Arithmetic Functions
********************

This chapter contains information about functions for doing basic arithmetic operations, such as splitting a float into its integer and fractional parts or retrieving the imaginary part of a complex value. These functions are declared in the header files math.h and complex.h.

Integers
========

The C language defines several integer data types: integer, short integer, long integer, and character, all in both signed and unsigned varieties. The GNU C compiler extends the language to contain long long integers as well.

The C integer types were intended to allow code to be portable among machines with different inherent data sizes (word sizes), so each type may have different ranges on different machines.  The problem with this is that a program often needs to be written for a particular range of integers, and sometimes must be written for a particular size of storage, regardless of what machine the program runs on.

To address this problem, theglibc contains C type definitions you can use to declare integers that meet your exact needs.  Because the glibcadj header files are customized to a specific machine, your program source code doesn't have to be.

These typedefs are in stdint.h.

If you require that an integer be represented in exactly N bits, use one of the following types, with the obvious mapping to bit size and signedness:

  * int8_t

  * int16_t

  * int32_t

  * int64_t

  * uint8_t

  * uint16_t

  * uint32_t

  * uint64_t

If your C compiler and target machine do not allow integers of a certain size, the corresponding above type does not exist.

If you don't need a specific storage size, but want the smallest data structure with at least N bits, use one of these:

  * int_least8_t

  * int_least16_t

  * int_least32_t

  * int_least64_t

  * uint_least8_t

  * uint_least16_t

  * uint_least32_t

  * uint_least64_t

If you don't need a specific storage size, but want the data structure that allows the fastest access while having at least N bits (and among data structures with the same access speed, the smallest one), use one of these:

  * int_fast8_t

  * int_fast16_t

  * int_fast32_t

  * int_fast64_t

  * uint_fast8_t

  * uint_fast16_t

  * uint_fast32_t

  * uint_fast64_t

If you want an integer with the widest range possible on the platform on which it is being used, use one of the following.  If you use these, you should write code that takes into account the variable size and range of the integer.

  * intmax_t

  * uintmax_t

Theglibc also provides macros that tell you the maximum and minimum possible values for each integer data type.  The macro names follow these examples: INT32_MAX, UINT8_MAX, INT_FAST32_MIN, INT_LEAST64_MIN, UINTMAX_MAX, INTMAX_MAX, INTMAX_MIN.  Note that there are no macros for unsigned integer minima.  These are always zero.  Similiarly, there are macros such as INTMAX_WIDTH for the width of these types. Those macros for integer type widths come from TS 18661-1:2014.

There are similar macros for use with C's built in integer types which should come with your C compiler.  These are described in Data Type Measurements.

Don't forget you can use the C sizeof function with any of these data types to get the number of bytes of storage each uses.

Integer Division
================

This section describes functions for performing integer division.  These functions are redundant when GNU CC is used, because 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
     ISO, stdlib.h 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)
     ISO, stdlib.h The 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
     ISO, stdlib.h 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)
     ISO, stdlib.h 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_t.

 -- Data Type: lldiv_t
     ISO, stdlib.h This is a structure type used to hold the result returned by the lldiv function.  It has the following members:

     long long int quot
          The quotient from the division.

     long long int rem
          The remainder from the division.

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

 -- Function: lldiv_t lldiv (long long int NUMERATOR, long long int DENOMINATOR)
     ISO, stdlib.h The lldiv function is like the div function, but the arguments are of type long long int and the result is returned as a structure of type lldiv_t.

     The lldiv function was added in ISO C99.

 -- Data Type: imaxdiv_t
     ISO, inttypes.h This is a structure type used to hold the result returned by the imaxdiv function.  It has the following members:

     intmax_t quot
          The quotient from the division.

     intmax_t rem
          The remainder from the division.

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

     See Integers for a description of the intmax_t type.

 -- Function: imaxdiv_t imaxdiv (intmax_t NUMERATOR, intmax_t DENOMINATOR)
     ISO, inttypes.h The imaxdiv function is like the div function, but the arguments are of type intmax_t and the result is returned as a structure of type imaxdiv_t.

     See Integers for a description of the intmax_t type.

     The imaxdiv function was added in ISO C99.

Floating Point Numbers
======================

Most computer hardware has support for two different kinds of numbers: integers (...-3, -2, -1, 0, 1, 2, 3...) and floating-point numbers.  Floating-point numbers have three parts: the mantissa, the exponent, and the sign bit.  The real number represented by a floating-point value is given by (s ? -1 : 1) mul 2^e mul M where s is the sign bit, e the exponent, and M the mantissa.  See Floating Point Concepts, for details.  (It is possible to have a different base for the exponent, but all modern hardware uses 2.)

Floating-point numbers can represent a finite subset of the real numbers.  While this subset is large enough for most purposes, it is important to remember that the only reals that can be represented exactly are rational numbers that have a terminating binary expansion shorter than the width of the mantissa.  Even simple fractions such as 1/5 can only be approximated by floating point.

Mathematical operations and functions frequently need to produce values that are not representable.  Often these values can be approximated closely enough for practical purposes, but sometimes they can't. Historically there was no way to tell when the results of a calculation were inaccurate.  Modern computers implement the IEEE 754 standard for numerical computations, which defines a framework for indicating to the program when the results of calculation are not trustworthy.  This framework consists of a set of exceptions that indicate why a result could not be represented, and the special values infinity and not a number (NaN).

Floating-Point Number Classification Functions
==============================================

ISO C99 defines macros that let you determine what sort of floating-point number a variable holds.

 -- Macro: int fpclassify (float-type X)
     ISO, math.h This is a generic macro which works on all floating-point types and which returns a value of type int.  The possible values are:

     FP_NAN
          C99, math.h The floating-point number X is ``Not a Number'' (see Infinity and NaN)

     FP_INFINITE
          C99, math.h The value of X is either plus or minus infinity (see Infinity and NaN)

     FP_ZERO
          C99, math.h The value of X is zero.  In floating-point formats like IEEE 754, where zero can be signed, this value is also returned if X is negative zero.

     FP_SUBNORMAL
          C99, math.h Numbers whose absolute value is too small to be represented in the normal format are represented in an alternate, denormalized format (see Floating Point Concepts).  This format is less precise but can represent values closer to zero.  fpclassify returns this value for values of X in this alternate format.

     FP_NORMAL
          C99, math.h This value is returned for all other values of X.  It indicates that there is nothing special about the number.

fpclassify is most useful if more than one property of a number must be tested.  There are more specific macros which only test one property at a time.  Generally these macros execute faster than fpclassify, since there is special hardware support for them. You should therefore use the specific macros whenever possible.

 -- Macro: int iscanonical (float-type X)
     ISO, math.h In some floating-point formats, some values have canonical (preferred) and noncanonical encodings (for IEEE interchange binary formats, all encodings are canonical).  This macro returns a nonzero value if X has a canonical encoding.  It is from TS 18661-1:2014.

     Note that some formats have multiple encodings of a value which are all equally canonical; iscanonical returns a nonzero value for all such encodings.  Also, formats may have encodings that do not correspond to any valid value of the type.  In ISO C terms these are trap representations; in theglibc, iscanonical returns zero for such encodings.

 -- Macro: int isfinite (float-type X)
     ISO, math.h This macro returns a nonzero value if X is finite: not plus or minus infinity, and not NaN.  It is equivalent to

          (fpclassify (x) != FP_NAN && fpclassify (x) != FP_INFINITE)

     isfinite is implemented as a macro which accepts any floating-point type.

 -- Macro: int isnormal (float-type X)
     ISO, math.h This macro returns a nonzero value if X is finite and normalized. It is equivalent to

          (fpclassify (x) == FP_NORMAL)

 -- Macro: int isnan (float-type X)
     ISO, math.h This macro returns a nonzero value if X is NaN.  It is equivalent to

          (fpclassify (x) == FP_NAN)

 -- Macro: int issignaling (float-type X)
     ISO, math.h This macro returns a nonzero value if X is a signaling NaN (sNaN).  It is from TS 18661-1:2014.

 -- Macro: int issubnormal (float-type X)
     ISO, math.h This macro returns a nonzero value if X is subnormal.  It is from TS 18661-1:2014.

 -- Macro: int iszero (float-type X)
     ISO, math.h This macro returns a nonzero value if X is zero.  It is from TS 18661-1:2014.

Another set of floating-point classification functions was provided by BSD.  Theglibc also supports these functions; however, we recommend that you use the ISO C99 macros in new code.  Those are standard and will be available more widely.  Also, since they are macros, you do not have to worry about the type of their argument.

 -- Function: int isinf (double X)
 -- Function: int isinff (float X)
 -- Function: int isinfl (long double X)
     BSD, math.h This function returns -1 if X represents negative infinity, 1 if X represents positive infinity, and 0 otherwise.

 -- Function: int isnan (double X)
 -- Function: int isnanf (float X)
 -- Function: int isnanl (long double X)
     BSD, math.h This function returns a nonzero value if X is a ``not a number'' value, and zero otherwise.

     NB: The isnan macro defined by ISO C99 overrides the BSD function.  This is normally not a problem, because the two routines behave identically.  However, if you really need to get the BSD function for some reason, you can write

          (isnan) (x)

 -- Function: int finite (double X)
 -- Function: int finitef (float X)
 -- Function: int finitel (long double X)
     BSD, math.h This function returns a nonzero value if X is neither infinite nor a ``not a number'' value, and zero otherwise.

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

Errors in Floating-Point Calculations
=====================================

FP Exceptions
-------------

The IEEE 754 standard defines five exceptions that can occur during a calculation.  Each corresponds to a particular sort of error, such as overflow.

When exceptions occur (when exceptions are raised, in the language of the standard), one of two things can happen.  By default the exception is simply noted in the floating-point status word, and the program continues as if nothing had happened.  The operation produces a default value, which depends on the exception (see the table below).  Your program can check the status word to find out which exceptions happened.

Alternatively, you can enable traps for exceptions.  In that case, when an exception is raised, your program will receive the SIGFPE signal.  The default action for this signal is to terminate the program.  See Signal Handling, for how you can change the effect of the signal.

The exceptions defined in IEEE 754 are:

Invalid Operation
     This exception is raised if the given operands are invalid for the operation to be performed.  Examples are (see IEEE 754, section 7):

      1. Addition or subtraction: infinity - infinity.  (But infinity + infinity = infinity).

      2. Multiplication: 0 mul infinity.

      3. Division: 0/0 or infinity/infinity.

      4. Remainder: x REM y, where y is zero or x is infinite.

      5. Square root if the operand is less than zero.  More generally, any mathematical function evaluated outside its domain produces this exception.

