Average Error: 30.7 → 0.1
Time: 19.2s
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 r2780798 = x;
        double r2780799 = r2780798 * r2780798;
        double r2780800 = 1.0;
        double r2780801 = r2780799 - r2780800;
        double r2780802 = sqrt(r2780801);
        double r2780803 = r2780798 + r2780802;
        double r2780804 = log(r2780803);
        return r2780804;
}
↓
double f(double x) {
        double r2780805 = x;
        double r2780806 = 1.0;
        double r2780807 = r2780806 + r2780805;
        double r2780808 = sqrt(r2780807);
        double r2780809 = r2780805 - r2780806;
        double r2780810 = sqrt(r2780809);
        double r2780811 = r2780808 * r2780810;
        double r2780812 = r2780805 + r2780811;
        double r2780813 = log(r2780812);
        return r2780813;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.7

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

  3. Applied *-un-lft-identity30.7

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

  4. Applied difference-of-squares30.7

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

  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 2019162 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))