Average Error: 31.0 → 0.2
Time: 13.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)
double f(double x) {
        double r1108398 = x;
        double r1108399 = r1108398 * r1108398;
        double r1108400 = 1.0;
        double r1108401 = r1108399 - r1108400;
        double r1108402 = sqrt(r1108401);
        double r1108403 = r1108398 + r1108402;
        double r1108404 = log(r1108403);
        return r1108404;
}
↓
double f(double x) {
        double r1108405 = x;
        double r1108406 = 0.5;
        double r1108407 = r1108406 / r1108405;
        double r1108408 = r1108405 - r1108407;
        double r1108409 = 0.125;
        double r1108410 = r1108409 / r1108405;
        double r1108411 = r1108405 * r1108405;
        double r1108412 = r1108410 / r1108411;
        double r1108413 = r1108408 - r1108412;
        double r1108414 = r1108405 + r1108413;
        double r1108415 = log(r1108414);
        return r1108415;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  2. Simplified31.0

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))