# atanh, atanhf, atanhl

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< c‎ | numeric‎ | math

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).

### Parameters

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

### Notes

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.

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <float.h>
    #include <errno.h>
    #include <fenv.h>
    #pragma STDC FENV_ACCESS ON
    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

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.5.3 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):

  - 7.12.5.3 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

asinh asinhf asinhl (C99)(C99)(C99) | computes inverse hyperbolic sine (arsinh(x)) (function)
acosh acoshf acoshl (C99)(C99)(C99) | computes inverse hyperbolic cosine (arcosh(x)) (function)
tanh tanhf tanhl (C99)(C99) | computes hyperbolic tangent (function)
catanh catanhf catanhl (C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function)
C++ documentation for atanh

### External links

Weisstein, Eric W. "Inverse Hyperbolic Tangent." From MathWorld--A Wolfram Web Resource.
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