Average Error: 32.2 → 0.3
Time: 8.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right) - \log x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right) - \log x\right)
double f(double x) {
        double r60030 = x;
        double r60031 = r60030 * r60030;
        double r60032 = 1.0;
        double r60033 = r60031 - r60032;
        double r60034 = sqrt(r60033);
        double r60035 = r60030 + r60034;
        double r60036 = log(r60035);
        return r60036;
}
↓
double f(double x) {
        double r60037 = 2.0;
        double r60038 = log(r60037);
        double r60039 = 0.09375;
        double r60040 = x;
        double r60041 = 4.0;
        double r60042 = pow(r60040, r60041);
        double r60043 = r60039 / r60042;
        double r60044 = 0.25;
        double r60045 = r60040 * r60040;
        double r60046 = r60044 / r60045;
        double r60047 = r60043 + r60046;
        double r60048 = log(r60040);
        double r60049 = r60047 - r60048;
        double r60050 = r60038 - r60049;
        return r60050;
}

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

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

  4. Final simplification0.3

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

Reproduce

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