Average Error: 31.3 → 0.1
Time: 13.3s
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 r1495801 = x;
        double r1495802 = r1495801 * r1495801;
        double r1495803 = 1.0;
        double r1495804 = r1495802 - r1495803;
        double r1495805 = sqrt(r1495804);
        double r1495806 = r1495801 + r1495805;
        double r1495807 = log(r1495806);
        return r1495807;
}
↓
double f(double x) {
        double r1495808 = x;
        double r1495809 = 1.0;
        double r1495810 = r1495809 + r1495808;
        double r1495811 = sqrt(r1495810);
        double r1495812 = r1495808 - r1495809;
        double r1495813 = sqrt(r1495812);
        double r1495814 = r1495811 * r1495813;
        double r1495815 = r1495808 + r1495814;
        double r1495816 = log(r1495815);
        return r1495816;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.3

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

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

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

  4. Applied difference-of-squares31.3

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

  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 2019146 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))