Average Error: 31.6 → 0.1
Time: 14.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r2822329 = x;
        double r2822330 = r2822329 * r2822329;
        double r2822331 = 1.0;
        double r2822332 = r2822330 - r2822331;
        double r2822333 = sqrt(r2822332);
        double r2822334 = r2822329 + r2822333;
        double r2822335 = log(r2822334);
        return r2822335;
}
↓
double f(double x) {
        double r2822336 = x;
        double r2822337 = 1.0;
        double r2822338 = sqrt(r2822337);
        double r2822339 = r2822336 + r2822338;
        double r2822340 = sqrt(r2822339);
        double r2822341 = r2822336 - r2822338;
        double r2822342 = sqrt(r2822341);
        double r2822343 = cbrt(r2822342);
        double r2822344 = r2822343 * r2822343;
        double r2822345 = r2822340 * r2822344;
        double r2822346 = r2822345 * r2822343;
        double r2822347 = r2822336 + r2822346;
        double r2822348 = log(r2822347);
        return r2822348;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-cube-cbrt0.1

    \[\leadsto \log \left(x + \sqrt{x + \sqrt{1}} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)}\right)\]

  8. Applied associate-*r*0.1

    \[\leadsto \log \left(x + \color{blue}{\left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}}\right)\]

  9. Final simplification0.1

    \[\leadsto \log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))