C Math tanh() Function

What You’ll Learn

The tanh() function computes the hyperbolic tangent of a real number. Built from exponentials—(ex − e−x) / (ex + e−x)—it passes through zero at x = 0 and smoothly saturates toward ±1 as |x| grows.

Definition and Usage

Hyperbolic tangent is defined as tanh(x) = (ex − e−x) / (ex + e−x), equivalently sinh(x) / cosh(x). Unlike circular tan(), which is unbounded and takes radians, tanh(x) accepts any real input and always returns a value strictly between −1 and 1.

The input x is any real number—not a degree or radian angle in the circular sense. Think of it as a hyperbolic parameter. At x = 0, numerator and denominator cancel to give tanh(0) = 0.

📝 Syntax

Standard C declaration:

double tanh(double x);

Related variants

float tanhf(float x);             /* <math.h> */
long double tanhl(long double x); /* <math.h> */

Parameters

• x — any real number (hyperbolic argument).

Return Value

• Hyperbolic tangent of x as double.
• tanh(0) returns 0.0.
• tanh(0.5) returns about 0.462.
• Result is always in the open interval (−1, 1).

Headers and linking

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

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Hyperbolic tangent pairs naturally with sinh(), cosh(), and atanh().

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

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

    printf("The hyperbolic tangent of %.2f is %.4f\n", x, result);

    return 0;
}

The hyperbolic tangent of 0.50 is 0.4621

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

int main(void) {
    printf("tanh(0) = %.1f\n", tanh(0.0));

    return 0;
}

tanh(0) = 0.0

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

int main(void) {
    double x = 0.5;

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

    return 0;
}

tanh( 0.5) = 0.462117
tanh(-0.5) = -0.462117

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

int main(void) {
    double x = 1.0;
    double ratio = sinh(x) / cosh(x);
    double library = tanh(x);

    printf("x = %.1f\n", x);
    printf("sinh/cosh: %.6f\n", ratio);
    printf("tanh():    %.6f\n", library);

    return 0;
}

x = 1.0
sinh/cosh: 0.761594
tanh():    0.761594

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

int main(void) {
    double vals[] = { 0.0, 0.5, 1.0, 1.5 };
    size_t i;

    printf("   x     tanh(x)   tan(x)\n");
    for (i = 0; i < 4; i++) {
        double x = vals[i];
        printf("%4.1f  %8.4f  %8.4f\n", x, tanh(x), tan(x));
    }

    return 0;
}

   x     tanh(x)   tan(x)
 0.0    0.0000    0.0000
 0.5    0.4621    0.5463
 1.0    0.7616    1.5574
 1.5    0.9051   14.1014

🚀 Common Use Cases

• Machine learning — activation functions and bounded outputs in neural networks.
• Statistics — Fisher’s z-transformation and logistic-related formulas.
• Signal processing — soft limiting and smooth saturation curves.
• Physics — rapidity in special relativity (hyperbolic angle).
• Pair with atanh() — invert tanh for values strictly between −1 and 1.

📝 Notes

• Defined as (ex − e−x) / (ex + e−x) or sinh(x) / cosh(x).
• Domain: all real numbers.
• Range: (−1, 1)—never exactly ±1 for finite x.
• tanh(0) = 0; odd function: tanh(-x) == -tanh(x).
• Not the same as tan()—no degree/radian conversion needed.
• As |x| grows, result approaches ±1 asymptotically.

⚡ Optimization

tanh() is implemented in the platform math library with careful handling of large |x| (where the result is already near ±1). Avoid rewriting it as sinh(x) / cosh(x) in hot loops—direct tanh is faster and more numerically stable. Use tanhf when float precision is sufficient to save memory bandwidth.

💡 Best Practices