Average Error: 32.0 → 0.1
Time: 6.3s
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 r3122853 = x;
        double r3122854 = r3122853 * r3122853;
        double r3122855 = 1.0;
        double r3122856 = r3122854 - r3122855;
        double r3122857 = sqrt(r3122856);
        double r3122858 = r3122853 + r3122857;
        double r3122859 = log(r3122858);
        return r3122859;
}
↓
double f(double x) {
        double r3122860 = x;
        double r3122861 = 1.0;
        double r3122862 = sqrt(r3122861);
        double r3122863 = r3122860 - r3122862;
        double r3122864 = sqrt(r3122863);
        double r3122865 = r3122860 + r3122862;
        double r3122866 = sqrt(r3122865);
        double r3122867 = r3122864 * r3122866;
        double r3122868 = r3122860 + r3122867;
        double r3122869 = log(r3122868);
        return r3122869;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))