Average Error: 32.4 → 0.4
Time: 4.1s
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 r80764 = x;
        double r80765 = r80764 * r80764;
        double r80766 = 1.0;
        double r80767 = r80765 - r80766;
        double r80768 = sqrt(r80767);
        double r80769 = r80764 + r80768;
        double r80770 = log(r80769);
        return r80770;
}
↓
double f(double x) {
        double r80771 = 2.0;
        double r80772 = log(r80771);
        double r80773 = x;
        double r80774 = log(r80773);
        double r80775 = 0.25;
        double r80776 = r80775 / r80773;
        double r80777 = r80776 / r80773;
        double r80778 = r80774 - r80777;
        double r80779 = 0.09375;
        double r80780 = 4.0;
        double r80781 = pow(r80773, r80780);
        double r80782 = r80779 / r80781;
        double r80783 = r80778 - r80782;
        double r80784 = r80772 + r80783;
        return r80784;
}

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