      6. Conversion of a floating-point number to an integer or decimal string, when the number cannot be represented in the target format (due to overflow, infinity, or NaN).

      7. Conversion of an unrecognizable input string.

      8. Comparison via predicates involving < or >, when one or other of the operands is NaN.  You can prevent this exception by using the unordered comparison functions instead; see FP Comparison Functions.

     If the exception does not trap, the result of the operation is NaN.

Division by Zero
     This exception is raised when a finite nonzero number is divided by zero.  If no trap occurs the result is either +infinity or -infinity, depending on the signs of the operands.

Overflow
     This exception is raised 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 (IEEE 754, section 7.3):

      1. Round to nearest carries all overflows to infinity with the sign of the intermediate result.

      2. Round toward 0 carries all overflows to the largest representable finite number with the sign of the intermediate result.

      3. Round toward -infinity carries positive overflows to the largest representable finite number and negative overflows to -infinity.

      4. Round toward infinity carries negative overflows to the most negative representable finite number and positive overflows to infinity.

     Whenever the overflow exception is raised, the inexact exception is also raised.

Underflow
     The underflow exception is raised when an intermediate result is too small to be calculated accurately, or if the operation's result rounded to the destination precision is too small to be normalized.

     When no trap is installed for the underflow exception, underflow is 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 imprecise small value, or zero if the destination precision cannot hold the small exact result.

Inexact
     This exception is signalled if a rounded result is not exact (such as when calculating the square root of two) or a result overflows without an overflow trap.

Infinity and NaN
----------------

IEEE 754 floating point numbers can represent positive or negative infinity, and NaN (not a number).  These three values arise from calculations whose result is undefined or cannot be represented accurately.  You can also deliberately set a floating-point variable to any of them, which is sometimes useful.  Some examples of calculations that produce infinity or NaN:

     1/0 = infinity
     log (0) = -infinity
     sqrt (-1) = NaN

When a calculation produces any of these values, an exception also occurs; see FP Exceptions.

The basic operations and math functions all accept infinity and NaN and produce sensible output.  Infinities propagate through calculations as one would expect: for example, 2 + infinity = infinity, 4/infinity = 0, atan (infinity) = pi/2.  NaN, on the other hand, infects any calculation that involves it.  Unless the calculation would produce the same result no matter what real value replaced NaN, the result is NaN.

In comparison operations, positive infinity is larger than all values except itself and NaN, and negative infinity is smaller than all values except itself and NaN.  NaN is unordered: it is not equal to, greater than, or less than anything, including itself. x == x is false if the value of x is NaN.  You can use this to test whether a value is NaN or not, but the recommended way to test for NaN is with the isnan function (see Floating Point Classes).  In addition, <, >, <=, and >= will raise an exception when applied to NaNs.

math.h defines macros that allow you to explicitly set a variable to infinity or NaN.

 -- Macro: float INFINITY
     ISO, math.h An expression representing positive infinity.  It is equal to the value produced  by mathematical operations like 1.0 / 0.0. -INFINITY represents negative infinity.

     You can test whether a floating-point value is infinite by comparing it to this macro.  However, this is not recommended; you should use the isfinite macro instead.  See Floating Point Classes.

     This macro was introduced in the ISO C99 standard.

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

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

 -- Macro: float SNANF
 -- Macro: double SNAN
 -- Macro: long double SNANL
 -- Macro: _FloatN SNANFN
 -- Macro: _FloatNx SNANFNx
     TS 18661-1:2014, math.h SNANFN, TS 18661-3:2015, math.h SNANFNx, TS 18661-3:2015, math.h These macros, defined by TS 18661-1:2014 and TS 18661-3:2015, are constant expressions for signaling NaNs.

 -- Macro: int FE_SNANS_ALWAYS_SIGNAL
     ISO, fenv.h This macro, defined by TS 18661-1:2014, is defined to 1 in fenv.h to indicate that functions and operations with signaling NaN inputs and floating-point results always raise the invalid exception and return a quiet NaN, even in cases (such as fmax, hypot and pow) where a quiet NaN input can produce a non-NaN result.  Because some compiler optimizations may not handle signaling NaNs correctly, this macro is only defined if compiler support for signaling NaNs is enabled.  That support can be enabled with the GCC option -fsignaling-nans.

IEEE 754 also allows for another unusual value: negative zero.  This value is produced when you divide a positive number by negative infinity, or when a negative result is smaller than the limits of representation.

Examining the FPU status word
-----------------------------

ISO C99 defines functions to query and manipulate the floating-point status word.  You can use these functions to check for untrapped exceptions when it's convenient, rather than worrying about them in the middle of a calculation.

These constants represent the various IEEE 754 exceptions.  Not all FPUs report all the different exceptions.  Each constant is defined if and only if the FPU you are compiling for supports that exception, so you can test for FPU support with #ifdef.  They are defined in fenv.h.

FE_INEXACT
     ISO, fenv.h The inexact exception.

FE_DIVBYZERO
     ISO, fenv.h The divide by zero exception.

FE_UNDERFLOW
     ISO, fenv.h The underflow exception.

FE_OVERFLOW
     ISO, fenv.h The overflow exception.

FE_INVALID
     ISO, fenv.h The invalid exception.

The macro FE_ALL_EXCEPT is the bitwise OR of all exception macros which are supported by the FP implementation.

These functions allow you to clear exception flags, test for exceptions, and save and restore the set of exceptions flagged.

 -- Function: int feclearexcept (int EXCEPTS)
     ISO, fenv.h This function clears all of the supported exception flags indicated by EXCEPTS.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int feraiseexcept (int EXCEPTS)
     ISO, fenv.h This function raises the supported exceptions indicated by EXCEPTS.  If more than one exception bit in EXCEPTS is set the order in which the exceptions are raised is undefined except that overflow (FE_OVERFLOW) or underflow (FE_UNDERFLOW) are raised before inexact (FE_INEXACT).  Whether for overflow or underflow the inexact exception is also raised is also implementation dependent.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int fesetexcept (int EXCEPTS)
     ISO, fenv.h This function sets the supported exception flags indicated by EXCEPTS, like feraiseexcept, but without causing enabled traps to be taken.  fesetexcept is from TS 18661-1:2014.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int fetestexcept (int EXCEPTS)
     ISO, fenv.h Test whether the exception flags indicated by the parameter EXCEPT are currently set.  If any of them are, a nonzero value is returned which specifies which exceptions are set.  Otherwise the result is zero.

To understand these functions, imagine that the status word is an integer variable named STATUS.  feclearexcept is then equivalent to status &= ~excepts and fetestexcept is equivalent to (status & excepts).  The actual implementation may be very different, of course.

Exception flags are only cleared when the program explicitly requests it, by calling feclearexcept.  If you want to check for exceptions from a set of calculations, you should clear all the flags first.  Here is a simple example of the way to use fetestexcept:

     {
       double f;
       int raised;
       feclearexcept (FE_ALL_EXCEPT);
       f = compute ();
       raised = fetestexcept (FE_OVERFLOW | FE_INVALID);
       if (raised & FE_OVERFLOW) { /* ... */ }
       if (raised & FE_INVALID) { /* ... */ }
       /* ... */
     }

You cannot explicitly set bits in the status word.  You can, however, save the entire status word and restore it later.  This is done with the following functions:

 -- Function: int fegetexceptflag (fexcept_t *FLAGP, int EXCEPTS)
     ISO, fenv.h This function stores in the variable pointed to by FLAGP an implementation-defined value representing the current setting of the exception flags indicated by EXCEPTS.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int fesetexceptflag (const fexcept_t *FLAGP, int EXCEPTS)
     ISO, fenv.h This function restores the flags for the exceptions indicated by EXCEPTS to the values stored in the variable pointed to by FLAGP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

Note that the value stored in fexcept_t bears no resemblance to the bit mask returned by fetestexcept.  The type may not even be an integer.  Do not attempt to modify an fexcept_t variable.

 -- Function: int fetestexceptflag (const fexcept_t *FLAGP, int EXCEPTS)
     ISO, fenv.h Test whether the exception flags indicated by the parameter EXCEPTS are set in the variable pointed to by FLAGP.  If any of them are, a nonzero value is returned which specifies which exceptions are set.  Otherwise the result is zero. fetestexceptflag is from TS 18661-1:2014.

Error Reporting by Mathematical Functions
-----------------------------------------

Many of the math functions are defined only over a subset of the real or complex numbers.  Even if they are mathematically defined, their result may be larger or smaller than the range representable by their return type without loss of accuracy.  These are known as domain errors, overflows, and underflows, respectively.  Math functions do several things when one of these errors occurs.  In this manual we will refer to the complete response as signalling a domain error, overflow, or underflow.

When a math function suffers a domain error, it raises the invalid exception and returns NaN.  It also sets errno to EDOM; this is for compatibility with old systems that do not support IEEE 754 exception handling.  Likewise, when overflow occurs, math functions raise the overflow exception and, in the default rounding mode, return infinity or -infinity as appropriate (in other rounding modes, the largest finite value of the appropriate sign is returned when appropriate for that rounding mode).  They also set errno to ERANGE if returning infinity or -infinity; errno may or may not be set to ERANGE when a finite value is returned on overflow.  When underflow occurs, the underflow exception is raised, and zero (appropriately signed) or a subnormal value, as appropriate for the mathematical result of the function and the rounding mode, is returned.  errno may be set to ERANGE, but this is not guaranteed; it is intended that theglibc should set it when the underflow is to an appropriately signed zero, but not necessarily for other underflows.

When a math function has an argument that is a signaling NaN, theglibc does not consider this a domain error, so errno is unchanged, but the invalid exception is still raised (except for a few functions that are specified to handle signaling NaNs differently).

Some of the math 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 complex math functions, such as csqrt, will return the appropriate complex value in this case.  The real-valued functions, such as sqrt, will signal a domain error.

Some older hardware does not support infinities.  On that hardware, overflows instead return a particular very large number (usually the largest representable number).  math.h defines macros you can use to test for overflow on both old and new hardware.

