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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’.
20.1 Integers | Basic integer types and concepts | |
20.2 Integer Division | Integer division with guaranteed rounding. | |
20.3 Floating Point Numbers | Basic concepts. IEEE 754. | |
20.4 Floating-Point Number Classification Functions | The five kinds of floating-point number. | |
20.5 Errors in Floating-Point Calculations | When something goes wrong in a calculation. | |
20.6 Rounding Modes | Controlling how results are rounded. | |
20.7 Floating-Point Control Functions | Saving and restoring the FPU's state. | |
20.8 Arithmetic Functions | Fundamental operations provided by the library. | |
20.9 Complex Numbers | The types. Writing complex constants. | |
20.10 Projections, Conjugates, and Decomposing of Complex Numbers | Projection, conjugation, decomposition. | |
20.11 Parsing of Numbers | Converting strings to numbers. | |
20.12 Old-fashioned System V number-to-string functions | An archaic way to convert numbers to strings. |
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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, the GNU C library contains C type definitions you can use to declare integers that meet your exact needs. Because the GNU C library header files are customized to a specific machine, your program source code doesn't have to be.
These typedef
s 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:
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:
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:
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.
The GNU C library 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.
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.
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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.
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.
This function div
computes the quotient and remainder from
the division of numerator by denominator, returning the
result in a structure of type div_t
.
If the result cannot be represented (as in a division by zero), the behavior is undefined.
Here is an example, albeit not a very useful one.
div_t result; result = div (20, -6); |
Now result.quot
is -3
and result.rem
is 2
.
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
.)
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
.
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
.)
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.
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.
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.
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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) · 2^e · M where s is the sign bit, e the exponent, and M the mantissa. See section Floating Point Representation 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).
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ISO C99 defines macros that let you determine what sort of floating-point number a variable holds.
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
The floating-point number x is “Not a Number” (see section Infinity and NaN)
FP_INFINITE
The value of x is either plus or minus infinity (see section Infinity and NaN)
FP_ZERO
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
Numbers whose absolute value is too small to be represented in the
normal format are represented in an alternate, denormalized format
(see section Floating Point Representation 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
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.
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.
This macro returns a nonzero value if x is finite and normalized. It is equivalent to
(fpclassify (x) == FP_NORMAL) |
This macro returns a nonzero value if x is NaN. It is equivalent to
(fpclassify (x) == FP_NAN) |
Another set of floating-point classification functions was provided by BSD. The GNU C library 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.
This function returns -1
if x represents negative infinity,
1
if x represents positive infinity, and 0
otherwise.
This function returns a nonzero value if x is a “not a number” value, and zero otherwise.
Note: 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) |
This function returns a nonzero value if x is finite or a “not a number” value, and zero otherwise.
Portability Note: The functions listed in this section are BSD extensions.
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20.5.1 FP Exceptions | IEEE 754 math exceptions and how to detect them. | |
20.5.2 Infinity and NaN | Special values returned by calculations. | |
20.5.3 Examining the FPU status word | Checking for exceptions after the fact. | |
20.5.4 Error Reporting by Mathematical Functions | How the math functions report errors. |
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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 section Signal Handling, for how you can change the effect of
the signal.
In the System V math library, the user-defined function matherr
is called when certain exceptions occur inside math library functions.
However, the Unix98 standard deprecates this interface. We support it
for historical compatibility, but recommend that you do not use it in
new programs.
The exceptions defined in IEEE 754 are:
This exception is raised if the given operands are invalid for the operation to be performed. Examples are (see IEEE 754, section 7):
If the exception does not trap, the result of the operation is NaN.
This exception is raised when a finite nonzero number is divided by zero. If no trap occurs the result is either +∞ or -∞, depending on the signs of the operands.
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):
Whenever the overflow exception is raised, the inexact exception is also raised.
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.
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.
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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 = ∞ log (0) = -∞ 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 + ∞ = ∞, 4/∞ = 0, atan (∞) = π/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 section Floating-Point Number Classification Functions). 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.
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 section Floating-Point Number Classification Functions.
This macro was introduced in the ISO C99 standard.
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’.)
