Average Error: 31.6 → 0.4
Time: 3.7s
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 r62804 = x;
        double r62805 = r62804 * r62804;
        double r62806 = 1.0;
        double r62807 = r62805 - r62806;
        double r62808 = sqrt(r62807);
        double r62809 = r62804 + r62808;
        double r62810 = log(r62809);
        return r62810;
}
↓
double f(double x) {
        double r62811 = 2.0;
        double r62812 = log(r62811);
        double r62813 = x;
        double r62814 = log(r62813);
        double r62815 = 0.25;
        double r62816 = r62815 / r62813;
        double r62817 = r62816 / r62813;
        double r62818 = r62814 - r62817;
        double r62819 = 0.09375;
        double r62820 = 4.0;
        double r62821 = pow(r62813, r62820);
        double r62822 = r62819 / r62821;
        double r62823 = r62818 - r62822;
        double r62824 = r62812 + r62823;
        return r62824;
}

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