acosh, acoshf, acoshl

Defined in header <math.h>
float       acoshf( float arg );
(1) (since C99)
double      acosh( double arg );
(2) (since C99)
long double acoshl( long double arg );
(3) (since C99)
Defined in header <tgmath.h>
#define acosh( arg )
(4) (since C99)
1-3) Computes the inverse hyperbolic cosine of arg.
4) Type-generic macro: If the argument has type long double, acoshl is called. Otherwise, if the argument has integer type or the type double, acosh is called. Otherwise, acoshf is called. If the argument is complex, then the macro invokes the corresponding complex function (cacoshf, cacosh, cacoshl).


arg - floating point value representing the area of a hyperbolic sector

Return value

If no errors occur, the inverse hyperbolic cosine of arg (cosh-1
(arg), or arcosh(arg)) on the interval [0, +∞], is returned.

If a domain error occurs, an implementation-defined value is returned (NaN where supported).

Error handling

Errors are reported as specified in math_errhandling.

If the argument is less than 1, a domain error occurs.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

  • if the argument is less than 1, FE_INVALID is raised an NaN is returned
  • if the argument is 1, +0 is returned
  • if the argument is +∞, +∞ is returned
  • if the argument is NaN, NaN is returned


Although the C standard names this function "arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "inverse hyperbolic cosine" (used by POSIX) or "area hyperbolic cosine".


#include <stdio.h>
#include <math.h>
#include <float.h>
#include <errno.h>
#include <fenv.h>
int main(void)
    printf("acosh(1) = %f\nacosh(10) = %f\n", acosh(1), acosh(10));
    printf("acosh(DBL_MAX) = %f\nacosh(Inf) = %f\n", acosh(DBL_MAX), acosh(INFINITY));
    //error handling
    errno = 0; feclearexcept(FE_ALL_EXCEPT);
    printf("acosh(0.5) = %f\n", acosh(0.5));
    if(errno == EDOM)         perror("    errno == EDOM");
    if(fetestexcept(FE_INVALID)) puts("    FE_INVALID raised");

Possible output:

acosh(1) = 0.000000
acosh(10) = 2.993223
acosh(DBL_MAX) = 710.475860
acosh(Inf) = inf
acosh(0.5) = -nan
    errno == EDOM: Numerical argument out of domain
    FE_INVALID raised


  • C11 standard (ISO/IEC 9899:2011):
    • The acosh functions (p: 240)
    • 7.25 Type-generic math <tgmath.h> (p: 373-375)
    • F.10.2.1 The acosh functions (p: 520)
  • C99 standard (ISO/IEC 9899:1999):
    • The acosh functions (p: 221)
    • 7.22 Type-generic math <tgmath.h> (p: 335-337)
    • F.9.2.1 The acosh functions (p: 457)

See also

computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x))
computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x))
computes hyperbolic cosine (\({\small\cosh{x} }\)cosh(x))
computes the complex arc hyperbolic cosine
C++ documentation for acosh
Weisstein, Eric W. "Inverse Hyperbolic Cosine." From MathWorld--A Wolfram Web Resource.

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