27. Mathematics <math.h>

The header <math.h> declares two types and many mathematical functions and defines several macros. Most synopses specify a family of functions consisting of a principal function with one or more double parameters, a double return value, or both; and other functions with the same name but with f and l suffixes, which are corresponding functions with float and long double parameters, return values, or both. 1 Integer arithmetic functions and conversion functions are discussed later.

The types

float_t
double_t

are floating types at least as wide as float and double, respectively, and such that double_t is at least as wide as float_t. If FLT_EVAL_METHOD equals 0, float_t and double_t are float and double, respectively; if FLT_EVAL_METHOD equals 1, they are both double; if FLT_EVAL_METHOD equals 2, they are both long double; and for other values of FLT_EVAL_METHOD, they are otherwise implementation-defined. 2

The macro

HUGE_VAL

expands to a positive double constant expression, not necessarily representable as a float. The macros

HUGE_VALF
HUGE_VALL

are respectively float and long double analogs of HUGE_VAL. 3

The macro

INFINITY

expands to a constant expression of type float representing positive or unsigned infinity, if available; else to a positive constant of type float that overflows at translation time. 4

The macro

NAN

is defined if and only if the implementation supports quiet NaNs for the float type. It expands to a constant expression of type float representing a quiet NaN.

The number classification macros

FP_INFINITE
FP_NAN
FP_NORMAL
FP_SUBNORMAL
FP_ZERO

represent the mutually exclusive kinds of floating-point values. They expand to integer constant expressions with distinct values. Additional implementation-defined floatingpoint classifications, with macro definitions beginning with FP_1 and an uppercase letter, may also be specified by the implementation.

The macro

FP_FAST_FMA

is optionally defined. If defined, it indicates that the fma function generally executes about as fast as, or faster than, a multiply and an add of double operands. 5 The macros

FP_FAST_FMAF
FP_FAST_FMAL

are, respectively, float and long double analogs of FP_FAST_FMA. If defined, these macros expand to the integer constant 1.

The macros

FP_ILOGB0
FP_ILOGBNAN

expand to integer constant expressions whose values are returned by ilogb(x) if x is zero or NaN, respectively. The value of FP_ILOGB0 shall be either INT_MIN or -INT_MAX. The value of FP_ILOGBNAN shall be either INT_MAX or INT_MIN.

The macros

MATH_ERRNO
MATH_ERREXCEPT

expand to the integer constants 1 and 2, respectively; the macro

math_errhandling

expands to an expression that has type int and the value MATH_ERRNO, MATH_ERREXCEPT or the bitwise OR of both. The value of math_errhandling is constant for the duration of the program. It is unspecified whether math_errhandling is a macro or an identifier with external linkage. If a macro definition is suppressed or a program defines an identifier with the name math_errhandling, the behavior is undefined. If the expression math_errhandling & MATH_ERREXCEPT can be nonzero, the implementation shall define the macros FE_DIVBYZERO, FE_INVALID and FE_OVERFLOW in <fenv.h>.

1

Particularly on systems with wide expression evaluation, a <math.h> function might pass arguments and return values in wider format than the synopsis prototype indicates.

2

The types float_t and double_t are intended to be the implementation’s most efficient types at least as wide as float and double, respectively. For FLT_EVAL_METHOD equal 0, 1 or 2, the type float_t is the narrowest type used by the implementation to evaluate floating expressions.

3

HUGE_VAL, HUGE_VALF and HUGE_VALL can be positive infinities in an implementation that supports infinities.

4

In this case, using INFINITY will violate the constraint in Constants and thus require a diagnostic.

5

Typically, the FP_FAST_FMA macro is defined if and only if the fma function is implemented directly with a hardware multiply-add instruction. Software implementations are expected to be substantially slower.

27.1. Treatment of error conditions

The behavior of each of the functions in <math.h> is specified for all representable values of its input arguments, except where stated otherwise. Each function shall execute as if it were a single operation without generating any externally visible exceptional conditions.

For all functions, a domain error occurs if an input argument is outside the domain over which the mathematical function is defined. The description of each function lists any required domain errors; an implementation may define additional domain errors, provided that such errors are consistent with the mathematical definition of the function. 6 On a domain error, the function returns an implementation-defined value; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value EDOM; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, the “invalid” floating-point exception is raised.

Similarly, a range error occurs if the mathematical result of the function cannot be represented in an object of the specified type, due to extreme magnitude.

