Average Error: 31.6 → 0.1
Time: 20.3s
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 r2236745 = x;
        double r2236746 = r2236745 * r2236745;
        double r2236747 = 1.0;
        double r2236748 = r2236746 - r2236747;
        double r2236749 = sqrt(r2236748);
        double r2236750 = r2236745 + r2236749;
        double r2236751 = log(r2236750);
        return r2236751;
}
↓
double f(double x) {
        double r2236752 = x;
        double r2236753 = 1.0;
        double r2236754 = r2236753 + r2236752;
        double r2236755 = sqrt(r2236754);
        double r2236756 = r2236752 - r2236753;
        double r2236757 = sqrt(r2236756);
        double r2236758 = r2236755 * r2236757;
        double r2236759 = r2236752 + r2236758;
        double r2236760 = log(r2236759);
        return r2236760;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  3. Applied difference-of-sqr-131.6

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))