Average Error: 31.6 → 0.4
Time: 5.3s
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 r70881 = x;
        double r70882 = r70881 * r70881;
        double r70883 = 1.0;
        double r70884 = r70882 - r70883;
        double r70885 = sqrt(r70884);
        double r70886 = r70881 + r70885;
        double r70887 = log(r70886);
        return r70887;
}
↓
double f(double x) {
        double r70888 = 2.0;
        double r70889 = log(r70888);
        double r70890 = x;
        double r70891 = log(r70890);
        double r70892 = 0.25;
        double r70893 = r70892 / r70890;
        double r70894 = r70893 / r70890;
        double r70895 = r70891 - r70894;
        double r70896 = 0.09375;
        double r70897 = 4.0;
        double r70898 = pow(r70890, r70897);
        double r70899 = r70896 / r70898;
        double r70900 = r70895 - r70899;
        double r70901 = r70889 + r70900;
        return r70901;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  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 2020021 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))