Average Error: 32.1 → 0.4
Time: 3.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)
double f(double x) {
        double r51280 = x;
        double r51281 = r51280 * r51280;
        double r51282 = 1.0;
        double r51283 = r51281 - r51282;
        double r51284 = sqrt(r51283);
        double r51285 = r51280 + r51284;
        double r51286 = log(r51285);
        return r51286;
}
↓
double f(double x) {
        double r51287 = 2.0;
        double r51288 = log(r51287);
        double r51289 = x;
        double r51290 = log(r51289);
        double r51291 = 0.25;
        double r51292 = r51291 / r51289;
        double r51293 = r51292 / r51289;
        double r51294 = r51290 - r51293;
        double r51295 = 0.09375;
        double r51296 = 4.0;
        double r51297 = pow(r51289, r51296);
        double r51298 = r51295 / r51297;
        double r51299 = r51294 - r51298;
        double r51300 = r51288 + r51299;
        return r51300;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)}\]

  4. Final simplification0.4

    \[\leadsto \log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))