atanh, atanhf, atanhl

Defined in header <math.h>
float       atanhf( float arg );
(1) (since C99)
double      atanh( double arg );
(2) (since C99)
long double atanhl( long double arg );
(3) (since C99)
Defined in header <tgmath.h>
#define atanh( arg )
(4) (since C99)
1-3) Computes the inverse hyperbolic tangent of arg.
4) Type-generic macro: If the argument has type long double, atanhl is called. Otherwise, if the argument has integer type or the type double, atanh is called. Otherwise, atanhf is called. If the argument is complex, then the macro invokes the corresponding complex function (catanhf, catanh, catanhl).


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

Return value

If no errors occur, the inverse hyperbolic tangent of arg (tanh-1
(arg), or artanh(arg)), is returned.

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

If a pole error occurs, ±HUGE_VAL, ±HUGE_VALF, or ±HUGE_VALL is returned (with the correct sign).

If a range error occurs due to underflow, the correct result (after rounding) is returned.

Error handling

Errors are reported as specified in math_errhandling.

If the argument is not on the interval [-1, +1], a range error occurs.

If the argument is ±1, a pole error occurs.

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

  • if the argument is ±0, it is returned unmodified
  • if the argument is ±1, ±∞ is returned and FE_DIVBYZERO is raised.
  • if |arg|>1, NaN is returned and FE_INVALID is raised.
  • if the argument is NaN, NaN is returned


Although the C standard names this function "arc hyperbolic tangent", 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 tangent" (used by POSIX) or "area hyperbolic tangent".

POSIX specifies that in case of underflow, arg is returned unmodified, and if that is not supported, an implementation-defined value no greater than DBL_MIN, FLT_MIN, and LDBL_MIN is returned.


#include <stdio.h>
#include <math.h>
#include <float.h>
#include <errno.h>
#include <fenv.h>
int main(void)
    printf("atanh(0) = %f\natanh(-0) = %f\n", atanh(0), atanh(-0.0));
    printf("atanh(0.9) = %f\n", atanh(0.9));
    //error handling
    errno = 0; feclearexcept(FE_ALL_EXCEPT);
    printf("atanh(-1) = %f\n", atanh(-1));
    if(errno == ERANGE) perror("    errno == ERANGE");
    if(fetestexcept(FE_DIVBYZERO)) puts("    FE_DIVBYZERO raised");

Possible output:

atanh(0) = 0.000000
atanh(-0) = -0.000000
atanh(0.9) = 1.472219
atanh(-1) = -inf
    errno == ERANGE: Numerical result out of range
    FE_DIVBYZERO raised


  • C11 standard (ISO/IEC 9899:2011):
    • The atanh functions (p: 241)
    • 7.25 Type-generic math <tgmath.h> (p: 373-375)
    • F.10.2.3 The atanh functions (p: 520)
  • C99 standard (ISO/IEC 9899:1999):
    • The atanh functions (p: 221-222)
    • 7.22 Type-generic math <tgmath.h> (p: 335-337)
    • F.9.2.3 The atanh functions (p: 457)

See also

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

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