C Math asinh() Function

What You’ll Learn

The asinh() function computes the inverse hyperbolic sine of a number. If you know a value x and want the hyperbolic angle t such that sinh(t) = x, asinh(x) gives you that answer—and unlike acosh(), it accepts any real number.

Definition and Usage

Hyperbolic sine is defined as sinh(t) = (et − e−t) / 2. The inverse asinh(x) answers: “What value of t produces this sinh output?”

Mathematically: asinh(x) = ln(x + √(x² + 1)). Because x² + 1 is always positive, asinh is defined for every real x—negative, zero, or positive.

📝 Syntax

Standard C declaration:

double asinh(double x);

Related variants

float asinhf(float x);            /* <math.h> */
long double asinhl(long double x); /* <math.h> */

Parameters

• x — any real number (no domain restriction).

Return Value

• Real number t such that sinh(t) = x.
• asinh(0) returns 0.
• Result grows slowly for large |x| (approximately ln(2|x|) when |x| is large).

Headers and linking

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

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Hyperbolic functions accept the full range of real inputs.

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

int main(void) {
    double x = 2.0;
    double result = asinh(x);

    printf("The inverse hyperbolic sine of %.2f is %.6f\n", x, result);

    return 0;
}

The inverse hyperbolic sine of 2.00 is 1.443635

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

int main(void) {
    double x = 1.5;

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

    return 0;
}

asinh(0)   = 0.000000
asinh(1.5) = 1.194763
asinh(-1.5) = -1.194763

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

int main(void) {
    double values[] = { -3.0, -1.0, 0.0, 1.0, 3.0 };
    size_t i;

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

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

    return 0;
}

x= -3.0  asinh(x)=-1.8184  sinh(asinh(x))=-3.000000
x= -1.0  asinh(x)=-0.8814  sinh(asinh(x))=-1.000000
x=  0.0  asinh(x)=0.0000  sinh(asinh(x))=0.000000
x=  1.0  asinh(x)=0.8814  sinh(asinh(x))=1.000000
x=  3.0  asinh(x)=1.8184  sinh(asinh(x))=3.000000

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

int main(void) {
    double x = 2.0;
    double lib = asinh(x);
    double manual = log(x + sqrt(x * x + 1.0));

    printf("asinh(%.1f)       = %.6f\n", x, lib);
    printf("ln(x+sqrt(x^2+1)) = %.6f\n", manual);

    return 0;
}

asinh(2.0)       = 1.443635
ln(x+sqrt(x^2+1)) = 1.443635

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

int main(void) {
    double x = 1000.0;
    double result = asinh(x);
    double approx = log(2.0 * x);

    printf("asinh(%.0f) = %.4f\n", x, result);
    printf("ln(2*x) approx = %.4f\n", approx);

    return 0;
}

asinh(1000) = 7.6009
ln(2*x) approx = 7.6009

🚀 Common Use Cases

• Special relativity — rapidity calculations involving hyperbolic functions.
• Statistics — Fisher’s z-transformation and related formulas.
• Electrical engineering — transmission-line and filter design equations.
• Scientific computing — solving equations where sinh(t) appears.
• Pair with sinh() — recover a hyperbolic angle from a known sinh value.

📝 Notes

• Domain is all real numbers—no minimum or maximum input.
• asinh(0) = 0; asinh(−x) = −asinh(x) (odd function).
• Not the same as asin() (circular inverse sine, domain [−1, 1]).
• Link with -lm when using GCC/Clang on Unix-like systems.
• Use asinhf or asinhl for float or long double data.

⚡ Optimization

asinh() is implemented in the platform math library and is already optimized. Cache results when the same value is computed repeatedly. For float pipelines, use asinhf() to avoid promotion to double.

💡 Best Practices