Average Error: 32.4 → 0.1
Time: 4.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r65328 = x;
        double r65329 = r65328 * r65328;
        double r65330 = 1.0;
        double r65331 = r65329 - r65330;
        double r65332 = sqrt(r65331);
        double r65333 = r65328 + r65332;
        double r65334 = log(r65333);
        return r65334;
}
↓
double f(double x) {
        double r65335 = x;
        double r65336 = 1.0;
        double r65337 = sqrt(r65336);
        double r65338 = r65335 + r65337;
        double r65339 = sqrt(r65338);
        double r65340 = r65335 - r65337;
        double r65341 = sqrt(r65340);
        double r65342 = sqrt(r65341);
        double r65343 = r65339 * r65342;
        double r65344 = r65343 * r65342;
        double r65345 = r65335 + r65344;
        double r65346 = log(r65345);
        return r65346;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))