C mathematical functions

C mathematical operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions.^[1][2] All functions use floating point numbers in one manner or another. Different C standards provide different, albeit backwards-compatible, sets of functions. Most of these functions are also available in the C++ standard library, though in different headers (the C headers are included as well, but only as a deprecated compatibility feature).

Overview of functions

Most of the mathematical functions are defined in math.h (cmath header in C++). The functions that operate on integers, such as abs, labs, div, and ldiv, are instead defined in the stdlib.h header (cstdlib header in C++).

Any functions that operate on angles use radians as the unit of angle.^[1]

Not all of these functions are available in the C89 version of the standard. For those that are, the functions accept only type double for the floating-point arguments, leading to expensive type conversions in code that otherwise used single-precision float values. In C99, this shortcoming was fixed by introducing new sets of functions that work on float and long double arguments. Those functions are identified by f and l suffixes respectively.^[3]

	Function	Description
	abs labs llabs	computes absolute value of an integer value
	fabs	computes absolute value of a floating point value
	div ldiv lldiv	computes the quotient and remainder of integer division
	fmod	remainder of the floating point division operation
	remainder	signed remainder of the division operation
	remquo	signed remainder as well as the three last bits of the division operation
	fma	fused multiply-add operation
	fmax	larger of two floating point values
	fmin	smaller of two floating point values
	fdim	positive difference of two floating point values
	nan nanf nanl	returns a not-a-number (NaN)
Exponential functions	exp	returns e raised to the given power
	exp2	returns 2 raised to the given power
	expm1	returns e raised to the given power, minus one
	log	computes natural logarithm (to base e)
	log2	computes binary logarithm (to base 2)
	log10	computes common logarithm (to base 10)
	log1p	computes natural logarithm (to base e) of 1 plus the given number
	ilogb	extracts exponent of the number
	logb	extracts exponent of the number
Power functions	sqrt	computes square root
	cbrt	computes cubic root
	hypot	computes square root of the sum of the squares of two given numbers
	pow	raises a number to the given power
Trigonometric functions	sin	computes sine
	cos	computes cosine
	tan	computes tangent
	asin	computes arc sine
	acos	computes arc cosine
	atan	computes arc tangent
	atan2	computes arc tangent, using signs to determine quadrants
Hyperbolic functions	sinh	computes hyperbolic sine
	cosh	computes hyperbolic cosine
	tanh	computes hyperbolic tangent
	asinh	computes hyperbolic arc sine
	acosh	computes hyperbolic arc cosine
	atanh	computes hyperbolic arc tangent
Error and gamma functions	erf	computes error function
	erfc	computes complementary error function
	lgamma	computes natural logarithm of the gamma function
	tgamma	computes gamma function
Nearest integer floating point operations	ceil	returns the nearest integer not less than the given value
	floor	returns the nearest integer not greater than the given value
	trunc	returns the nearest integer not greater in magnitude than the given value
	round lround llround	returns the nearest integer, rounding away from zero in halfway cases
	nearbyint	returns the nearest integer using current rounding mode
	rint lrint llrint	returns the nearest integer using current rounding mode with exception if the result differs
Floating point manipulation functions	frexp	decomposes a number into significand and a power of 2
	ldexp	multiplies a number by 2 raised to a power
	modf	decomposes a number into integer and fractional parts
	scalbn scalbln	multiplies a number by FLT_RADIX raised to a power
	nextafter nexttoward	returns next representable floating point value towards the given value
	copysign	copies the sign of a floating point value
Classification	fpclassify	categorizes the given floating point value
	isfinite	checks if the given number has finite value
	isinf	checks if the given number is infinite
	isnan	checks if the given number is NaN
	isnormal	checks if the given number is normal
	signbit	checks if the given number is negative

Floating point environment

C99 adds several functions and types for fine-grained control of floating point environment.^[3] These functions can be used to control a variety of settings that affect floating-point computations, for example, the rounding mode, on what conditions exceptions occur, when numbers are flushed to zero, etc. The floating point environment functions and types are defined in fenv.h header (cfenv in C++).

Function	Description
feclearexcept	clears exceptions (C99)
fegetenv	stores current floating-point environment (C99)
fegetexceptflag	stores current status flags (C99)
fegetround	retrieves current rounding direction (C99)
feholdexcept	saves current floating-point environment and clears all exceptions (C99)
feraiseexcept	raises a floating-point exception (C99)
fesetenv	sets current floating-point environment (C99)
fesetexceptflag	sets current status flags (C99)
fesetround	sets current rounding direction (C99)
fetestexcept	tests whether certain exceptions have been raised (C99)
feupdateenv	restores floating-point environment, but keeps current exceptions (C99)

Complex numbers

C99 adds a new _Complex keyword (and complex convenience macro) that provides support for complex numbers. Any floating point type can be modified with complex, and is then defined as a pair of floating point numbers. Note that C99 and C++ do not implement complex numbers in a code-compatible way - the latter instead provides the class std::complex.