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. Negative zero behaves identically to zero in all
calculations, unless you explicitly test the sign bit with
signbit
or copysign
.
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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
The inexact exception.
FE_DIVBYZERO
The divide by zero exception.
FE_UNDERFLOW
The underflow exception.
FE_OVERFLOW
The overflow exception.
FE_INVALID
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.
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.
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.
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:
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.
excepts) 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.
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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. 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 return ∞ or
-∞ as appropriate. They also set errno to
ERANGE
. When underflow occurs, the underflow exception is
raised, and zero (appropriately signed) is returned. errno may be
set to ERANGE
, but this is not guaranteed.
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.
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.
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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:
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
.
All results are rounded to the smallest representable value which is greater than the result.
All results are rounded to the largest representable value which is less than the result.
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
Round to nearest.
FE_UPWARD
Round toward +∞.
FE_DOWNWARD
Round toward -∞.
FE_TOWARDZERO
Round toward zero.
Underflow is an unusual case. Normally, IEEE 754 floating point
numbers are always normalized (see section Floating Point Representation 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/-∞. Negative zero behaves identically to zero except
when the copysign
or signbit
functions are used to check
the sign bit directly.
At any time one of the above four rounding modes is selected. You can find out which one with this function:
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:
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, 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.
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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.
Note: 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:
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.
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:
fenv_t
objects, which were initialized previously by a
call to fegetenv
or feholdexcept
.
FE_DFL_ENV
which represents the floating-point
environment as it was available at program start.
FE_
and
having type fenv_t *
.
If possible, the GNU C Library 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:
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.
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.
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.
This functions enables traps for each of the exceptions as indicated by the parameter except. The individual excepetions are described in Examining the FPU status word. 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.
This functions disables traps for each of the exceptions as indicated by the parameter except. The individual excepetions are described in Examining the FPU status word. 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.
The function returns a bitmask of all currently enabled exceptions. It
returns -1
in case of failure.
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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.
20.8.1 Absolute Value | Absolute values of integers and floats. | |
20.8.2 Normalization Functions | Extracting exponents and putting them back. | |
20.8.3 Rounding Functions | Rounding floats to integers. | |
20.8.4 Remainder Functions | Remainders on division, precisely defined. | |
20.8.5 Setting and modifying single bits of FP values | Sign bit adjustment. Adding epsilon. | |
20.8.6 Floating-Point Comparison Functions | Comparisons without risk of exceptions. | |
20.8.7 Miscellaneous FP arithmetic functions | Max, min, positive difference, multiply-add. |
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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’;
fabs
, fabsf
and fabsl
are declared in ‘math.h’.
cabs
, cabsf
and cabsl
are declared in ‘complex.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.
This function returns the absolute value of the floating-point number number.
These functions return the absolute value of the complex number z (see section 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 Exponentiation and Logarithms.
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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 Representation Concepts. These functions are required to have equivalent behavior even if the representation does not use a radix of 2, but of course they are unlikely to be particularly efficient in those cases.
All these functions are declared in ‘math.h’.
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 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
.
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 scalb
function is the BSD name for ldexp
.
scalbn
is identical to scalb
, except that the exponent
n is an int
instead of a floating-point number.
scalbln
is identical to scalb
, except that the exponent
n is a long int
instead of a floating-point number.
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.
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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.
These functions round x upwards to the nearest integer,
returning that value as a double
. Thus, ceil (1.5)
is 2.0
.
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
.
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
.
These functions round x to an integer value according to the current rounding mode. See section Floating Point Parameters, for information about the various rounding modes. The default rounding mode is to round to the nearest integer; some machines support other modes, but round-to-nearest is always used unless you explicitly select another.
If x was not initially an integer, these functions raise the inexact exception.
These functions return the same value as the rint
functions, but
do not raise the inexact exception if x is not an integer.
These functions are similar to rint
, but they round halfway
cases away from zero instead of to the nearest even integer.
These functions are just like rint
, but they return a
long int
instead of a floating-point number.
These functions are just like rint
, but they return a
long long int
instead of a floating-point number.
These functions are just like round
, but they return a
long int
instead of a floating-point number.
