Average Error: 32.1 → 0.1
Time: 6.9s
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 r49870 = x;
        double r49871 = r49870 * r49870;
        double r49872 = 1.0;
        double r49873 = r49871 - r49872;
        double r49874 = sqrt(r49873);
        double r49875 = r49870 + r49874;
        double r49876 = log(r49875);
        return r49876;
}
↓
double f(double x) {
        double r49877 = x;
        double r49878 = 1.0;
        double r49879 = sqrt(r49878);
        double r49880 = r49877 + r49879;
        double r49881 = sqrt(r49880);
        double r49882 = r49877 - r49879;
        double r49883 = sqrt(r49882);
        double r49884 = r49881 * r49883;
        double r49885 = r49877 + r49884;
        double r49886 = log(r49885);
        return r49886;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  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 2019208 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))