A floating result overflows if the magnitude of the mathematical result is finite but so large that the mathematical result cannot be represented without extraordinary roundoff error in an object of the specified type. If a floating result overflows and default rounding is in effect, or if the mathematical result is an exact infinity (for example log(0.0)), then the function returns the value of the macro HUGE_VAL, HUGE_VALF or HUGE_VALL according to the return type, with the same sign as the correct value of the function; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value ERANGE; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, the “divide-by-zero” floating-point exception is raised if the mathematical result is an exact infinity and the “overflow” floating-point exception is raised otherwise.

The result underflows if the magnitude of the mathematical result is so small that the mathematical result cannot be represented, without extraordinary roundoff error, in an object of the specified type. 7 If the result underflows, the function returns an implementation-defined value whose magnitude is no greater than the smallest normalized positive number in the specified type; if the integer expression math_errhandling & MATH_ERRNO is nonzero, whether errno acquires the value ERANGE is implementation-defined; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, whether the “underflow” floating-point exception is raised is implementation-defined.

6

In an implementation that supports infinities, this allows an infinity as an argument to be a domain error if the mathematical domain of the function does not include the infinity.

7

The term underflow here is intended to encompass both “gradual underflow” as in IEC 60559 and also “flush-to-zero” underflow.

27.2. The FP_CONTRACT pragma

Synopsis

#include <math.h>
#pragma STDC FP_CONTRACT on-off-switch


Description

The FP_CONTRACT pragma can be used to allow (if the state is “on”) or disallow (if the state is “off”) the implementation to contract expressions (Expressions). Each pragma can occur either outside external declarations or preceding all explicit declarations and statements inside a compound statement. When outside external declarations, the pragma takes effect from its occurrence until another FP_CONTRACT pragma is encountered, or until the end of the translation unit. When inside a compound statement, the pragma takes effect from its occurrence until another FP_CONTRACT pragma is encountered (including within a nested compound statement), or until the end of the compound statement; at the end of a compound statement the state for the pragma is restored to its condition just before the compound statement. If this pragma is used in any other context, the behavior is undefined. The default state (“on” or “off”) for the pragma is implementation-defined.

27.3. Classification macros

In the synopses in this subclause, real-floating indicates that the argument shall be an expression of real floating type.

27.3.1. The fpclassify macro

Synopsis

#include <math.h>
int fpclassify(real-floating x);


Description

The fpclassify macro classifies its argument value as NaN, infinite, normal, subnormal, zero, or into another implementation-defined category. First, an argument represented in a format wider than its semantic type is converted to its semantic type. Then classification is based on the type of the argument. 8

Returns

The fpclassify macro returns the value of the number classification macro appropriate to the value of its argument.

EXAMPLE The fpclassify macro might be implemented in terms of ordinary functions as

#define fpclassify(x) \
((sizeof (x) == sizeof (float)) ? _ _fpclassifyf(x) : \
(sizeof (x) == sizeof (double)) ? _ _fpclassifyd(x) : \
_ _fpclassifyl(x))

8

Since an expression can be evaluated with more range and precision than its type has, it is important to know the type that classification is based on. For example, a normal long double value might become subnormal when converted to double, and zero when converted to float.

27.3.2. The isfinite macro

Synopsis

#include <math.h>
int isfinite(real-floating x);


Description

The isfinite macro determines whether its argument has a finite value (zero, subnormal, or normal, and not infinite or NaN). First, an argument represented in a format wider than its semantic type is converted to its semantic type. Then determination is based on the type of the argument.

Returns

The isfinite macro returns a nonzero value if and only if its argument has a finite value.

27.3.3. The isinf macro

Synopsis

#include <math.h>
int isinf(real-floating x);


Description

The isinf macro determines whether its argument value is an infinity (positive or negative). First, an argument represented in a format wider than its semantic type is converted to its semantic type. Then determination is based on the type of the argument.

Returns

The isinf macro returns a nonzero value if and only if its argument has an infinite value.

27.3.4. The isnan macro

Synopsis

#include <math.h>
int isnan(real-floating x);


Description

The isnan macro determines whether its argument value is a NaN. First, an argument represented in a format wider than its semantic type is converted to its semantic type. Then determination is based on the type of the argument. 9

Returns

The isnan macro returns a nonzero value if and only if its argument has a NaN value.

