Average Error: 31.7 → 0.1
Time: 9.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x - \sqrt{1}} \cdot \sqrt{x + \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x - \sqrt{1}} \cdot \sqrt{x + \sqrt{1}}\right)
double f(double x) {
        double r2542854 = x;
        double r2542855 = r2542854 * r2542854;
        double r2542856 = 1.0;
        double r2542857 = r2542855 - r2542856;
        double r2542858 = sqrt(r2542857);
        double r2542859 = r2542854 + r2542858;
        double r2542860 = log(r2542859);
        return r2542860;
}
↓
double f(double x) {
        double r2542861 = x;
        double r2542862 = 1.0;
        double r2542863 = sqrt(r2542862);
        double r2542864 = r2542861 - r2542863;
        double r2542865 = sqrt(r2542864);
        double r2542866 = r2542861 + r2542863;
        double r2542867 = sqrt(r2542866);
        double r2542868 = r2542865 * r2542867;
        double r2542869 = r2542861 + r2542868;
        double r2542870 = log(r2542869);
        return r2542870;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))