 -- Macro: double HUGE_VAL
 -- Macro: float HUGE_VALF
 -- Macro: long double HUGE_VALL
 -- Macro: _FloatN HUGE_VAL_FN
 -- Macro: _FloatNx HUGE_VAL_FNx
     ISO, math.h HUGE_VAL_FN, TS 18661-3:2015, math.h HUGE_VAL_FNx, TS 18661-3:2015, math.h An expression representing a particular very large number.  On machines that use IEEE 754 floating point format, HUGE_VAL is infinity. On other machines, it's typically the largest positive number that can be represented.

     Mathematical functions return the appropriately typed version of HUGE_VAL or -HUGE_VAL when the result is too large to be represented.

Rounding Modes
==============

Floating-point calculations are carried out internally with extra precision, and then rounded to fit into the destination type.  This ensures that results are as precise as the input data.  IEEE 754 defines four possible rounding modes:

Round to nearest.
     This is the default mode.  It should be used unless there is a specific need for one of the others.  In this mode results are rounded to the nearest representable value.  If the result is midway between two representable values, the even representable is chosen. Even here means the lowest-order bit is zero.  This rounding mode prevents statistical bias and guarantees numeric stability: round-off errors in a lengthy calculation will remain smaller than half of FLT_EPSILON.

Round toward plus Infinity.
     All results are rounded to the smallest representable value which is greater than the result.

Round toward minus Infinity.
     All results are rounded to the largest representable value which is less than the result.

Round toward zero.
     All results are rounded to the largest representable value whose magnitude is less than that of the result.  In other words, if the result is negative it is rounded up; if it is positive, it is rounded down.

fenv.h defines constants which you can use to refer to the various rounding modes.  Each one will be defined if and only if the FPU supports the corresponding rounding mode.

FE_TONEAREST
     ISO, fenv.h Round to nearest.

FE_UPWARD
     ISO, fenv.h Round toward +infinity.

FE_DOWNWARD
     ISO, fenv.h Round toward -infinity.

FE_TOWARDZERO
     ISO, fenv.h Round toward zero.

Underflow is an unusual case.  Normally, IEEE 754 floating point numbers are always normalized (see Floating Point Concepts). Numbers smaller than 2^r (where r is the minimum exponent, FLT_MIN_RADIX-1 for FLOAT) cannot be represented as normalized numbers.  Rounding all such numbers to zero or 2^r would cause some algorithms to fail at 0.  Therefore, they are left in denormalized form.  That produces loss of precision, since some bits of the mantissa are stolen to indicate the decimal point.

If a result is too small to be represented as a denormalized number, it is rounded to zero.  However, the sign of the result is preserved; if the calculation was negative, the result is negative zero. Negative zero can also result from some operations on infinity, such as 4/-infinity.

At any time, one of the above four rounding modes is selected.  You can find out which one with this function:

 -- Function: int fegetround (void)
     ISO, fenv.h Returns the currently selected rounding mode, represented by one of the values of the defined rounding mode macros.

To change the rounding mode, use this function:

 -- Function: int fesetround (int ROUND)
     ISO, fenv.h Changes the currently selected rounding mode to ROUND.  If ROUND does not correspond to one of the supported rounding modes nothing is changed.  fesetround returns zero if it changed the rounding mode, or a nonzero value if the mode is not supported.

You should avoid changing the rounding mode if possible.  It can be an expensive operation; also, some hardware requires you to compile your program differently for it to work.  The resulting code may run slower. See your compiler documentation for details.

Floating-Point Control Functions
================================

IEEE 754 floating-point implementations allow the programmer to decide whether traps will occur for each of the exceptions, by setting bits in the control word.  In C, traps result in the program receiving the SIGFPE signal; see Signal Handling.

NB: IEEE 754 says that trap handlers are given details of the exceptional situation, and can set the result value.  C signals do not provide any mechanism to pass this information back and forth. Trapping exceptions in C is therefore not very useful.

It is sometimes necessary to save the state of the floating-point unit while you perform some calculation.  The library provides functions which save and restore the exception flags, the set of exceptions that generate traps, and the rounding mode.  This information is known as the floating-point environment.

The functions to save and restore the floating-point environment all use a variable of type fenv_t to store information.  This type is defined in fenv.h.  Its size and contents are implementation-defined.  You should not attempt to manipulate a variable of this type directly.

To save the state of the FPU, use one of these functions:

 -- Function: int fegetenv (fenv_t *ENVP)
     ISO, fenv.h Store the floating-point environment in the variable pointed to by ENVP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int feholdexcept (fenv_t *ENVP)
     ISO, fenv.h Store the current floating-point environment in the object pointed to by ENVP.  Then clear all exception flags, and set the FPU to trap no exceptions.  Not all FPUs support trapping no exceptions; if feholdexcept cannot set this mode, it returns nonzero value.  If it succeeds, it returns zero.

The functions which restore the floating-point environment can take these kinds of arguments:

  * Pointers to fenv_t objects, which were initialized previously by a call to fegetenv or feholdexcept.

  * The special macro FE_DFL_ENV which represents the floating-point environment as it was available at program start.

  * Implementation defined macros with names starting with FE_ and having type fenv_t *.

     If possible, theglibc defines a macro FE_NOMASK_ENV which represents an environment where every exception raised causes a trap to occur.  You can test for this macro using #ifdef.  It is only defined if _GNU_SOURCE is defined.

     Some platforms might define other predefined environments.

To set the floating-point environment, you can use either of these functions:

 -- Function: int fesetenv (const fenv_t *ENVP)
     ISO, fenv.h Set the floating-point environment to that described by ENVP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int feupdateenv (const fenv_t *ENVP)
     ISO, fenv.h Like fesetenv, this function sets the floating-point environment to that described by ENVP.  However, if any exceptions were flagged in the status word before feupdateenv was called, they remain flagged after the call.  In other words, after feupdateenv is called, the status word is the bitwise OR of the previous status word and the one saved in ENVP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

TS 18661-1:2014 defines additional functions to save and restore floating-point control modes (such as the rounding mode and whether traps are enabled) while leaving other status (such as raised flags) unchanged.

The special macro FE_DFL_MODE may be passed to fesetmode.  It represents the floating-point control modes at program start.

 -- Function: int fegetmode (femode_t *MODEP)
     ISO, fenv.h Store the floating-point control modes in the variable pointed to by MODEP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

 -- Function: int fesetmode (const femode_t *MODEP)
     ISO, fenv.h Set the floating-point control modes to those described by MODEP.

     The function returns zero in case the operation was successful, a non-zero value otherwise.

To control for individual exceptions if raising them causes a trap to occur, you can use the following two functions.

Portability Note: These functions are all GNU extensions.

 -- Function: int feenableexcept (int EXCEPTS)
     GNU, fenv.h This function enables traps for each of the exceptions as indicated by the parameter EXCEPTS.  The individual exceptions are described in Status bit operations.  Only the specified exceptions are enabled, the status of the other exceptions is not changed.

     The function returns the previous enabled exceptions in case the operation was successful, -1 otherwise.

 -- Function: int fedisableexcept (int EXCEPTS)
     GNU, fenv.h This function disables traps for each of the exceptions as indicated by the parameter EXCEPTS.  The individual exceptions are described in Status bit operations.  Only the specified exceptions are disabled, the status of the other exceptions is not changed.

     The function returns the previous enabled exceptions in case the operation was successful, -1 otherwise.

 -- Function: int fegetexcept (void)
     GNU, fenv.h The function returns a bitmask of all currently enabled exceptions.  It returns -1 in case of failure.

Arithmetic Functions
====================

The C library provides functions to do basic operations on floating-point numbers.  These include absolute value, maximum and minimum, normalization, bit twiddling, rounding, and a few others.

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 if 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, labs and llabs are in stdlib.h; imaxabs is declared in inttypes.h; the fabs functions are declared in math.h; the cabs functions are declared in complex.h.

 -- Function: int abs (int NUMBER)
 -- Function: long int labs (long int NUMBER)
 -- Function: long long int llabs (long long int NUMBER)
 -- Function: intmax_t imaxabs (intmax_t NUMBER)
     ISO, stdlib.h imaxabs, ISO, inttypes.h These functions return 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.

     llabs and imaxdiv are new to ISO C99.

     See Integers for a description of the intmax_t type.

 -- Function: double fabs (double NUMBER)
 -- Function: float fabsf (float NUMBER)
 -- Function: long double fabsl (long double NUMBER)
 -- Function: _FloatN fabsfN (_FloatN NUMBER)
 -- Function: _FloatNx fabsfNx (_FloatNx NUMBER)
     ISO, math.h fabsfN, TS 18661-3:2015, math.h fabsfNx, TS 18661-3:2015, math.h This function returns the absolute value of the floating-point number NUMBER.

 -- Function: double cabs (complex double Z)
 -- Function: float cabsf (complex float Z)
 -- Function: long double cabsl (complex long double Z)
 -- Function: _FloatN cabsfN (complex _FloatN Z)
 -- Function: _FloatNx cabsfNx (complex _FloatNx Z)
     ISO, complex.h cabsfN, TS 18661-3:2015, complex.h cabsfNx, TS 18661-3:2015, complex.h These functions return the absolute  value of the complex number Z (see Complex Numbers).  The absolute value of a complex number is:

          sqrt (creal (Z) * creal (Z) + cimag (Z) * cimag (Z))

     This function should always be used instead of the direct formula because it takes special care to avoid losing precision.  It may also take advantage of hardware support for this operation.  See hypot in Exponents and Logarithms.

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 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.

All these functions are declared in math.h.

 -- Function: double frexp (double VALUE, int *EXPONENT)
 -- Function: float frexpf (float VALUE, int *EXPONENT)
 -- Function: long double frexpl (long double VALUE, int *EXPONENT)
 -- Function: _FloatN frexpfN (_FloatN VALUE, int *EXPONENT)
 -- Function: _FloatNx frexpfNx (_FloatNx VALUE, int *EXPONENT)
     ISO, math.h frexpfN, TS 18661-3:2015, math.h frexpfNx, TS 18661-3:2015, math.h These functions are 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 its magnitude 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)
 -- Function: float ldexpf (float VALUE, int EXPONENT)
 -- Function: long double ldexpl (long double VALUE, int EXPONENT)
 -- Function: _FloatN ldexpfN (_FloatN VALUE, int EXPONENT)
 -- Function: _FloatNx ldexpfNx (_FloatNx VALUE, int EXPONENT)
     ISO, math.h ldexpfN, TS 18661-3:2015, math.h ldexpfNx, TS 18661-3:2015, math.h These functions return 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.  See also the ISO C function logb which originally also appeared in BSD. The _FloatN and _FloatN variants of the following functions come from TS 18661-3:2015.