9

For the isnan macro, the type for determination does not matter unless the implementation supports NaNs in the evaluation type but not in the semantic type.

27.3.5. The isnormal macro

Synopsis

#include <math.h>
int isnormal(real-floating x);


Description

The isnormal macro determines whether its argument value is normal (neither zero, subnormal, infinite, nor NaN). First, an argument represented in a format wider than its semantic type is converted to its semantic type. Then determination is based on the type of the argument.

Returns

The isnormal macro returns a nonzero value if and only if its argument has a normal value.

27.3.6. The signbit macro

Synopsis

#include <math.h>
int signbit(real-floating x);


Description

The signbit macro determines whether the sign of its argument value is negative. 10

Returns

The signbit macro returns a nonzero value if and only if the sign of its argument value is negative.

10

The signbit macro reports the sign of all values, including infinities, zeros, and NaNs. If zero is unsigned, it is treated as positive.

27.4. Trigonometric functions

27.4.1. The acos functions

Synopsis

#include <math.h>
double acos(double x);
float acosf(float x);
long double acosl(long double x);


Description

The acos functions compute the principal value of the arc cosine of x. A domain error occurs for arguments not in the interval [-1, +1].

Returns

The acos functions return arccos x in the interval [$$0, \pi$$] radians.

27.4.2. The asin functions

Synopsis

#include <math.h>
double asin(double x);
float asinf(float x);
long double asinl(long double x);


Description

The asin functions compute the principal value of the arc sine of x. A domain error occurs for arguments not in the interval [-1, +1].

Returns

The asin functions return arcsin x in the interval [$$-\pi/2, +\pi/2$$] radians.

27.4.3. The atan functions

Synopsis

#include <math.h>
double atan(double x);
float atanf(float x);
long double atanl(long double x);


Description

The atan functions compute the principal value of the arc tangent of x.

Returns

The atan functions return arctan x in the interval [$$-\pi/2, +\pi/2$$] radians.

27.4.4. The atan2 functions

Synopsis

#include <math.h>
double atan2(double y, double x);
float atan2f(float y, float x);
long double atan2l(long double y, long double x);


Description

The atan2 functions compute the value of the arc tangent of y/x, using the signs of both arguments to determine the quadrant of the return value. A domain error may occur if both arguments are zero.

Returns

The atan2 functions return arctan y/x in the interval [$$-\pi, +\pi$$] radians.

27.4.5. The cos functions

Synopsis

#include <math.h>
double cos(double x);
float cosf(float x);
long double cosl(long double x);


Description

The cos functions compute the cosine of x (measured in radians).

Returns

The cos functions return cos x.

27.4.6. The sin functions

Synopsis

#include <math.h>
double sin(double x);
float sinf(float x);
long double sinl(long double x);


Description

The sin functions compute the sine of x (measured in radians).

Returns

The sin functions return sin x.

27.4.7. The tan functions

Synopsis

#include <math.h>
double tan(double x);
float tanf(float x);
long double tanl(long double x);


Description

The tan functions return the tangent of x (measured in radians).

Returns

The tan functions return tan x.

27.5. Hyperbolic functions

27.5.1. The acosh functions

Synopsis

#include <math.h>
double acosh(double x);
float acoshf(float x);
long double acoshl(long double x);


Description

The acosh functions compute the (nonnegative) arc hyperbolic cosine of x. A domain error occurs for arguments less than 1.

Returns

The acosh functions return arcosh x in the interval [$$0, +\infty$$].

27.5.2. The asinh functions

Synopsis

#include <math.h>
double asinh(double x);
float asinhf(float x);
long double asinhl(long double x);


Description

The asinh functions compute the arc hyperbolic sine of x.

Returns

The asinh functions return arsinh x.

27.5.3. The atanh functions

Synopsis

#include <math.h>
double atanh(double x);
float atanhf(float x);
long double atanhl(long double x);


Description

The atanh functions compute the arc hyperbolic tangent of x. A domain error occurs for arguments not in the interval [-1, +1]. A range error may occur if the argument equals -1 or +1.

Returns

The atanh functions return atanh x.

27.5.4. The cosh functions

Synopsis

#include <math.h>
double cosh(double x);
float coshf(float x);
long double coshl(long double x);


Description

The cosh functions compute the hyperbolic cosine of x. A range error occurs if the magnitude of x is too large.

