1 @node Arithmetic, Date and Time, Mathematics, Top
2 @c %MENU% Low level arithmetic functions
3 @chapter Arithmetic Functions
5 This chapter contains information about functions for doing basic
6 arithmetic operations, such as splitting a float into its integer and
7 fractional parts or retrieving the imaginary part of a complex value.
8 These functions are declared in the header files @file{math.h} and
12 * Floating Point Numbers:: Basic concepts. IEEE 754.
13 * Floating Point Classes:: The five kinds of floating-point number.
14 * Floating Point Errors:: When something goes wrong in a calculation.
15 * Rounding:: Controlling how results are rounded.
16 * Control Functions:: Saving and restoring the FPU's state.
17 * Arithmetic Functions:: Fundamental operations provided by the library.
18 * Complex Numbers:: The types. Writing complex constants.
19 * Operations on Complex:: Projection, conjugation, decomposition.
20 * Integer Division:: Integer division with guaranteed rounding.
21 * Parsing of Numbers:: Converting strings to numbers.
22 * System V Number Conversion:: An archaic way to convert numbers to strings.
25 @node Floating Point Numbers
26 @section Floating Point Numbers
27 @cindex floating point
29 @cindex IEEE floating point
31 Most computer hardware has support for two different kinds of numbers:
32 integers (@math{@dots{}-3, -2, -1, 0, 1, 2, 3@dots{}}) and
33 floating-point numbers. Floating-point numbers have three parts: the
34 @dfn{mantissa}, the @dfn{exponent}, and the @dfn{sign bit}. The real
35 number represented by a floating-point value is given by
37 $(s \mathrel? -1 \mathrel: 1) \cdot 2^e \cdot M$
40 @math{(s ? -1 : 1) @mul{} 2^e @mul{} M}
42 where @math{s} is the sign bit, @math{e} the exponent, and @math{M}
43 the mantissa. @xref{Floating Point Concepts}, for details. (It is
44 possible to have a different @dfn{base} for the exponent, but all modern
45 hardware uses @math{2}.)
47 Floating-point numbers can represent a finite subset of the real
48 numbers. While this subset is large enough for most purposes, it is
49 important to remember that the only reals that can be represented
50 exactly are rational numbers that have a terminating binary expansion
51 shorter than the width of the mantissa. Even simple fractions such as
52 @math{1/5} can only be approximated by floating point.
54 Mathematical operations and functions frequently need to produce values
55 that are not representable. Often these values can be approximated
56 closely enough for practical purposes, but sometimes they can't.
57 Historically there was no way to tell when the results of a calculation
58 were inaccurate. Modern computers implement the @w{IEEE 754} standard
59 for numerical computations, which defines a framework for indicating to
60 the program when the results of calculation are not trustworthy. This
61 framework consists of a set of @dfn{exceptions} that indicate why a
62 result could not be represented, and the special values @dfn{infinity}
63 and @dfn{not a number} (NaN).
65 @node Floating Point Classes
66 @section Floating-Point Number Classification Functions
67 @cindex floating-point classes
68 @cindex classes, floating-point
71 @w{ISO C 9x} defines macros that let you determine what sort of
72 floating-point number a variable holds.
76 @deftypefn {Macro} int fpclassify (@emph{float-type} @var{x})
77 This is a generic macro which works on all floating-point types and
78 which returns a value of type @code{int}. The possible values are:
82 The floating-point number @var{x} is ``Not a Number'' (@pxref{Infinity
85 The value of @var{x} is either plus or minus infinity (@pxref{Infinity
88 The value of @var{x} is zero. In floating-point formats like @w{IEEE
89 754}, where zero can be signed, this value is also returned if
90 @var{x} is negative zero.
92 Numbers whose absolute value is too small to be represented in the
93 normal format are represented in an alternate, @dfn{denormalized} format
94 (@pxref{Floating Point Concepts}). This format is less precise but can
95 represent values closer to zero. @code{fpclassify} returns this value
96 for values of @var{x} in this alternate format.
98 This value is returned for all other values of @var{x}. It indicates
99 that there is nothing special about the number.
104 @code{fpclassify} is most useful if more than one property of a number
105 must be tested. There are more specific macros which only test one
106 property at a time. Generally these macros execute faster than
107 @code{fpclassify}, since there is special hardware support for them.
108 You should therefore use the specific macros whenever possible.
112 @deftypefn {Macro} int isfinite (@emph{float-type} @var{x})
113 This macro returns a nonzero value if @var{x} is finite: not plus or
114 minus infinity, and not NaN. It is equivalent to
117 (fpclassify (x) != FP_NAN && fpclassify (x) != FP_INFINITE)
120 @code{isfinite} is implemented as a macro which accepts any
126 @deftypefn {Macro} int isnormal (@emph{float-type} @var{x})
127 This macro returns a nonzero value if @var{x} is finite and normalized.
131 (fpclassify (x) == FP_NORMAL)
137 @deftypefn {Macro} int isnan (@emph{float-type} @var{x})
138 This macro returns a nonzero value if @var{x} is NaN. It is equivalent
142 (fpclassify (x) == FP_NAN)
146 Another set of floating-point classification functions was provided by
147 BSD. The GNU C library also supports these functions; however, we
148 recommend that you use the C9x macros in new code. Those are standard
149 and will be available more widely. Also, since they are macros, you do
150 not have to worry about the type of their argument.
154 @deftypefun int isinf (double @var{x})
157 @deftypefunx int isinff (float @var{x})
160 @deftypefunx int isinfl (long double @var{x})
161 This function returns @code{-1} if @var{x} represents negative infinity,
162 @code{1} if @var{x} represents positive infinity, and @code{0} otherwise.
167 @deftypefun int isnan (double @var{x})
170 @deftypefunx int isnanf (float @var{x})
173 @deftypefunx int isnanl (long double @var{x})
174 This function returns a nonzero value if @var{x} is a ``not a number''
175 value, and zero otherwise.
177 @strong{Note:} The @code{isnan} macro defined by @w{ISO C 9x} overrides
178 the BSD function. This is normally not a problem, because the two
179 routines behave identically. However, if you really need to get the BSD
180 function for some reason, you can write
189 @deftypefun int finite (double @var{x})
192 @deftypefunx int finitef (float @var{x})
195 @deftypefunx int finitel (long double @var{x})
196 This function returns a nonzero value if @var{x} is finite or a ``not a
197 number'' value, and zero otherwise.
202 @deftypefun double infnan (int @var{error})
203 This function is provided for compatibility with BSD. Its argument is
204 an error code, @code{EDOM} or @code{ERANGE}; @code{infnan} returns the
205 value that a math function would return if it set @code{errno} to that
206 value. @xref{Math Error Reporting}. @code{-ERANGE} is also acceptable
207 as an argument, and corresponds to @code{-HUGE_VAL} as a value.
209 In the BSD library, on certain machines, @code{infnan} raises a fatal
210 signal in all cases. The GNU library does not do likewise, because that
211 does not fit the @w{ISO C} specification.
214 @strong{Portability Note:} The functions listed in this section are BSD
218 @node Floating Point Errors
219 @section Errors in Floating-Point Calculations
222 * FP Exceptions:: IEEE 754 math exceptions and how to detect them.
223 * Infinity and NaN:: Special values returned by calculations.
224 * Status bit operations:: Checking for exceptions after the fact.
225 * Math Error Reporting:: How the math functions report errors.
229 @subsection FP Exceptions
233 @cindex division by zero
234 @cindex inexact exception
235 @cindex invalid exception
236 @cindex overflow exception
237 @cindex underflow exception
239 The @w{IEEE 754} standard defines five @dfn{exceptions} that can occur
240 during a calculation. Each corresponds to a particular sort of error,
243 When exceptions occur (when exceptions are @dfn{raised}, in the language
244 of the standard), one of two things can happen. By default the
245 exception is simply noted in the floating-point @dfn{status word}, and
246 the program continues as if nothing had happened. The operation
247 produces a default value, which depends on the exception (see the table
248 below). Your program can check the status word to find out which
251 Alternatively, you can enable @dfn{traps} for exceptions. In that case,
252 when an exception is raised, your program will receive the @code{SIGFPE}
253 signal. The default action for this signal is to terminate the
254 program. @xref{Signal Handling}, for how you can change the effect of
258 In the System V math library, the user-defined function @code{matherr}
259 is called when certain exceptions occur inside math library functions.
260 However, the Unix98 standard deprecates this interface. We support it
261 for historical compatibility, but recommend that you do not use it in
265 The exceptions defined in @w{IEEE 754} are:
268 @item Invalid Operation
269 This exception is raised if the given operands are invalid for the
270 operation to be performed. Examples are
271 (see @w{IEEE 754}, @w{section 7}):
274 Addition or subtraction: @math{@infinity{} - @infinity{}}. (But
275 @math{@infinity{} + @infinity{} = @infinity{}}).
277 Multiplication: @math{0 @mul{} @infinity{}}.
279 Division: @math{0/0} or @math{@infinity{}/@infinity{}}.
281 Remainder: @math{x} REM @math{y}, where @math{y} is zero or @math{x} is
284 Square root if the operand is less then zero. More generally, any
285 mathematical function evaluated outside its domain produces this
288 Conversion of a floating-point number to an integer or decimal
289 string, when the number cannot be represented in the target format (due
290 to overflow, infinity, or NaN).
