Average Error: 31.6 → 0.1
Time: 15.7s
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 r2798255 = x;
        double r2798256 = r2798255 * r2798255;
        double r2798257 = 1.0;
        double r2798258 = r2798256 - r2798257;
        double r2798259 = sqrt(r2798258);
        double r2798260 = r2798255 + r2798259;
        double r2798261 = log(r2798260);
        return r2798261;
}
↓
double f(double x) {
        double r2798262 = x;
        double r2798263 = 1.0;
        double r2798264 = sqrt(r2798263);
        double r2798265 = r2798262 + r2798264;
        double r2798266 = sqrt(r2798265);
        double r2798267 = r2798262 - r2798264;
        double r2798268 = sqrt(r2798267);
        double r2798269 = cbrt(r2798268);
        double r2798270 = r2798269 * r2798269;
        double r2798271 = r2798266 * r2798270;
        double r2798272 = r2798271 * r2798269;
        double r2798273 = r2798262 + r2798272;
        double r2798274 = log(r2798273);
        return r2798274;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

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)))))