Returns

The cosh functions return cosh x.

27.5.5. The sinh functions

Synopsis

#include <math.h>
double sinh(double x);
float sinhf(float x);
long double sinhl(long double x);


Description

The sinh functions compute the hyperbolic sine of x. A range error occurs if the magnitude of x is too large.

Returns

The sinh functions return sinh x.

27.5.6. The tanh functions

Synopsis

#include <math.h>
double tanh(double x);
float tanhf(float x);
long double tanhl(long double x);


Description

The tanh functions compute the hyperbolic tangent of x.

Returns

The tanh functions return tanh x.

27.6. Exponential and logarithmic functions

27.6.1. The exp functions

Synopsis

#include <math.h>
double exp(double x);
float expf(float x);
long double expl(long double x);


Description

The exp functions compute the base-e exponential of x. A range error occurs if the magnitude of x is too large.

Returns

The exp functions return $$e^x$$.

27.6.2. The exp2 functions

Synopsis

#include <math.h>
double exp2(double x);
float exp2f(float x);
long double exp2l(long double x);


Description The exp2 functions compute the base-2 exponential of x. A range error occurs if the magnitude of x is too large.

Returns

The exp2 functions return $$2^x$$.

27.6.3. The expm1 functions

Synopsis

#include <math.h>
double expm1(double x);
float expm1f(float x);
long double expm1l(long double x);


Description

The expm1 functions compute the base-e exponential of the argument, minus 1. A range error occurs if x is too large. 11

Returns

The expm1 functions return $$e^x-1$$.

11

For small magnitude x, expm1(x) is expected to be more accurate than exp(x) - 1.

27.6.4. The frexp functions

Synopsis

#include <math.h>
double frexp(double value, int *exp);
float frexpf(float value, int *exp);
long double frexpl(long double value, int *exp);


Description

The frexp functions break a floating-point number into a normalized fraction and an integral power of 2. They store the integer in the int object pointed to by exp.

Returns

If value is not a floating-point number, the results are unspecified. Otherwise, the frexp functions return the value x, such that x has a magnitude in the interval [1/2, 1) or zero, and value equals $$x *2^{*exp}$$. If value is zero, both parts of the result are zero.

27.6.5. The ilogb functions

Synopsis

#include <math.h>
int ilogb(double x);
int ilogbf(float x);
int ilogbl(long double x);


Description

The ilogb functions extract the exponent of x as a signed int value. If x is zero they compute the value FP_ILOGB0; if x is infinite they compute the value INT_MAX; if x is a NaN they compute the value FP_ILOGBNAN; otherwise, they are equivalent to calling the corresponding logb function and casting the returned value to type int. A domain error or range error may occur if x is zero, infinite, or NaN. If the correct value is outside the range of the return type, the numeric result is unspecified.

Returns

The ilogb functions return the exponent of x as a signed int value.

Forward references: the logb functions (The logb functions).

27.6.6. The ldexp functions

Synopsis

#include <math.h>
double ldexp(double x, int exp);
float ldexpf(float x, int exp);
long double ldexpl(long double x, int exp);


Description

The ldexp functions multiply a floating-point number by an integral power of 2. A range error may occur.

Returns

The ldexp functions return $$x * 2^{exp}$$.

27.6.7. The log functions

Synopsis

#include <math.h>
double log(double x);
float logf(float x);
long double logl(long double x);


Description

The log functions compute the base-e (natural) logarithm of x. A domain error occurs if the argument is negative. A range error may occur if the argument is zero.

Returns

The log functions return $$log_e x$$.

27.6.8. The log10 functions

Synopsis

#include <math.h>
double log10(double x);
float log10f(float x);
long double log10l(long double x);


Description

The log10 functions compute the base-10 (common) logarithm of x. A domain error occurs if the argument is negative. A range error may occur if the argument is zero.

Returns

The log10 functions return $$log_{10} x$$.

27.6.9. The log1p functions

Synopsis

#include <math.h>
double log1p(double x);
float log1pf(float x);
long double log1pl(long double x);


Description

The log1p functions compute the base-e (natural) logarithm of 1 plus the argument. 12 A domain error occurs if the argument is less than -1. A range error may occur if the argument equals 11.

Returns

The log1p functions return $$log_e (1 + x)$$.

12

For small magnitude x, log1p(x) is expected to be more accurate than log(1 + x).