 -- Function: double scalb (double VALUE, double EXPONENT)
 -- Function: float scalbf (float VALUE, float EXPONENT)
 -- Function: long double scalbl (long double VALUE, long double EXPONENT)
     BSD, math.h The scalb function is the BSD name for ldexp.

 -- Function: double scalbn (double X, int N)
 -- Function: float scalbnf (float X, int N)
 -- Function: long double scalbnl (long double X, int N)
 -- Function: _FloatN scalbnfN (_FloatN X, int N)
 -- Function: _FloatNx scalbnfNx (_FloatNx X, int N)
     BSD, math.h scalbnfN, TS 18661-3:2015, math.h scalbnfNx, TS 18661-3:2015, math.h scalbn is identical to scalb, except that the exponent N is an int instead of a floating-point number.

 -- Function: double scalbln (double X, long int N)
 -- Function: float scalblnf (float X, long int N)
 -- Function: long double scalblnl (long double X, long int N)
 -- Function: _FloatN scalblnfN (_FloatN X, long int N)
 -- Function: _FloatNx scalblnfNx (_FloatNx X, long int N)
     BSD, math.h scalblnfN, TS 18661-3:2015, math.h scalblnfNx, TS 18661-3:2015, math.h scalbln is identical to scalb, except that the exponent N is a long int instead of a floating-point number.

 -- Function: double significand (double X)
 -- Function: float significandf (float X)
 -- Function: long double significandl (long double X)
     BSD, math.h significand returns the mantissa of X scaled to the range [1, 2). It is equivalent to scalb (X, (double) -ilogb (X)).

     This function exists mainly for use in certain standardized tests of IEEE 754 conformance.

Rounding Functions
------------------

The functions listed here perform operations such as rounding and truncation of floating-point values.  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.

The fromfp functions use the following macros, from TS 18661-1:2014, to specify the direction of rounding.  These correspond to the rounding directions defined in IEEE 754-2008.

FP_INT_UPWARD
     ISO, math.h Round toward +infinity.

FP_INT_DOWNWARD
     ISO, math.h Round toward -infinity.

FP_INT_TOWARDZERO
     ISO, math.h Round toward zero.

FP_INT_TONEARESTFROMZERO
     ISO, math.h Round to nearest, ties round away from zero.

FP_INT_TONEAREST
     ISO, math.h Round to nearest, ties round to even.

 -- Function: double ceil (double X)
 -- Function: float ceilf (float X)
 -- Function: long double ceill (long double X)
 -- Function: _FloatN ceilfN (_FloatN X)
 -- Function: _FloatNx ceilfNx (_FloatNx X)
     ISO, math.h ceilfN, TS 18661-3:2015, math.h ceilfNx, TS 18661-3:2015, math.h These functions round X upwards to the nearest integer, returning that value as a double.  Thus, ceil (1.5) is 2.0.

 -- Function: double floor (double X)
 -- Function: float floorf (float X)
 -- Function: long double floorl (long double X)
 -- Function: _FloatN floorfN (_FloatN X)
 -- Function: _FloatNx floorfNx (_FloatNx X)
     ISO, math.h floorfN, TS 18661-3:2015, math.h floorfNx, TS 18661-3:2015, math.h These functions round 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 trunc (double X)
 -- Function: float truncf (float X)
 -- Function: long double truncl (long double X)
 -- Function: _FloatN truncfN (_FloatN X)
 -- Function: _FloatNx truncfNx (_FloatNx X)
     ISO, math.h truncfN, TS 18661-3:2015, math.h truncfNx, TS 18661-3:2015, math.h The trunc functions round X towards zero to the nearest integer (returned in floating-point format).  Thus, trunc (1.5) is 1.0 and trunc (-1.5) is -1.0.

 -- Function: double rint (double X)
 -- Function: float rintf (float X)
 -- Function: long double rintl (long double X)
 -- Function: _FloatN rintfN (_FloatN X)
 -- Function: _FloatNx rintfNx (_FloatNx X)
     ISO, math.h rintfN, TS 18661-3:2015, math.h rintfNx, TS 18661-3:2015, math.h These functions round X to an integer value according to the current rounding mode.  See 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 explicitly select another.

     If X was not initially an integer, these functions raise the inexact exception.

 -- Function: double nearbyint (double X)
 -- Function: float nearbyintf (float X)
 -- Function: long double nearbyintl (long double X)
 -- Function: _FloatN nearbyintfN (_FloatN X)
 -- Function: _FloatNx nearbyintfNx (_FloatNx X)
     ISO, math.h nearbyintfN, TS 18661-3:2015, math.h nearbyintfNx, TS 18661-3:2015, math.h These functions return the same value as the rint functions, but do not raise the inexact exception if X is not an integer.

 -- Function: double round (double X)
 -- Function: float roundf (float X)
 -- Function: long double roundl (long double X)
 -- Function: _FloatN roundfN (_FloatN X)
 -- Function: _FloatNx roundfNx (_FloatNx X)
     ISO, math.h roundfN, TS 18661-3:2015, math.h roundfNx, TS 18661-3:2015, math.h These functions are similar to rint, but they round halfway cases away from zero instead of to the nearest integer (or other current rounding mode).

 -- Function: double roundeven (double X)
 -- Function: float roundevenf (float X)
 -- Function: long double roundevenl (long double X)
 -- Function: _FloatN roundevenfN (_FloatN X)
 -- Function: _FloatNx roundevenfNx (_FloatNx X)
     ISO, math.h roundevenfN, TS 18661-3:2015, math.h roundevenfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, are similar to round, but they round halfway cases to even instead of away from zero.

 -- Function: long int lrint (double X)
 -- Function: long int lrintf (float X)
 -- Function: long int lrintl (long double X)
 -- Function: long int lrintfN (_FloatN X)
 -- Function: long int lrintfNx (_FloatNx X)
     ISO, math.h lrintfN, TS 18661-3:2015, math.h lrintfNx, TS 18661-3:2015, math.h These functions are just like rint, but they return a long int instead of a floating-point number.

 -- Function: long long int llrint (double X)
 -- Function: long long int llrintf (float X)
 -- Function: long long int llrintl (long double X)
 -- Function: long long int llrintfN (_FloatN X)
 -- Function: long long int llrintfNx (_FloatNx X)
     ISO, math.h llrintfN, TS 18661-3:2015, math.h llrintfNx, TS 18661-3:2015, math.h These functions are just like rint, but they return a long long int instead of a floating-point number.

 -- Function: long int lround (double X)
 -- Function: long int lroundf (float X)
 -- Function: long int lroundl (long double X)
 -- Function: long int lroundfN (_FloatN X)
 -- Function: long int lroundfNx (_FloatNx X)
     ISO, math.h lroundfN, TS 18661-3:2015, math.h lroundfNx, TS 18661-3:2015, math.h These functions are just like round, but they return a long int instead of a floating-point number.

 -- Function: long long int llround (double X)
 -- Function: long long int llroundf (float X)
 -- Function: long long int llroundl (long double X)
 -- Function: long long int llroundfN (_FloatN X)
 -- Function: long long int llroundfNx (_FloatNx X)
     ISO, math.h llroundfN, TS 18661-3:2015, math.h llroundfNx, TS 18661-3:2015, math.h These functions are just like round, but they return a long long int instead of a floating-point number.

 -- Function: intmax_t fromfp (double X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpf (float X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpl (long double X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpfN (_FloatN X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpfNx (_FloatNx X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfp (double X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpf (float X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpl (long double X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpfN (_FloatN X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpfNx (_FloatNx X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpx (double X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpxf (float X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpxl (long double X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpxfN (_FloatN X, int ROUND, unsigned int WIDTH)
 -- Function: intmax_t fromfpxfNx (_FloatNx X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpx (double X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpxf (float X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpxl (long double X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpxfN (_FloatN X, int ROUND, unsigned int WIDTH)
 -- Function: uintmax_t ufromfpxfNx (_FloatNx X, int ROUND, unsigned int WIDTH)
     ISO, math.h fromfpfN, TS 18661-3:2015, math.h fromfpfNx, TS 18661-3:2015, math.h ufromfpfN, TS 18661-3:2015, math.h ufromfpfNx, TS 18661-3:2015, math.h fromfpxfN, TS 18661-3:2015, math.h fromfpxfNx, TS 18661-3:2015, math.h ufromfpxfN, TS 18661-3:2015, math.h ufromfpxfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, convert a floating-point number to an integer according to the rounding direction ROUND (one of the FP_INT_* macros).  If the integer is outside the range of a signed or unsigned (depending on the return type of the function) type of width WIDTH bits (or outside the range of the return type, if WIDTH is larger), or if X is infinite or NaN, or if WIDTH is zero, a domain error occurs and an unspecified value is returned.  The functions with an x in their names raise the inexact exception when a domain error does not occur and the argument is not an integer; the other functions do not raise the inexact exception.

 -- Function: double modf (double VALUE, double *INTEGER-PART)
 -- Function: float modff (float VALUE, float *INTEGER-PART)
 -- Function: long double modfl (long double VALUE, long double *INTEGER-PART)
 -- Function: _FloatN modffN (_FloatN VALUE, _FloatN *INTEGER-PART)
 -- Function: _FloatNx modffNx (_FloatNx VALUE, _FloatNx *INTEGER-PART)
     ISO, math.h modffN, TS 18661-3:2015, math.h modffNx, TS 18661-3:2015, math.h These functions break 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, and the integer part is always rounded toward 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.

Remainder Functions
-------------------

The functions in this section compute the remainder on division of two floating-point numbers.  Each is a little different; pick the one that suits your problem.

 -- Function: double fmod (double NUMERATOR, double DENOMINATOR)
 -- Function: float fmodf (float NUMERATOR, float DENOMINATOR)
 -- Function: long double fmodl (long double NUMERATOR, long double DENOMINATOR)
 -- Function: _FloatN fmodfN (_FloatN NUMERATOR, _FloatN DENOMINATOR)
 -- Function: _FloatNx fmodfNx (_FloatNx NUMERATOR, _FloatNx DENOMINATOR)
     ISO, math.h fmodfN, TS 18661-3:2015, math.h fmodfNx, TS 18661-3:2015, math.h These functions compute 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 signals a domain error.

