Average Error: 32.2 → 0.3
Time: 3.5s
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 r69836 = x;
        double r69837 = r69836 * r69836;
        double r69838 = 1.0;
        double r69839 = r69837 - r69838;
        double r69840 = sqrt(r69839);
        double r69841 = r69836 + r69840;
        double r69842 = log(r69841);
        return r69842;
}
↓
double f(double x) {
        double r69843 = 2.0;
        double r69844 = log(r69843);
        double r69845 = x;
        double r69846 = log(r69845);
        double r69847 = 0.25;
        double r69848 = r69847 / r69845;
        double r69849 = r69848 / r69845;
        double r69850 = r69846 - r69849;
        double r69851 = 0.09375;
        double r69852 = 4.0;
        double r69853 = pow(r69845, r69852);
        double r69854 = r69851 / r69853;
        double r69855 = r69850 - r69854;
        double r69856 = r69844 + r69855;
        return r69856;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))