27.6.10. The log2 functions

Synopsis

#include <math.h>
double log2(double x);
float log2f(float x);
long double log2l(long double x);


Description

The log2 functions compute the base-2 logarithm of x. A domain error occurs if the argument is less than zero. A range error may occur if the argument is zero.

Returns

The log2 functions return $$log_2 x$$.

27.6.11. The logb functions

Synopsis

#include <math.h>
double logb(double x);
float logbf(float x);
long double logbl(long double x);


Description

The logb functions extract the exponent of x, as a signed integer value in floating-point format. If x is subnormal it is treated as though it were normalized; thus, for positive finite x,

$1~\leq~x~*~FLT\_RADIX^{-logb(x)}~<~FLT\_RADIX$

A domain error or range error may occur if the argument is zero.

Returns

The logb functions return the signed exponent of x.

27.6.12. The modf functions

Synopsis

#include <math.h>
double modf(double value, double *iptr);
float modff(float value, float *iptr);
long double modfl(long double value, long double *iptr);


Description

The modf functions break the argument value into integral and fractional parts, each of which has the same type and sign as the argument. They store the integral part (in floating-point format) in the object pointed to by iptr.

Returns

The modf functions return the signed fractional part of value.

27.6.13. The scalbn and scalbln functions

Synopsis

#include <math.h>
double scalbn(double x, int n);
float scalbnf(float x, int n);
long double scalbnl(long double x, int n);
double scalbln(double x, long int n);
float scalblnf(float x, long int n);
long double scalblnl(long double x, long int n);


Description

The scalbn and scalbln functions compute $$x*FLT\_RADIX^n$$ efficiently, not normally by computing $$FLT\_RADIX^n$$ explicitly. A range error may occur.

Returns

The scalbn and scalbln functions return $$x*FLT\_RADIX^n$$.

27.7. Power and absolute-value functions

27.7.1. The cbrt functions

Synopsis

#include <math.h>
double cbrt(double x);
float cbrtf(float x);
long double cbrtl(long double x);


Description

The cbrt functions compute the real cube root of x.

Returns

The cbrt functions return $$x^{1/3}$$.

27.7.2. The fabs functions

Synopsis

#include <math.h>
double fabs(double x);
float fabsf(float x);
long double fabsl(long double x);


Description

The fabs functions compute the absolute value of a floating-point number x.

Returns

The fabs functions return $$|x|$$.

27.7.3. The hypot functions

Synopsis

#include <math.h>
double hypot(double x, double y);
float hypotf(float x, float y);
long double hypotl(long double x, long double y);


Description

The hypot functions compute the square root of the sum of the squares of x and y, without undue overflow or underflow. A range error may occur.

Returns

The hypot functions return $$\sqrt{x^2+y^2}$$.

27.7.4. The pow functions

Synopsis

#include <math.h>
double pow(double x, double y);
float powf(float x, float y);
long double powl(long double x, long double y);


Description

The pow functions compute x raised to the power y. A domain error occurs if x is finite and negative and y is finite and not an integer value. A range error may occur. A domain error may occur if x is zero and y is zero. A domain error or range error may occur if x is zero and y is less than zero.

Returns

The pow functions return $$x^y$$.

27.7.5. The sqrt functions

Synopsis

#include <math.h>
double sqrt(double x);
float sqrtf(float x);
long double sqrtl(long double x);


Description

The sqrt functions compute the nonnegative square root of x. A domain error occurs if the argument is less than zero.

Returns

The sqrt functions return $$\sqrt{x}$$.

27.8. Error and gamma functions

27.8.1. The erf functions

Synopsis

#include <math.h>
double erf(double x);
float erff(float x);
long double erfl(long double x);


Description

The erf functions compute the error function of x.

Returns

The erf functions return erf $$x = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$$

27.8.2. The erfc functions

Synopsis

#include <math.h>
double erfc(double x);
float erfcf(float x);
long double erfcl(long double x);


Description

The erfc functions compute the complementary error function of x. A range error occurs if x is too large.

Returns

The erfc functions return erfc $$x = 1 - erf x = \frac{2}{\sqrt{\pi}}\int_0^{\infty} e^{-t^2} dt$$

27.8.3. The lgamma functions

Synopsis

#include <math.h>
double lgamma(double x);
float lgammaf(float x);
long double lgammal(long double x);


Description

The lgamma functions compute the natural logarithm of the absolute value of gamma of x. A range error occurs if x is too large. A range error may occur if x is a negative integer or zero.