 -- Function: double remainder (double NUMERATOR, double DENOMINATOR)
 -- Function: float remainderf (float NUMERATOR, float DENOMINATOR)
 -- Function: long double remainderl (long double NUMERATOR, long double DENOMINATOR)
 -- Function: _FloatN remainderfN (_FloatN NUMERATOR, _FloatN DENOMINATOR)
 -- Function: _FloatNx remainderfNx (_FloatNx NUMERATOR, _FloatNx DENOMINATOR)
     ISO, math.h remainderfN, TS 18661-3:2015, math.h remainderfNx, TS 18661-3:2015, math.h These functions are like fmod except that they round the internal quotient N to the nearest integer instead of towards zero to an integer.  For example, remainder (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 remainder (NUMERATOR, DENOMINATOR) is always either DENOMINATOR, minus DENOMINATOR, or zero.

     If DENOMINATOR is zero, remainder signals a domain error.

 -- Function: double drem (double NUMERATOR, double DENOMINATOR)
 -- Function: float dremf (float NUMERATOR, float DENOMINATOR)
 -- Function: long double dreml (long double NUMERATOR, long double DENOMINATOR)
     BSD, math.h This function is another name for remainder.

Setting and modifying single bits of FP values
----------------------------------------------

There are some operations that are too complicated or expensive to perform by hand on floating-point numbers.  ISO C99 defines functions to do these operations, which mostly involve changing single bits.

 -- Function: double copysign (double X, double Y)
 -- Function: float copysignf (float X, float Y)
 -- Function: long double copysignl (long double X, long double Y)
 -- Function: _FloatN copysignfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx copysignfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h copysignfN, TS 18661-3:2015, math.h copysignfNx, TS 18661-3:2015, math.h These functions return X but with the sign of Y.  They work even if X or Y are NaN or zero.  Both of these can carry a sign (although not all implementations support it) and this is one of the few operations that can tell the difference.

     copysign never raises an exception.

     This function is defined in IEC 559 (and the appendix with recommended functions in IEEE 754/IEEE 854).

 -- Function: int signbit (float-type X)
     ISO, math.h signbit is a generic macro which can work on all floating-point types.  It returns a nonzero value if the value of X has its sign bit set.

     This is not the same as x < 0.0, because IEEE 754 floating point allows zero to be signed.  The comparison -0.0 < 0.0 is false, but signbit (-0.0) will return a nonzero value.

 -- Function: double nextafter (double X, double Y)
 -- Function: float nextafterf (float X, float Y)
 -- Function: long double nextafterl (long double X, long double Y)
 -- Function: _FloatN nextafterfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx nextafterfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h nextafterfN, TS 18661-3:2015, math.h nextafterfNx, TS 18661-3:2015, math.h The nextafter function returns the next representable neighbor of X in the direction towards Y.  The size of the step between X and the result depends on the type of the result.  If X = Y the function simply returns Y.  If either value is NaN, NaN is returned.  Otherwise a value corresponding to the value of the least significant bit in the mantissa is added or subtracted, depending on the direction. nextafter will signal overflow or underflow if the result goes outside of the range of normalized numbers.

     This function is defined in IEC 559 (and the appendix with recommended functions in IEEE 754/IEEE 854).

 -- Function: double nexttoward (double X, long double Y)
 -- Function: float nexttowardf (float X, long double Y)
 -- Function: long double nexttowardl (long double X, long double Y)
     ISO, math.h These functions are identical to the corresponding versions of nextafter except that their second argument is a long double.

 -- Function: double nextup (double X)
 -- Function: float nextupf (float X)
 -- Function: long double nextupl (long double X)
 -- Function: _FloatN nextupfN (_FloatN X)
 -- Function: _FloatNx nextupfNx (_FloatNx X)
     ISO, math.h nextupfN, TS 18661-3:2015, math.h nextupfNx, TS 18661-3:2015, math.h The nextup function returns the next representable neighbor of X in the direction of positive infinity.  If X is the smallest negative subnormal number in the type of X the function returns -0.  If X = 0 the function returns the smallest positive subnormal number in the type of X.  If X is NaN, NaN is returned. If X is +infinity, +infinity is returned. nextup is from TS 18661-1:2014 and TS 18661-3:2015. nextup never raises an exception except for signaling NaNs.

 -- Function: double nextdown (double X)
 -- Function: float nextdownf (float X)
 -- Function: long double nextdownl (long double X)
 -- Function: _FloatN nextdownfN (_FloatN X)
 -- Function: _FloatNx nextdownfNx (_FloatNx X)
     ISO, math.h nextdownfN, TS 18661-3:2015, math.h nextdownfNx, TS 18661-3:2015, math.h The nextdown function returns the next representable neighbor of X in the direction of negative infinity.  If X is the smallest positive subnormal number in the type of X the function returns +0.  If X = 0 the function returns the smallest negative subnormal number in the type of X.  If X is NaN, NaN is returned. If X is -infinity, -infinity is returned. nextdown is from TS 18661-1:2014 and TS 18661-3:2015. nextdown never raises an exception except for signaling NaNs.

 -- Function: double nan (const char *TAGP)
 -- Function: float nanf (const char *TAGP)
 -- Function: long double nanl (const char *TAGP)
 -- Function: _FloatN nanfN (const char *TAGP)
 -- Function: _FloatNx nanfNx (const char *TAGP)
     ISO, math.h nanfN, TS 18661-3:2015, math.h nanfNx, TS 18661-3:2015, math.h The nan function returns a representation of NaN, provided that NaN is supported by the target platform. nan ("N-CHAR-SEQUENCE") is equivalent to strtod ("NAN(N-CHAR-SEQUENCE)").

     The argument TAGP is used in an unspecified manner.  On IEEE 754 systems, there are many representations of NaN, and TAGP selects one.  On other systems it may do nothing.

 -- Function: int canonicalize (double *CX, const double *X)
 -- Function: int canonicalizef (float *CX, const float *X)
 -- Function: int canonicalizel (long double *CX, const long double *X)
 -- Function: int canonicalizefN (_FloatN *CX, const _FloatN *X)
 -- Function: int canonicalizefNx (_FloatNx *CX, const _FloatNx *X)
     ISO, math.h canonicalizefN, TS 18661-3:2015, math.h canonicalizefNx, TS 18661-3:2015, math.h In some floating-point formats, some values have canonical (preferred) and noncanonical encodings (for IEEE interchange binary formats, all encodings are canonical).  These functions, defined by TS 18661-1:2014 and TS 18661-3:2015, attempt to produce a canonical version of the floating-point value pointed to by X; if that value is a signaling NaN, they raise the invalid exception and produce a quiet NaN.  If a canonical value is produced, it is stored in the object pointed to by CX, and these functions return zero.  Otherwise (if a canonical value could not be produced because the object pointed to by X is not a valid representation of any floating-point value), the object pointed to by CX is unchanged and a nonzero value is returned.

     Note that some formats have multiple encodings of a value which are all equally canonical; when such an encoding is used as an input to this function, any such encoding of the same value (or of the corresponding quiet NaN, if that value is a signaling NaN) may be produced as output.

 -- Function: double getpayload (const double *X)
 -- Function: float getpayloadf (const float *X)
 -- Function: long double getpayloadl (const long double *X)
 -- Function: _FloatN getpayloadfN (const _FloatN *X)
 -- Function: _FloatNx getpayloadfNx (const _FloatNx *X)
     ISO, math.h getpayloadfN, TS 18661-3:2015, math.h getpayloadfNx, TS 18661-3:2015, math.h IEEE 754 defines the payload of a NaN to be an integer value encoded in the representation of the NaN.  Payloads are typically propagated from NaN inputs to the result of a floating-point operation.  These functions, defined by TS 18661-1:2014 and TS 18661-3:2015, return the payload of the NaN pointed to by X (returned as a positive integer, or positive zero, represented as a floating-point number); if X is not a NaN, they return an unspecified value.  They raise no floating-point exceptions even for signaling NaNs.

 -- Function: int setpayload (double *X, double PAYLOAD)
 -- Function: int setpayloadf (float *X, float PAYLOAD)
 -- Function: int setpayloadl (long double *X, long double PAYLOAD)
 -- Function: int setpayloadfN (_FloatN *X, _FloatN PAYLOAD)
 -- Function: int setpayloadfNx (_FloatNx *X, _FloatNx PAYLOAD)
     ISO, math.h setpayloadfN, TS 18661-3:2015, math.h setpayloadfNx, TS 18661-3:2015, math.h These functions, defined by TS 18661-1:2014 and TS 18661-3:2015, set the object pointed to by X to a quiet NaN with payload PAYLOAD and a zero sign bit and return zero.  If PAYLOAD is not a positive-signed integer that is a valid payload for a quiet NaN of the given type, the object pointed to by X is set to positive zero and a nonzero value is returned.  They raise no floating-point exceptions.

 -- Function: int setpayloadsig (double *X, double PAYLOAD)
 -- Function: int setpayloadsigf (float *X, float PAYLOAD)
 -- Function: int setpayloadsigl (long double *X, long double PAYLOAD)
 -- Function: int setpayloadsigfN (_FloatN *X, _FloatN PAYLOAD)
 -- Function: int setpayloadsigfNx (_FloatNx *X, _FloatNx PAYLOAD)
     ISO, math.h setpayloadsigfN, TS 18661-3:2015, math.h setpayloadsigfNx, TS 18661-3:2015, math.h These functions, defined by TS 18661-1:2014 and TS 18661-3:2015, set the object pointed to by X to a signaling NaN with payload PAYLOAD and a zero sign bit and return zero.  If PAYLOAD is not a positive-signed integer that is a valid payload for a signaling NaN of the given type, the object pointed to by X is set to positive zero and a nonzero value is returned.  They raise no floating-point exceptions.

Floating-Point Comparison Functions
-----------------------------------

The standard C comparison operators provoke exceptions when one or other of the operands is NaN.  For example,

     int v = a < 1.0;

will raise an exception if A is NaN.  (This does not happen with == and !=; those merely return false and true, respectively, when NaN is examined.)  Frequently this exception is undesirable.  ISO C99 therefore defines comparison functions that do not raise exceptions when NaN is examined.  All of the functions are implemented as macros which allow their arguments to be of any floating-point type.  The macros are guaranteed to evaluate their arguments only once.  TS 18661-1:2014 adds such a macro for an equality comparison that does raise an exception for a NaN argument; it also adds functions that provide a total ordering on all floating-point values, including NaNs, without raising any exceptions even for signaling NaNs.