292 Conversion of an unrecognizable input string.
294 Comparison via predicates involving @math{<} or @math{>}, when one or
295 other of the operands is NaN. You can prevent this exception by using
296 the unordered comparison functions instead; see @ref{FP Comparison Functions}.
299 If the exception does not trap, the result of the operation is NaN.
301 @item Division by Zero
302 This exception is raised when a finite nonzero number is divided
303 by zero. If no trap occurs the result is either @math{+@infinity{}} or
304 @math{-@infinity{}}, depending on the signs of the operands.
307 This exception is raised whenever the result cannot be represented
308 as a finite value in the precision format of the destination. If no trap
309 occurs the result depends on the sign of the intermediate result and the
310 current rounding mode (@w{IEEE 754}, @w{section 7.3}):
313 Round to nearest carries all overflows to @math{@infinity{}}
314 with the sign of the intermediate result.
316 Round toward @math{0} carries all overflows to the largest representable
317 finite number with the sign of the intermediate result.
319 Round toward @math{-@infinity{}} carries positive overflows to the
320 largest representable finite number and negative overflows to
324 Round toward @math{@infinity{}} carries negative overflows to the
325 most negative representable finite number and positive overflows
326 to @math{@infinity{}}.
329 Whenever the overflow exception is raised, the inexact exception is also
333 The underflow exception is raised when an intermediate result is too
334 small to be calculated accurately, or if the operation's result rounded
335 to the destination precision is too small to be normalized.
337 When no trap is installed for the underflow exception, underflow is
338 signaled (via the underflow flag) only when both tininess and loss of
339 accuracy have been detected. If no trap handler is installed the
340 operation continues with an imprecise small value, or zero if the
341 destination precision cannot hold the small exact result.
344 This exception is signalled if a rounded result is not exact (such as
345 when calculating the square root of two) or a result overflows without
349 @node Infinity and NaN
350 @subsection Infinity and NaN
355 @w{IEEE 754} floating point numbers can represent positive or negative
356 infinity, and @dfn{NaN} (not a number). These three values arise from
357 calculations whose result is undefined or cannot be represented
358 accurately. You can also deliberately set a floating-point variable to
359 any of them, which is sometimes useful. Some examples of calculations
360 that produce infinity or NaN:
364 @math{1/0 = @infinity{}}
365 @math{log (0) = -@infinity{}}
366 @math{sqrt (-1) = NaN}
370 $${1\over0} = \infty$$
372 $$\sqrt{-1} = \hbox{NaN}$$
375 When a calculation produces any of these values, an exception also
376 occurs; see @ref{FP Exceptions}.
378 The basic operations and math functions all accept infinity and NaN and
379 produce sensible output. Infinities propagate through calculations as
380 one would expect: for example, @math{2 + @infinity{} = @infinity{}},
381 @math{4/@infinity{} = 0}, atan @math{(@infinity{}) = @pi{}/2}. NaN, on
382 the other hand, infects any calculation that involves it. Unless the
383 calculation would produce the same result no matter what real value
384 replaced NaN, the result is NaN.
386 In comparison operations, positive infinity is larger than all values
387 except itself and NaN, and negative infinity is smaller than all values
388 except itself and NaN. NaN is @dfn{unordered}: it is not equal to,
389 greater than, or less than anything, @emph{including itself}. @code{x ==
390 x} is false if the value of @code{x} is NaN. You can use this to test
391 whether a value is NaN or not, but the recommended way to test for NaN
392 is with the @code{isnan} function (@pxref{Floating Point Classes}). In
393 addition, @code{<}, @code{>}, @code{<=}, and @code{>=} will raise an
394 exception when applied to NaNs.
396 @file{math.h} defines macros that allow you to explicitly set a variable
401 @deftypevr Macro float INFINITY
402 An expression representing positive infinity. It is equal to the value
403 produced by mathematical operations like @code{1.0 / 0.0}.
404 @code{-INFINITY} represents negative infinity.
406 You can test whether a floating-point value is infinite by comparing it
407 to this macro. However, this is not recommended; you should use the
408 @code{isfinite} macro instead. @xref{Floating Point Classes}.
410 This macro was introduced in the @w{ISO C 9X} standard.
415 @deftypevr Macro float NAN
416 An expression representing a value which is ``not a number''. This
417 macro is a GNU extension, available only on machines that support the
418 ``not a number'' value---that is to say, on all machines that support
421 You can use @samp{#ifdef NAN} to test whether the machine supports
422 NaN. (Of course, you must arrange for GNU extensions to be visible,
423 such as by defining @code{_GNU_SOURCE}, and then you must include
427 @w{IEEE 754} also allows for another unusual value: negative zero. This
428 value is produced when you divide a positive number by negative
429 infinity, or when a negative result is smaller than the limits of
430 representation. Negative zero behaves identically to zero in all
431 calculations, unless you explicitly test the sign bit with
432 @code{signbit} or @code{copysign}.
434 @node Status bit operations
435 @subsection Examining the FPU status word
437 @w{ISO C 9x} defines functions to query and manipulate the
438 floating-point status word. You can use these functions to check for
439 untrapped exceptions when it's convenient, rather than worrying about
440 them in the middle of a calculation.
442 These constants represent the various @w{IEEE 754} exceptions. Not all
443 FPUs report all the different exceptions. Each constant is defined if
444 and only if the FPU you are compiling for supports that exception, so
445 you can test for FPU support with @samp{#ifdef}. They are defined in
452 The inexact exception.
456 The divide by zero exception.
460 The underflow exception.
464 The overflow exception.
468 The invalid exception.
471 The macro @code{FE_ALL_EXCEPT} is the bitwise OR of all exception macros
472 which are supported by the FP implementation.
474 These functions allow you to clear exception flags, test for exceptions,
475 and save and restore the set of exceptions flagged.
479 @deftypefun void feclearexcept (int @var{excepts})
480 This function clears all of the supported exception flags indicated by
486 @deftypefun int fetestexcept (int @var{excepts})
487 Test whether the exception flags indicated by the parameter @var{except}
488 are currently set. If any of them are, a nonzero value is returned
489 which specifies which exceptions are set. Otherwise the result is zero.
492 To understand these functions, imagine that the status word is an
493 integer variable named @var{status}. @code{feclearexcept} is then
494 equivalent to @samp{status &= ~excepts} and @code{fetestexcept} is
495 equivalent to @samp{(status & excepts)}. The actual implementation may
496 be very different, of course.
498 Exception flags are only cleared when the program explicitly requests it,
499 by calling @code{feclearexcept}. If you want to check for exceptions
500 from a set of calculations, you should clear all the flags first. Here
501 is a simple example of the way to use @code{fetestexcept}:
507 feclearexcept (FE_ALL_EXCEPT);
509 raised = fetestexcept (FE_OVERFLOW | FE_INVALID);
510 if (raised & FE_OVERFLOW) @{ /* ... */ @}
511 if (raised & FE_INVALID) @{ /* ... */ @}
516 You cannot explicitly set bits in the status word. You can, however,
517 save the entire status word and restore it later. This is done with the
522 @deftypefun void fegetexceptflag (fexcept_t *@var{flagp}, int @var{excepts})
523 This function stores in the variable pointed to by @var{flagp} an
524 implementation-defined value representing the current setting of the
525 exception flags indicated by @var{excepts}.
530 @deftypefun void fesetexceptflag (const fexcept_t *@var{flagp}, int
532 This function restores the flags for the exceptions indicated by
533 @var{excepts} to the values stored in the variable pointed to by
537 Note that the value stored in @code{fexcept_t} bears no resemblance to
538 the bit mask returned by @code{fetestexcept}. The type may not even be
539 an integer. Do not attempt to modify an @code{fexcept_t} variable.
541 @node Math Error Reporting
542 @subsection Error Reporting by Mathematical Functions
543 @cindex errors, mathematical
547 Many of the math functions are defined only over a subset of the real or
548 complex numbers. Even if they are mathematically defined, their result
549 may be larger or smaller than the range representable by their return
550 type. These are known as @dfn{domain errors}, @dfn{overflows}, and
551 @dfn{underflows}, respectively. Math functions do several things when
552 one of these errors occurs. In this manual we will refer to the
553 complete response as @dfn{signalling} a domain error, overflow, or
556 When a math function suffers a domain error, it raises the invalid
557 exception and returns NaN. It also sets @var{errno} to @code{EDOM};
558 this is for compatibility with old systems that do not support @w{IEEE
559 754} exception handling. Likewise, when overflow occurs, math
560 functions raise the overflow exception and return @math{@infinity{}} or
561 @math{-@infinity{}} as appropriate. They also set @var{errno} to
562 @code{ERANGE}. When underflow occurs, the underflow exception is
563 raised, and zero (appropriately signed) is returned. @var{errno} may be
564 set to @code{ERANGE}, but this is not guaranteed.
566 Some of the math functions are defined mathematically to result in a
567 complex value over parts of their domains. The most familiar example of
568 this is taking the square root of a negative number. The complex math
569 functions, such as @code{csqrt}, will return the appropriate complex value
570 in this case. The real-valued functions, such as @code{sqrt}, will
571 signal a domain error.