Returns

The lgamma functions return $$log_e |\Gamma (x)|$$.

27.8.4. The tgamma functions

Synopsis

#include <math.h>
double tgamma(double x);
float tgammaf(float x);
long double tgammal(long double x);


Description

The tgamma functions compute the gamma function of x. A domain error or range error may occur if x is a negative integer or zero. A range error may occur if the magnitude of x is too large or too small.

Returns

The tgamma functions return $$\Gamma (x)$$.

27.9. Nearest integer functions

27.9.1. The ceil functions

Synopsis

#include <math.h>
double ceil(double x);
float ceilf(float x);
long double ceill(long double x);


Description

The ceil functions compute the smallest integer value not less than x.

The ceil functions return $$\lceil x\rceil$$, expressed as a floating-point number.

27.9.2. The floor functions

Synopsis

#include <math.h>
double floor(double x);
float floorf(float x);
long double floorl(long double x);


Description

The floor functions compute the largest integer value not greater than x.

Returns

The floor functions return $$\lfloor x\rfloor$$, expressed as a floating-point number.

27.9.3. The nearbyint functions

Synopsis

#include <math.h>
double nearbyint(double x);
float nearbyintf(float x);
long double nearbyintl(long double x);


Description

The nearbyint functions round their argument to an integer value in floating-point format, using the current rounding direction and without raising the “inexact” floatingpoint exception.

Returns

The nearbyint functions return the rounded integer value.

27.9.4. The rint functions

Synopsis

#include <math.h>
double rint(double x);
float rintf(float x);
long double rintl(long double x);


Description

The rint functions differ from the nearbyint functions (The nearbyint functions) only in that the rint functions may raise the “inexact” floating-point exception if the result differs in value from the argument.

Returns

The rint functions return the rounded integer value.

27.9.5. The lrint and llrint functions

Synopsis

#include <math.h>
long int lrint(double x);
long int lrintf(float x);
long int lrintl(long double x);
long long int llrint(double x);
long long int llrintf(float x);
long long int llrintl(long double x);


Description

The lrint and llrint functions round their argument to the nearest integer value, rounding according to the current rounding direction. If the rounded value is outside the range of the return type, the numeric result is unspecified and a domain error or range error may occur.

Returns

The lrint and llrint functions return the rounded integer value.

27.9.6. The round functions

Synopsis

#include <math.h>
double round(double x);
float roundf(float x);
long double roundl(long double x);


Description

The round functions round their argument to the nearest integer value in floating-point format, rounding halfway cases away from zero, regardless of the current rounding direction.

Returns

The round functions return the rounded integer value.

27.9.7. The lround and llround functions

Synopsis

#include <math.h>
long int lround(double x);
long int lroundf(float x);
long int lroundl(long double x);
long long int llround(double x);
long long int llroundf(float x);
long long int llroundl(long double x);


Description

The lround and llround functions round their argument to the nearest integer value, rounding halfway cases away from zero, regardless of the current rounding direction. If the rounded value is outside the range of the return type, the numeric result is unspecified and a domain error or range error may occur.

Returns

The lround and llround functions return the rounded integer value.

27.9.8. The trunc functions

Synopsis

#include <math.h>
double trunc(double x);
float truncf(float x);
long double truncl(long double x);


Description

The trunc functions round their argument to the integer value, in floating format, nearest to but no larger in magnitude than the argument.

Returns

The trunc functions return the truncated integer value.

27.10. Remainder functions

27.10.1. The fmod functions

Synopsis

#include <math.h>
double fmod(double x, double y);
float fmodf(float x, float y);
long double fmodl(long double x, long double y);


Description

The fmod functions compute the floating-point remainder of $$x/y$$.

Returns

The fmod functions return the value $$x - ny$$, for some integer n such that, if y is nonzero, the result has the same sign as x and magnitude less than the magnitude of y. If y is zero, whether a domain error occurs or the fmod functions return zero is implementationdefined.

27.10.2. The remainder functions

Synopsis

#include <math.h>
double remainder(double x, double y);
float remainderf(float x, float y);
long double remainderl(long double x, long double y);


Description

The remainder functions compute the remainder $$x REM y$$ required by IEC 60559. 13

Returns

The remainder functions return $$x REM y$$. If y is zero, whether a domain error occurs or the functions return zero is implementation defined.

