Average Error: 30.5 → 0.1
Time: 12.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
double f(double x) {
        double r7424868 = x;
        double r7424869 = r7424868 * r7424868;
        double r7424870 = 1.0;
        double r7424871 = r7424869 - r7424870;
        double r7424872 = sqrt(r7424871);
        double r7424873 = r7424868 + r7424872;
        double r7424874 = log(r7424873);
        return r7424874;
}
↓
double f(double x) {
        double r7424875 = x;
        double r7424876 = 1.0;
        double r7424877 = r7424876 + r7424875;
        double r7424878 = sqrt(r7424877);
        double r7424879 = r7424875 - r7424876;
        double r7424880 = sqrt(r7424879);
        double r7424881 = r7424878 * r7424880;
        double r7424882 = r7424875 + r7424881;
        double r7424883 = log(r7424882);
        return r7424883;
}


\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)

Error

Bits error versus x

Derivation

  1. Initial program 30.5

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

  3. Applied *-un-lft-identity30.5

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

  4. Applied difference-of-squares30.5

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

  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 2019101 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))