573 Some older hardware does not support infinities. On that hardware,
574 overflows instead return a particular very large number (usually the
575 largest representable number). @file{math.h} defines macros you can use
576 to test for overflow on both old and new hardware.
580 @deftypevr Macro double HUGE_VAL
583 @deftypevrx Macro float HUGE_VALF
586 @deftypevrx Macro {long double} HUGE_VALL
587 An expression representing a particular very large number. On machines
588 that use @w{IEEE 754} floating point format, @code{HUGE_VAL} is infinity.
589 On other machines, it's typically the largest positive number that can
592 Mathematical functions return the appropriately typed version of
593 @code{HUGE_VAL} or @code{@minus{}HUGE_VAL} when the result is too large
598 @section Rounding Modes
600 Floating-point calculations are carried out internally with extra
601 precision, and then rounded to fit into the destination type. This
602 ensures that results are as precise as the input data. @w{IEEE 754}
603 defines four possible rounding modes:
606 @item Round to nearest.
607 This is the default mode. It should be used unless there is a specific
608 need for one of the others. In this mode results are rounded to the
609 nearest representable value. If the result is midway between two
610 representable values, the even representable is chosen. @dfn{Even} here
611 means the lowest-order bit is zero. This rounding mode prevents
612 statistical bias and guarantees numeric stability: round-off errors in a
613 lengthy calculation will remain smaller than half of @code{FLT_EPSILON}.
615 @c @item Round toward @math{+@infinity{}}
616 @item Round toward plus Infinity.
617 All results are rounded to the smallest representable value
618 which is greater than the result.
620 @c @item Round toward @math{-@infinity{}}
621 @item Round toward minus Infinity.
622 All results are rounded to the largest representable value which is less
625 @item Round toward zero.
626 All results are rounded to the largest representable value whose
627 magnitude is less than that of the result. In other words, if the
628 result is negative it is rounded up; if it is positive, it is rounded
633 @file{fenv.h} defines constants which you can use to refer to the
634 various rounding modes. Each one will be defined if and only if the FPU
635 supports the corresponding rounding mode.
648 Round toward @math{+@infinity{}}.
654 Round toward @math{-@infinity{}}.
658 @vindex FE_TOWARDZERO
663 Underflow is an unusual case. Normally, @w{IEEE 754} floating point
664 numbers are always normalized (@pxref{Floating Point Concepts}).
665 Numbers smaller than @math{2^r} (where @math{r} is the minimum exponent,
666 @code{FLT_MIN_RADIX-1} for @var{float}) cannot be represented as
667 normalized numbers. Rounding all such numbers to zero or @math{2^r}
668 would cause some algorithms to fail at 0. Therefore, they are left in
669 denormalized form. That produces loss of precision, since some bits of
670 the mantissa are stolen to indicate the decimal point.
672 If a result is too small to be represented as a denormalized number, it
673 is rounded to zero. However, the sign of the result is preserved; if
674 the calculation was negative, the result is @dfn{negative zero}.
675 Negative zero can also result from some operations on infinity, such as
676 @math{4/-@infinity{}}. Negative zero behaves identically to zero except
677 when the @code{copysign} or @code{signbit} functions are used to check
678 the sign bit directly.
680 At any time one of the above four rounding modes is selected. You can
681 find out which one with this function:
685 @deftypefun int fegetround (void)
686 Returns the currently selected rounding mode, represented by one of the
687 values of the defined rounding mode macros.
691 To change the rounding mode, use this function:
695 @deftypefun int fesetround (int @var{round})
696 Changes the currently selected rounding mode to @var{round}. If
697 @var{round} does not correspond to one of the supported rounding modes
698 nothing is changed. @code{fesetround} returns a nonzero value if it
699 changed the rounding mode, zero if the mode is not supported.
702 You should avoid changing the rounding mode if possible. It can be an
703 expensive operation; also, some hardware requires you to compile your
704 program differently for it to work. The resulting code may run slower.
705 See your compiler documentation for details.
706 @c This section used to claim that functions existed to round one number
707 @c in a specific fashion. I can't find any functions in the library
710 @node Control Functions
711 @section Floating-Point Control Functions
713 @w{IEEE 754} floating-point implementations allow the programmer to
714 decide whether traps will occur for each of the exceptions, by setting
715 bits in the @dfn{control word}. In C, traps result in the program
716 receiving the @code{SIGFPE} signal; see @ref{Signal Handling}.
718 @strong{Note:} @w{IEEE 754} says that trap handlers are given details of
719 the exceptional situation, and can set the result value. C signals do
720 not provide any mechanism to pass this information back and forth.
721 Trapping exceptions in C is therefore not very useful.
723 It is sometimes necessary to save the state of the floating-point unit
724 while you perform some calculation. The library provides functions
725 which save and restore the exception flags, the set of exceptions that
726 generate traps, and the rounding mode. This information is known as the
727 @dfn{floating-point environment}.
729 The functions to save and restore the floating-point environment all use
730 a variable of type @code{fenv_t} to store information. This type is
731 defined in @file{fenv.h}. Its size and contents are
732 implementation-defined. You should not attempt to manipulate a variable
733 of this type directly.
735 To save the state of the FPU, use one of these functions:
739 @deftypefun void fegetenv (fenv_t *@var{envp})
740 Store the floating-point environment in the variable pointed to by
746 @deftypefun int feholdexcept (fenv_t *@var{envp})
747 Store the current floating-point environment in the object pointed to by
748 @var{envp}. Then clear all exception flags, and set the FPU to trap no
749 exceptions. Not all FPUs support trapping no exceptions; if
750 @code{feholdexcept} cannot set this mode, it returns zero. If it
751 succeeds, it returns a nonzero value.
754 The functions which restore the floating-point environment can take two
759 Pointers to @code{fenv_t} objects, which were initialized previously by a
760 call to @code{fegetenv} or @code{feholdexcept}.
763 The special macro @code{FE_DFL_ENV} which represents the floating-point
764 environment as it was available at program start.
766 Implementation defined macros with names starting with @code{FE_}.
768 @vindex FE_NOMASK_ENV
769 If possible, the GNU C Library defines a macro @code{FE_NOMASK_ENV}
770 which represents an environment where every exception raised causes a
771 trap to occur. You can test for this macro using @code{#ifdef}. It is
772 only defined if @code{_GNU_SOURCE} is defined.
774 Some platforms might define other predefined environments.
778 To set the floating-point environment, you can use either of these
783 @deftypefun void fesetenv (const fenv_t *@var{envp})
784 Set the floating-point environment to that described by @var{envp}.
789 @deftypefun void feupdateenv (const fenv_t *@var{envp})
790 Like @code{fesetenv}, this function sets the floating-point environment
791 to that described by @var{envp}. However, if any exceptions were
792 flagged in the status word before @code{feupdateenv} was called, they
793 remain flagged after the call. In other words, after @code{feupdateenv}
794 is called, the status word is the bitwise OR of the previous status word
795 and the one saved in @var{envp}.
798 @node Arithmetic Functions
799 @section Arithmetic Functions
801 The C library provides functions to do basic operations on
802 floating-point numbers. These include absolute value, maximum and minimum,
803 normalization, bit twiddling, rounding, and a few others.
806 * Absolute Value:: Absolute values of integers and floats.
807 * Normalization Functions:: Extracting exponents and putting them back.
808 * Rounding Functions:: Rounding floats to integers.
809 * Remainder Functions:: Remainders on division, precisely defined.
810 * FP Bit Twiddling:: Sign bit adjustment. Adding epsilon.
811 * FP Comparison Functions:: Comparisons without risk of exceptions.
812 * Misc FP Arithmetic:: Max, min, positive difference, multiply-add.
816 @subsection Absolute Value
817 @cindex absolute value functions
819 These functions are provided for obtaining the @dfn{absolute value} (or
820 @dfn{magnitude}) of a number. The absolute value of a real number
821 @var{x} is @var{x} if @var{x} is positive, @minus{}@var{x} if @var{x} is
822 negative. For a complex number @var{z}, whose real part is @var{x} and
823 whose imaginary part is @var{y}, the absolute value is @w{@code{sqrt
824 (@var{x}*@var{x} + @var{y}*@var{y})}}.
828 Prototypes for @code{abs}, @code{labs} and @code{llabs} are in @file{stdlib.h};
829 @code{imaxabs} is declared in @file{inttypes.h};
830 @code{fabs}, @code{fabsf} and @code{fabsl} are declared in @file{math.h}.
831 @code{cabs}, @code{cabsf} and @code{cabsl} are declared in @file{complex.h}.
835 @deftypefun int abs (int @var{number})
838 @deftypefunx {long int} labs (long int @var{number})
841 @deftypefunx {long long int} llabs (long long int @var{number})
844 @deftypefunx intmax_t imaxabs (intmax_t @var{number})
845 These functions return the absolute value of @var{number}.
847 Most computers use a two's complement integer representation, in which
848 the absolute value of @code{INT_MIN} (the smallest possible @code{int})
849 cannot be represented; thus, @w{@code{abs (INT_MIN)}} is not defined.
851 @code{llabs} and @code{imaxdiv} are new to @w{ISO C 9x}.
