# 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 <errno.h>
    #include <fenv.h>
    #include <float.h>
    #include <math.h>
    #include <stdio.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

- C23 standard (ISO/IEC 9899:2024):

  - 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)

- C17 standard (ISO/IEC 9899:2018):

  - 7.12.5.3 The atanh functions (p: TBD)

  - 7.25 Type-generic math <tgmath.h> (p: TBD)

  - F.10.2.3 The atanh functions (p: TBD)

- 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

asinhasinhfasinhl (C99)(C99)(C99) | computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) (function)
acoshacoshfacoshl (C99)(C99)(C99) | computes inverse hyperbolic cosine (\({\small\operatorname{arcosh}{x} }\)arcosh(x)) (function)
tanhtanhftanhl (C99)(C99) | computes hyperbolic tangent (\({\small\tanh{x} }\)tanh(x)) (function)
catanhcatanhfcatanhl (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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