Average Error: 32.5 → 0.1
Time: 2.6s
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 r72856 = x;
        double r72857 = r72856 * r72856;
        double r72858 = 1.0;
        double r72859 = r72857 - r72858;
        double r72860 = sqrt(r72859);
        double r72861 = r72856 + r72860;
        double r72862 = log(r72861);
        return r72862;
}
↓
double f(double x) {
        double r72863 = x;
        double r72864 = 1.0;
        double r72865 = sqrt(r72864);
        double r72866 = r72863 + r72865;
        double r72867 = sqrt(r72866);
        double r72868 = r72863 - r72865;
        double r72869 = sqrt(r72868);
        double r72870 = r72867 * r72869;
        double r72871 = r72863 + r72870;
        double r72872 = log(r72871);
        return r72872;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2019353 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))