 -- Macro: int isgreater (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether the argument X is greater than Y.  It is equivalent to (X) > (Y), but no exception is raised if X or Y are NaN.

 -- Macro: int isgreaterequal (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether the argument X is greater than or equal to Y.  It is equivalent to (X) >= (Y), but no exception is raised if X or Y are NaN.

 -- Macro: int isless (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether the argument X is less than Y. It is equivalent to (X) < (Y), but no exception is raised if X or Y are NaN.

 -- Macro: int islessequal (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether the argument X is less than or equal to Y.  It is equivalent to (X) <= (Y), but no exception is raised if X or Y are NaN.

 -- Macro: int islessgreater (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether the argument X is less or greater than Y.  It is equivalent to (X) < (Y) || (X) > (Y) (although it only evaluates X and Y once), but no exception is raised if X or Y are NaN.

     This macro is not equivalent to X != Y, because that expression is true if X or Y are NaN.

 -- Macro: int isunordered (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether its arguments are unordered.  In other words, it is true if X or Y are NaN, and false otherwise.

 -- Macro: int iseqsig (real-floating X, real-floating Y)
     ISO, math.h This macro determines whether its arguments are equal.  It is equivalent to (X) == (Y), but it raises the invalid exception and sets errno to EDOM if either argument is a NaN.

 -- Function: int totalorder (const double *X, const double *Y)
 -- Function: int totalorderf (const float *X, const float *Y)
 -- Function: int totalorderl (const long double *X, const long double *Y)
 -- Function: int totalorderfN (const _FloatN *X, const _FloatN *Y)
 -- Function: int totalorderfNx (const _FloatNx *X, const _FloatNx *Y)
     TS 18661-1:2014, math.h totalorderfN, TS 18661-3:2015, math.h totalorderfNx, TS 18661-3:2015, math.h These functions determine whether the total order relationship, defined in IEEE 754-2008, is true for *X and *Y, returning nonzero if it is true and zero if it is false.  No exceptions are raised even for signaling NaNs.  The relationship is true if they are the same floating-point value (including sign for zero and NaNs, and payload for NaNs), or if *X comes before *Y in the following order: negative quiet NaNs, in order of decreasing payload; negative signaling NaNs, in order of decreasing payload; negative infinity; finite numbers, in ascending order, with negative zero before positive zero; positive infinity; positive signaling NaNs, in order of increasing payload; positive quiet NaNs, in order of increasing payload.

 -- Function: int totalordermag (const double *X, const double *Y)
 -- Function: int totalordermagf (const float *X, const float *Y)
 -- Function: int totalordermagl (const long double *X, const long double *Y)
 -- Function: int totalordermagfN (const _FloatN *X, const _FloatN *Y)
 -- Function: int totalordermagfNx (const _FloatNx *X, const _FloatNx *Y)
     TS 18661-1:2014, math.h totalordermagfN, TS 18661-3:2015, math.h totalordermagfNx, TS 18661-3:2015, math.h These functions determine whether the total order relationship, defined in IEEE 754-2008, is true for the absolute values of *X and *Y, returning nonzero if it is true and zero if it is false. No exceptions are raised even for signaling NaNs.

Not all machines provide hardware support for these operations.  On machines that don't, the macros can be very slow.  Therefore, you should not use these functions when NaN is not a concern.

NB: There are no macros isequal or isunequal. They are unnecessary, because the == and != operators do not throw an exception if one or both of the operands are NaN.

Miscellaneous FP arithmetic functions
-------------------------------------

The functions in this section perform miscellaneous but common operations that are awkward to express with C operators.  On some processors these functions can use special machine instructions to perform these operations faster than the equivalent C code.

 -- Function: double fmin (double X, double Y)
 -- Function: float fminf (float X, float Y)
 -- Function: long double fminl (long double X, long double Y)
 -- Function: _FloatN fminfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx fminfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h fminfN, TS 18661-3:2015, math.h fminfNx, TS 18661-3:2015, math.h The fmin function returns the lesser of the two values X and Y.  It is similar to the expression

          ((x) < (y) ? (x) : (y))

     except that X and Y are only evaluated once.

     If an argument is NaN, the other argument is returned.  If both arguments are NaN, NaN is returned.

 -- Function: double fmax (double X, double Y)
 -- Function: float fmaxf (float X, float Y)
 -- Function: long double fmaxl (long double X, long double Y)
 -- Function: _FloatN fmaxfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx fmaxfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h fmaxfN, TS 18661-3:2015, math.h fmaxfNx, TS 18661-3:2015, math.h The fmax function returns the greater of the two values X and Y.

     If an argument is NaN, the other argument is returned.  If both arguments are NaN, NaN is returned.

 -- Function: double fminmag (double X, double Y)
 -- Function: float fminmagf (float X, float Y)
 -- Function: long double fminmagl (long double X, long double Y)
 -- Function: _FloatN fminmagfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx fminmagfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h fminmagfN, TS 18661-3:2015, math.h fminmagfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, return whichever of the two values X and Y has the smaller absolute value.  If both have the same absolute value, or either is NaN, they behave the same as the fmin functions.

 -- Function: double fmaxmag (double X, double Y)
 -- Function: float fmaxmagf (float X, float Y)
 -- Function: long double fmaxmagl (long double X, long double Y)
 -- Function: _FloatN fmaxmagfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx fmaxmagfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h fmaxmagfN, TS 18661-3:2015, math.h fmaxmagfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014, return whichever of the two values X and Y has the greater absolute value.  If both have the same absolute value, or either is NaN, they behave the same as the fmax functions.

 -- Function: double fdim (double X, double Y)
 -- Function: float fdimf (float X, float Y)
 -- Function: long double fdiml (long double X, long double Y)
 -- Function: _FloatN fdimfN (_FloatN X, _FloatN Y)
 -- Function: _FloatNx fdimfNx (_FloatNx X, _FloatNx Y)
     ISO, math.h fdimfN, TS 18661-3:2015, math.h fdimfNx, TS 18661-3:2015, math.h The fdim function returns the positive difference between X and Y.  The positive difference is X - Y if X is greater than Y, and 0 otherwise.

     If X, Y, or both are NaN, NaN is returned.

 -- Function: double fma (double X, double Y, double Z)
 -- Function: float fmaf (float X, float Y, float Z)
 -- Function: long double fmal (long double X, long double Y, long double Z)
 -- Function: _FloatN fmafN (_FloatN X, _FloatN Y, _FloatN Z)
 -- Function: _FloatNx fmafNx (_FloatNx X, _FloatNx Y, _FloatNx Z)
     ISO, math.h fmafN, TS 18661-3:2015, math.h fmafNx, TS 18661-3:2015, math.h The fma function performs floating-point multiply-add.  This is the operation (X mul Y) + Z, but the intermediate result is not rounded to the destination type.  This can sometimes improve the precision of a calculation.

     This function was introduced because some processors have a special instruction to perform multiply-add.  The C compiler cannot use it directly, because the expression x*y + z is defined to round the intermediate result.  fma lets you choose when you want to round only once.

     On processors which do not implement multiply-add in hardware, fma can be very slow since it must avoid intermediate rounding. math.h defines the symbols FP_FAST_FMA, FP_FAST_FMAF, and FP_FAST_FMAL when the corresponding version of fma is no slower than the expression x*y + z. In theglibc, this always means the operation is implemented in hardware.

 -- Function: float fadd (double X, double Y)
 -- Function: float faddl (long double X, long double Y)
 -- Function: double daddl (long double X, long double Y)
 -- Function: _FloatM fMaddfN (_FloatN X, _FloatN Y)
 -- Function: _FloatM fMaddfNx (_FloatNx X, _FloatNx Y)
 -- Function: _FloatMx fMxaddfN (_FloatN X, _FloatN Y)
 -- Function: _FloatMx fMxaddfNx (_FloatNx X, _FloatNx Y)
     TS 18661-1:2014, math.h fMaddfN, TS 18661-3:2015, math.h fMaddfNx, TS 18661-3:2015, math.h fMxaddfN, TS 18661-3:2015, math.h fMxaddfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, return X + Y, rounded once to the return type of the function without any intermediate rounding to the type of the arguments.

 -- Function: float fsub (double X, double Y)
 -- Function: float fsubl (long double X, long double Y)
 -- Function: double dsubl (long double X, long double Y)
 -- Function: _FloatM fMsubfN (_FloatN X, _FloatN Y)
 -- Function: _FloatM fMsubfNx (_FloatNx X, _FloatNx Y)
 -- Function: _FloatMx fMxsubfN (_FloatN X, _FloatN Y)
 -- Function: _FloatMx fMxsubfNx (_FloatNx X, _FloatNx Y)
     TS 18661-1:2014, math.h fMsubfN, TS 18661-3:2015, math.h fMsubfNx, TS 18661-3:2015, math.h fMxsubfN, TS 18661-3:2015, math.h fMxsubfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, return X - Y, rounded once to the return type of the function without any intermediate rounding to the type of the arguments.

 -- Function: float fmul (double X, double Y)
 -- Function: float fmull (long double X, long double Y)
 -- Function: double dmull (long double X, long double Y)
 -- Function: _FloatM fMmulfN (_FloatN X, _FloatN Y)
 -- Function: _FloatM fMmulfNx (_FloatNx X, _FloatNx Y)
 -- Function: _FloatMx fMxmulfN (_FloatN X, _FloatN Y)
 -- Function: _FloatMx fMxmulfNx (_FloatNx X, _FloatNx Y)
     TS 18661-1:2014, math.h fMmulfN, TS 18661-3:2015, math.h fMmulfNx, TS 18661-3:2015, math.h fMxmulfN, TS 18661-3:2015, math.h fMxmulfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, return X * Y, rounded once to the return type of the function without any intermediate rounding to the type of the arguments.

 -- Function: float fdiv (double X, double Y)
 -- Function: float fdivl (long double X, long double Y)
 -- Function: double ddivl (long double X, long double Y)
 -- Function: _FloatM fMdivfN (_FloatN X, _FloatN Y)
 -- Function: _FloatM fMdivfNx (_FloatNx X, _FloatNx Y)
 -- Function: _FloatMx fMxdivfN (_FloatN X, _FloatN Y)
 -- Function: _FloatMx fMxdivfNx (_FloatNx X, _FloatNx Y)
     TS 18661-1:2014, math.h fMdivfN, TS 18661-3:2015, math.h fMdivfNx, TS 18661-3:2015, math.h fMxdivfN, TS 18661-3:2015, math.h fMxdivfNx, TS 18661-3:2015, math.h These functions, from TS 18661-1:2014 and TS 18661-3:2015, return X / Y, rounded once to the return type of the function without any intermediate rounding to the type of the arguments.

