Average Error: 31.9 → 0.1
Time: 7.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)
double f(double x) {
        double r95770 = x;
        double r95771 = r95770 * r95770;
        double r95772 = 1.0;
        double r95773 = r95771 - r95772;
        double r95774 = sqrt(r95773);
        double r95775 = r95770 + r95774;
        double r95776 = log(r95775);
        return r95776;
}
↓
double f(double x) {
        double r95777 = x;
        double r95778 = 1.0;
        double r95779 = sqrt(r95778);
        double r95780 = r95777 + r95779;
        double r95781 = sqrt(r95780);
        double r95782 = r95777 - r95779;
        double r95783 = sqrt(r95782);
        double r95784 = r95781 * r95783;
        double r95785 = r95777 + r95784;
        double r95786 = log(r95785);
        return r95786;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  3. Applied add-sqr-sqrt31.9

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

  4. Applied difference-of-squares31.9

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))