Average Error: 31.5 → 0.1
Time: 8.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)\right)
double f(double x) {
        double r68903 = x;
        double r68904 = r68903 * r68903;
        double r68905 = 1.0;
        double r68906 = r68904 - r68905;
        double r68907 = sqrt(r68906);
        double r68908 = r68903 + r68907;
        double r68909 = log(r68908);
        return r68909;
}
↓
double f(double x) {
        double r68910 = x;
        double r68911 = 1.0;
        double r68912 = sqrt(r68911);
        double r68913 = r68910 + r68912;
        double r68914 = sqrt(r68913);
        double r68915 = sqrt(r68914);
        double r68916 = r68910 - r68912;
        double r68917 = sqrt(r68916);
        double r68918 = sqrt(r68917);
        double r68919 = r68915 * r68918;
        double r68920 = r68919 * r68919;
        double r68921 = r68910 + r68920;
        double r68922 = log(r68921);
        return r68922;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  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 add-sqr-sqrt0.1

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

  10. Applied sqrt-prod0.1

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

  11. Applied unswap-sqr0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))