Average Error: 32.4 → 0.4
Time: 6.1s
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 r63847 = x;
        double r63848 = r63847 * r63847;
        double r63849 = 1.0;
        double r63850 = r63848 - r63849;
        double r63851 = sqrt(r63850);
        double r63852 = r63847 + r63851;
        double r63853 = log(r63852);
        return r63853;
}
↓
double f(double x) {
        double r63854 = 2.0;
        double r63855 = log(r63854);
        double r63856 = x;
        double r63857 = log(r63856);
        double r63858 = 0.25;
        double r63859 = r63858 / r63856;
        double r63860 = r63859 / r63856;
        double r63861 = r63857 - r63860;
        double r63862 = 0.09375;
        double r63863 = 4.0;
        double r63864 = pow(r63856, r63863);
        double r63865 = r63862 / r63864;
        double r63866 = r63861 - r63865;
        double r63867 = r63855 + r63866;
        return r63867;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  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 2020059 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))