856 @deftypefun double fabs (double @var{number})
859 @deftypefunx float fabsf (float @var{number})
862 @deftypefunx {long double} fabsl (long double @var{number})
863 This function returns the absolute value of the floating-point number
869 @deftypefun double cabs (complex double @var{z})
872 @deftypefunx float cabsf (complex float @var{z})
875 @deftypefunx {long double} cabsl (complex long double @var{z})
876 These functions return the absolute value of the complex number @var{z}
877 (@pxref{Complex Numbers}). The absolute value of a complex number is:
880 sqrt (creal (@var{z}) * creal (@var{z}) + cimag (@var{z}) * cimag (@var{z}))
883 This function should always be used instead of the direct formula
884 because it takes special care to avoid losing precision. It may also
885 take advantage of hardware support for this operation. See @code{hypot}
886 in @ref{Exponents and Logarithms}.
889 @node Normalization Functions
890 @subsection Normalization Functions
891 @cindex normalization functions (floating-point)
893 The functions described in this section are primarily provided as a way
894 to efficiently perform certain low-level manipulations on floating point
895 numbers that are represented internally using a binary radix;
896 see @ref{Floating Point Concepts}. These functions are required to
897 have equivalent behavior even if the representation does not use a radix
898 of 2, but of course they are unlikely to be particularly efficient in
902 All these functions are declared in @file{math.h}.
906 @deftypefun double frexp (double @var{value}, int *@var{exponent})
909 @deftypefunx float frexpf (float @var{value}, int *@var{exponent})
912 @deftypefunx {long double} frexpl (long double @var{value}, int *@var{exponent})
913 These functions are used to split the number @var{value}
914 into a normalized fraction and an exponent.
916 If the argument @var{value} is not zero, the return value is @var{value}
917 times a power of two, and is always in the range 1/2 (inclusive) to 1
918 (exclusive). The corresponding exponent is stored in
919 @code{*@var{exponent}}; the return value multiplied by 2 raised to this
920 exponent equals the original number @var{value}.
922 For example, @code{frexp (12.8, &exponent)} returns @code{0.8} and
923 stores @code{4} in @code{exponent}.
925 If @var{value} is zero, then the return value is zero and
926 zero is stored in @code{*@var{exponent}}.
931 @deftypefun double ldexp (double @var{value}, int @var{exponent})
934 @deftypefunx float ldexpf (float @var{value}, int @var{exponent})
937 @deftypefunx {long double} ldexpl (long double @var{value}, int @var{exponent})
938 These functions return the result of multiplying the floating-point
939 number @var{value} by 2 raised to the power @var{exponent}. (It can
940 be used to reassemble floating-point numbers that were taken apart
943 For example, @code{ldexp (0.8, 4)} returns @code{12.8}.
946 The following functions, which come from BSD, provide facilities
947 equivalent to those of @code{ldexp} and @code{frexp}.
951 @deftypefun double logb (double @var{x})
954 @deftypefunx float logbf (float @var{x})
957 @deftypefunx {long double} logbl (long double @var{x})
958 These functions return the integer part of the base-2 logarithm of
959 @var{x}, an integer value represented in type @code{double}. This is
960 the highest integer power of @code{2} contained in @var{x}. The sign of
961 @var{x} is ignored. For example, @code{logb (3.5)} is @code{1.0} and
962 @code{logb (4.0)} is @code{2.0}.
964 When @code{2} raised to this power is divided into @var{x}, it gives a
965 quotient between @code{1} (inclusive) and @code{2} (exclusive).
967 If @var{x} is zero, the return value is minus infinity if the machine
968 supports infinities, and a very small number if it does not. If @var{x}
969 is infinity, the return value is infinity.
971 For finite @var{x}, the value returned by @code{logb} is one less than
972 the value that @code{frexp} would store into @code{*@var{exponent}}.
977 @deftypefun double scalb (double @var{value}, int @var{exponent})
980 @deftypefunx float scalbf (float @var{value}, int @var{exponent})
983 @deftypefunx {long double} scalbl (long double @var{value}, int @var{exponent})
984 The @code{scalb} function is the BSD name for @code{ldexp}.
989 @deftypefun {long long int} scalbn (double @var{x}, int n)
992 @deftypefunx {long long int} scalbnf (float @var{x}, int n)
995 @deftypefunx {long long int} scalbnl (long double @var{x}, int n)
996 @code{scalbn} is identical to @code{scalb}, except that the exponent
997 @var{n} is an @code{int} instead of a floating-point number.
1002 @deftypefun {long long int} scalbln (double @var{x}, long int n)
1005 @deftypefunx {long long int} scalblnf (float @var{x}, long int n)
1008 @deftypefunx {long long int} scalblnl (long double @var{x}, long int n)
1009 @code{scalbln} is identical to @code{scalb}, except that the exponent
1010 @var{n} is a @code{long int} instead of a floating-point number.
1015 @deftypefun {long long int} significand (double @var{x})
1018 @deftypefunx {long long int} significandf (float @var{x})
1021 @deftypefunx {long long int} significandl (long double @var{x})
1022 @code{significand} returns the mantissa of @var{x} scaled to the range
1024 It is equivalent to @w{@code{scalb (@var{x}, (double) -ilogb (@var{x}))}}.
1026 This function exists mainly for use in certain standardized tests
1027 of @w{IEEE 754} conformance.
1030 @node Rounding Functions
1031 @subsection Rounding Functions
1032 @cindex converting floats to integers
1035 The functions listed here perform operations such as rounding and
1036 truncation of floating-point values. Some of these functions convert
1037 floating point numbers to integer values. They are all declared in
1040 You can also convert floating-point numbers to integers simply by
1041 casting them to @code{int}. This discards the fractional part,
1042 effectively rounding towards zero. However, this only works if the
1043 result can actually be represented as an @code{int}---for very large
1044 numbers, this is impossible. The functions listed here return the
1045 result as a @code{double} instead to get around this problem.
1049 @deftypefun double ceil (double @var{x})
1052 @deftypefunx float ceilf (float @var{x})
1055 @deftypefunx {long double} ceill (long double @var{x})
1056 These functions round @var{x} upwards to the nearest integer,
1057 returning that value as a @code{double}. Thus, @code{ceil (1.5)}
1063 @deftypefun double floor (double @var{x})
1066 @deftypefunx float floorf (float @var{x})
1069 @deftypefunx {long double} floorl (long double @var{x})
1070 These functions round @var{x} downwards to the nearest
1071 integer, returning that value as a @code{double}. Thus, @code{floor
1072 (1.5)} is @code{1.0} and @code{floor (-1.5)} is @code{-2.0}.
1077 @deftypefun double trunc (double @var{x})
1080 @deftypefunx float truncf (float @var{x})
1083 @deftypefunx {long double} truncl (long double @var{x})
1084 @code{trunc} is another name for @code{floor}
1089 @deftypefun double rint (double @var{x})
1092 @deftypefunx float rintf (float @var{x})
1095 @deftypefunx {long double} rintl (long double @var{x})
1096 These functions round @var{x} to an integer value according to the
1097 current rounding mode. @xref{Floating Point Parameters}, for
1098 information about the various rounding modes. The default
1099 rounding mode is to round to the nearest integer; some machines
1100 support other modes, but round-to-nearest is always used unless
1101 you explicitly select another.
1103 If @var{x} was not initially an integer, these functions raise the
1109 @deftypefun double nearbyint (double @var{x})
1112 @deftypefunx float nearbyintf (float @var{x})
1115 @deftypefunx {long double} nearbyintl (long double @var{x})
1116 These functions return the same value as the @code{rint} functions, but
1117 do not raise the inexact exception if @var{x} is not an integer.
1122 @deftypefun double round (double @var{x})
1125 @deftypefunx float roundf (float @var{x})
1128 @deftypefunx {long double} roundl (long double @var{x})
1129 These functions are similar to @code{rint}, but they round halfway
1130 cases away from zero instead of to the nearest even integer.
1135 @deftypefun {long int} lrint (double @var{x})
1138 @deftypefunx {long int} lrintf (float @var{x})
1141 @deftypefunx {long int} lrintl (long double @var{x})
1142 These functions are just like @code{rint}, but they return a
1143 @code{long int} instead of a floating-point number.
1148 @deftypefun {long long int} llrint (double @var{x})
1151 @deftypefunx {long long int} llrintf (float @var{x})
1154 @deftypefunx {long long int} llrintl (long double @var{x})
1155 These functions are just like @code{rint}, but they return a
1156 @code{long long int} instead of a floating-point number.
1161 @deftypefun {long int} lround (double @var{x})
1164 @deftypefunx {long int} lroundf (float @var{x})
1167 @deftypefunx {long int} lroundl (long double @var{x})
1168 These functions are just like @code{round}, but they return a
1169 @code{long int} instead of a floating-point number.
1174 @deftypefun {long long int} llround (double @var{x})
1177 @deftypefunx {long long int} llroundf (float @var{x})
1180 @deftypefunx {long long int} llroundl (long double @var{x})
1181 These functions are just like @code{round}, but they return a
1182 @code{long long int} instead of a floating-point number.
