Average Error: 31.3 → 0.0
Time: 19.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x - 1} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x - 1} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)\right)
double f(double x) {
        double r1725714 = x;
        double r1725715 = r1725714 * r1725714;
        double r1725716 = 1.0;
        double r1725717 = r1725715 - r1725716;
        double r1725718 = sqrt(r1725717);
        double r1725719 = r1725714 + r1725718;
        double r1725720 = log(r1725719);
        return r1725720;
}
↓
double f(double x) {
        double r1725721 = x;
        double r1725722 = 1.0;
        double r1725723 = r1725721 - r1725722;
        double r1725724 = sqrt(r1725723);
        double r1725725 = r1725722 + r1725721;
        double r1725726 = sqrt(r1725725);
        double r1725727 = cbrt(r1725726);
        double r1725728 = r1725724 * r1725727;
        double r1725729 = r1725727 * r1725727;
        double r1725730 = r1725728 * r1725729;
        double r1725731 = r1725721 + r1725730;
        double r1725732 = log(r1725731);
        return r1725732;
}

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 difference-of-sqr-131.3

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

  4. Applied sqrt-prod0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied associate-*l*0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))