Complex Numbers
===============

ISO C99 introduces support for complex numbers in C.  This is done with a new type qualifier, complex.  It is a keyword if and only if complex.h has been included.  There are three complex types, corresponding to the three real types:  float complex, double complex, and long double complex.

Likewise, on machines that have support for _FloatN or _FloatNx enabled, the complex types _FloatN complex and _FloatNx complex are also available if complex.h has been included; see Mathematics.

To construct complex numbers you need a way to indicate the imaginary part of a number.  There is no standard notation for an imaginary floating point constant.  Instead, complex.h defines two macros that can be used to create complex numbers.

 -- Macro: const float complex _Complex_I
     C99, complex.h This macro is a representation of the complex number ``0+1i''. Multiplying a real floating-point value by _Complex_I gives a complex number whose value is purely imaginary.  You can use this to construct complex constants:

          3.0 + 4.0i = 3.0 + 4.0 * _Complex_I

     Note that _Complex_I * _Complex_I has the value -1, but the type of that value is complex.

_Complex_I is a bit of a mouthful.  complex.h also defines a shorter name for the same constant.

 -- Macro: const float complex I
     C99, complex.h This macro has exactly the same value as _Complex_I.  Most of the time it is preferable.  However, it causes problems if you want to use the identifier I for something else.  You can safely write

          #include <complex.h>
          #undef I

     if you need I for your own purposes.  (In that case we recommend you also define some other short name for _Complex_I, such as J.)

Projections, Conjugates, and Decomposing of Complex Numbers
===========================================================

ISO C99 also defines functions that perform basic operations on complex numbers, such as decomposition and conjugation.  The prototypes for all these functions are in complex.h.  All functions are available in three variants, one for each of the three complex types.

 -- Function: double creal (complex double Z)
 -- Function: float crealf (complex float Z)
 -- Function: long double creall (complex long double Z)
 -- Function: _FloatN crealfN (complex _FloatN Z)
 -- Function: _FloatNx crealfNx (complex _FloatNx Z)
     ISO, complex.h crealfN, TS 18661-3:2015, complex.h crealfNx, TS 18661-3:2015, complex.h These functions return the real part of the complex number Z.

 -- Function: double cimag (complex double Z)
 -- Function: float cimagf (complex float Z)
 -- Function: long double cimagl (complex long double Z)
 -- Function: _FloatN cimagfN (complex _FloatN Z)
 -- Function: _FloatNx cimagfNx (complex _FloatNx Z)
     ISO, complex.h cimagfN, TS 18661-3:2015, complex.h cimagfNx, TS 18661-3:2015, complex.h These functions return the imaginary part of the complex number Z.

 -- Function: complex double conj (complex double Z)
 -- Function: complex float conjf (complex float Z)
 -- Function: complex long double conjl (complex long double Z)
 -- Function: complex _FloatN conjfN (complex _FloatN Z)
 -- Function: complex _FloatNx conjfNx (complex _FloatNx Z)
     ISO, complex.h conjfN, TS 18661-3:2015, complex.h conjfNx, TS 18661-3:2015, complex.h These functions return the conjugate value of the complex number Z.  The conjugate of a complex number has the same real part and a negated imaginary part.  In other words, conj(a + bi) = a + -bi.

 -- Function: double carg (complex double Z)
 -- Function: float cargf (complex float Z)
 -- Function: long double cargl (complex long double Z)
 -- Function: _FloatN cargfN (complex _FloatN Z)
 -- Function: _FloatNx cargfNx (complex _FloatNx Z)
     ISO, complex.h cargfN, TS 18661-3:2015, complex.h cargfNx, TS 18661-3:2015, complex.h These functions return the argument of the complex number Z. The argument of a complex number is the angle in the complex plane between the positive real axis and a line passing through zero and the number.  This angle is measured in the usual fashion and ranges from -pi to pi.

     carg has a branch cut along the negative real axis.

 -- Function: complex double cproj (complex double Z)
 -- Function: complex float cprojf (complex float Z)
 -- Function: complex long double cprojl (complex long double Z)
 -- Function: complex _FloatN cprojfN (complex _FloatN Z)
 -- Function: complex _FloatNx cprojfNx (complex _FloatNx Z)
     ISO, complex.h cprojfN, TS 18661-3:2015, complex.h cprojfNx, TS 18661-3:2015, complex.h These functions return the projection of the complex value Z onto the Riemann sphere.  Values with an infinite imaginary part are projected to positive infinity on the real axis, even if the real part is NaN.  If the real part is infinite, the result is equivalent to

          INFINITY + I * copysign (0.0, cimag (z))

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 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
-------------------

The str functions are declared in stdlib.h and those beginning with wcs are declared in wchar.h.  One might wonder about the use of restrict in the prototypes of the functions in this section.  It is seemingly useless but the ISO C standard uses it (for the functions defined there) so we have to do it as well.

 -- Function: long int strtol (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, stdlib.h 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 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 36. 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 Range of Type), as appropriate for the sign of the value.  It also sets errno to ERANGE to indicate there was overflow.

     You should not check for errors by examining the return value of strtol, because the string might be a valid representation of 0l, LONG_MAX, or LONG_MIN.  Instead, check whether TAILPTR points to what you expect after the number (e.g. '\0' if the string should end after the number).  You also need to clear errno before the call and check it afterward, in case there was overflow.

     There is an example at the end of this section.

 -- Function: long int wcstol (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstol function is equivalent to the strtol function in nearly all aspects but handles wide character strings.

     The wcstol function was introduced in Amendment 1 of ISO C90.

 -- Function: unsigned long int strtoul (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, stdlib.h The strtoul (``string-to-unsigned-long'') function is like strtol except it converts to an unsigned long int value. The syntax is the same as described above for strtol.  The value returned on overflow is ULONG_MAX (see Range of Type).

     If STRING depicts a negative number, strtoul acts the same as STRTOL but casts the result to an unsigned integer.  That means for example that strtoul on "-1" returns ULONG_MAX and an input more negative than LONG_MIN returns (ULONG_MAX + 1) / 2.

     strtoul sets errno to EINVAL if BASE is out of range, or ERANGE on overflow.

 -- Function: unsigned long int wcstoul (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstoul function is equivalent to the strtoul function in nearly all aspects but handles wide character strings.

     The wcstoul function was introduced in Amendment 1 of ISO C90.

 -- Function: long long int strtoll (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, stdlib.h The strtoll function is like strtol except that it returns a long long int value, and accepts numbers with a correspondingly larger range.

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

     The strtoll function was introduced in ISO C99.

 -- Function: long long int wcstoll (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstoll function is equivalent to the strtoll function in nearly all aspects but handles wide character strings.

     The wcstoll function was introduced in Amendment 1 of ISO C90.

 -- Function: long long int strtoq (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     BSD, stdlib.h strtoq (``string-to-quad-word'') is the BSD name for strtoll.

 -- Function: long long int wcstoq (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     GNU, wchar.h The wcstoq function is equivalent to the strtoq function in nearly all aspects but handles wide character strings.

     The wcstoq function is a GNU extension.

 -- Function: unsigned long long int strtoull (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, stdlib.h The strtoull function is related to strtoll the same way strtoul is related to strtol.

     The strtoull function was introduced in ISO C99.

 -- Function: unsigned long long int wcstoull (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstoull function is equivalent to the strtoull function in nearly all aspects but handles wide character strings.

     The wcstoull function was introduced in Amendment 1 of ISO C90.

 -- Function: unsigned long long int strtouq (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     BSD, stdlib.h strtouq is the BSD name for strtoull.

 -- Function: unsigned long long int wcstouq (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     GNU, wchar.h The wcstouq function is equivalent to the strtouq function in nearly all aspects but handles wide character strings.

     The wcstouq function is a GNU extension.

 -- Function: intmax_t strtoimax (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, inttypes.h The strtoimax function is like strtol except that it returns a intmax_t value, and accepts numbers of a corresponding range.

     If the string has valid syntax for an integer but the value is not representable because of overflow, strtoimax returns either INTMAX_MAX or INTMAX_MIN (see Integers), as appropriate for the sign of the value.  It also sets errno to ERANGE to indicate there was overflow.

     See Integers for a description of the intmax_t type.  The strtoimax function was introduced in ISO C99.

 -- Function: intmax_t wcstoimax (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstoimax function is equivalent to the strtoimax function in nearly all aspects but handles wide character strings.

     The wcstoimax function was introduced in ISO C99.

 -- Function: uintmax_t strtoumax (const char *restrict STRING, char **restrict TAILPTR, int BASE)
     ISO, inttypes.h The strtoumax function is related to strtoimax the same way that strtoul is related to strtol.

     See Integers for a description of the intmax_t type.  The strtoumax function was introduced in ISO C99.

 -- Function: uintmax_t wcstoumax (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR, int BASE)
     ISO, wchar.h The wcstoumax function is equivalent to the strtoumax function in nearly all aspects but handles wide character strings.

     The wcstoumax function was introduced in ISO C99.

 -- Function: long int atol (const char *STRING)
     ISO, stdlib.h 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)
     ISO, stdlib.h This function is like atol, except that it returns an int. The atoi function is also considered obsolete; use strtol instead.

 -- Function: long long int atoll (const char *STRING)
     ISO, stdlib.h This function is similar to atol, except it returns a long long int.

     The atoll function was introduced in ISO C99.  It too is obsolete (despite having just been added); use strtoll instead.

All the functions mentioned in this section so far do not handle alternative representations of characters as described in the locale data.  Some locales specify thousands separator and the way they have to be used which can help to make large numbers more readable.  To read such numbers one has to use the scanf functions with the ' flag.

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;
         /* Advance past it.  */
         string = tail;
       }

       return sum;
     }

Parsing of Floats
-----------------

The str functions are declared in stdlib.h and those beginning with wcs are declared in wchar.h.  One might wonder about the use of restrict in the prototypes of the functions in this section.  It is seemingly useless but the ISO C standard uses it (for the functions defined there) so we have to do it as well.