1188 @deftypefun double modf (double @var{value}, double *@var{integer-part})
1191 @deftypefunx float modff (float @var{value}, float *@var{integer-part})
1194 @deftypefunx {long double} modfl (long double @var{value}, long double *@var{integer-part})
1195 These functions break the argument @var{value} into an integer part and a
1196 fractional part (between @code{-1} and @code{1}, exclusive). Their sum
1197 equals @var{value}. Each of the parts has the same sign as @var{value},
1198 and the integer part is always rounded toward zero.
1200 @code{modf} stores the integer part in @code{*@var{integer-part}}, and
1201 returns the fractional part. For example, @code{modf (2.5, &intpart)}
1202 returns @code{0.5} and stores @code{2.0} into @code{intpart}.
1205 @node Remainder Functions
1206 @subsection Remainder Functions
1208 The functions in this section compute the remainder on division of two
1209 floating-point numbers. Each is a little different; pick the one that
1214 @deftypefun double fmod (double @var{numerator}, double @var{denominator})
1217 @deftypefunx float fmodf (float @var{numerator}, float @var{denominator})
1220 @deftypefunx {long double} fmodl (long double @var{numerator}, long double @var{denominator})
1221 These functions compute the remainder from the division of
1222 @var{numerator} by @var{denominator}. Specifically, the return value is
1223 @code{@var{numerator} - @w{@var{n} * @var{denominator}}}, where @var{n}
1224 is the quotient of @var{numerator} divided by @var{denominator}, rounded
1225 towards zero to an integer. Thus, @w{@code{fmod (6.5, 2.3)}} returns
1226 @code{1.9}, which is @code{6.5} minus @code{4.6}.
1228 The result has the same sign as the @var{numerator} and has magnitude
1229 less than the magnitude of the @var{denominator}.
1231 If @var{denominator} is zero, @code{fmod} signals a domain error.
1236 @deftypefun double drem (double @var{numerator}, double @var{denominator})
1239 @deftypefunx float dremf (float @var{numerator}, float @var{denominator})
1242 @deftypefunx {long double} dreml (long double @var{numerator}, long double @var{denominator})
1243 These functions are like @code{fmod} except that they rounds the
1244 internal quotient @var{n} to the nearest integer instead of towards zero
1245 to an integer. For example, @code{drem (6.5, 2.3)} returns @code{-0.4},
1246 which is @code{6.5} minus @code{6.9}.
1248 The absolute value of the result is less than or equal to half the
1249 absolute value of the @var{denominator}. The difference between
1250 @code{fmod (@var{numerator}, @var{denominator})} and @code{drem
1251 (@var{numerator}, @var{denominator})} is always either
1252 @var{denominator}, minus @var{denominator}, or zero.
1254 If @var{denominator} is zero, @code{drem} signals a domain error.
1259 @deftypefun double remainder (double @var{numerator}, double @var{denominator})
1262 @deftypefunx float remainderf (float @var{numerator}, float @var{denominator})
1265 @deftypefunx {long double} remainderl (long double @var{numerator}, long double @var{denominator})
1266 This function is another name for @code{drem}.
1269 @node FP Bit Twiddling
1270 @subsection Setting and modifying single bits of FP values
1271 @cindex FP arithmetic
1273 There are some operations that are too complicated or expensive to
1274 perform by hand on floating-point numbers. @w{ISO C 9x} defines
1275 functions to do these operations, which mostly involve changing single
1280 @deftypefun double copysign (double @var{x}, double @var{y})
1283 @deftypefunx float copysignf (float @var{x}, float @var{y})
1286 @deftypefunx {long double} copysignl (long double @var{x}, long double @var{y})
1287 These functions return @var{x} but with the sign of @var{y}. They work
1288 even if @var{x} or @var{y} are NaN or zero. Both of these can carry a
1289 sign (although not all implementations support it) and this is one of
1290 the few operations that can tell the difference.
1292 @code{copysign} never raises an exception.
1293 @c except signalling NaNs
1295 This function is defined in @w{IEC 559} (and the appendix with
1296 recommended functions in @w{IEEE 754}/@w{IEEE 854}).
1301 @deftypefun int signbit (@emph{float-type} @var{x})
1302 @code{signbit} is a generic macro which can work on all floating-point
1303 types. It returns a nonzero value if the value of @var{x} has its sign
1306 This is not the same as @code{x < 0.0}, because @w{IEEE 754} floating
1307 point allows zero to be signed. The comparison @code{-0.0 < 0.0} is
1308 false, but @code{signbit (-0.0)} will return a nonzero value.
1313 @deftypefun double nextafter (double @var{x}, double @var{y})
1316 @deftypefunx float nextafterf (float @var{x}, float @var{y})
1319 @deftypefunx {long double} nextafterl (long double @var{x}, long double @var{y})
1320 The @code{nextafter} function returns the next representable neighbor of
1321 @var{x} in the direction towards @var{y}. The size of the step between
1322 @var{x} and the result depends on the type of the result. If
1323 @math{@var{x} = @var{y}} the function simply returns @var{x}. If either
1324 value is @code{NaN}, @code{NaN} is returned. Otherwise
1325 a value corresponding to the value of the least significant bit in the
1326 mantissa is added or subtracted, depending on the direction.
1327 @code{nextafter} will signal overflow or underflow if the result goes
1328 outside of the range of normalized numbers.
1330 This function is defined in @w{IEC 559} (and the appendix with
1331 recommended functions in @w{IEEE 754}/@w{IEEE 854}).
1336 @deftypefun double nexttoward (double @var{x}, long double @var{y})
1339 @deftypefunx float nexttowardf (float @var{x}, long double @var{y})
1342 @deftypefunx {long double} nexttowardl (long double @var{x}, long double @var{y})
1343 These functions are identical to the corresponding versions of
1344 @code{nextafter} except that their second argument is a @code{long
1351 @deftypefun double nan (const char *@var{tagp})
1354 @deftypefunx float nanf (const char *@var{tagp})
1357 @deftypefunx {long double} nanl (const char *@var{tagp})
1358 The @code{nan} function returns a representation of NaN, provided that
1359 NaN is supported by the target platform.
1360 @code{nan ("@var{n-char-sequence}")} is equivalent to
1361 @code{strtod ("NAN(@var{n-char-sequence})")}.
1363 The argument @var{tagp} is used in an unspecified manner. On @w{IEEE
1364 754} systems, there are many representations of NaN, and @var{tagp}
1365 selects one. On other systems it may do nothing.
1368 @node FP Comparison Functions
1369 @subsection Floating-Point Comparison Functions
1370 @cindex unordered comparison
1372 The standard C comparison operators provoke exceptions when one or other
1373 of the operands is NaN. For example,
1380 will raise an exception if @var{a} is NaN. (This does @emph{not}
1381 happen with @code{==} and @code{!=}; those merely return false and true,
1382 respectively, when NaN is examined.) Frequently this exception is
1383 undesirable. @w{ISO C 9x} therefore defines comparison functions that
1384 do not raise exceptions when NaN is examined. All of the functions are
1385 implemented as macros which allow their arguments to be of any
1386 floating-point type. The macros are guaranteed to evaluate their
1387 arguments only once.
1391 @deftypefn Macro int isgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1392 This macro determines whether the argument @var{x} is greater than
1393 @var{y}. It is equivalent to @code{(@var{x}) > (@var{y})}, but no
1394 exception is raised if @var{x} or @var{y} are NaN.
1399 @deftypefn Macro int isgreaterequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1400 This macro determines whether the argument @var{x} is greater than or
1401 equal to @var{y}. It is equivalent to @code{(@var{x}) >= (@var{y})}, but no
1402 exception is raised if @var{x} or @var{y} are NaN.
1407 @deftypefn Macro int isless (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1408 This macro determines whether the argument @var{x} is less than @var{y}.
1409 It is equivalent to @code{(@var{x}) < (@var{y})}, but no exception is
1410 raised if @var{x} or @var{y} are NaN.
1415 @deftypefn Macro int islessequal (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1416 This macro determines whether the argument @var{x} is less than or equal
1417 to @var{y}. It is equivalent to @code{(@var{x}) <= (@var{y})}, but no
1418 exception is raised if @var{x} or @var{y} are NaN.
1423 @deftypefn Macro int islessgreater (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1424 This macro determines whether the argument @var{x} is less or greater
1425 than @var{y}. It is equivalent to @code{(@var{x}) < (@var{y}) ||
1426 (@var{x}) > (@var{y})} (although it only evaluates @var{x} and @var{y}
1427 once), but no exception is raised if @var{x} or @var{y} are NaN.
1429 This macro is not equivalent to @code{@var{x} != @var{y}}, because that
1430 expression is true if @var{x} or @var{y} are NaN.
1435 @deftypefn Macro int isunordered (@emph{real-floating} @var{x}, @emph{real-floating} @var{y})
1436 This macro determines whether its arguments are unordered. In other
1437 words, it is true if @var{x} or @var{y} are NaN, and false otherwise.
1440 Not all machines provide hardware support for these operations. On
1441 machines that don't, the macros can be very slow. Therefore, you should
1442 not use these functions when NaN is not a concern.
1444 @strong{Note:} There are no macros @code{isequal} or @code{isunequal}.
