C complex math support

The Microsoft C Runtime library (CRT) provides complex math library functions, including all of the ones required by ISO C99. The compiler doesn't directly support a complex or _Complex keyword, therefore the Microsoft implementation uses structure types to represent complex numbers.

These functions are implemented to balance performance with correctness. Because producing the correctly rounded result may be prohibitively expensive, these functions are designed to efficiently produce a close approximation to the correctly rounded result. In most cases, the result produced is within +/-1 unit of least precision (ULP) of the correctly rounded result, though there may be cases where there's greater inaccuracy.

The complex math routines rely on the floating point math library functions for their implementation. These functions have different implementations for different CPU architectures. For example, the 32-bit x86 CRT may have a different implementation than the 64-bit x64 CRT. In addition, some of the functions may have multiple implementations for a given CPU architecture. The most efficient implementation is selected dynamically at run-time depending on the instruction sets supported by the CPU. For example, in the 32-bit x86 CRT, some functions have both an x87 implementation and an SSE2 implementation. When running on a CPU that supports SSE2, the faster SSE2 implementation is used. When running on a CPU that doesn't support SSE2, the slower x87 implementation is used. Because different implementations of the math library functions may use different CPU instructions and different algorithms to produce their results, the functions may produce different results across CPUs. In most cases, the results are within +/-1 ULP of the correctly rounded result, but the actual results may vary across CPUs.

Types used in complex math

The Microsoft implementation of the complex.h header defines these types as equivalents for the C99 standard native complex types:

Standard type Microsoft type
float complex or float _Complex _Fcomplex
double complex or double _Complex _Dcomplex
long double complex or long double _Complex _Lcomplex

The math.h header defines a separate type, struct _complex, used for the _cabs function. The struct _complex type isn't used by the equivalent complex math functions cabs, cabsf, cabsl.

Complex constants and macros

I is defined as the complex type _Fcomplex initialized by { 0.0f, 1.0f }.

Trigonometric functions

Function Description
cacos, cacosf, cacosl Compute the complex arc cosine of a complex number
casin, casinf, casinl Compute the complex arc sine of a complex number
catan, catanf, catanl Compute the complex arc tangent of a complex number
ccos, ccosf, ccosl Compute the complex cosine of a complex number
csin, csinf, csinl Compute the complex sine of a complex number
ctan, ctanf, ctanl Compute the complex tangent of a complex number

Hyperbolic functions

Function Description
cacosh, cacoshf, cacoshl Compute the complex arc hyperbolic cosine of a complex number
casinh, casinhf, casinhl Compute the complex arc hyperbolic sine of a complex number
catanh, catanhf, catanhl Compute the complex arc hyperbolic tangent of a complex number
ccosh, ccoshf, ccoshl Compute the complex hyperbolic cosine of a complex number
csinh, csinhf, csinhl Compute the complex hyperbolic sine of a complex number
ctanh, ctanhf, ctanhl Compute the complex hyperbolic tangent of a complex number

Exponential and logarithmic functions

Function Description
cexp, cexpf, cexpl Compute the complex base-e exponential of a complex number
clog, clogf, clogl Compute the complex natural (base-e) logarithm of a complex number
clog10, clog10f, clog10l Compute the complex base-10 logarithm of a complex number

Power and absolute-value functions

Function Description
cabs, cabsf, cabsl Compute the complex absolute value (also called the norm, modulus, or magnitude) of a complex number
cpow, cpowf, cpowl Compute the complex power function
csqrt, csqrtf, csqrtl Compute the complex square root of a complex number

Manipulation functions

Function Description
_Cbuild, _FCbuild, _LCbuild Construct a complex number from real and imaginary parts
carg, cargf, cargl Compute the argument (also called the phase angle) of a complex number
cimag, cimagf, cimagl Compute the imaginary part of a complex number
conj, conjf, conjl Compute the complex conjugate of a complex number
cproj, cprojf, cprojl Compute a projection of a complex number onto the Riemann sphere
creal, crealf, creall Compute the real part of a complex number
norm, normf, norml Compute the squared magnitude of a complex number

Operation functions

Because complex numbers aren't a native type in the Microsoft compiler, the standard arithmetic operators aren't defined on complex types. For convenience, these complex math library functions are provided to enable limited manipulation of complex numbers in user code:

Function Description
_Cmulcc, _FCmulcc, _LCmulcc Multiply two complex numbers
_Cmulcr, _FCmulcr, _LCmulcr Multiply a complex and a floating-point number

See also

Type-generic math
Universal C runtime routines by category