 -- Function: double strtod (const char *restrict STRING, char **restrict TAILPTR)
     ISO, stdlib.h 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 Classification of Characters).  These are discarded.

       * An optional plus or minus sign (+ or -).

       * A floating point number in decimal or hexadecimal format.  The decimal format is:

            - A nonempty sequence of digits optionally containing a decimal-point character---normally ., but it depends on the locale (see General Numeric).

            - An optional exponent part, consisting of a character e or E, an optional sign, and a sequence of digits.

          The hexadecimal format is as follows:

            - A 0x or 0X followed by a nonempty sequence of hexadecimal digits optionally containing a decimal-point character---normally ., but it depends on the locale (see General Numeric).

            - An optional binary-exponent part, consisting of a character p or P, 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 outside the range of a double, strtod will signal overflow or underflow as described in Math Error Reporting.

     strtod recognizes four special input strings.  The strings "inf" and "infinity" are converted to infinity, or to the largest representable value if the floating-point format doesn't support infinities.  You can prepend a "+" or "-" to specify the sign.  Case is ignored when scanning these strings.

     The strings "nan" and "nan(CHARS...)" are converted to NaN.  Again, case is ignored.  If CHARS... are provided, they are used in some unspecified fashion to select a particular representation of NaN (there can be several).

     Since zero is a valid result as well as the value returned on error, you should check for errors in the same way as for strtol, by examining errno and TAILPTR.

 -- Function: float strtof (const char *STRING, char **TAILPTR)
 -- Function: long double strtold (const char *STRING, char **TAILPTR)
     ISO, stdlib.h These functions are analogous to strtod, but return float and long double values respectively.  They report errors in the same way as strtod.  strtof can be substantially faster than strtod, but has less precision; conversely, strtold can be much slower but has more precision (on systems where long double is a separate type).

     These functions have been GNU extensions and are new to ISO C99.

 -- Function: _FloatN strtofN (const char *STRING, char **TAILPTR)
 -- Function: _FloatNx strtofNx (const char *STRING, char **TAILPTR)
     ISO/IEC TS 18661-3, stdlib.h These functions are like strtod, except for the return type.

     They were introduced in ISO/IEC TS 18661-3 and are available on machines that support the related types; see Mathematics.

 -- Function: double wcstod (const wchar_t *restrict STRING, wchar_t **restrict TAILPTR)
 -- Function: float wcstof (const wchar_t *STRING, wchar_t **TAILPTR)
 -- Function: long double wcstold (const wchar_t *STRING, wchar_t **TAILPTR)
 -- Function: _FloatN wcstofN (const wchar_t *STRING, wchar_t **TAILPTR)
 -- Function: _FloatNx wcstofNx (const wchar_t *STRING, wchar_t **TAILPTR)
     ISO, wchar.h wcstofN, GNU, wchar.h wcstofNx, GNU, wchar.h The wcstod, wcstof, wcstol, wcstofN, and wcstofNx functions are equivalent in nearly all aspects to the strtod, strtof, strtold, strtofN, and strtofNx functions, but they handle wide character strings.

     The wcstod function was introduced in Amendment 1 of ISO C90.  The wcstof and wcstold functions were introduced in ISO C99.

     The wcstofN and wcstofNx functions are not in any standard, but are added to provide completeness for the non-deprecated interface of wide character string to floating-point conversion functions.  They are only available on machines that support the related types; see Mathematics.

 -- Function: double atof (const char *STRING)
     ISO, stdlib.h 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.

Theglibc also provides _l versions of these functions, which take an additional argument, the locale to use in conversion.

See also Parsing of Integers.

Printing of Floats
==================

The strfrom functions are declared in stdlib.h.

 -- Function: int strfromd (char *restrict STRING, size_t SIZE, const char *restrict FORMAT, double VALUE)
 -- Function: int strfromf (char *restrict STRING, size_t SIZE, const char *restrict FORMAT, float VALUE)
 -- Function: int strfroml (char *restrict STRING, size_t SIZE, const char *restrict FORMAT, long double VALUE)
     ISO/IEC TS 18661-1, stdlib.h The functions strfromd (``string-from-double''), strfromf (``string-from-float''), and strfroml (``string-from-long-double'') convert the floating-point number VALUE to a string of characters and stores them into the area pointed to by STRING.  The conversion writes at most SIZE characters and respects the format specified by FORMAT.

     The format string must start with the character %.  An optional precision follows, which starts with a period, ., and may be followed by a decimal integer, representing the precision.  If a decimal integer is not specified after the period, the precision is taken to be zero.  The character * is not allowed.  Finally, the format string ends with one of the following conversion specifiers: a, A, e, E, f, F, g or G (see Table of Output Conversions).  Invalid format strings result in undefined behavior.

     These functions return the number of characters that would have been written to STRING had SIZE been sufficiently large, not counting the terminating null character.  Thus, the null-terminated output has been completely written if and only if the returned value is less than SIZE.

     These functions were introduced by ISO/IEC TS 18661-1.

 -- Function: int strfromfN (char *restrict STRING, size_t SIZE, const char *restrict FORMAT, _FloatN VALUE)
 -- Function: int strfromfNx (char *restrict STRING, size_t SIZE, const char *restrict FORMAT, _FloatNx VALUE)
     ISO/IEC TS 18661-3, stdlib.h These functions are like strfromd, except for the type of value.

     They were introduced in ISO/IEC TS 18661-3 and are available on machines that support the related types; see Mathematics.

Old-fashioned System V number-to-string functions
=================================================

The old System V C library provided three functions to convert numbers to strings, with unusual and hard-to-use semantics.  Theglibc also provides these functions and some natural extensions.

These functions are only available in theglibc and on systems descended from AT&T Unix.  Therefore, unless these functions do precisely what you need, it is better to use sprintf, which is standard.

All these functions are defined in stdlib.h.

 -- Function: char * ecvt (double VALUE, int NDIGIT, int *DECPT, int *NEG)
     SVID, stdlib.h Unix98, stdlib.h prelim:ecvt asunsafe acsafe The function ecvt converts the floating-point number VALUE to a string with at most NDIGIT decimal digits.  The returned string contains no decimal point or sign.  The first digit of the string is non-zero (unless VALUE is actually zero) and the last digit is rounded to nearest.  *DECPT is set to the index in the string of the first digit after the decimal point. *NEG is set to a nonzero value if VALUE is negative, zero otherwise.

     If NDIGIT decimal digits would exceed the precision of a double it is reduced to a system-specific value.

     The returned string is statically allocated and overwritten by each call to ecvt.

     If VALUE is zero, it is implementation defined whether *DECPT is 0 or 1.

     For example: ecvt (12.3, 5, &d, &n) returns "12300" and sets D to 2 and N to 0.

 -- Function: char * fcvt (double VALUE, int NDIGIT, int *DECPT, int *NEG)
     SVID, stdlib.h Unix98, stdlib.h prelim:fcvt The function fcvt is like ecvt, but NDIGIT specifies the number of digits after the decimal point.  If NDIGIT is less than zero, VALUE is rounded to the NDIGIT+1'th place to the left of the decimal point.  For example, if NDIGIT is -1, VALUE will be rounded to the nearest 10.  If NDIGIT is negative and larger than the number of digits to the left of the decimal point in VALUE, VALUE will be rounded to one significant digit.

     If NDIGIT decimal digits would exceed the precision of a double it is reduced to a system-specific value.

     The returned string is statically allocated and overwritten by each call to fcvt.

 -- Function: char * gcvt (double VALUE, int NDIGIT, char *BUF)
     SVID, stdlib.h Unix98, stdlib.h gcvt is functionally equivalent to sprintf(buf, "%*g", ndigit, value).  It is provided only for compatibility's sake.  It returns BUF.

     If NDIGIT decimal digits would exceed the precision of a double it is reduced to a system-specific value.

As extensions, theglibc provides versions of these three functions that take long double arguments.

 -- Function: char * qecvt (long double VALUE, int NDIGIT, int *DECPT, int *NEG)
     GNU, stdlib.h prelim:qecvt asunsafe acsafe This function is equivalent to ecvt except that it takes a long double for the first parameter and that NDIGIT is restricted by the precision of a long double.

 -- Function: char * qfcvt (long double VALUE, int NDIGIT, int *DECPT, int *NEG)
     GNU, stdlib.h prelim:qfcvt This function is equivalent to fcvt except that it takes a long double for the first parameter and that NDIGIT is restricted by the precision of a long double.

 -- Function: char * qgcvt (long double VALUE, int NDIGIT, char *BUF)
     GNU, stdlib.h This function is equivalent to gcvt except that it takes a long double for the first parameter and that NDIGIT is restricted by the precision of a long double.

The ecvt and fcvt functions, and their long double equivalents, all return a string located in a static buffer which is overwritten by the next call to the function.  Theglibc provides another set of extended functions which write the converted string into a user-supplied buffer.  These have the conventional _r suffix.

gcvt_r is not necessary, because gcvt already uses a user-supplied buffer.

 -- Function: int ecvt_r (double VALUE, int NDIGIT, int *DECPT, int *NEG, char *BUF, size_t LEN)
     GNU, stdlib.h The ecvt_r function is the same as ecvt, except that it places its result into the user-specified buffer pointed to by BUF, with length LEN.  The return value is -1 in case of an error and zero otherwise.

     This function is a GNU extension.

 -- Function: int fcvt_r (double VALUE, int NDIGIT, int *DECPT, int *NEG, char *BUF, size_t LEN)
     SVID, stdlib.h Unix98, stdlib.h The fcvt_r function is the same as fcvt, except that it places its result into the user-specified buffer pointed to by BUF, with length LEN.  The return value is -1 in case of an error and zero otherwise.

     This function is a GNU extension.

 -- Function: int qecvt_r (long double VALUE, int NDIGIT, int *DECPT, int *NEG, char *BUF, size_t LEN)
     GNU, stdlib.h The qecvt_r function is the same as qecvt, except that it places its result into the user-specified buffer pointed to by BUF, with length LEN.  The return value is -1 in case of an error and zero otherwise.

     This function is a GNU extension.

 -- Function: int qfcvt_r (long double VALUE, int NDIGIT, int *DECPT, int *NEG, char *BUF, size_t LEN)
     GNU, stdlib.h The qfcvt_r function is the same as qfcvt, except that it places its result into the user-specified buffer pointed to by BUF, with length LEN.  The return value is -1 in case of an error and zero otherwise.

     This function is a GNU extension.