1445 They are unnecessary, because the @code{==} and @code{!=} operators do
1446 @emph{not} throw an exception if one or both of the operands are NaN.
1448 @node Misc FP Arithmetic
1449 @subsection Miscellaneous FP arithmetic functions
1452 @cindex positive difference
1453 @cindex multiply-add
1455 The functions in this section perform miscellaneous but common
1456 operations that are awkward to express with C operators. On some
1457 processors these functions can use special machine instructions to
1458 perform these operations faster than the equivalent C code.
1462 @deftypefun double fmin (double @var{x}, double @var{y})
1465 @deftypefunx float fminf (float @var{x}, float @var{y})
1468 @deftypefunx {long double} fminl (long double @var{x}, long double @var{y})
1469 The @code{fmin} function returns the lesser of the two values @var{x}
1470 and @var{y}. It is similar to the expression
1472 ((x) < (y) ? (x) : (y))
1474 except that @var{x} and @var{y} are only evaluated once.
1476 If an argument is NaN, the other argument is returned. If both arguments
1477 are NaN, NaN is returned.
1482 @deftypefun double fmax (double @var{x}, double @var{y})
1485 @deftypefunx float fmaxf (float @var{x}, float @var{y})
1488 @deftypefunx {long double} fmaxl (long double @var{x}, long double @var{y})
1489 The @code{fmax} function returns the greater of the two values @var{x}
1492 If an argument is NaN, the other argument is returned. If both arguments
1493 are NaN, NaN is returned.
1498 @deftypefun double fdim (double @var{x}, double @var{y})
1501 @deftypefunx float fdimf (float @var{x}, float @var{y})
1504 @deftypefunx {long double} fdiml (long double @var{x}, long double @var{y})
1505 The @code{fdim} function returns the positive difference between
1506 @var{x} and @var{y}. The positive difference is @math{@var{x} -
1507 @var{y}} if @var{x} is greater than @var{y}, and @math{0} otherwise.
1509 If @var{x}, @var{y}, or both are NaN, NaN is returned.
1514 @deftypefun double fma (double @var{x}, double @var{y}, double @var{z})
1517 @deftypefunx float fmaf (float @var{x}, float @var{y}, float @var{z})
1520 @deftypefunx {long double} fmal (long double @var{x}, long double @var{y}, long double @var{z})
1522 The @code{fma} function performs floating-point multiply-add. This is
1523 the operation @math{(@var{x} @mul{} @var{y}) + @var{z}}, but the
1524 intermediate result is not rounded to the destination type. This can
1525 sometimes improve the precision of a calculation.
1527 This function was introduced because some processors have a special
1528 instruction to perform multiply-add. The C compiler cannot use it
1529 directly, because the expression @samp{x*y + z} is defined to round the
1530 intermediate result. @code{fma} lets you choose when you want to round
1534 On processors which do not implement multiply-add in hardware,
1535 @code{fma} can be very slow since it must avoid intermediate rounding.
1536 @file{math.h} defines the symbols @code{FP_FAST_FMA},
1537 @code{FP_FAST_FMAF}, and @code{FP_FAST_FMAL} when the corresponding
1538 version of @code{fma} is no slower than the expression @samp{x*y + z}.
1539 In the GNU C library, this always means the operation is implemented in
1543 @node Complex Numbers
1544 @section Complex Numbers
1546 @cindex complex numbers
1548 @w{ISO C 9x} introduces support for complex numbers in C. This is done
1549 with a new type qualifier, @code{complex}. It is a keyword if and only
1550 if @file{complex.h} has been included. There are three complex types,
1551 corresponding to the three real types: @code{float complex},
1552 @code{double complex}, and @code{long double complex}.
1554 To construct complex numbers you need a way to indicate the imaginary
1555 part of a number. There is no standard notation for an imaginary
1556 floating point constant. Instead, @file{complex.h} defines two macros
1557 that can be used to create complex numbers.
1559 @deftypevr Macro {const float complex} _Complex_I
1560 This macro is a representation of the complex number ``@math{0+1i}''.
1561 Multiplying a real floating-point value by @code{_Complex_I} gives a
1562 complex number whose value is purely imaginary. You can use this to
1563 construct complex constants:
1566 @math{3.0 + 4.0i} = @code{3.0 + 4.0 * _Complex_I}
1569 Note that @code{_Complex_I * _Complex_I} has the value @code{-1}, but
1570 the type of that value is @code{complex}.
1573 @c Put this back in when gcc supports _Imaginary_I. It's too confusing.
1576 Without an optimizing compiler this is more expensive than the use of
1577 @code{_Imaginary_I} but with is better than nothing. You can avoid all
1578 the hassles if you use the @code{I} macro below if the name is not
1581 @deftypevr Macro {const float imaginary} _Imaginary_I
1582 This macro is a representation of the value ``@math{1i}''. I.e., it is
1586 _Imaginary_I * _Imaginary_I = -1
1590 The result is not of type @code{float imaginary} but instead @code{float}.
1591 One can use it to easily construct complex number like in
1594 3.0 - _Imaginary_I * 4.0
1598 which results in the complex number with a real part of 3.0 and a
1599 imaginary part -4.0.
1604 @code{_Complex_I} is a bit of a mouthful. @file{complex.h} also defines
1605 a shorter name for the same constant.
1607 @deftypevr Macro {const float complex} I
1608 This macro has exactly the same value as @code{_Complex_I}. Most of the
1609 time it is preferable. However, it causes problems if you want to use
1610 the identifier @code{I} for something else. You can safely write
1613 #include <complex.h>
1618 if you need @code{I} for your own purposes. (In that case we recommend
1619 you also define some other short name for @code{_Complex_I}, such as
1623 If the implementation does not support the @code{imaginary} types
1624 @code{I} is defined as @code{_Complex_I} which is the second best
1625 solution. It still can be used in the same way but requires a most
1626 clever compiler to get the same results.
1630 @node Operations on Complex
1631 @section Projections, Conjugates, and Decomposing of Complex Numbers
1632 @cindex project complex numbers
1633 @cindex conjugate complex numbers
1634 @cindex decompose complex numbers
1637 @w{ISO C 9x} also defines functions that perform basic operations on
1638 complex numbers, such as decomposition and conjugation. The prototypes
1639 for all these functions are in @file{complex.h}. All functions are
1640 available in three variants, one for each of the three complex types.
1644 @deftypefun double creal (complex double @var{z})
1647 @deftypefunx float crealf (complex float @var{z})
1650 @deftypefunx {long double} creall (complex long double @var{z})
1651 These functions return the real part of the complex number @var{z}.
1656 @deftypefun double cimag (complex double @var{z})
1659 @deftypefunx float cimagf (complex float @var{z})
1662 @deftypefunx {long double} cimagl (complex long double @var{z})
1663 These functions return the imaginary part of the complex number @var{z}.
1668 @deftypefun {complex double} conj (complex double @var{z})
1671 @deftypefunx {complex float} conjf (complex float @var{z})
1674 @deftypefunx {complex long double} conjl (complex long double @var{z})
1675 These functions return the conjugate value of the complex number
1676 @var{z}. The conjugate of a complex number has the same real part and a
1677 negated imaginary part. In other words, @samp{conj(a + bi) = a + -bi}.
1682 @deftypefun double carg (complex double @var{z})
1685 @deftypefunx float cargf (complex float @var{z})
1688 @deftypefunx {long double} cargl (complex long double @var{z})
1689 These functions return the argument of the complex number @var{z}.
1690 The argument of a complex number is the angle in the complex plane
1691 between the positive real axis and a line passing through zero and the
1692 number. This angle is measured in the usual fashion and ranges from @math{0}
1695 @code{carg} has a branch cut along the positive real axis.
1700 @deftypefun {complex double} cproj (complex double @var{z})
1703 @deftypefunx {complex float} cprojf (complex float @var{z})
1706 @deftypefunx {complex long double} cprojl (complex long double @var{z})
1707 These functions return the projection of the complex value @var{z} onto
1708 the Riemann sphere. Values with a infinite imaginary part are projected
1709 to positive infinity on the real axis, even if the real part is NaN. If
1710 the real part is infinite, the result is equivalent to
1713 INFINITY + I * copysign (0.0, cimag (z))
1717 @node Integer Division
1718 @section Integer Division
1719 @cindex integer division functions
1721 This section describes functions for performing integer division. These
1722 functions are redundant when GNU CC is used, because in GNU C the
1723 @samp{/} operator always rounds towards zero. But in other C
1724 implementations, @samp{/} may round differently with negative arguments.
1725 @code{div} and @code{ldiv} are useful because they specify how to round
1726 the quotient: towards zero. The remainder has the same sign as the
1729 These functions are specified to return a result @var{r} such that the value
1730 @code{@var{r}.quot*@var{denominator} + @var{r}.rem} equals
1734 To use these facilities, you should include the header file
1735 @file{stdlib.h} in your program.
1739 @deftp {Data Type} div_t
1740 This is a structure type used to hold the result returned by the @code{div}
1741 function. It has the following members:
1745 The quotient from the division.
1748 The remainder from the division.
1754 @deftypefun div_t div (int @var{numerator}, int @var{denominator})
1755 This function @code{div} computes the quotient and remainder from
1756 the division of @var{numerator} by @var{denominator}, returning the
1757 result in a structure of type @code{div_t}.