These functions are just like round
, but they return a
long long int
instead of a floating-point number.
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
.
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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.
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.
These functions are like fmod
except that they rounds the
internal quotient n to the nearest integer instead of towards zero
to an integer. For example, drem (6.5, 2.3)
returns -0.4
,
which is 6.5
minus 6.9
.
The absolute value of the result is less than or equal to half the
absolute value of the denominator. The difference between
fmod (numerator, denominator)
and drem
(numerator, denominator)
is always either
denominator, minus denominator, or zero.
If denominator is zero, drem
signals a domain error.
This function is another name for drem
.
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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.
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).
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.
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).
These functions are identical to the corresponding versions of
nextafter
except that their second argument is a long
double
.
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.
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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.
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.
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.
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.
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.
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.
This macro determines whether its arguments are unordered. In other words, it is true if x or y are NaN, and false otherwise.
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.
Note: 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.
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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.
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.
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.
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.
The fma
function performs floating-point multiply-add. This is
the operation (x · 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 the GNU C library, this always means the operation is implemented in
hardware.
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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
.
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.
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 = |
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.
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
.)
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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.
These functions return the real part of the complex number z.
These functions return the imaginary part of the complex number z.
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’.
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 0 to 2π.
carg
has a branch cut along the positive real axis.
These functions return the projection of the complex value z onto the Riemann sphere. Values with a 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)) |
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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.
20.11.1 Parsing of Integers | Functions for conversion of integer values. | |
20.11.2 Parsing of Floats | Functions for conversion of floating-point values. |
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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.
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:
isspace
function
(see section Classification of Characters). These are discarded.
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
.
strtol
stores a pointer to this tail in
*tailptr
.
If the string is empty, contains only whitespace, or does not contain an
initial substring that has the expected syntax for an integer in the
specified base, no conversion is performed. In this case,
strtol
returns a value of zero and the value stored in
*tailptr
is the value of string.
In a locale other than the standard "C"
locale, this function
may recognize additional implementation-dependent syntax.
If the string has valid syntax for an integer but the value is not
representable because of overflow, strtol
returns either
LONG_MAX
or LONG_MIN
(see section Range of an Integer Type), as
appropriate for the sign of the value. It also sets errno
to ERANGE
to indicate there was overflow.
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.
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.
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 section Range of an Integer 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.
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.
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
LONG_LONG_MAX
or LONG_LONG_MIN
(see section Range of an Integer Type), as
appropriate for the sign of the value. It also sets errno
to
ERANGE
to indicate there was overflow.
The strtoll
function was introduced in ISO C99.
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.
strtoq
(“string-to-quad-word”) is the BSD name for strtoll
.
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.
The strtoull
function is related to strtoll
the same way
strtoul
is related to strtol
.
The strtoull
function was introduced in ISO C99.
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.
strtouq
is the BSD name for strtoull
.
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.
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 section 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.
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.
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.
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.
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.
This function is like atol
, except that it returns an int
.
The atoi
function is also considered obsolete; use strtol
instead.
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; } |
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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.
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:
isspace
function
(see section Classification of Characters). These are discarded.
The hexadecimal format is as follows:
*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 Error Reporting by Mathematical Functions.
strtod
recognizes four special input strings. The strings
"inf"
and "infinity"
are converted to ∞,
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.
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.
The wcstod
, wcstof
, and wcstol
functions are
equivalent in nearly all aspect to the strtod
, strtof
, and
strtold
functions but it handles wide character string.
The wcstod
function was introduced in Amendment 1 of ISO
C90. The wcstof
and wcstold
functions were introduced in
ISO C99.
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.
The GNU C library also provides ‘_l’ versions of these functions, which take an additional argument, the locale to use in conversion. See section Parsing of Integers.
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The old System V C library provided three functions to convert numbers to strings, with unusual and hard-to-use semantics. The GNU C library also provides these functions and some natural extensions.
These functions are only available in glibc 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’.
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
.
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
.
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, the GNU C library provides versions of these three
functions that take long double
arguments.
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
.
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
.
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. The GNU C library
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.
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.
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.
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.
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.
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