13

“When $$y\neq 0$$, the remainder $$r = x REM y$$ is defined regardless of the rounding mode by the mathematical relation $$r = x - ny$$, where $$n$$ is the integer nearest the exact value of $$x/y$$; whenever $$| n - x/y | = 1/2$$, then $$n$$ is even. Thus, the remainder is always exact. If $$r = 0$$, its sign shall be that of $$x$$.” This definition is applicable for all implementations.

27.10.3. The remquo functions

Synopsis

#include <math.h>
double remquo(double x, double y, int *quo);
float remquof(float x, float y, int *quo);
long double remquol(long double x, long double y, int *quo);


Description

The remquo functions compute the same remainder as the remainder functions. In the object pointed to by quo they store a value whose sign is the sign of x/y and whose magnitude is congruent modulo $$2^n$$ to the magnitude of the integral quotient of x/y, where $$n$$ is an implementation-defined integer greater than or equal to 3.

Returns

The remquo functions return $$x REM y$$. If y is zero, the value stored in the object pointed to by quo is unspecified and whether a domain error occurs or the functions return zero is implementation defined.

27.11. Manipulation functions

27.11.1. The copysign functions

Synopsis

#include <math.h>
double copysign(double x, double y);
float copysignf(float x, float y);
long double copysignl(long double x, long double y);


Description

The copysign functions produce a value with the magnitude of x and the sign of y. They produce a NaN (with the sign of y) if x is a NaN. On implementations that represent a signed zero but do not treat negative zero consistently in arithmetic operations, the copysign functions regard the sign of zero as positive.

Returns

The copysign functions return a value with the magnitude of x and the sign of y.

27.11.2. The nan functions

Synopsis

#include <math.h>
double nan(const char *tagp);
float nanf(const char *tagp);
long double nanl(const char *tagp);


Description

The call nan("*n-char-sequence*") is equivalent to strtod("NAN(*n-char-sequence*)", (char**) NULL); the call nan("") is equivalent to strtod("NAN()", (char**) NULL). If tagp does not point to an n-char sequence or an empty string, the call is equivalent to strtod("NAN", (char**) NULL). Calls to nanf and nanl are equivalent to the corresponding calls to strtof and strtold.

Returns

The nan functions return a quiet NaN, if available, with content indicated through tagp. If the implementation does not support quiet NaNs, the functions return zero.

Forward references: the strtod, strtof and strtold functions (The strtod, strtof and strtold functions).

27.11.3. The nextafter functions

Synopsis

#include <math.h>
double nextafter(double x, double y);
float nextafterf(float x, float y);
long double nextafterl(long double x, long double y);


Description

The nextafter functions determine the next representable value, in the type of the function, after x in the direction of y, where x and y are first converted to the type of the function. 14 The nextafter functions return y if x equals y. A range error may occur if the magnitude of x is the largest finite value representable in the type and the result is infinite or not representable in the type.

Returns

The nextafter functions return the next representable value in the specified format after x in the direction of y.

14

The argument values are converted to the type of the function, even by a macro implementation of the function.

27.11.4. The nexttoward functions

Synopsis

#include <math.h>
double nexttoward(double x, long double y);
float nexttowardf(float x, long double y);
long double nexttowardl(long double x, long double y);


Description

The nexttoward functions are equivalent to the nextafter functions except that the second parameter has type long double and the functions return y converted to the type of the function if x equals y. 15

15

The result of the nexttoward functions is determined in the type of the function, without loss of range or precision in a floating second argument.

27.12. Maximum, minimum and positive difference functions

27.12.1. The fdim functions

Synopsis

#include <math.h>
double fdim(double x, double y);
float fdimf(float x, float y);
long double fdiml(long double x, long double y);


Description

The fdim functions determine the positive difference between their arguments:

$\begin{split}\left\{\begin{array}{ll} x-y & \quad\text{if x\geq y}\\ +0 & \quad\text{if x\leq y} \end{array}\right.\end{split}$

A range error may occur.

Returns

The fdim functions return the positive difference value.

27.12.2. The fmax functions

Synopsis

#include <math.h>
double fmax(double x, double y);
float fmaxf(float x, float y);
long double fmaxl(long double x, long double y);


Description

The fmax functions determine the maximum numeric value of their arguments. 16

Returns

The fmax functions return the maximum numeric value of their arguments.