1759 If the result cannot be represented (as in a division by zero), the
1760 behavior is undefined.
1762 Here is an example, albeit not a very useful one.
1766 result = div (20, -6);
1770 Now @code{result.quot} is @code{-3} and @code{result.rem} is @code{2}.
1775 @deftp {Data Type} ldiv_t
1776 This is a structure type used to hold the result returned by the @code{ldiv}
1777 function. It has the following members:
1781 The quotient from the division.
1784 The remainder from the division.
1787 (This is identical to @code{div_t} except that the components are of
1788 type @code{long int} rather than @code{int}.)
1793 @deftypefun ldiv_t ldiv (long int @var{numerator}, long int @var{denominator})
1794 The @code{ldiv} function is similar to @code{div}, except that the
1795 arguments are of type @code{long int} and the result is returned as a
1796 structure of type @code{ldiv_t}.
1801 @deftp {Data Type} lldiv_t
1802 This is a structure type used to hold the result returned by the @code{lldiv}
1803 function. It has the following members:
1806 @item long long int quot
1807 The quotient from the division.
1809 @item long long int rem
1810 The remainder from the division.
1813 (This is identical to @code{div_t} except that the components are of
1814 type @code{long long int} rather than @code{int}.)
1819 @deftypefun lldiv_t lldiv (long long int @var{numerator}, long long int @var{denominator})
1820 The @code{lldiv} function is like the @code{div} function, but the
1821 arguments are of type @code{long long int} and the result is returned as
1822 a structure of type @code{lldiv_t}.
1824 The @code{lldiv} function was added in @w{ISO C 9x}.
1829 @deftp {Data Type} imaxdiv_t
1830 This is a structure type used to hold the result returned by the @code{imaxdiv}
1831 function. It has the following members:
1835 The quotient from the division.
1838 The remainder from the division.
1841 (This is identical to @code{div_t} except that the components are of
1842 type @code{intmax_t} rather than @code{int}.)
1847 @deftypefun imaxdiv_t imaxdiv (intmax_t @var{numerator}, intmax_t @var{denominator})
1848 The @code{imaxdiv} function is like the @code{div} function, but the
1849 arguments are of type @code{intmax_t} and the result is returned as
1850 a structure of type @code{imaxdiv_t}.
1852 The @code{imaxdiv} function was added in @w{ISO C 9x}.
1856 @node Parsing of Numbers
1857 @section Parsing of Numbers
1858 @cindex parsing numbers (in formatted input)
1859 @cindex converting strings to numbers
1860 @cindex number syntax, parsing
1861 @cindex syntax, for reading numbers
1863 This section describes functions for ``reading'' integer and
1864 floating-point numbers from a string. It may be more convenient in some
1865 cases to use @code{sscanf} or one of the related functions; see
1866 @ref{Formatted Input}. But often you can make a program more robust by
1867 finding the tokens in the string by hand, then converting the numbers
1871 * Parsing of Integers:: Functions for conversion of integer values.
1872 * Parsing of Floats:: Functions for conversion of floating-point
1876 @node Parsing of Integers
1877 @subsection Parsing of Integers
1880 These functions are declared in @file{stdlib.h}.
1884 @deftypefun {long int} strtol (const char *@var{string}, char **@var{tailptr}, int @var{base})
1885 The @code{strtol} (``string-to-long'') function converts the initial
1886 part of @var{string} to a signed integer, which is returned as a value
1887 of type @code{long int}.
1889 This function attempts to decompose @var{string} as follows:
1893 A (possibly empty) sequence of whitespace characters. Which characters
1894 are whitespace is determined by the @code{isspace} function
1895 (@pxref{Classification of Characters}). These are discarded.
1898 An optional plus or minus sign (@samp{+} or @samp{-}).
1901 A nonempty sequence of digits in the radix specified by @var{base}.
1903 If @var{base} is zero, decimal radix is assumed unless the series of
1904 digits begins with @samp{0} (specifying octal radix), or @samp{0x} or
1905 @samp{0X} (specifying hexadecimal radix); in other words, the same
1906 syntax used for integer constants in C.
1908 Otherwise @var{base} must have a value between @code{2} and @code{35}.
1909 If @var{base} is @code{16}, the digits may optionally be preceded by
1910 @samp{0x} or @samp{0X}. If base has no legal value the value returned
1911 is @code{0l} and the global variable @code{errno} is set to @code{EINVAL}.
1914 Any remaining characters in the string. If @var{tailptr} is not a null
1915 pointer, @code{strtol} stores a pointer to this tail in
1916 @code{*@var{tailptr}}.
1919 If the string is empty, contains only whitespace, or does not contain an
1920 initial substring that has the expected syntax for an integer in the
1921 specified @var{base}, no conversion is performed. In this case,
1922 @code{strtol} returns a value of zero and the value stored in
1923 @code{*@var{tailptr}} is the value of @var{string}.
1925 In a locale other than the standard @code{"C"} locale, this function
1926 may recognize additional implementation-dependent syntax.
1928 If the string has valid syntax for an integer but the value is not
1929 representable because of overflow, @code{strtol} returns either
1930 @code{LONG_MAX} or @code{LONG_MIN} (@pxref{Range of Type}), as
1931 appropriate for the sign of the value. It also sets @code{errno}
1932 to @code{ERANGE} to indicate there was overflow.
1934 You should not check for errors by examining the return value of
1935 @code{strtol}, because the string might be a valid representation of
1936 @code{0l}, @code{LONG_MAX}, or @code{LONG_MIN}. Instead, check whether
1937 @var{tailptr} points to what you expect after the number
1938 (e.g. @code{'\0'} if the string should end after the number). You also
1939 need to clear @var{errno} before the call and check it afterward, in
1940 case there was overflow.
1942 There is an example at the end of this section.
1947 @deftypefun {unsigned long int} strtoul (const char *@var{string}, char **@var{tailptr}, int @var{base})
1948 The @code{strtoul} (``string-to-unsigned-long'') function is like
1949 @code{strtol} except it returns an @code{unsigned long int} value. If
1950 the number has a leading @samp{-} sign, the return value is negated.
1951 The syntax is the same as described above for @code{strtol}. The value
1952 returned on overflow is @code{ULONG_MAX} (@pxref{Range of
1955 @code{strtoul} sets @var{errno} to @code{EINVAL} if @var{base} is out of
1956 range, or @code{ERANGE} on overflow.
1961 @deftypefun {long long int} strtoll (const char *@var{string}, char **@var{tailptr}, int @var{base})
1962 The @code{strtoll} function is like @code{strtol} except that it returns
1963 a @code{long long int} value, and accepts numbers with a correspondingly
1966 If the string has valid syntax for an integer but the value is not
1967 representable because of overflow, @code{strtoll} returns either
1968 @code{LONG_LONG_MAX} or @code{LONG_LONG_MIN} (@pxref{Range of Type}), as
1969 appropriate for the sign of the value. It also sets @code{errno} to
1970 @code{ERANGE} to indicate there was overflow.
1972 The @code{strtoll} function was introduced in @w{ISO C 9x}.
1977 @deftypefun {long long int} strtoq (const char *@var{string}, char **@var{tailptr}, int @var{base})
1978 @code{strtoq} (``string-to-quad-word'') is the BSD name for @code{strtoll}.
1983 @deftypefun {unsigned long long int} strtoull (const char *@var{string}, char **@var{tailptr}, int @var{base})
1984 The @code{strtoull} function is like @code{strtoul} except that it
1985 returns an @code{unsigned long long int}. The value returned on overflow
1986 is @code{ULONG_LONG_MAX} (@pxref{Range of Type}).
1988 The @code{strtoull} function was introduced in @w{ISO C 9x}.
1993 @deftypefun {unsigned long long int} strtouq (const char *@var{string}, char **@var{tailptr}, int @var{base})
1994 @code{strtouq} is the BSD name for @code{strtoull}.
1999 @deftypefun {long int} atol (const char *@var{string})
2000 This function is similar to the @code{strtol} function with a @var{base}
2001 argument of @code{10}, except that it need not detect overflow errors.
2002 The @code{atol} function is provided mostly for compatibility with
2003 existing code; using @code{strtol} is more robust.
2008 @deftypefun int atoi (const char *@var{string})
2009 This function is like @code{atol}, except that it returns an @code{int}.
2010 The @code{atoi} function is also considered obsolete; use @code{strtol}
2016 @deftypefun {long long int} atoll (const char *@var{string})
2017 This function is similar to @code{atol}, except it returns a @code{long
2020 The @code{atoll} function was introduced in @w{ISO C 9x}. It too is
2021 obsolete (despite having just been added); use @code{strtoll} instead.
2024 @c !!! please fact check this paragraph -zw
2029 @cindex parsing numbers and locales
2030 @cindex locales, parsing numbers and
2031 Some locales specify a printed syntax for numbers other than the one
2032 that these functions understand. If you need to read numbers formatted
2033 in some other locale, you can use the @code{strtoX_l} functions. Each
2034 of the @code{strtoX} functions has a counterpart with @samp{_l} added to
2035 its name. The @samp{_l} counterparts take an additional argument: a
2036 pointer to an @code{locale_t} structure, which describes how the numbers
2037 to be read are formatted. @xref{Locales}.
