Average Error: 31.5 → 0.0
Time: 9.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\left(\sqrt{x - \sqrt{1}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}}}\right) \cdot \left(\sqrt[3]{\sqrt{x + \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}}}\right) + x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\left(\sqrt{x - \sqrt{1}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}}}\right) \cdot \left(\sqrt[3]{\sqrt{x + \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}}}\right) + x\right)
double f(double x) {
        double r79945 = x;
        double r79946 = r79945 * r79945;
        double r79947 = 1.0;
        double r79948 = r79946 - r79947;
        double r79949 = sqrt(r79948);
        double r79950 = r79945 + r79949;
        double r79951 = log(r79950);
        return r79951;
}
↓
double f(double x) {
        double r79952 = x;
        double r79953 = 1.0;
        double r79954 = sqrt(r79953);
        double r79955 = r79952 - r79954;
        double r79956 = sqrt(r79955);
        double r79957 = r79952 + r79954;
        double r79958 = sqrt(r79957);
        double r79959 = cbrt(r79958);
        double r79960 = r79956 * r79959;
        double r79961 = r79959 * r79959;
        double r79962 = r79960 * r79961;
        double r79963 = r79962 + r79952;
        double r79964 = log(r79963);
        return r79964;
}

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-prod0.0

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

  6. Simplified0.0

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

  8. Applied add-cube-cbrt0.0

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

  9. Applied associate-*l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))