16

NaN arguments are treated as missing data: if one argument is a NaN and the other numeric, then the fmax functions choose the numeric value. See F.9.9.2.

27.12.3. The fmin functions

Synopsis

#include <math.h>
double fmin(double x, double y);
float fminf(float x, float y);
long double fminl(long double x, long double y);


Description

The fmin functions determine the minimum numeric value of their arguments. 17

Returns

The fmin functions return the minimum numeric value of their arguments.

17

The fmin functions are analogous to the fmax functions in their treatment of NaNs.

27.13.1. The fma functions

Synopsis

#include <math.h>
double fma(double x, double y, double z);
float fmaf(float x, float y, float z);
long double fmal(long double x, long double y, long double z);


Description

The fma functions compute (x * y) + z, rounded as one ternary operation: they compute the value (as if) to infinite precision and round once to the result format, according to the rounding mode characterized by the value of FLT_ROUNDS. A range error may occur.

Returns

The fma functions return (x * y) + z, rounded as one ternary operation.

27.14. Comparison macros

The relational and equality operators support the usual mathematical relationships between numeric values. For any ordered pair of numeric values exactly one of the relationships — less, greater and equal — is true. Relational operators may raise the “invalid” floating-point exception when argument values are NaNs. For a NaN and a numeric value, or for two NaNs, just the unordered relationship is true. 18 The following subclauses provide macros that are quiet (non floating-point exception raising) versions of the relational operators, and other comparison macros that facilitate writing efficient code that accounts for NaNs without suffering the “invalid” floating-point exception. In the synopses in this subclause, real-floating indicates that the argument shall be an expression of real floating type.

18

IEC 60559 requires that the built-in relational operators raise the “invalid” floating-point exception if the operands compare unordered, as an error indicator for programs written without consideration of NaNs; the result in these cases is false.

27.14.1. The isgreater macro

Synopsis

#include <math.h>
int isgreater(real-floating x, real-floating y);


Description

The isgreater macro determines whether its first argument is greater than its second argument. The value of isgreater(x, y) is always equal to (x) > (y); however, unlike (x) > (y), isgreater(x, y) does not raise the “invalid” floating-point exception when x and y are unordered.

Returns

The isgreater macro returns the value of (x) > (y).

27.14.2. The isgreaterequal macro

Synopsis

#include <math.h>
int isgreaterequal(real-floating x, real-floating y);


Description

The isgreaterequal macro determines whether its first argument is greater than or equal to its second argument. The value of isgreaterequal(x, y) is always equal to (x) >= (y); howev er, unlike (x) >= (y), isgreaterequal(x, y) does not raise the “invalid” floating-point exception when x and y are unordered.

Returns

The isgreaterequal macro returns the value of (x) >= (y).

27.14.3. The isless macro

Synopsis

#include <math.h>
int isless(real-floating x, real-floating y);


Description

The isless macro determines whether its first argument is less than its second argument. The value of isless(x, y) is always equal to (x) < (y); however, unlike (x) < (y), isless(x, y) does not raise the “invalid” floating-point exception when x and y are unordered.

Returns

The isless macro returns the value of (x) < (y).

27.14.4. The islessequal macro

Synopsis

#include <math.h>
int islessequal(real-floating x, real-floating y);


Description

The islessequal macro determines whether its first argument is less than or equal to its second argument. The value of islessequal(x, y) is always equal to (x) <= (y); however, unlike (x) <= (y), islessequal(x, y) does not raise the “invalid” floating-point exception when x and y are unordered.

Returns

The islessequal macro returns the value of (x) <= (y).

27.14.5. The islessgreater macro

Synopsis

#include <math.h>
int islessgreater(real-floating x, real-floating y);


Description

The islessgreater macro determines whether its first argument is less than or greater than its second argument. The islessgreater(x, y) macro is similar to (x) < (y) || (x) > (y); however, islessgreater(x, y) does not raise the “invalid” floating-point exception when x and y are unordered (nor does it evaluate x and y twice).

Returns

The islessgreater macro returns the value of (x) < (y) || (x) > (y).

27.14.6. The isunordered macro

Synopsis

#include <math.h>
int isunordered(real-floating x, real-floating y);


Description

The isunordered macro determines whether its arguments are unordered.

Returns

The isunordered` macro returns 1 if its arguments are unordered and 0 otherwise.