines `float` to have a mantissa of 23 bits while `double` has 53 bits and every additional bit of precision can require additional computation.

If the string has valid syntax for a floating-point number but the value is not representable because of overflow, `strtof` returns either positive or negative `HUGE_VALf` (see section Mathematics), depending on the sign of the value.

This function is a GNU extension.

Function: long double strtold (const char *string, char **tailptr)
This function is similar to the `strtod` function but it returns a `long double` value instead of a `double` value. It should be used when high precision is needed. On systems which define a ```long double``` type (i.e., on which it is not the same as `double`) running this function might take significantly more time since more bits of precision are required.

If the string has valid syntax for a floating-point number but the value is not representable because of overflow, `strtold` returns either positive or negative `HUGE_VALl` (see section Mathematics), depending on the sign of the value.

This function is a GNU extension.

As for the integer parsing functions there are additional functions which will handle numbers represented using the grouping scheme of the current locale (see section Parsing of Integers).

Function: double atof (const char *string)
This function is similar to the `strtod` function, except that it need not detect overflow and underflow errors. The `atof` function is provided mostly for compatibility with existing code; using `strtod` is more robust.