Average Error: 31.4 → 0.1
Time: 16.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x - 1} \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x - 1} \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right)\right)
double f(double x) {
        double r3490646 = x;
        double r3490647 = r3490646 * r3490646;
        double r3490648 = 1.0;
        double r3490649 = r3490647 - r3490648;
        double r3490650 = sqrt(r3490649);
        double r3490651 = r3490646 + r3490650;
        double r3490652 = log(r3490651);
        return r3490652;
}
↓
double f(double x) {
        double r3490653 = x;
        double r3490654 = 1.0;
        double r3490655 = r3490653 - r3490654;
        double r3490656 = sqrt(r3490655);
        double r3490657 = r3490654 + r3490653;
        double r3490658 = sqrt(r3490657);
        double r3490659 = sqrt(r3490658);
        double r3490660 = r3490659 * r3490659;
        double r3490661 = r3490656 * r3490660;
        double r3490662 = r3490653 + r3490661;
        double r3490663 = log(r3490662);
        return r3490663;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.4

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

  3. Applied *-un-lft-identity31.4

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

  4. Applied difference-of-squares31.4

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))