Average Error: 32.2 → 0.3
Time: 6.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{\frac{0.25}{x}}{x}\right) - \log x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{\frac{0.25}{x}}{x}\right) - \log x\right)
double f(double x) {
        double r52051 = x;
        double r52052 = r52051 * r52051;
        double r52053 = 1.0;
        double r52054 = r52052 - r52053;
        double r52055 = sqrt(r52054);
        double r52056 = r52051 + r52055;
        double r52057 = log(r52056);
        return r52057;
}
↓
double f(double x) {
        double r52058 = 2.0;
        double r52059 = log(r52058);
        double r52060 = 0.09375;
        double r52061 = x;
        double r52062 = 4.0;
        double r52063 = pow(r52061, r52062);
        double r52064 = r52060 / r52063;
        double r52065 = 0.25;
        double r52066 = r52065 / r52061;
        double r52067 = r52066 / r52061;
        double r52068 = r52064 + r52067;
        double r52069 = log(r52061);
        double r52070 = r52068 - r52069;
        double r52071 = r52059 - r52070;
        return r52071;
}

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

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

  4. Final simplification0.3

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

Reproduce

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