Average Error: 31.3 → 0.0
Time: 16.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\right)
double f(double x) {
        double r1140766 = x;
        double r1140767 = r1140766 * r1140766;
        double r1140768 = 1.0;
        double r1140769 = r1140767 - r1140768;
        double r1140770 = sqrt(r1140769);
        double r1140771 = r1140766 + r1140770;
        double r1140772 = log(r1140771);
        return r1140772;
}
↓
double f(double x) {
        double r1140773 = x;
        double r1140774 = 1.0;
        double r1140775 = r1140774 + r1140773;
        double r1140776 = sqrt(r1140775);
        double r1140777 = r1140773 - r1140774;
        double r1140778 = sqrt(r1140777);
        double r1140779 = r1140776 * r1140778;
        double r1140780 = cbrt(r1140779);
        double r1140781 = r1140780 * r1140780;
        double r1140782 = r1140780 * r1140781;
        double r1140783 = r1140773 + r1140782;
        double r1140784 = log(r1140783);
        return r1140784;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied *-un-lft-identity31.3

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

  4. Applied difference-of-squares31.3

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-cube-cbrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))