Average Error: 31.1 → 0.1
Time: 12.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r2560771 = x;
        double r2560772 = r2560771 * r2560771;
        double r2560773 = 1.0;
        double r2560774 = r2560772 - r2560773;
        double r2560775 = sqrt(r2560774);
        double r2560776 = r2560771 + r2560775;
        double r2560777 = log(r2560776);
        return r2560777;
}
↓
double f(double x) {
        double r2560778 = x;
        double r2560779 = 1.0;
        double r2560780 = r2560779 + r2560778;
        double r2560781 = sqrt(r2560780);
        double r2560782 = r2560778 - r2560779;
        double r2560783 = sqrt(r2560782);
        double r2560784 = r2560781 * r2560783;
        double r2560785 = r2560778 + r2560784;
        double r2560786 = log(r2560785);
        return r2560786;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.1

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

  3. Applied *-un-lft-identity31.1

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

  4. Applied difference-of-squares31.1

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))