2039 @strong{Portability Note:} These functions are all GNU extensions. You
2040 can also use @code{scanf} or its relatives, which have the @samp{'} flag
2041 for parsing numeric input according to the current locale
2042 (@pxref{Numeric Input Conversions}). This feature is standard.
2044 Here is a function which parses a string as a sequence of integers and
2045 returns the sum of them:
2049 sum_ints_from_string (char *string)
2057 /* @r{Skip whitespace by hand, to detect the end.} */
2058 while (isspace (*string)) string++;
2062 /* @r{There is more nonwhitespace,} */
2063 /* @r{so it ought to be another number.} */
2066 next = strtol (string, &tail, 0);
2067 /* @r{Add it in, if not overflow.} */
2069 printf ("Overflow\n");
2072 /* @r{Advance past it.} */
2080 @node Parsing of Floats
2081 @subsection Parsing of Floats
2084 These functions are declared in @file{stdlib.h}.
2088 @deftypefun double strtod (const char *@var{string}, char **@var{tailptr})
2089 The @code{strtod} (``string-to-double'') function converts the initial
2090 part of @var{string} to a floating-point number, which is returned as a
2091 value of type @code{double}.
2093 This function attempts to decompose @var{string} as follows:
2097 A (possibly empty) sequence of whitespace characters. Which characters
2098 are whitespace is determined by the @code{isspace} function
2099 (@pxref{Classification of Characters}). These are discarded.
2102 An optional plus or minus sign (@samp{+} or @samp{-}).
2105 A nonempty sequence of digits optionally containing a decimal-point
2106 character---normally @samp{.}, but it depends on the locale
2107 (@pxref{General Numeric}).
2110 An optional exponent part, consisting of a character @samp{e} or
2111 @samp{E}, an optional sign, and a sequence of digits.
2114 Any remaining characters in the string. If @var{tailptr} is not a null
2115 pointer, a pointer to this tail of the string is stored in
2116 @code{*@var{tailptr}}.
2119 If the string is empty, contains only whitespace, or does not contain an
2120 initial substring that has the expected syntax for a floating-point
2121 number, no conversion is performed. In this case, @code{strtod} returns
2122 a value of zero and the value returned in @code{*@var{tailptr}} is the
2123 value of @var{string}.
2125 In a locale other than the standard @code{"C"} or @code{"POSIX"} locales,
2126 this function may recognize additional locale-dependent syntax.
2128 If the string has valid syntax for a floating-point number but the value
2129 is outside the range of a @code{double}, @code{strtod} will signal
2130 overflow or underflow as described in @ref{Math Error Reporting}.
2132 @code{strtod} recognizes four special input strings. The strings
2133 @code{"inf"} and @code{"infinity"} are converted to @math{@infinity{}},
2134 or to the largest representable value if the floating-point format
2135 doesn't support infinities. You can prepend a @code{"+"} or @code{"-"}
2136 to specify the sign. Case is ignored when scanning these strings.
2138 The strings @code{"nan"} and @code{"nan(@var{chars...})"} are converted
2139 to NaN. Again, case is ignored. If @var{chars...} are provided, they
2140 are used in some unspecified fashion to select a particular
2141 representation of NaN (there can be several).
2143 Since zero is a valid result as well as the value returned on error, you
2144 should check for errors in the same way as for @code{strtol}, by
2145 examining @var{errno} and @var{tailptr}.
2150 @deftypefun float strtof (const char *@var{string}, char **@var{tailptr})
2153 @deftypefunx {long double} strtold (const char *@var{string}, char **@var{tailptr})
2154 These functions are analogous to @code{strtod}, but return @code{float}
2155 and @code{long double} values respectively. They report errors in the
2156 same way as @code{strtod}. @code{strtof} can be substantially faster
2157 than @code{strtod}, but has less precision; conversely, @code{strtold}
2158 can be much slower but has more precision (on systems where @code{long
2159 double} is a separate type).
2161 These functions are GNU extensions.
2166 @deftypefun double atof (const char *@var{string})
2167 This function is similar to the @code{strtod} function, except that it
2168 need not detect overflow and underflow errors. The @code{atof} function
2169 is provided mostly for compatibility with existing code; using
2170 @code{strtod} is more robust.
2173 The GNU C library also provides @samp{_l} versions of thse functions,
2174 which take an additional argument, the locale to use in conversion.
2175 @xref{Parsing of Integers}.
2177 @node System V Number Conversion
2178 @section Old-fashioned System V number-to-string functions
2180 The old @w{System V} C library provided three functions to convert
2181 numbers to strings, with unusual and hard-to-use semantics. The GNU C
2182 library also provides these functions and some natural extensions.
2184 These functions are only available in glibc and on systems descended
2185 from AT&T Unix. Therefore, unless these functions do precisely what you
2186 need, it is better to use @code{sprintf}, which is standard.
2188 All these functions are defined in @file{stdlib.h}.
2191 @comment SVID, Unix98
2192 @deftypefun {char *} ecvt (double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg})
2193 The function @code{ecvt} converts the floating-point number @var{value}
2194 to a string with at most @var{ndigit} decimal digits.
2195 The returned string contains no decimal point or sign. The first
2196 digit of the string is non-zero (unless @var{value} is actually zero)
2197 and the last digit is rounded to nearest. @var{decpt} is set to the
2198 index in the string of the first digit after the decimal point.
2199 @var{neg} is set to a nonzero value if @var{value} is negative, zero
2202 The returned string is statically allocated and overwritten by each call
2205 If @var{value} is zero, it's implementation defined whether @var{decpt} is
2206 @code{0} or @code{1}.
2208 For example: @code{ecvt (12.3, 5, &decpt, &neg)} returns @code{"12300"}
2209 and sets @var{decpt} to @code{2} and @var{neg} to @code{0}.
2213 @comment SVID, Unix98
2214 @deftypefun {char *} fcvt (double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg})
2215 The function @code{fcvt} is like @code{ecvt}, but @var{ndigit} specifies
2216 the number of digits after the decimal point. If @var{ndigit} is less
2217 than zero, @var{value} is rounded to the @math{@var{ndigit}+1}'th place to the
2218 left of the decimal point. For example, if @var{ndigit} is @code{-1},
2219 @var{value} will be rounded to the nearest 10. If @var{ndigit} is
2220 negative and larger than the number of digits to the left of the decimal
2221 point in @var{value}, @var{value} will be rounded to one significant digit.
2223 The returned string is statically allocated and overwritten by each call
2228 @comment SVID, Unix98
2229 @deftypefun {char *} gcvt (double @var{value}, int @var{ndigit}, char *@var{buf})
2230 @code{gcvt} is functionally equivalent to @samp{sprintf(buf, "%*g",
2231 ndigit, value}. It is provided only for compatibility's sake. It
2235 As extensions, the GNU C library provides versions of these three
2236 functions that take @code{long double} arguments.
2240 @deftypefun {char *} qecvt (long double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg})
2241 This function is equivalent to @code{ecvt} except that it
2242 takes a @code{long double} for the first parameter.
2247 @deftypefun {char *} qfcvt (long double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg})
2248 This function is equivalent to @code{fcvt} except that it
2249 takes a @code{long double} for the first parameter.
2254 @deftypefun {char *} qgcvt (long double @var{value}, int @var{ndigit}, char *@var{buf})
2255 This function is equivalent to @code{gcvt} except that it
2256 takes a @code{long double} for the first parameter.
2261 The @code{ecvt} and @code{fcvt} functions, and their @code{long double}
2262 equivalents, all return a string located in a static buffer which is
2263 overwritten by the next call to the function. The GNU C library
2264 provides another set of extended functions which write the converted
2265 string into a user-supplied buffer. These have the conventional
2268 @code{gcvt_r} is not necessary, because @code{gcvt} already uses a
2269 user-supplied buffer.
2273 @deftypefun {char *} ecvt_r (double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
2274 The @code{ecvt_r} function is the same as @code{ecvt}, except
2275 that it places its result into the user-specified buffer pointed to by
2276 @var{buf}, with length @var{len}.
2278 This function is a GNU extension.
2282 @comment SVID, Unix98
2283 @deftypefun {char *} fcvt_r (double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
2284 The @code{fcvt_r} function is the same as @code{fcvt}, except
2285 that it places its result into the user-specified buffer pointed to by
2286 @var{buf}, with length @var{len}.
2288 This function is a GNU extension.
2293 @deftypefun {char *} qecvt_r (long double @var{value}, int @var{ndigit}, int *@var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
2294 The @code{qecvt_r} function is the same as @code{qecvt}, except
2295 that it places its result into the user-specified buffer pointed to by
2296 @var{buf}, with length @var{len}.
2298 This function is a GNU extension.
2303 @deftypefun {char *} qfcvt_r (long double @var{value}, int @var{ndigit}, int @var{decpt}, int *@var{neg}, char *@var{buf}, size_t @var{len})
2304 The @code{qfcvt_r} function is the same as @code{qfcvt}, except
2305 that it places its result into the user-specified buffer pointed to by
2306 @var{buf}, with length @var{len}.
2308 This function is a GNU extension.