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 NaN (not-a-number)
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^[4]
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 absolute value 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 argument has finite value
	`isinf`	checks if the argument is infinite
	`isnan`	checks if the argument is NaN
	`isnormal`	checks if the argument is normal
	`signbit`	checks if the sign of the argument 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; only available if the <complex.h> header is included) 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 the <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".^[5] 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)
Error functions	`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.^[6]

Function	Description
`rand`	generates a pseudo-random number between 0 and `RAND_MAX`, inclusive.
`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.

Implementations

Under POSIX systems like Linux and BSD, 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, github, used almost as is by Windows
Intel C++ Compiler libm
Red Hat's libm (Newlib)
Sun's FDLIBM, which was used as the basis for FreeBSD's msun and OpenBSD's libm, both of which in turn were the basis of Julia's OpenLibm
musl's libm, based on the BSD libms and other projects like Arm
LLVM's libm, which is correctly rounded (i.e. errors from the mathematically correct result are lower than 0.5 unit in the last place)^[7]
Arénaire project's CRlibm (correctly rounded libm), and its successor MetaLibm, which uses Remez algorithm to automatically generate approximations that are formally proven.
Rutger's RLIBM, which provides correctly rounded functions in single precision.^[8]

Implementations not necessarily under a name of libm include:

Arm's optimized math routines
GCE-Math is a version of C/C++ math functions written for C++ constexpr (compile-time calculation)
CORE-MATH, correctly rounded for single and double precision.
SIMD (vectorized) math libraries include SLEEF, Yeppp!, and Agner Fog's VCL, plus a few closed-source ones like SVML and DirectXMath.^[9]

References

^ ^a ^b ISO/IEC 9899:1999 specification (PDF). p. 212, § 7.12.
^ 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.
^ ^a ^b Prata, Stephen (2004). C primer plus. Sams Publishing. Appendix B, Section VIII: C99 Numeric Computational Enhancements. ISBN 0-672-32696-5.
^ Notationally, it may seem convenient to use pow(x,2) or pow(x,3) to compute squares or cubes. However, this is not advisable in time-critical code. Unless an implementation takes special care of these cases at compile time, x*x or x*x*x will execute much faster. Also, sqrt(x) and cbrt(x) should be preferred over pow(x,.5) or pow(x,1./3).
^ man cerf(3), man cerfc(3), see e.g. https://linux.die.net/man/3/cerf.
^ "The GNU C Library – ISO Random". Retrieved 18 July 2018.
^ "Math Functions — The LLVM C Library". libc.llvm.org.
^ "RLibm: Rutgers Architecture and Programming Languages Lab's Correctly Rounded Libm". people.cs.rutgers.edu.
^ Cordes, Peter. "intel - Where is Clang's '_mm256_pow_ps' intrinsic?". Stack Overflow.

External links

math.h: mathematical declarations – Base Definitions Reference, The Single UNIX Specification, Version 4 from The Open Group
C reference for math functions

[c99-1] ISO/IEC 9899:1999 specification (PDF). p. 212, § 7.12.

[c_primer-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.

[c_primer_c99-3] Prata, Stephen (2004). C primer plus. Sams Publishing. Appendix B, Section VIII: C99 Numeric Computational Enhancements. ISBN 0-672-32696-5.

[4] Notationally, it may seem convenient to use pow(x,2) or pow(x,3) to compute squares or cubes. However, this is not advisable in time-critical code. Unless an implementation takes special care of these cases at compile time, x*x or x*x*x will execute much faster. Also, sqrt(x) and cbrt(x) should be preferred over pow(x,.5) or pow(x,1./3).

[5] rf(3), man cerfc(3), see e.g. https://linux.die.net/man/3/cerf.

[6] "The GNU C Library – ISO Random". Retrieved 18 July 2018.

[7] "Math Functions — The LLVM C Library". libc.llvm.org.

[8] "RLibm: Rutgers Architecture and Programming Languages Lab's Correctly Rounded Libm". people.cs.rutgers.edu.

[9] Cordes, Peter. "intel - Where is Clang's '_mm256_pow_ps' intrinsic?". Stack Overflow.

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