C Math atanh() Function

What You’ll Learn

The atanh() function computes the inverse hyperbolic tangent of a number. If you know a value x between −1 and 1 and want the hyperbolic angle t such that tanh(t) = x, atanh(x) gives you that answer.

Definition and Usage

Hyperbolic tangent is defined as tanh(t) = sinh(t) / cosh(t). Its output is always between −1 and 1. The inverse atanh(x) recovers the value t such that tanh(t) = x.

Mathematically, for |x| < 1: atanh(x) = ½ ln((1 + x) / (1 − x)). Because tanh never reaches ±1, atanh cannot accept |x| ≥ 1 as finite input.

📝 Syntax

Standard C declaration:

double atanh(double x);

Related variants

float atanhf(float x);            /* <math.h> */
long double atanhl(long double x); /* <math.h> */

Parameters

• x — hyperbolic tangent value; must satisfy |x| < 1.

Return Value

• Real number t such that tanh(t) = x.
• atanh(0) returns 0.
• If |x| ≥ 1, returns NaN and may set errno to EDOM.

Headers and linking

• #include <math.h>
• Compile: gcc atanh.c -std=c11 -o atanh -lm

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Keep |x| < 1 for valid results.

#include <stdio.h>
#include <math.h>

int main(void) {
    double x = 0.5;
    double result = atanh(x);

    printf("Inverse hyperbolic tangent of %.1f: %.6f\n", x, result);

    return 0;
}

Inverse hyperbolic tangent of 0.5: 0.549306

#include <stdio.h>
#include <math.h>

int main(void) {
    double x = 0.5;

    printf("atanh(0)    = %.6f\n", atanh(0.0));
    printf("atanh(%.1f) = %.6f\n", x, atanh(x));
    printf("atanh(-%.1f) = %.6f\n", x, atanh(-x));

    return 0;
}

atanh(0)    = 0.000000
atanh(0.5) = 0.549306
atanh(-0.5) = -0.549306

#include <math.h>
#include <stdio.h>

int main(void) {
    double bad = 1.0;
    double result = atanh(bad);

    if (isnan(result)) {
        printf("atanh(%.1f) is undefined (NaN)\n", bad);
        printf("Input must satisfy |x| < 1.0\n");
    } else {
        printf("atanh(%.1f) = %.6f\n", bad, result);
    }

    return 0;
}

atanh(1.0) is undefined (NaN)
Input must satisfy |x| < 1.0

#include <stdio.h>
#include <math.h>

int main(void) {
    double values[] = { -0.8, -0.3, 0.0, 0.3, 0.8 };
    size_t i;

    for (i = 0; i < sizeof values / sizeof values[0]; i++) {
        double x = values[i];
        double t = atanh(x);
        double back = tanh(t);

        printf("x=%5.1f  atanh(x)=%.4f  tanh(atanh(x))=%.6f\n",
               x, t, back);
    }

    return 0;
}

x= -0.8  atanh(x)=-1.0986  tanh(atanh(x))=-0.800000
x= -0.3  atanh(x)=-0.3095  tanh(atanh(x))=-0.300000
x=  0.0  atanh(x)=0.0000  tanh(atanh(x))=0.000000
x=  0.3  atanh(x)=0.3095  tanh(atanh(x))=0.300000
x=  0.8  atanh(x)=1.0986  tanh(atanh(x))=0.800000

#include <math.h>
#include <stdio.h>

int main(void) {
    double x = 0.5;
    double lib = atanh(x);
    double manual = 0.5 * log((1.0 + x) / (1.0 - x));

    printf("atanh(%.1f)              = %.6f\n", x, lib);
    printf("0.5*log((1+x)/(1-x))     = %.6f\n", manual);

    return 0;
}

atanh(0.5)              = 0.549306
0.5*log((1+x)/(1-x))     = 0.549306

🚀 Common Use Cases

• Statistics — Fisher’s z-transform uses atanh on correlation coefficients.
• Machine learning — inverse of sigmoid-like functions bounded in (−1, 1).
• Physics — equations involving hyperbolic flow or relativistic velocity composition.
• Scientific computing — solving equations where tanh(t) appears.
• Pair with tanh() — recover a hyperbolic angle from a known tanh value.

📝 Notes

• Domain is the open interval (−1, 1)—endpoints ±1 are invalid.
• Returns the inverse hyperbolic tangent, not the forward tanh value.
• Not the same as atan() (circular inverse tangent).
• Link with -lm when using GCC/Clang on Unix-like systems.
• Use atanhf or atanhl for float or long double data.

⚡ Optimization

atanh() is implemented in the platform math library and is already optimized. Validate input range once before batch processing. For float pipelines, use atanhf() to avoid unnecessary promotion to double.

💡 Best Practices