Average Error: 30.6 → 0.3
Time: 12.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\right)
double f(double x) {
        double r2103497 = x;
        double r2103498 = r2103497 * r2103497;
        double r2103499 = 1.0;
        double r2103500 = r2103498 - r2103499;
        double r2103501 = sqrt(r2103500);
        double r2103502 = r2103497 + r2103501;
        double r2103503 = log(r2103502);
        return r2103503;
}
↓
double f(double x) {
        double r2103504 = x;
        double r2103505 = 0.125;
        double r2103506 = r2103505 / r2103504;
        double r2103507 = r2103504 * r2103504;
        double r2103508 = r2103506 / r2103507;
        double r2103509 = r2103504 - r2103508;
        double r2103510 = 0.5;
        double r2103511 = r2103510 / r2103504;
        double r2103512 = r2103509 - r2103511;
        double r2103513 = r2103504 + r2103512;
        double r2103514 = log(r2103513);
        return r2103514;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.6

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)}\right)\]

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(\left(x - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))