Average Error: 32.4 → 0.1
Time: 4.8s
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 r85764 = x;
        double r85765 = r85764 * r85764;
        double r85766 = 1.0;
        double r85767 = r85765 - r85766;
        double r85768 = sqrt(r85767);
        double r85769 = r85764 + r85768;
        double r85770 = log(r85769);
        return r85770;
}
↓
double f(double x) {
        double r85771 = x;
        double r85772 = 1.0;
        double r85773 = sqrt(r85772);
        double r85774 = r85771 + r85773;
        double r85775 = sqrt(r85774);
        double r85776 = r85771 - r85773;
        double r85777 = sqrt(r85776);
        double r85778 = sqrt(r85777);
        double r85779 = r85775 * r85778;
        double r85780 = r85779 * r85778;
        double r85781 = r85771 + r85780;
        double r85782 = log(r85781);
        return r85782;
}

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)))))