C Math acosh() Function

What You’ll Learn

The acosh() function computes the inverse hyperbolic cosine of a number. If you know a value x and want the hyperbolic angle t such that cosh(t) = x, acosh(x) gives you that answer—used in science, engineering, and advanced geometry.

Definition and Usage

Hyperbolic cosine is defined as cosh(t) = (et + e−t) / 2. The inverse acosh(x) answers: “What value of t produces this cosh output?” Because cosh(t) ≥ 1 for all real t, acosh only accepts x ≥ 1.

Mathematically, for x ≥ 1: acosh(x) = ln(x + √(x² − 1)). The C library computes this precisely so you do not need to code the logarithm yourself.

📝 Syntax

Standard C declaration:

double acosh(double x);

Related variants

float acoshf(float x);            /* <math.h> */
long double acoshl(long double x);  /* <math.h> */

Parameters

• x — hyperbolic cosine value; must be ≥ 1.

Return Value

• Non-negative real number: the value t such that cosh(t) = x.
• acosh(1) returns 0.
• If x < 1, returns NaN and may set errno to EDOM.

Headers and linking

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

Examples Gallery

Compile every example with gcc file.c -std=c11 -o out -lm. Hyperbolic functions appear in physics and engineering formulas.

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

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

    printf("acosh(%.1f) = %.6f\n", x, result);

    return 0;
}

acosh(2.0) = 1.316958

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

int main(void) {
    printf("acosh(1) = %.6f\n", acosh(1.0));
    printf("cosh(0) = %.6f  (should equal 1)\n", cosh(0.0));

    return 0;
}

acosh(1) = 0.000000
cosh(0) = 1.000000  (should equal 1)

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

int main(void) {
    double bad = 0.5;
    double result = acosh(bad);

    if (isnan(result)) {
        printf("acosh(%.1f) is undefined (NaN)\n", bad);
        printf("Input must be >= 1.0\n");
    } else {
        printf("acosh(%.1f) = %.6f\n", bad, result);
    }

    return 0;
}

acosh(0.5) is undefined (NaN)
Input must be >= 1.0

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

int main(void) {
    double values[] = { 1.0, 1.5, 2.0, 5.0, 10.0 };
    size_t i;

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

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

    return 0;
}

x=  1.0  acosh(x)=0.0000  cosh(acosh(x))=1.000000
x=  1.5  acosh(x)=0.9624  cosh(acosh(x))=1.500000
x=  2.0  acosh(x)=1.3170  cosh(acosh(x))=2.000000
x=  5.0  acosh(x)=2.2924  cosh(acosh(x))=5.000000
x= 10.0  acosh(x)=2.9932  cosh(acosh(x))=10.000000

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

int main(void) {
    double x = 3.0;
    double lib = acosh(x);
    double manual = log(x + sqrt(x * x - 1.0));

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

    return 0;
}

acosh(3.0)     = 1.762747
ln(x+sqrt(x^2-1)) = 1.762747

🚀 Common Use Cases

• Special relativity — rapidity and hyperbolic angle calculations.
• Catenary curves — shapes of hanging cables involve cosh and its inverse.
• Signal processing — hyperbolic transforms in certain filter designs.
• Scientific computing — solving equations where cosh(t) appears.
• Pair with cosh() — recover a hyperbolic angle from a known cosh value.

📝 Notes

• Domain is x ≥ 1; acosh(1) = 0.
• Values below 1 return NaN—there is no real inverse in that range.
• Not the same as acos() (circular inverse cosine).
• Link with -lm when using GCC/Clang on Unix-like systems.
• Use acoshf or acoshl for float or long double data.

⚡ Optimization

acosh() is implemented in the platform math library and is already optimized. Avoid recomputing it in loops when the input is unchanged. For very hot paths with float data, consider acoshf() to match your precision needs and reduce conversion overhead.

💡 Best Practices