All operations on complex numbers are defined in complex.h header. As with the real-valued functions, an f or l suffix denotes the float complex or long double complex variant of the function.

	Function	Description
Basic operations	cabs	computes absolute value (C99)
	carg	computes argument of a complex number (C99)
	cimag	computes imaginary part of a complex number (C99)
	creal	computes real part of a complex number (C99)
	conj	computes complex conjugate (C99)
	cproj	computes complex projection into the Riemann sphere (C99)
Exponentiation operations	cexp	computes complex exponential (C99)
	clog	computes complex logarithm (C99)
	csqrt	computes complex square root (C99)
	cpow	computes complex power (C99)
Trigonometric operations	csin	computes complex sine (C99)
	ccos	computes complex cosine (C99)
	ctan	computes complex tangent (C99)
	casin	computes complex arc sine (C99)
	cacos	computes complex arc cosine (C99)
	catan	computes complex arc tangent (C99)
Hyperbolic operations	csinh	computes complex hyperbolic sine (C99)
	ccosh	computes complex hyperbolic cosine (C99)
	ctanh	computes complex hyperbolic tangent (C99)
	casinh	computes complex hyperbolic arc sine (C99)
	cacosh	computes complex hyperbolic arc cosine (C99)
	catanh	computes complex hyperbolic arc tangent (C99)

A few more complex functions are "reserved for future use in C99".^[4] Implementations are provided by open-source projects that are not part of the standard library.

	Function	Description
Error functions	cerf	computes the complex error function (C99)
	cerfc	computes the complex complementary error function (C99)

Type-generic functions

The header tgmath.h defines a type-generic macro for each mathematical function defined in math.h and complex.h. This adds a limited support for function overloading of the mathematical functions: the same function name can be used with different types of parameters; the actual function will be selected at compile-time according to the types of the parameters.

Each type-generic macro that corresponds to a function that is defined for both real and complex numbers encapsulates a total of 6 different functions: float, double and long double, and their complex variants. The type-generic macros that correspond to a function that is defined for only real numbers encapsulates a total of 3 different functions: float, double and long double variants of the function.

The C++ language includes native support for function overloading and thus does not provide the tgmath.h header even as a compatibility feature.

Random number generation

The header stdlib.h (cstdlib in C++) defines several functions that can be used for statistically random number generation.^[5]

Function	Description
rand	generates a pseudo-random number between 0 and RAND_MAX.
srand	initializes a pseudo-random number generator
arc4random	generates a pseudo-random number between 0 and UINT32_MAX, usually using a better algorithm than rand
arc4random_uniform	generates a pseudo-random number between 0 and a maximum value.
arc4random_buf	fill a buffer with a pseudo-random bitstream.
arc4random_stir	initializes a pseudo-random number generator.

The arc4random family of random number functions are not defined in POSIX standard, but is found in some common libc implementations. It used to refer to the keystream generator of a leaked version of RC4 cipher (hence "alleged RC4"), but different algorithms, usually from other ciphers like ChaCha20, have been implemented since using the same name.

The quality of randomness from rand are usually too weak to be even considered statistically random, and it requires explicit seeding. It is usually advised to use arc4random instead of rand when possible. Some C libraries implement rand using arc4random_uniform internally.

libm

Under Linux and FreeBSD, the mathematical functions (as declared in math.h) are bundled separately in the mathematical library libm. Therefore, if any of those functions are used, the linker must be given the directive -lm.

There are various libm implementations, including:

• GNU libc's libm
• AMD's libm
• Red Hat's libm
• Sun's FDLIBM, which was used as the basis for FreeBSD's msun, which in turn is the basis of Julia's OpenLibm
• Arénaire project's CRlibm (Correctly Rounded libm)

See also

• C99 floating point support

References

1. ISO/IEC 9899:1999 specification (PDF). p. 212, § 7.12.
2. Prata, Stephen (2004). C primer plus. Sams Publishing. Appendix B, Section V: The Standard ANSI C Library with C99 Additions. ISBN 0-672-32696-5.
3. Prata, Stephen (2004). C primer plus. Sams Publishing. Appendix B, Section VIII: C99 Numeric Computational Enhancements. ISBN 0-672-32696-5.
4. man cerf(3), man cerfc(3), see e.g. http://linux.die.net/man/3/cerf.
5. "GNU C Library -- Mathematics". Retrieved 2 November 2011.

External links

• math.h: mathematical declarations – Base Definitions Reference, The Single UNIX® Specification, Issue 7 from The Open Group
• C reference for math functions