Average Error: 31.1 → 0.1
Time: 12.6s
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 r2021813 = x;
        double r2021814 = r2021813 * r2021813;
        double r2021815 = 1.0;
        double r2021816 = r2021814 - r2021815;
        double r2021817 = sqrt(r2021816);
        double r2021818 = r2021813 + r2021817;
        double r2021819 = log(r2021818);
        return r2021819;
}
↓
double f(double x) {
        double r2021820 = x;
        double r2021821 = 1.0;
        double r2021822 = r2021821 + r2021820;
        double r2021823 = sqrt(r2021822);
        double r2021824 = r2021820 - r2021821;
        double r2021825 = sqrt(r2021824);
        double r2021826 = r2021823 * r2021825;
        double r2021827 = r2021820 + r2021826;
        double r2021828 = log(r2021827);
        return r2021828;
}

Error

Bits error versus x

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Results

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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)))))